Analysis and Design of a Linear Tubular Electric Machine for Free

Analysis and Design of a Linear Tubular Electric Machine for Free
Sede Amministrativa: Università degli Studi di Padova
Dipartimento di Ingegneria Industriale
SCUOLA DI DOTTORATO DI RICERCA IN: INGEGNERIA INDUSTRIALE
INDIRIZZO: INGEGNERIA DELL’ENERGIA
CICLO XXVI
Analysis and Design of a Linear Tubular
Electric Machine for Free-piston Stirling
Micro-cogeneration Systems
Direttore della Scuola: Ch.mo Prof. Paolo Colombo
Coordinatore d’indirizzo: Ch.ma Prof.ssa Luisa Rossetto
Supervisore: Ch.mo Prof. Giovanni Attilio Martinelli
Co-supervisori: Ch.mo Prof. Mauro Andriollo
Ch.mo Prof. Andrea Tortella
Dottorando: Luca Dall’Ora
31 gennaio 2014
ii
Abstract
The UE investments for the renewable source development, in order to achieve the set
goals (Kyoto protocol and “20-20-20” targets), push to investigate in new technologies
and to develop the existing. In this context, the cogeneration (CHP) plays a fundamental
role, and in particular, the micro-CHP has wide development margins.
Among the different cogeneration process, the systems driven by a free-piston Stirling
engine are one of the most significant challenges in the research area. In such systems,
the thermal energy, coming from primary energy source (for example renewable
energy), is converted into mechanical energy through a Stirling engine, and then a linear
generator converts the mechanical energy into electrical energy, finally, the generator is
connected to the electric grid or to the load by means of an electric converter. The use
of the linear generator, instead of the traditional systems of linear to alternating motion
conversion (rod-crank system), allows achieving several advantages, including:
improving the system reliability, noise and cost reduction. Finally, this kind of system,
if well-designed, allows improving the system efficiency.
In this thesis a linear generator, directly coupled to a free-piston Stirling engine in a
CHP system, was developed and analysed. It was found, after a first phase of the study
and literature review, that the most convenient choice, from the technical and economic
point of view, is a single-phase tubular permanent magnet linear generator. In particular,
the magnets are made of plasto-neodymium, while, for the realization of the stator
magnetic circuit, due to the geometrical complexity, soft magnetic composites (SMC)
materials have been considered.
In order to determine the generator performance, an analysis method based on FEAs
was developed. This simplified method (HFEA) allows the study and the comparison of
different magnetization patterns and current supply strategies. The proposed
methodology exploits the representation of the magnetization spatial harmonics through
an analytical processing that allows taking into account different magnetization profile
of the permanent magnets. Thus, it was possible to reconstruct the most important
quantities, such as the flux density and the flux linkage, superposing the effect of each
harmonic obtained through the Fourier analysis. Furthermore, a procedure, able to
reproduce the effects of magnetic saturation of the mover, generally not negligible in
such kind of machines, was developed. For this purpose, an appropriate surface current
iii
distribution on the yoke of the mover was introduced, in order to reproduce the
demagnetizing effect due to the saturation.
By means of the air gap flux density, the force provided by linear generator was
calculated, while, by means of the flux density sampled on suitable points on the stator
and mover yokes, the iron losses were estimated and then the machine efficiency. By
means of the flux linkage the emf provided by linear generator was determined.
The results show a very good agreement with corresponding FEAs. The proposed
analysis method allows carrying out a parametric analysis with a lower computational
effort. Thanks to this feature, different magnetization patterns, supply strategies and
SMC materials can be compared in order to optimize the machine design.
A prototype based on the design guidelines was built; then, a procedure based on
experimental measurement was developed to characterize the electromagnetic
parameters. To determine the magnetization profile of the magnets, the flux density on
the mover surface was carried out by means of a Gaussmeter. As regards the SMC
materials that compose the stator core, a calculation method was developed from
suitable experimental elaborations, in order to determine the most important magnetic
properties, such as the BH curve and core loss coefficients.
From experimental results, it can be noted that the actual characteristics are poorer than
those provided by the manufactured datasheets, likely due to the manufacturing
processes and spurious air gaps between the SMC modules. The update electromagnetic
parameters are used to determine the actual performance of the machine, particularly to
estimate the efficiency, the emf and the force.
Finally, a simplified model of the cogeneration system was developed in order to
predict the dynamic behaviour and particularly, the actual values of the speed, output
power and efficiency. This model allows developing the control strategy of the linear
generator acting on the electric converter.
iv
Sommario
Gli investimenti da parte dei paesi della UE per lo sviluppo delle fonti rinnovabili, al
fine di raggiungere gli obiettivi preposti (Protocollo di Kioto e Piano “20-20-20”),
spingono a ricercare nuove tecnologie e a sviluppare quelle già esistenti. In questo
ambito, la cogenerazione ricopre un ruolo fondamentale, ed in particolare, la microcogenerazione, anche in ambito domestico, presenta ancora ampi margini di sviluppo.
Tra le diverse modalità di cogenerazione, i sistemi free-piston azionati da motori
Stirling rappresentano una tra le sfide più importanti nell’ambito della ricerca. In tali
sistemi l’energia termica, proveniente da una fonte di energia primaria (ad esempio di
tipo rinnovabile), è convertita in energia meccanica attraverso un motore Stirling;
successivamente un generatore lineare converte l’energia meccanica in elettrica; ed
infine, un convertitore elettrico interfaccia il generatore con un carico o con la rete
elettrica. L’utilizzo del generatore lineare, al posto dei tradizionali sistemi di
conversione del moto da lineare ad alternato (sistema biella-manovella), consente di
ottenere diversi benefici, tra cui: migliorare l’affidabilità del sistema, limitare il rumore
e ridurre i costi. Infine, questo sistema, se ben progettato, permette di aumentare il
rendimento dell’impianto di cogenerazione.
In questo lavoro di tesi è stato progettato ed analizzato un generatore lineare da
accoppiare direttamente ad un motore Stirling free-piston in un sistema di
cogenerazione domestico. Dopo una prima fase di studio e ricerca bibliografica, è stato
riscontrato che la scelta più conveniente dal punto di vista tecnico-economico, risulta
essere quella di un generatore monofase tubolare a magneti permanenti. In particolare, i
magneti sono a base di plasto-neodimio, mentre, per la realizzazione del circuito
magnetico di statore, data la complessità geometrica, si sono considerati materiali
magnetici compositi (SMC).
Al fine di determinare le prestazioni del generatore, è stata sviluppata una metodologia
di analisi, basata su analisi FEM, che permettesse lo studio ed il confronto di diverse
tipologie di magnetizzazione, in particolare radiale e Halbach, e considerare diverse
strategie di alimentazione. Tale metodologia (HFEA) sfrutta la rappresentazione delle
armoniche spaziali di magnetizzazione attraverso un'elaborazione di tipo analitico che
permette di riprodurre diverse funzioni di magnetizzazione dei magneti permanenti. In
tal modo, è stato possibile ricostruire le principali grandezze di interesse (induzione e
v
flusso concatenato) sovrapponendo l’effetto di ciascuna armonica ricavata attraverso
l'analisi di Fourier. Inoltre, è stata sviluppata una procedura in grado di riprodurre gli
effetti della saturazione magnetica del traslatore, generalmente non trascurabile in
questo tipo di macchine. A questo proposito sono state introdotte opportune
distribuzioni superficiali di corrente sul giogo del traslatore che riproducono l'effetto
smagnetizzante dovuto alla saturazione.
Dai valori dell’induzione al traferro, è possibile risalire al valore della forza esplicata
dal generatore; mentre, dai valori dell’induzione, campionati in opportuni punti
all’interno dei nuclei magnetici dello statore e del traslatore, è stato possibile stimare le
perdite nel ferro e quindi il rendimento della macchina. Dal profilo del flusso
concatenato è stato possibile stimare l’andamento della forza elettromotrice generata.
I risultati hanno evidenziato un ottimo accordo con le corrispondenti analisi agli
elementi finiti, con il vantaggio rispetto a queste di poter effettuare analisi parametriche
che coinvolgono diverse grandezze progettuali con tempi di calcolo inferiori. Grazie a
questa caratteristica, è stato possibile confrontare le prestazioni con varie tipologie di
magnetizzazioni, strategie di alimentazione e caratteristiche di materiali SMC,
effettuando una prima ottimizzazione del progetto della macchina.
Dopo aver realizzato un prototipo sulla base delle indicazioni progettuali, è stata
sviluppata una procedura sperimentale atta a determinare i parametri elettromagnetici
del generatore, per verificarne la congruenza con le ipotesi progettuali e mettere a punto
quindi i modelli per le analisi successive. Per determinare il profilo di magnetizzazione
dei magneti sono state effettuate delle misure di induzione sulla superficie del materiale
magnetico mediante l’uso di un Gaussmetro. Invece, per i materiali SCM che
costituiscono lo statore della macchina, a partire da opportune elaborazioni
sperimentali, è stato sviluppato un metodo di calcolo in grado di determinare le
principali proprietà magnetiche (permeabilità, caratteristica BH e coefficienti delle
perdite nel ferro per isteresi, correnti parassite ed eccesso).
Dai risultati sperimentali si nota che le caratteristiche reali sono meno performanti
rispetto a quelle fornite dai datasheet dei costruttori, molto probabilmente a causa dei
processi di lavorazione e dei traferri spuri presenti tra i moduli di SMC. I valori delle
grandezze magnetiche aggiornate sono stati utilizzati per stimare le reali condizioni di
esercizio della macchina, in particolare per determinare il valore del rendimento, della
tensione e della forza sviluppata dalla macchina.
È poi stato sviluppato un modello semplificato che consente di riprodurre il
comportamento dinamico del sistema di cogenerazione completo e di ricavare il profilo
vi
effettivo di velocità, la potenza erogata e il rendimento complessivo del sistema.
Considerando l'interazione tra i singoli componenti, l'applicazione di questo modello
risulta di estrema importanza per eseguire l’analisi e permette di mettere a punto la
strategia di controllo del generatore agendo sul convertitore di interfacciamento con il
carico.
vii
viii
Contents
1 Introduction ............................................................................................................... 1
1.1 Stirling engine applications ............................................................................ 1
1.1.1
Cogeneration system ..................................................................... 1
1.1.2
Solar energy converters ................................................................ 7
1.1.3
Automotive applications ............................................................. 10
1.1.4
Wave energy converters.............................................................. 11
1.2 Linear generators .......................................................................................... 14
1.3 Scope of research .......................................................................................... 16
1.4 Methodology adopted ................................................................................... 17
1.5 Thesis structure ............................................................................................. 18
2 Stirling engines ........................................................................................................ 19
2.1 Introduction ................................................................................................... 19
2.2 Development of Stirling engines .................................................................. 21
2.2.1
First era of Stirling engines ......................................................... 21
2.2.2
Second era of Stirling engines .................................................... 22
2.2.3
Stirling engines for industries ..................................................... 22
2.2.4
Stirling engines for rural and remote areas ................................. 22
2.2.5
Stirling engines optimization ...................................................... 23
2.3 Development of solar-powered Stirling engines .......................................... 25
2.3.1
Solar-powered Stirling engines in the first era ........................... 25
2.3.2
Solar-powered Stirling engines in the second era ....................... 26
2.3.3
Solar-powered Stirling engines optimization ............................. 29
2.3.4
Development of LTD Stirling engines........................................ 29
2.4 Mechanical configurations of Stirling engines ............................................. 30
2.5 Principle of operation.................................................................................... 34
2.6 Analytical models of Stirling engine ............................................................ 39
2.6.1
Empirical modelling ................................................................... 39
2.6.2
Thermodynamic modelling ......................................................... 40
2.7 Optimization of the design of Stirling engines ............................................. 47
ix
3
Linear generators .................................................................................................... 51
3.1 Linear generator typologies ........................................................................... 51
3.1.1
Induction machines ..................................................................... 51
3.1.2
Synchronous machines ................................................................ 52
3.1.3
Permanent magnet machines ....................................................... 52
3.2 Variable reluctance permanent magnet machines ......................................... 53
3.2.1
Transversal flux machines ........................................................... 55
3.2.2
Vernier hybrid machines ............................................................. 56
3.3 Principle of operation .................................................................................... 57
3.4 Dynamic equations ........................................................................................ 58
3.5 Structures of linear machine .......................................................................... 59
3.6 Control of permanent magnet linear generators ............................................ 60
3.7 History and applications of linear generators ................................................ 62
4
Magnetic materials utilization in linear electric machines .................................. 65
4.1 Introduction ................................................................................................... 65
4.2 Hard ferromagnetic materials – Permanent magnets .................................... 71
4.3 The effect of the temperature variation ......................................................... 76
4.4 Manufacturing and magnetization process .................................................... 77
4.5 Soft ferromagnetic materials – Iron core materials ....................................... 80
4.6 Soft magnetic composite materials ............................................................... 83
4.6.1
Powder characteristics ................................................................. 86
4.6.2
Manufacturing process ................................................................ 89
4.7 Analytical analysis of magnetic circuits ........................................................ 91
5
4.7.1
Equivalent electric circuit............................................................ 91
4.7.2
Limitation of the magnetic circuit approach ............................... 95
Semi-analytical model of a linear machine ........................................................... 99
5.1 Introduction ................................................................................................... 99
5.2 Preliminary design....................................................................................... 103
5.3 Simplified FEA model ................................................................................ 105
5.4 Working principle of simplified model ....................................................... 111
5.5 Analysis take into account the mover saturation ......................................... 117
5.6 Core losses and efficiency estimation ......................................................... 124
5.7 Model validation ......................................................................................... 126
x
5.8 Example of applications.............................................................................. 128
5.9 Conclusion .................................................................................................. 133
6 Characterization of electromagnetic parameters .............................................. 135
6.1 Introduction ................................................................................................. 135
6.2 Determination of the permanent magnet parameters .................................. 138
6.3 Determination of SMC magnetic curve ...................................................... 139
6.4 Determination of SMC core losses coefficients ......................................... 145
6.5 Evolution of the actual TPMLG performance ............................................ 150
6.6 Conclusion .................................................................................................. 152
7 Dynamic model of Stirling free-piston cogeneration system ............................ 155
7.1 Introduction ................................................................................................. 155
7.2 Dynamic model of cogeneration system ..................................................... 158
7.3 Stirling engine model .................................................................................. 161
7.4 TPMLG model ............................................................................................ 166
7.5 Electronic power converter model .............................................................. 167
7.6 Control system ............................................................................................ 167
7.7 Dynamic analysis of the cogeneration system ............................................ 169
7.8 Conclusion .................................................................................................. 175
8 Conclusions and future work ............................................................................... 177
References .................................................................................................................... 181
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xii
Chapter 1
Introduction
The depletion of fossil energy resources and the global warming involve searching
alternative solutions to develop clean renewable energy resource. The rapid
development of renewable energies represents a challenge for the scientific community
to develop new technology or improve the old ones. Among the many technologies,
permanent magnet linear generators (PMLG) represent, of course, an important case of
study. PMLGs convert mechanical energy into electric energy. They find application in
all the fields that involves linear motion, such as: cogeneration systems that use freepiston combustion engine (FPCE), wave energy converters and hybrid automotive field.
The choice of a linear motion generator overcomes a possible mechanical linkage
system (crankshaft), which is source of losses, noise and failures.
In this chapter some linear generator applications will be presented, in order to identify
the technical specifications for the machine design. Then, main linear generator features
and layouts will be discussed. Finally, the research scope and the adopted methodology
will be described in order to pointing out the original contribution to actual knowledge.
1.1
Stirling engine applications
Applications of the Stirling engine range from mechanical propulsion to heating and
cooling to electrical generation systems. Particularly, Stirling engines can find
application in cogeneration systems, as solar and wave energy converters and in the
hybrid electric vehicles.
1.1.1 Cogeneration systems
The domestic sector accounts for a considerable portion of the final energy consumption
of most developed countries. Within the European Union (EU), approximately 25% of
2
Chapter 1
the total annual final energy consumption is used in dwellings [1]. A precise breakdown
of how this energy is consumed on a country by country basis is difficult to ascertain,
especially considering that such a breakdown is highly dependent on factors such as
climate, technology used and other conditions which are variable from country to
country [2]. However, an approximate breakdown is given by the European
Environmental Agency (EEA) in [3], which indicates that a substantial amount of this
energy, about 69%, goes into space heating with the remaining balance accounted for by
water heating (15%), providing electricity for appliances and lighting equipment (11%)
and cooking activities (5%). In warm climates such as that present in southern European
countries, space cooling is the predominant load [4]. Ensuring the efficient provision
and use of energy in dwellings is therefore an essential aspect in attaining the overall
20% improvement in energy efficiency target set out by the Presidency Conclusions of
the European Council [5] for 2020.
In order to reach the energy-efficiency improvement targets set by the EU, a concrete
and perhaps active approach is therefore required [6]. Most of the energy efficiency
measures which have been implemented so far are considered to be a first step towards
decreasing energy demand in the domestic sector; a step in the right direction but still
far away from achieving the desired target. One way in which the individual targets can
be achieved is through policy measures specifically aimed at achieving higher
penetrations of renewable energy technologies or efficient micro-generation
technologies in dwellings [48].
The term “cogeneration”, often called combined heat and power (CHP), means the
simultaneous production of two type of energy from the only one primary energy
source. The most common cogeneration system produces both heat and electricity at the
same time.
Unlike conventional heat and power supplies to buildings in which each is satisfied
separately by two distinct energy flow processes, cogeneration systems, as shown in
Figure 1.1, make use of a single process whereby the ‘waste heat’ from electricity
generation is recovered and re-utilised to satisfy a heating demand [8], hence improving
the fuel usage efficiency. As shown in Figure 1.1, the efficiency to produce electrical
energy and heat separately is less than 50%, instead the (global) efficiency of a CHP
system is around 80 %.
Introduction
3
Losses (44%)
Primary
fuel
source
Power
plant
Electricity (20%)
Heat (29%)
Boiler
Losses (7%)
Primary
fuel
source
Electricity + Heat
(80%)
CHP
Losses (20%)
Figure 1.1. Separate generation vs. cogeneration (efficiency values are indicative).
Cogeneration can make use of the ‘waste heat’ exhausted from practically any energy
conversion system including the traditional internal combustion engines (ICE), Stirling
engines, to the more advanced technologies such as micro-gas turbines and fuel cells.
The idea of re-utilising waste heat from an electricity producing process is not a new
concept. In 1882, Thomas Edison’s first commercial power station in Pearl Street used
to supply both power and heat [9] to a neighbouring block in Manhattan. Similarly,
district heating in northern and central Europe has been successfully implemented for
many years, with the result that an average 13% of the gross electricity generation in the
EU is currently produced by cogeneration as shown in
Figure 1.2.
Figure 1.2. Share of electricity produced by cogeneration in the EU (data published by Eurostat for 2009
[10]).
From Figure 1.2 it can be observed how the average share of electricity produced from
cogeneration, in the 7 countries which border the Mediterranean Sea and which have a
predominant warm climate with short mild winters, is still a small part of the overall
cogenerated electricity produced in the EU. Like any other technology, the feasibility
and market interest in cogeneration depends on the payback period. When assessing the
payback period an important role is played by the operational or usage time, as this has
a direct effect on the ability of a system to offer reasonable and adequate return on
4
Chapter 1
investments. Although plant configuration schemes such as additional storage can be
used to introduce a degree of flexibility in micro-cogeneration operation [11], the
operational or usage time is still a function of the electrical and thermal demand. In
warm climates with the exception of the provision of hot water, the thermal demand is
very low due to the low number of heating days. Consequently, the investment required
for the installation of a micro-cogeneration system is difficult to justify [12].
A solution to this problem is discuss in [13], that suggest to extend the concept of
cogeneration to tri-generation by including a thermally driven cooling load. Unlike
cogeneration, tri-generation also known as Combined Cooling, Heat and Power
(CCHP), is the process whereby the ‘waste heat’ recovered from an energy conversion
process is used either to supply a heating demand such as in a normal cogeneration
system or to supply a thermally activated cooling (TAC) device [14] capable of
satisfying a cooling demand.
There are various types of TAC devices which can be used in a tri-generation system,
single-effect absorption chillers are the preferred type, due to low temperature heat
source at which ‘waste heat’ is exhausted from CHP units used in micro-cogeneration
systems.
In warm climates this provides a more consistent requirement for heat, thus increasing
the system utilisation time. This makes the tri-generation process particularly suitable
for use in warm climates given that it can be used both during the summer and the
winter period [12]. Tri-generation offers an interesting avenue for efficiently delivering
power, heating and cooling to residential buildings in warmer climates.
Further increase in the use of cogeneration in such countries or indeed introducing large
scale cogeneration in countries with no prior experience of district heating is however
hindered by the fact that district heating requires huge investments costs and
implementing such a scheme requires appropriate location conditions (e.g. dense
population) and extensive infrastructural work.
One way how these limitations can be overcome is by extending cogeneration to include
micro-scale cogeneration. The idea of micro-scale cogeneration or micro-cogeneration
differs from district heating or centralised cogeneration, in that, micro-scale
cogeneration typically refers to single systems capable of supplying heat and power to
individual buildings such as households, small commercial premises, hotels etc. which
may or may not be connected to the centralised grid network. Sizes for microcogeneration are considerably smaller than typical cogeneration or district heating
systems, running into only few kWel. Although the official definition given in Directive
Introduction
5
2004/8/EC on the promotion of cogeneration based on a useful heat demand in the
internal energy market, the relevant EU directive on cogeneration, qualifies microcogeneration as systems with an electrical power output of less than 50 kWel.
One of the most promising systems for cogeneration is powered by free-piston Stirling
engine (FPSE), shown in Figure 1.3.
Mechanical
energy
Thermal
energy
FPSE
Electrical
energy
PMLG
(a)
(b)
Figure 1.3. Energies transformation process of a free-piston Stirling engine cogeneration system (a),
cross-section showing major components (b).
Despite its relatively low efficiency, it has a lot advantages. The Stirling engine is an
internal combustion engine, for this reason the working fluid is enclosed, thus guarantee
no pollution, reliability and overall, the possibility to use different kind of primary
energy sources, such as fossil fuels (oil, coal, gas, etc.) or renewable source (biomass,
biogas, solar energy, etc.).
In response to the short-comings of the conventional kinematic Stirling engine, William
Beale invented the free-piston Stirling engine (FPSE) in 1964 [15]. All kinematic
Stirling machines, no matter what the arrangement, face a number of very significant
engineering design challenges which may be so great as to prevent any commercial
success. These challenges include the requirement for long-lived non-contact bearings
and seals, the difficulty of sealing and pressurizing the machine, the need for lubrication
and to separate the lubricant from the heat exchangers, modulation and control. The
promise of the free piston machine has been a hermetically sealed, near fixed-frequency
device operating in a non-contact mode providing very long life, with no need for
6
Chapter 1
lubricants, and direct alternating current output from an integral linear alternator. Figure
1.4 describes the road map to commercialization for some of Sunpower’s technology [16].
Figure 1.4. Free-piston Machine Commercialization Roadmap (Sunpower).
In the idealised form of Stirling engine, heat is supplied to as well as rejected from the
engine during isothermal expansion and compression processes, respectively. The cycle
is completed by two constant volume processes, during which the heat transported by
working fluid is stored and released by a regenerative heat exchanger. Theoretically, the
Stirling cycle is the unique that is able to reach the maximum efficiency achieved by the
Carnot cycle. Practically, the Stirling engine can attain around 60 – 70 % of the Carnot
efficiency, in the case it operates at high wall temperature (600 – 700 °C) [17].
As suggested in [18], one of the critical tasks in the linear machines operation is the
initial process of engine starting. A linear engine cannot be started by an ordinary
starting motor because it has not a flywheel, crankshaft or any other mechanical
coupling which can accept the rotating push of the starter engine. This problem could be
solved running linear generator as a linear motor to produce the required reciprocating
motion, supplying the linear machine with a suitable current supply strategy
(Rectangular Current Commutation).
Paper [19] describe different topologies of linear generator for Stirling cogeneration
systems.
Introduction
7
1.1.2 Solar energy converters
Solar energy is one of the most important renewable energy sources. The yearly solar
fluxes is estimated around 3.85 1024 J. Insolation in most countries is from 150 to 300
W/m2 that means from 1.3 to 2.5 MWh/m2/year [20]. In 2012, the word’s human
primary energy consumption was 539 1018 J [21], more than 7000 times less of the
yearly word’s solar energy.
Solar cogeneration systems convert the thermal energy of solar radiation to mechanical
energy (by means of Stirling engine) and then to electrical energy (by means of linear
generator). The conversion principle is the same utilized in the conventional power
plants (oil, coal, gas, etc.).
There are different typologies of solar cogeneration systems, the main are: parabolic
trough, linear Fresnel reflectors, solar tower and dish Stirling; the main features of these
systems are reported in Table 1.1 [22]. The solar irradiance has enough potential to
produce the hot fluid with temperatures up to 165 °C using solar evacuated collectors
and up to 400 °C using a concentrating mirror system (usually parabolic troughs).
Table 1.1. Main features of concentrated solar-thermal-electric technologies.
The most promising system is dish Stirling (shown in Figure 1.5), thanks its higher
efficiency. The dish Stirling systems use a mirror array to reflect and concentrate solar
insolation to a receiver, in order to achieve the temperature required efficiency convert
heat to work.
The ideal concentrator shape is a paraboloid of revolution which concentrates sunlight
on the focus of paraboloid. Concentration ratio is typically over 2000 and, hence, the
concentrator must track the sun in two axes. At the focus is a receiver which is heated
up to over 700 °C [22].
8
Chapter 1
Figure 1.5. Solar dash Stirling cogeneration system.
The solar-to-electric conversion efficiency of Dish Stirling systems can achieve 30%
[23], which represent the highest value of all solar energy converters.
The advantages of dish Stirling technology are [24]:
i.
small land requirement (2.5 Ha/MW), due to high efficiency;
ii.
short construction times (10 months for a 20 MW plant);
iii.
low maintenance and operating costs,
iv.
more stable output than photovoltaic systems.
Paper [24] reports the study of a low-cost solar cogeneration system. This system,
cheaper than dish Stirling, is conceived to operate with collector temperature in the
range of 120° C to 150° C, which is consistent with the use of stationary solar thermal
collectors employing low-concentration nonimaging reflectors [25]. A non-tracking
system avoids the cost and maintenance issues associated with tracking collectors with
high concentration ratios. A Stirling engine converts the delivery heat provided by solar
collector into mechanical power. Even if the efficiency (around 10% [18]) is less than
dish Stirling systems, due to low-temperature operating conditions, the lower costs and
reduced maintenance make this solution interesting in the micro-cogeneration systems.
The complete system cost (collector and Stirling engine) is around 0.8 $/W [24].
Since during two-thirds of the day, solar energy is not available, solar/fuel hybrids are
needed in case of continuative energy production. Improving the solar system with an
external fuel source is not a problem for Stirling engine, as it is able to use all kinds of
fuel with any quality.
The efficiency of a solar-thermal collector
(
where
, as measured experimentally, is given by:
)
is the maximum collector efficiency,
coefficients [W/m2K] and [W/m2K2] respectively,
(
)
and
(1.1)
are the thermal loss
is the power density of incident
Introduction
9
sunlight [W/m2],
is the mean temperature of the collector [K] and
is the ambient
temperature [K], respectively.
As regards the engine, the efficiency
is given by:
(
)
(1.2)
where
is the fraction of the theoretical Carnot efficiency that the engine can
achieves and
is the cold side working temperature of the Stirling [K], respectively.
The total conversion system efficiency , is then given by:
(1.3)
For a representative system, the efficiencies of the collector, engine, and system are
plotted as a function of temperature in Figure 1.6 [20].
Figure 1.6. Efficiency as a function of temperature for a representative system. The parameters used are:
= 1000 W/m2, = 77.3%,
= 1.09 W/m2K,
= 0.0094 W/m2K2, = 27 °C, = 27 °C, and
= 66%. The dot indicates the point of optimal system efficiency.
To minimize cost per watt of output electricity, it is desirable to operate a system of
given cost at the temperature corresponding to peak system efficiency. This temperature
is a function of collector properties as well as ambient temperature and intensity of
sunlight. The heat engine can be designed to regulate its loading to maintain optimum
collector temperature and system efficiency. Figure 1.6 shows that the system efficiency
is rather flat over a range of temperatures near the extremes.
10
Chapter 1
An operating temperature of 142 °C permits a maximum thermodynamic (Carnot)
efficiency of 31.6%, assuming the sink temperature is 27 °C. We might reasonably
expect the Stirling engine and generator to achieve a thermal-electric efficiency of about
18.3%, roughly 66% of the Carnot efficiency, while the collector operates at a thermal
efficiency of about 52.3%. Thus, the estimated overall efficiency of the system would
be about 9.6%.
1.1.3 Automotive applications
Linear generator finds application as well as in the hybrid electric vehicles, particularly
in the series topology. The topology of these vehicles is reported in Figure 1.7.
Figure 1.7. Series hybrid electric vehicle topology.
One of the main disadvantages of Stirling engines is the very low dynamic speed: they
take a lot of time (several tens of seconds) to achieve the steady state condition. This is
the reason whereby they cannot use in traditional vehicles, whereas they represent a
very interesting choice in series hybrid electric vehicles. In facts, in this kind of hybrid
electric vehicles, free-piston Stirling can be used to provide the mean value of power
amount, while the ripple of power can be supply by batteries and/or ultracapacitors. The
Stirling engine is designed in order to working at its maximum efficiency value. As
explained in [26], the high degree of efficiency of free-piston linear generators at all
load levels is inherent in their design. It is achieved by keeping the system frequency
constant and adapting to the power demanded by varying stroke and compression. One
Introduction
11
of the main advantages of this system is the possibility to operate with different fuel
typologies, such as conventional fuels (petro, diesel and gas) as well as alternative fuel
(sun fuel, natural fuel, hydrogen, etc.).
An innovative construction technology, called linear generator of second generation, is
reported in [27]. This construction is implemented as a double chamber set-up and an
iron-less mover configuration. This is a set-up that uses two stators arranged around one
mover (Figure 1.8). This set-up allows for an optimal usage of the whole installation
space available and, therefore, for reaching the highest power density possible.
Figure 1.8. Linear generator of second generation topology.
The frequency-dependent losses that usually occur in the mover’s core no longer occur
in this configuration, due to the fact that the mover no longer contains iron. Thus, the
mover losses can be reduced considerably, resulting in an efficiency improving and in a
reduction of cooling efforts. Another advantage of this configuration is the mover mass
reduction, a nice feature to reach a high dynamic performance.
1.1.4 Wave energy converters
The wave energy is generally provided by ocean tides. Oceanic waves could be
considered as concentrated form of solar energy. Waves are produced by wind that is
created by pressure difference in the Earth’s atmosphere, which is created by different
solar heating. The energy transferred from wind to water is in the form of potential
energy and kinetic energy. The amount of see wave energy is related to wind speed, the
amount of time that the wind blowing and the distance over which the waves travel
[28].
The World Energy Council has estimated the global ocean wave energy over 2 TW.,
which means 17,500 TWh/year. From this, it has been estimated that practical amount
12
Chapter 1
of see energy that could be converted is around 2,000 TWh/year, similar to current
installed nuclear power. Most of this energy (around 95%) is offshore.
The power potential for waves can be described as units of power per meter of wave
crest length. The greatest power achievable is about 100 kW/m in the Atlantic Ocean,
Southern Ocean and Cape Horm [29].
There are several compelling arguments for using the wave energy technology: first, it
is one of the lowest cost renewable energy sources, thanks to its high power density;
second, the wave energy is more predictable than other renewable energy source, such
as solar and wind energy; then, the it is one of the most environmentally ways to
generate electricity, hence it does not produce any waste; finally, the wave energy
conversion devices can be located far enough away from the shore (offshore) in order
they are generally not visible.
The fundamental principle behind absorbing energy from a water wave is that energy
must be removed from that wave. It therefore follows that the resultant wave, after
passing the wave energy device, is either reduced or cancelled altogether. For the device
to remove energy from the wave, it is necessary for the resultant wave to be smaller
than the incoming wave, and hence the two waves interfere destructively. If a device is
to be a good wave absorber it must therefore inherently be a good wave maker.
A symmetrical body constrained such that it may only oscillate in one plane, either
perpendicular or parallel to the water surface, is only able to absorb a maximum of 50%
of the energy contained in an incident wave [30]. It hence shows the effect of a wave
energy converter (WEC) being allowed to move in two degrees of freedom. The wave
approaching the device is unaltered from the original wave, as the effects of the body
corresponding to its degrees of freedom cancel each other out. After passing the WEC,
however, these effects are summated and equal in magnitude to the original wave. Thus,
in order to absorb from the approaching wave the theoretical ideal of 100 % energy, a
device with two degrees of freedom is required.
Different topologies for wave energy converter are reported in Figure 1.9. The simplest
way to convert the wave energy is through a floating buoy lifted-dropped by the waves
connected directly to the linear generator fixed on a concrete foundation, which stands
on the bottom of the sea (as shown in Figure 1.9a) [31].
The power generation is provided by linear oscillating of buoy between wave top and
wave bottom. By optimizing the shape and operation of the buoy, up to 90% efficiency
can be achieved with such devices.
Introduction
13
Figure 1.9. Various solutions for placing the linear generator WEC components.
Linear generator extracts the power of the waves in the form of a reciprocating motion a
low speed and high torque. The linear generator is connected to the grid by a power
electronic converter, which is able to process the electric waveform.
Since the linear generator is located at the sea bottom, in a hard to access place, it has to
be maintenance free. Furthermore, in order to have as low mass and volume as possible,
high values of force density must achieve.
The main features of linear generator depend on the characteristics of the waves in the
specified area, generally the common values are [28, 32]:

power of a few kWs;

speed up to 1 m/s;

stroke under 1 m.
These features are achieved by means the use of permanent magnets and tubular linear
generator topology.
In literature mainly the following linear generators are proposed for wave energy
converting power plants:

linear permanent magnet synchronous machines, with surface as well as buried
permanent magnets [33];

Vernier hybrid linear machines [34, 35];

air cored permanent magnet tubular linear machines [36, 37].
14
1.2
Chapter 1
Linear generators
Electrical machines are almost exclusively of a rotary configuration, primarily due to
their application. Notable exceptions to this, where some research has been performed,
are for transportation purposes (levitating high speed trains, 'Maglev'), projectile
launchers or small biomedical actuators. These are all high speed operations, with the
electrical machine acting as an actuator / motor.
For the applications presented in the previous chapters, the concept of a linear machine
offers the opportunity to simplify the mechanics of the system if it can be used as a slow
speed generator. The guiding principles for rotary and linear machines are identical,
simply in a different plane of motion. As such, all equations describing a rotary machine
are equally applicable to a linear machine, with the slight modification of some terms,
such as torque to force. Conceptually, a linear machine can be thought of as the
circumference of its rotary counterpart flattened out.
The power absorbed by a linear machine is equal to the product of the reactive force it
can deliver and the velocity at which it is displaced. In order to develop greater power
for a given velocity, therefore, it is necessary to be able to exert a greater force and thus
a high current.
Due to strictly dynamic constraints, the mover iron core mass should be smaller as
possible, consequently the mover yoke saturation is generally not negligible [16].
Of the various linear machine topologies which might be employed to convert the
kinetic energy into electric energy, tubular configuration is the most interesting solution
since it has zero net radial force between mover and stator, no end-windings and
volumetrically efficient. One of the disadvantages of tubular topology is the lamination.
Because the magnetic flux lines are parallel to the movement direction, mounting the
lamination perpendicular to the machine shaft would mean that the magnetic flux
should pass from one lamination to the next trough the insulation, resulting a large
equivalent air-gap and poor performance, due to high losses. For this reason the
lamination should be placed parallel to the translator shaft, raising the problem of
aligning them in the air-gap, where they would come in contact, while at the outer part
of the stator a gap would appear between adjacent laminations, resulting in a
complicated construction process with high costs. Different solutions were taken into
account to solve this problem. The first possibility is to use, for the stator core, of
molded powder materials, such as soft magnetic composites (SMC). These kinds of
materials allow a simple construction of stator core by join different pieces of SMC
material. This solution allows, using a simple ring structure as coil, to achieve a high
Introduction
15
modularity solution. Another way to solve the problem is to replace the tubular
topology with a structure with more statoric parts (from 2 to 8), mounted around the
translator (Figure 1.10) [38]. Increasing the number of stator parts, the approximation to
the tubular structure is more accurate, but it involves a higher cost.
Figure 1.10. Four-sided linear generators.
The permanent magnets allow achieving high efficiency, power density and lighter
moving part. However, the uses of PMs require a more sophisticated temperature study.
There are different ways to magnetize the permanent magnet: radial magnetization,
Halbach magnetization and quasi-Halbach magnetization are the most common
solutions.

Radial magnetization: is the simplest kind of magnetization. The advantages are
the production low-cost cost and the high modularity, the drawbacks are the high
values of cogging force and the high value of the back iron mover saturation;

Halbach magnetization: this kind of magnetization is provided by a radial and
axial sinusoidal magnetization (as shown in Figure 1.11a), this involves a lower
harmonic content. In this case it is generally possible to remove the ferromagnetic back iron as the axially magnetized magnets essentially provide a
return path for the radial air-gap flux, for this reason the moving part results
lighter and thus it is possible achieving higher mechanical dynamic. On the other
hand, Halbach magnetization does not allow to achieve the same values of
linkage flux with the same amount of PM material respect to radial
magnetization, furthermore this solution is generally more expensive than the
previous one;
16
Chapter 1

Quasi-Halbach magnetization: this kind of magnetization is provided by a radial
and axial square wave magnetization pattern (as shown in Figure 1.11b). This
solution finds place between the previous two, as it should be possible to remove
the ferro-magnetic back iron and it results in a quite simple construction process
as well.
mr
mz
(a)
mz
mr
(b)
Figure 1.11. Halbach magnetization (a) and quasi-Halbach magnetization (b).
1.3
Scope of research
The scope of this research can be summarised as analysis and optimization of linear
generators for renewable energy applications, particularly focused on free-piston
Stirling engine (FPSE) cogeneration systems. Another topic of this research activity is
to develop the dynamic model of the whole cogeneration system, composed by external
combustion engine (ECE), linear generator and power electric converter. More
specifically, the aim of this thesis is, firstly, to develop a model of linear generator able
to describe the behaviour of the machine, in terms of force, flux linkage, electromotive
force, losses and efficiency. Then, a parametric analysis, in terms of magnetization
pattern, current supply strategy and iron core materials, is carried out in order to
optimize the linear generator performance. Finally, a dynamic model of the whole
cogeneration system is developed with the aims to describe the dynamic operating
condition of the system and achieve the best operation condition, in terms of mechanical
parameter (i.e. spring constant) values, oscillating frequency and control strategy. The
mathematical model was checked out by some experimental tests, aims to identify more
accurately the material magnetic properties, since results evidenced remarkably poorer
characteristics with respect to the material reference datasheets, likely due to
manufacturing processes.
Introduction
1.4
17
Methodology adopted
Analytical models [39-41] would allow to achieve a good parameter estimation, but the
complexity of geometry and the several parameter quantity that have to change advise
against the use of such kind of models for optimization point of view. Besides, such
kinds of models often do not take into account the iron core saturation phenomena, but
usually not negligible. On the other hand, finite element analysis (FEA) generally
requires a greater computational effort and time consumption for preliminary machine
design point of view.
The solution to such problem is given by a semi-analytic model [39], in which a suitable
number of FEA is carried out in order to study any linear generator configuration. In
order to reduce the computational effort, an approach combining harmonic
decomposition and automated magnetostatic FEA sequences (HFEA) is proposed in this
thesis. The proposed method allows to simulate the PM motion, different magnetization
profiles and different supply strategies, by analysing fixed magnetization space
harmonic distributions. The mover saturation, generally not negligible, is taking into
account by a suitable number of current sheets on the mover yoke surface, which
reproduce the equivalent magnetic permeability decay due to magnetic saturation. The
field solution is obtained by superposing the effect of each magnetization space
harmonic, the effect of current coil and the effect of equivalent current sheet. The mover
axial force is calculated through Maxwell stress tensor evaluation, by radial and axial
component of the flux density on the air gap.
The proposed method allows, once carried out FEAs for different magnetization
harmonic order and stored the radial and axial component of the air gap flux density, to
evaluate the machine performance for different magnetization and supply strategy.
Another feature of the proposed method is the possibility to carry out a performance
optimization of the machine, adjusting the magnetization profile as well as the current
supply waveform.
The efficiency can be estimated as well by evaluating the flux density in some point
inside the stator and mover iron cores, in order to calculate the iron core losses.
Finally, a dynamic model of the whole cogeneration system, arranged by free-piston
engine (Stirling engine), linear generator and possibly by an electronic power converter,
was developed. The dynamic model has a double purpose: one the one hand allows to
study the dynamic performances of the system, on the other hand allows optimizing the
system in order to achieve the best performances under given constraints. The latter
18
Chapter 1
point is an important challenge because the cogeneration system design should be done
on the Stirling engine – linear generator system due to their very strict connection.
1.5
Thesis structure
In the chapter 2 a literature review is presented. The state of art is important in order to
understand in which point the research activity have to focus. Chapter 3 describes the
Stirling engine features and the working principle. Particularly, it focuses the attention
on different engine typologies in order to understand which is the best configuration.
Chapter 3 presents the advantages and disadvantage of a Stirling engine. Free-piston
configuration is described. Chapter 4 describes the linear generator theory. Different
typologies and configurations are discussed. The main features of tubular permanent
magnet linear generator are pointed out. Chapter 5 presents the research activity
focusing the attention on linear generator models utilised to design and optimize the
machine. The electromagnetic parameters take into account for a parametric
optimization are the magnetization profile, the current supply strategy and the ferromagnetic materials utilised for iron cores. Then some experimental tests are carried out
in order to describe the real behaviour of the permanent magnets and of the iron core
materials, which generally may differ to manufacturer data-sheet values. Finally a
dynamic model is developed in order to describe the dynamic behaviour of the whole
cogeneration system and carry out an optimization. Chapter 6 concludes the thesis by
providing an overview of the work carried out in this research listing also possible
future work.
Chapter 2
Stirling engines
2.1
Introduction
The Stirling engine (shown in Figure 2.1), sometimes called hot-air engine, converts
thermal energy into mechanical energy. It was invented by Robert Stirling in 1816. It is
a closed cycle regenerative engine that compresses a fixed amount of working gas at
low temperature and then, after heating, expands the gas at high temperature. By
cooling the expanded gas it completes the cycle. Since the expansion at high
temperature gives a larger amount of work than is consumed by the compression at low
temperature, the result is a net mechanical work.
In order to obtain two regions at different temperatures (low and high), a piston called
displacer divide the working space of the engine in two spaces: compression and
expansion space. As the high pressure gas expands, it moves another piston called
power piston, hence doing work. After the gas is expanded, the power piston
compresses it by moving inward.
There are three heat exchangers between the compression (cold) and expansion (hot)
spaces: cooler, regenerator and heater. The cooler and the heater are used for more
effective cooling and heating of the working gas. They bring the gas at compression and
expansion temperatures respectively, before it enters these spaces. The regenerator is a
heat store, generally made of fine wires in the form of matrix. It captures the heat from
the expanded hot gas that flows from the expansion space to the compression space and
returns it to the compressed cold gas when it flows back to the expansion space from
compression space. This method allows saving a considerable amount of heat energy
and thus increasing the engine efficiency.
20
Chapter 2
Figure 2.1. Stirling Engine cross-section.
Stirling engines working theoretically on the Stirling cycle, or its modifications, in
which compressible fluids, such as air, hydrogen, helium, nitrogen or even vapours, are
used as working fluids. The Stirling engine offers possibility for having high efficiency
engine with less exhaust emissions in comparison with the internal combustion engine.
The earlier Stirling engines were huge and inefficient. However, over a period of time, a
number of new Stirling engine models have been developed to improve the deficiencies.
The modern Stirling engine is more efficient than the early engines and can use any
high temperature heat source. Since the Stirling engine is an external combustion
engine, most sources of heat can power it (multi-fuel working), including combustion of
any combustible material, field waste, rice husk or the like, biomass methane and solar
energy. In principle, the Stirling engine is simple in design and construction, and can be
operated easily.
The Stirling engine could be used in many applications and is suitable where [42]:

multi-fuel characteristic is required;

a very good cooling source is available;

quiet operation is required;

relatively low speed operation is permitted;

constant power output operation is permitted;

slow changing of engine power output is permitted;

a long warm-up period is permitted.
From its invention, Stirling engine was used in several different parts of the world,
especially in Europe and in the USA, to run a wide variety of appliances including
Stirling engines
21
water pumps, cooling fans, sewing machines, laboratory centrifuges and mixers and
organ air pumps.
In the beginning of the before century, the Stirling engine was surpassated by the
internal combustion engine, which can be easily linked to almost any kind of motor
driving systems thanks to its high specific power rate.
A new interest in the Stirling engines was initiated in 1937 by Philips Company. It
started a research program to update the old hot-air engine, to produce an electric power
generator which could run electronic equipment in remote places. They found out that
using, as working fluid, lighter gases then air, such as helium or hydrogen, increase the
efficiency and the specific power of the engine.
Further research on Stirling engine was carried out by General Motors in 1958 for space
and underwater power systems and vehicle applications. Also the NASA developed
several kinds of Stirling engines for aerospace applications. Nowadays, Sunpower Inc.
is producing and developing several kinds of Stirling engines.
2.2
Development of Stirling engines
2.2.1
First era of Stirling engines
The Stirling engine was the first invented regenerative cycle heat engine. Robert Stirling
patented the Stirling engine in 1816 (patent no. 4081). Engines based upon his invention
were built in many forms and sizes until the turn of the century. Because Stirling
engines were simple and safe to operate, ran almost silently on any combustible fuel,
and were clean and efficient compared to steam engines, they were quite popular [43].
These Stirling engines were small and the power produced from the engine was low
(100 W to 4 kW).
In 1853, John Ericsson built a large marine Stirling engine having four 4.2 m diameter
pistons with a stroke of 1.5 m producing a brake power of 220 kW at 9 rpm [44]. The
first era of the Stirling engine was terminated by the rapid development of the internal
combustion engine and electric motor.
22
Chapter 2
2.2.2
Second era of Stirling engines
The second era of the Stirling engine began around 1937 [43], when the Stirling engine
was brought to a high state of technological development by the Philips Research
Laboratory in Eindhoven, Holland, and has progressed continuously since that time.
Initial work was focused on the development of small thermal-power electric generators
for radios and similar equipment used in remote areas [2, 4].
New materials were one of the keys to Stirling engine success. The Philips research
team used new materials, such as stainless steel [43]. Another key to success was a
better knowledge of thermal and fluid physics than in the first era. The specific power
of the small ‘102C’ engine of 1952 was 30 times that of the old Stirling engines [46].
The progress in further development made by Philips and many other industrial
laboratories, together with the need for more energy resources, has sustained the second
era of Stirling engine development until today [43].
2.2.3
Stirling engines for industries
Intensive research by Philips and industrial laboratories led to the development of small
Stirling engines with high efficiencies of 30% or more. In 1954, Philips developed an
engine using hydrogen as a working fluid. This engine produced 30 kW for a maximum
cycle temperature of 977 K at 36% thermal efficiency. The efficiency of the same
engine was later improved to 38%. The experimental studies of engines of various sizes
up to 336 kW were studied [45].
Other attempts to further develop Stirling engines under license of Philips, were carried
out by General Motors from 1958 to 1970 [51]. Other licenses were granted by Philips
to United Stirling AB of Malmo, Sweden in 1968 and to the West German consortium
of MAN and MWM in 1967 [51]. In 1973, the Philips/Ford 4-125 experimental
automotive Stirling engine accomplished a specific power of over 300 times that of the
early Stirling engines [44].
2.2.4
Stirling engines for rural and remote areas
Trayser and Eibling [47] carried out a design study to determine the technical feasibility
of developing a 50 W portable solar-powered generator for use in remote areas. The
results of their study indicate that it is possible to build a solar-powered lightweight
portable, reliable, Stirling engine at a reasonable cost.
Stirling engines
23
Gupta et al. [7] developed 1 and 1.9 kW solar-powered reciprocating engines for rural
applications. Engine efficiencies were found to be between 5.5 and 5.7% and overall
efficiency was found to be 2.02% [49].
Pearch et al. [50] proposed and analysed a 1 kW domestic, combined heat and power
(DCHP) system. The results show that 30% of a home’s electrical demand could be
generated and electricity cost could be reduced by about 25%.
Podesser [51] designed, constructed and operated a Stirling engine, heated by the flue
gas of a biomass furnace, for electricity production in rural villages. With a working gas
pressure of 33 bar at 600 rpm and a shaft power of 3.2 kW, an overall efficiency of 25%
was obtained. He expected to extend the shaft power to 30 kW in the next step.
Dixit and Ghodke [52] designed compact power generating systems capable of using
the combination of a wide variety of solid fuels as a local power source. The system was
a heat pipe-based, biomass energy-driven Stirling engine. The macroscopic thermal
design of the engine along with the calculation of various energy losses was reported.
2.2.5
Stirling engine optimization
Usually the design point of a Stirling engine will be somewhere between the two limits
of: maximum efficiency point and maximum power point. Markman et al. [53]
conducted an experiment using the beta-configuration of the Stirling engine to
determine the parameters of a 200 W Stirling engine by measuring the thermal-flux and
mechanical-power losses. The aim of the project was to optimize and increase the
engine efficiency.
Orunov et al. [54] presented a method to calculate the optimum parameters of a singlecylinder Stirling engine. They concluded that mass and size characteristics of the engine
could be improved by using the correct choice of the optimal parameters which would
result in larger efficiency.
Abdalla and Yacoub [55] studied the feasibility of using waste heat from a refuse
incinerator with a Stirling engine. Heat from incineration was used to power the
desalination plant and the Stirling engine. Using saline feed raw water as the cooling
water and by assuming 50% heat recovery efficiency, they claimed that the engine
efficiency could be improved and a thermal efficiency of 27% was obtained.
Nakajima et al. [56] developed a 10 g micro Stirling engine with an approximately 0.05
cm3 piston swept volume. An engine output power of 10 mW at 10 Hz was reported.
The problems of scaling down were discussed.
24
Chapter 2
Aramtummaphon [57] tested an open cycle Stirling engines by using steam heated from
producer gas. The first engine produced an indicated power of about 1.36 kW at a
maximum speed of 950 rpm, while the second engine, improved from the first one,
produced an indicated power of about 2.92 kW at a maximum speed of 2200 rpm.
Hirata et al. [58] evaluated the performance of a small 100 W displacer-type Stirling
engine Ecoboy-SCM81. An analysis model using an isothermal method considering a
pressure loss in the regenerator, a buffer space loss, and a mechanical loss for the
prototype engine was developed to improve the engine performance. After the
effectiveness of the analysis model was evaluated, some improvements for the
prototype engine were discussed.
Costea and Feidt [59] studied the effect of the variation of the overall heat transfer
coefficient on the optimum state and on the optimum distribution of the heat transfer
surface conductance or area of the Stirling engine heat exchanger. The results pointed
out either an optimum variation range for some model parameters, or some significant
differences of the power output, source and sink temperature differences, heat transfer
characteristic values with respect to each of the studied cases.
Wu et al. [60] analyzed the optimal performance of a Stirling engine. The influences of
heat transfer, regeneration time and imperfect regeneration on the optimal performance
of the irreversible Stirling engine cycle were discussed. The results of their work
provided a new theoretical basis for evaluating performance and improving Stirling
engines.
Wu et al. [61] studied the optimal performance of forward and reverse quantum Stirling
cycles. The finite time thermodynamic performance bound, optimization criteria and
sensitivity analysis were presented. The results showed that the quantum Stirling cycle
was different from the classical thermodynamic one. This was because of the different
characters of the working fluids.
Wu et al. [62] studied the finite-time exergo-economic optimal performance of a
quantum Stirling engine. The maximum exergo-economic profit, the optimal thermal
efficiency and power output corresponding to performance bound of an endo-reversible
quantum Stirling engine were presented. The result of this work showed a profit bound
for designing a real Stirling engine working with a quantum fluid.
Gu et al. [63] attempted to design a high efficiency Stirling engine using a composite
working fluid, e.g. two-component fluid: gaseous carrier and phase-change component
and single multi-phase fluid, together with supercritical heat recovery process. The
results were compared with those of common Stirling engines. The criteria for the
Stirling engines
25
choice of working fluid were discussed. Calculation by using sulphur hexafluoride as
the working fluid was given as an example to show the thermal efficiency and optimum
condensing pressure and temperature.
Winkler and Lorenz [64] described the integration of thin tubular solid oxide fuel cells
(SOFCs) and heat engine system. The heat engines investigated were microturbines and
Stirling engines. A high system efficiency, low specific volumes, and a small available
unit of solid oxide fuel cells was expected from Stirling engine system. Further
development for industrial projects was recommended.
Hsu et al. [65] studied the integrated system of a free-piston Stirling engine and an
incinerator. The performance of a free-piston Stirling engine was investigated using the
averaged heat transfer model. The efficiency and the optimal power output, including
the effect induced by internal and external irreversibility, were described.
Petrescu et al. [66] presented a method for calculating the efficiency and power of a
Stirling engine. The method was based on the first law of thermodynamics for processes
with finite speed and the direct method for closed systems. The results showed good
agreement with the actual engine performance obtained from 12 different Stirling
engines over a range from economy to maximum power output.
2.3
Development of solar-powered Stirling engines
2.3.1
Solar-powered Stirling engines in the first era
In 1864, Ericsson [67] invented a solar-powered hot air engine using a reflector to heat
the displacer cylinder hot-end. Jordan and Ibele [68] reported that between 1864 and
1870, Ericsson used parabolic trough collectors to heat steam and used steam to drive
his engine [69]. In 1870, the Stirling engine was adapted by Ericsson to operate with
solar energy ([70] cited in Ref. [71]).
Spencer [69] reported that in 1872, Ericsson built an open-cycle hot-air engine using a
spherical mirror concentrator. This engine was the first solar-powered hot air engine. It
was also reported that the engine could work at 420 rpm at noon on a clear sky day in
New York .
Meinel and Meinel [72] commented on the conclusions made by Ericsson pointing out
that solar-powered engines would be economical only in remote areas where sunshine
was available and pointing out their cost was 10 times higher than conventional engines.
The amount of solar-powered Stirling engines built in the first era was quiet small.
26
Chapter 2
Reader and Hooper [73] reported that in 1908 a solar-powered Stirling engine was
proposed for a water pumping system.
2.3.2
Solar-powered Stirling engines in the second era
During 1950–1955, Ghai and Khanna worked with an open cycle solar-powered Stirling
engine using a parabolic collector in India [45, 49, 71]. The solar energy was focused on
the metal engine head but they had problems with heat loss.
Jordan and Ibele [68] described the 100 W solar-powered Stirling engine for water
pumping.
Ghai [74] pointed out the point of economy and technical simplicity of a solar powered
device even though its competitor was the internal combustion engine.
Later works [75–77] related to solar-powered Stirling engines and heat pipes were
previously reviewed by Spencer [49]. Other works concerning the different varieties
and arrangements of the cylinder and displacer including construction and operation of
solar-powered Stirling engines [71–76] have been reported by Daniels [71]. More
details of solar-powered Stirling engines can be found from Jordan and Ibele [68] and
Jordan [77].

Stirling engines with transparent quartz window: Daniels [71] and Spencer
[49] described many research works on solar-powered Stirling engines with
transparent quartz windows and related works [47, 79–83]. The problems of this
engine could be with the heat transfer and fouling effects. However, Walpita
[45] proposed a design for a solar receiver made from a spiral steel tube of 3.175
mm outside diameter for a solar- powered Stirling engine. The heat transfer from
solar radiation to the working fluid was analysed and an optimum heat transfer
area was obtained.

Stirling engines with concentrating collector: the review work on a 15 W solarpowered Stirling engine with concentrating collector was described by Daniels
[71]. Other works on Stirling engine with concentrating collectors [83–91] have
been comprehensively reviewed by Spencer [90].
Ahmed et al. [92] reported briefly the operation of a 50 kW solar-powered
Stirling engine for electricity production using a single membrane dish
concentrator and hydrogen as a working gas. They described the problems of the
tracking system due to errors in design and difficulties in starting during the
winter season due to improper control part selection.
Stirling engines
27
Childs et al. [93] presented an innovative concept to determine the costeffectiveness of new approaches to solar-powered desalting technology. These
approaches combined modern solar conversion technology with newly
developed, hydraulic-driven pumping and energy recovery technology for solarpowered desalting. A solar dish concentrator-Stirling engine electric module,
having overall efficiency of 22% for 10 h/day average production, was reported.
Audy et al. [94] reported a solar dynamic power system using a Stirling engine
for space station applications. Theoretical models for four different
representative orbit configurations were developed. The simulation results were
compared to those of a solar dynamic power module using a Brayton gas
turbine. Moreover, they showed that the complex unsteady behaviour with either
the Brayton cycle or Stirling cycle can be simplified on the basis of
parameterizations and energy balances.

Solar dish/engine technology: solar dish/engine systems convert solar energy to
mechanical energy and then electrical energy. In order to obtain the required
temperature for efficient energy conversion, solar dish/engine systems use a
mirror array to track the sun. These systems can be characterized by efficiency,
modularity, autonomous operation and the capability to work with either a
conventional fuel or solar energy. Among many solar technologies, these
systems have been accepted to be the systems with the highest solar-to-electrical
conversion efficiency [95].
High-temperature and high-pressure Stirling engines working with hydrogen or
helium are normally used in solar dish/Stirling system engines. Modern high
performance Stirling engines usually operate with a working fluid temperature
of over 973 K and a pressure as high as 200 bar. The efficiencies of conversion
from heat to electricity of the best Stirling engines are about 40% [44, 96, 97].
At this moment the kinematic Stirling engines, the Kockums (United Stirling) 495 25-kWe, Stirling Thermal Motors STM 4-120 25-kWe, and the SOLO 161
11-kWe are the examples for the engines used for dish/Stirling systems.
Solar dish/engine technology is one of the oldest solar technologies. In the late
1970s and early 1980s modern solar dish/engine technology was developed by
Advanco Corporation, United Stirling AB, McDonnell Douglas Aerospace
Corporation (MDA), the US Department of Energy (DOE), and NASA’s Jet
Propulsion Laboratory.
28
Chapter 2
It was reported that [95, 98] the Advanco Vanguard system, 25-kWe nominal
output module, using the United Stirling Power Conversion Unit (PUC),
obtained a solar to electric conversion efficiency of 29.4%. MDA attempted to
commercialize a system consisting of their own designed dish and the United
Stirling PCU. Before the program was cancelled in 1986, MDA produced eight
prototype systems. The rights to the MDA hardware were sold later to Southern
California Edison (SCE). In 1988, an annual efficiency of over 23% was
expected to be obtained without outages [99–102].
The Dish/Stirling Joint Venture Program (DSJVP) was initiated in 1991 [103].
The aim of the program was to develop a 5–10-kWe dish/Stirling system for
applications in remote areas. The Utility Scale Joint Venture Program (USJVP)
for 25-kWe dish/engine system was started in late 1993 [105]. The comparably
priced systems obtained by the lower-cost stretch-membrane design and its
improved operational flexibility were projected by SAIC [106].
The Advanced Dish Development System (ADDS) project plan and technical
approach were reported by Diver et al. [106]. The aims of the project were to
develop and validate a 9-kWe dish/Stirling solar power system for remote power
markets.
The system was composed of the WGAssociates solar concentrator and controls,
and the SOLO 161 Stirling power conversion unit. The main system
components, features, test results and project status were also reported.
Davenport et al. [107] reported the operational results and experiences from a
prototype of the SunDish system at the Salt River Project (SRP). This project
was executed through the cooperation of SRP, SAIC, STM, and DOE. The
methane gas collected from a landfill was used as fuel when solar energy was
not available. They also discussed the design changes and system improvements
resulting from operation with the prototype of the SunDish system.
Davenport et al. [108] reported the operation of the second-generation of
dish/Stirling
power
systems
(SAIC/STM
SunDish
systems).
Many
improvements to both the engine and dish subsystems were made to increase
reliability, to improve system performance, to simplify installation, and to
correct problems encountered during operation. They reported that the power
output was improved from below 20 kW in 1998 to over 23 kW in 2002. An
instantaneous peak power of 23.3 kW and efficiency of 26% were observed.
Stirling engines
2.3.3
29
Solar-powered Stirling engine optimization
When a solar collector system is used as a heat input source for power generation, the
solar collector and working conditions giving the optimum values of the cost of the
system and the optimum power output must be considered. Some theoretical work to
optimize solar-powered Stirling engine design was carried out by Umarov et al. [109,
110], however the applications on the engine were not shown in the research papers.
Howell and Bannerot [111] determined the optimum value of the outlet temperature of
the solar collector to maximize the work output of Carnot, Stirling, Ericsson, and
Brayton cycle engines powered by a solar collector. Eldighidy et al. [112] optimized the
conditions for maximum solar energy absorbed by a flat-plate collector used with a
plane reflector. A simple flat-plate collector/flat-sheet reflector was analysed
completely. Later, Eldighily [113] theoretically investigated the optimum outlet
temperatures of the solar collector for the maximum work output for an Otto airstandard cycle with ideal regeneration. This work may be applied to an air-standard
Stirling cycle.
Gordon [114] examined the accuracy of the energetic optimization of solar-driven heat
engines. The results were obtained for the two limiting cases of maximum efficiency
and maximum power. Altfeld et al. [115, 116] minimized the sum of exergy losses,
including exergy losses by absorption of radiation at the absorber temperature level by
maximizing the net exergy flow. The optimum designs of the absorbers and flow ducts
were presented.
Costea et al. [117] studied the effect of pressure losses and actual heat transfer on solar
Stirling engine cycle performance. The results indicated that when the engine was
operated at the optimum temperature, the real cycle efficiency was approximately half
the ideal cycle efficiency.
Chen et al. [118] proposed a non-image focusing heliostat consisting of a number of
grouped slave mirrors for solar-powered Stirling engines. An experiment with a low
power Stirling engine was reported. They proposed that a solar-powered engine of 20–
50 kW was most in demand and would be less costly under stationary conditions.
2.3.4
Development of LTD Stirling engines
Haneman [119] studied the possibility of using air with low temperature sources. An
unusual engine, in which the exhaust heat was still sufficiently hot to be useful for other
purposes, was constructed. In practice, such an engine would produce only little useful
30
Chapter 2
work relative to the collector system size, and would give little gain compared to the
additional maintenance required [120].
A simply constructed low temperature heat engine modelled on the Stirling engine
configurations was patented in 1983 by White [121]. White suggested improving
performance by pressurizing the displacer chamber. Efficiencies were claimed to be
around 30%, but this can be regarded as quite high for a low temperature engine.
In 1984, O’Hare [122] patented a device passing cooled and heated streams of air
through a heat exchanger for changing the pressure of air inside the bellows. The
practical usefulness of this device was not shown in detail as in the case of Haneman’s
work.
Kolin [67] experimented with a number of LTD Stirling engines, over a period of many
years. In 1983, he presented a model that worked on a temperature difference between
the hot and cold ends of the displacer cylinder of as low as 15 °C. After Kolin published
his work, Senft [43] made an in-depth study of the Ringbom engine and its derivatives,
including the LTD engine. Senft’s research in LTD Stirling engines resulted in the most
interesting engine, which had an ultra-low temperature difference of 0.5 °C. It is very
difficult to create any development better than this result. Senft’s work [123] showed
the principal motivation for Stirling and general heat engines, their target being to
develop an engine operating with a temperature difference of 2 °C or lower.
Senft [43] described the design and testing of a small LTD Ringbom Stirling engine
powered by a 60° conical reflector. He reported that the 60° test conical reflector
producing a hot-end temperature of 93 °C under running conditions, worked very well.
2.4
Mechanical configurations of Stirling engines
Stirling engines can be classified by their drive method: constrained drive (kinematic
Stirling engines) and unconstrained drive (free-piston Stirling engines). In the first one
typology, the motion of the power piston and the displacer are constrained by cranks
and connecting rods. Instead, in the free-piston Stirling engines, the power piston and
the displacer are moved by the variation of the pressure of the working gas.
The kinematic Stirling engine can, in turn, be classified in alpha, beta and gamma
configuration (Figure 2.2). Each configuration has the same thermodynamic cycle but
has different mechanical design characteristics [124].

alpha-configuration: the displacer is not used. Two pistons, called the hot and
cold pistons, are used on either side of the heater, regenerator, and cooler. These
Stirling engines
31
pistons move uniformly in the same direction to provide constant-volume
heating or cooling processes of the working fluid. When all the working fluid
has been transferred into one cylinder, one piston will be fixed and the other
piston moves to expand or compress the working fluid. The expansion work is
done by the hot piston while the compression work is done by the cold piston.

beta-configuration: the displacer and the power piston are incorporated in the
same cylinder. The displacer moves working fluid between the hot space and the
cold space of the cylinder through the heater, regenerator, and cooler. The power
piston, located at the cold space of the cylinder, compresses the working fluid
when the working fluid is in the cold space and expands the working fluid when
the working fluid is moved into the hot space.

gamma-configuration: it uses separated cylinders for the displacer and the
power pistons, with the power cylinder connected to the displacer cylinder. The
displacer moves working fluid between the hot space and the cold space of the
displacer cylinder through the heater, regenerator, and cooler. In this
configuration, the power piston both compresses and expands the working fluid.
The
gamma-configuration
with
double-acting
piston
arrangement
has
theoretically the highest possible mechanical efficiency. This configuration also
shows good self-pressurization [124]. However, the engine cylinder should be
designed in vertical type rather than horizontal in order to reduce bushing
friction [126].
(a)
(b)
(c)
Figure 2.2. Stirling engine configurations: alpha (a), beta (b) and gamma (c).

Free-piston Stirling Engine (FPSE): In addition to general characteristics of
the Stirling engine, the free-piston Stirling engine, shown in Figure 3.3, has its
own unique features that make it more appropriate for the use under certain
circumstances. These characteristics are:
32
Chapter 2
 it does not required an external high pressure seal, since it can be
hermetically sealed to the environment;
 it can be produced at low cost, since it has only two moving parts and
has no complicated drive mechanism;
 the power piston and displacer have no loads thanks to the lack of
connecting rods, and as a consequence, the engine can operate for a long
life without needing lubrication.
 the FPSE can be made to operate very quietly since all the motions and
forces hence the resulting vibrations are in the axial direction to the
engine and may be easily isolated.
The FPSE has advantages for operation in remote locations with its potential to
run for a long time free from maintenance. For such purpose, FPSE finds
applications in all those fields in which maintenance is a problem, such as wave
energy converters, army and aerospace applications.
Figure 2.3. Free-piston Stirling engine.

Low temperature differential (LTD): it can be run with small temperature
difference between the hot and cold ends of the displacer cylinder [46]. It is
different from other types of Stirling-cycle engines, which have a greater
temperature difference between the two ends, and therefore the power developed
from the engine can be greater. A LTD Stirling engine is shown in Figure 2.4.
Stirling engines
33
(a)
(b)
Figure 2.4. LTD Stirling Engine:the scheme (a) and a prototype (b).
LTD engines may be of two designs. The first uses single-crank operation where
only the power piston is connected to the flywheel, called the Ringbom engine.
This type of engine, that has been appearing more frequently, is based on the
Ringbom principle. A short, large-diameter displacer rod in a precise-machined
fitted guide has been used to replace the displacer connecting rod [67]. The other
design is called a kinematic engine, where both the displacer and the power
piston are connected to the flywheel. The kinematic engine with a normal 90°
phase angle is a gamma-configuration engine [67].
Some characteristics of the LTD Stirling engine are as follows [67]:
 displacer to power piston swept volumes ratio is large;
 diameter of displacer cylinder and displacer is large;
 displacer is short;
 effective heat transfer surfaces on both end plates of the displacer
cylinder are large;
 displacer stroke is small;
 dwell period at the end of the displacer stroke is rather longer than the
normal Stirling engine;
 operating speed is low.
LTD Stirling engines provide value as demonstration units, but they
immediately become of interest when considering the possibility of power
generation from many low temperature waste heat sources in which the
temperature is less than 100 °C [83]. A calculation using the Carnot cycle
formula shows that an engine operating with a source temperature of 100 °C and
34
Chapter 2
a sink temperature of 35 °C gives a maximum theoretical thermal efficiency
(Carnot efficiency) of about 17.42%. If an engine could be built for achieving
50% of the maximum thermal efficiency, it would have about 8.71% overall
Carnot efficiency. Even the calculated thermal efficiency seems rather low, but
LTD Stirling engines could be used with free or cheap low temperature sources.
This engine should be selected when the low cost engines are put into
consideration.
Although the specific power developed by LTD Stirling engines is low,
lightweight and cheap materials such as plastics can be used as engine parts.
2.5
Principle of operation
The Stirling hot air engine is a simple type of engine that uses a compressible fluid as
the working fluid. Because the working fluid is in a closed system, there are no
problems with contamination and working fluid costs. Heat transfer to the working fluid
is very important. High mass flow is needed for good heat transfer. The working fluid
should be that of low viscosity to reduce pumping losses. Using higher pressure or
lower viscosity, or combinations thereof, could reduce the high mass flow required. The
Stirling engine could theoretically be a very efficient engine in upgrading from heat to
mechanical work with the Carnot efficiency. The thermal limit of the operation of the
Stirling engine depends on the material used for construction. Engine efficiency ranges
from about 30 to 40% resulting from a typical temperature range of 923–1073 K, and a
normal operating speed range from 2000 to 4000 rpm [124].
The ideal Stirling cycle has three theoretical advantages. First, the thermal efficiency of
the cycle with ideal regeneration is equal to the Carnot cycle. During the transfer
strokes, the regenerator, which is a typical temporary energy storage, rapidly absorbs
and releases heat to the working fluid which is passing through. Therefore, the quantity
of heat taken from the external heat source is reduced, this results in improving the
thermal efficiency (Figure 2.5).
The second advantage, over the Carnot cycle, is obtained by substitution of two
isentropic processes with two constant-volume processes. This results in increasing the
p–v diagram area. Therefore, a reasonable amount of work from the Stirling cycle is
obtained without the necessity to use very high pressures and large swept volumes, as in
the Carnot cycle. The Stirling cycle compared with the Carnot cycle between the same
given limits of pressure, volume, and temperature, is also shown in Fig. 2.5. The shaded
Stirling engines
35
areas 2C-2-3 and 1-4C-4 indicate the additional work available by replacing two
isentropic processes with two constant-volume processes. The Carnot cycle isothermal
processes (1-2C and 3-4C) are, respectively, extended to process 1–2 and 3–4. The
amount of available work is increased in the same proportion as the heat supplied to—
and rejected from—the Stirling cycle [44].
2
Carnot cycle
2C
1
(a)
Stirling cycle
V
(b)
Figure 2.5. Comparison between Stirling and Carnot cycles: p-V (a) and T-s (b) diagrams.
The third advantage has recently been discovered. Compared with all reciprocal piston
heat engines working at the same temperature limits, the same volume ratios, the same
mass of ideal working fluid, the same external pressure, and mechanism of the same
overall effectiveness, the ideal Stirling engine has the maximum possible mechanical
efficiency [43]. These three advantages reveal that the Stirling engine is a theoretical
equivalent of all heat engines [43].
The Stirling engine operation can be described as follow (Figure 2.6):

Isothermal compression process 1–2 (heat transfer from working fluid at
low temperature to an external sink) (Figure 2.6a): After the displacer has
pushed the working fluid into the cold space of the cylinder, where it was
cooled, it was then held stationary at its top dead center (TDC). This indicated
the state 1 and the pressure at this state is p1. The power piston is then being
pushed from bottom dead center (BDC) to TDC by flywheel momentum helped
by partial vacuum created by the cooling working fluid. The working fluid is in
the cold space and is under compression by power piston, which is approaching
TDC, and compressing working fluid from 1 to 2 at constant temperature. The
work done on the working fluid indicated by the area under process 1–2.

Constant-volume heating process 2–3 (heat transfer to the working fluid
from regenerator) (Figure 2.6b): The displacer is moving from TDC to BDC
36
Chapter 2
and transferring working fluid from the cold space to the hot space, while the
power piston remains stationary at its TDC, awaiting increase in pressure as a
result of expanding working fluid. The displacer is pushing the working fluid
into the hot space, passing through a regenerator which has stored heat, and
already a certain amount is being heated. Heat given up by the regenerator raises
the temperature and pressure of the working fluid from 2 to 3 at constant
volume. Heat stored in the regenerator is added to the working fluid.

Isothermal expansion process 3–4 (heat transfer to the working fluid at high
temperature supplied by an external source) (Figure 2.6c): After the displacer
has pushed all the working fluid into the hot space, with a corresponding
increase in pressure to the maximum, it is then kept at rest at its BDC. The
working fluid is in the hot space and is expanding to pressure p4, while a
constant temperature process 3–4 is maintained applied at the hot space. The
power piston is being pushed from TDC to BDC by the increased pressure, and
is applying force to the flywheel, thus creating mechanical energy. This energy
will be utilized throughout the remaining processes of the cycle. The work done
by the working fluid is indicated by the area under process 3–4.

Constant-volume cooling process 4–1 (heat transfer from the working fluid
to the regenerator) (Figure 2.6d): After the power piston has reached its BDC
and has supplied its energy to the flywheel, it remains stationary and is ready to
travel back to TDC under flywheel momentum and the sucking action of the
partial vacuum created by the falling pressure. The displacer is moving from
BDC to TDC and is transferring working fluid to the cold space where the
pressure will fall and a partial vacuum is created, through the regenerator,
causing a fall in temperature and pressure of the working fluid from 4 to 1 at
constant volume. Heat is transferred from the working fluid to the regenerator.
Stirling engines
37
Figure 2.6. Stirling engine operation process: isothermal compression (a), constant-volume
heating (b), isothermal expansion (c), constant-volume cooling (d).
For an air-standard Stirling cycle, the amounts of heat added
and rejected
per
unit mass of working fluid [J/kg] are as follows [111]:
where
and
[J/kgK],
(
)
(2.1)
(
)
(2.2)
is the fractional deviation from ideal regeneration (
for ideal regeneration),
for no regeneration
is the specific heat capacity at constant volume
is the source (heater) temperature in the Stirling cycle [K],
(cooler) temperature [K],
is the gas constant [J/kgK],
and
is the sink
are the specific
volumes of the constant-volume regeneration processes of the cycle [m3/kg],
respectively.
The Stirling cycle efficiency
can be expressed as [111]:
(
)
(
)
(2.3)
(
)
38
where
Chapter 2
and
.
Equation (2.3) shows that the efficiency is rated to source and sick temperatures and the
volume compression ratio
, particularly the efficiency increase if increase the
temperature differential between heater and cooler or the volume compression ratio.
Beale [15] noted that the power output of several Stirling engines observed could be
calculated approximately from the equation:
(2.4)
where
is the engine power output [W],
cycle frequency [Hz], and
the mean cycle pressure [bar],
the
is displacement of power piston [cm3]. The Beale formula
can be used for all configurations and for various sizes of Stirling engines.
Equation (2.4) may be written in a general form as follows:
(2.5)
The resulting dimensionless parameter
⁄
is called the Beale number. It is
clear that the Beale number is a function of both source and sinks temperatures. The
solid line in Figure 2.7 indicates the relationship between the Beale number and source
temperature. The upper bound represents the high efficiency, well-designed engines
with low sink temperatures, while the lower bound represents the moderate efficiency,
less well-designed engines with high sink temperatures [15].
Figure 2.7. Beale number as a function of source temperature [44].
Stirling engines
2.6
39
Analytical models of Stirling engine
Models used for calculation of the working process and performance of Stirling engines can
be classified as of empirical, analytical and numerical types.
2.6.1
Empirical modelling
A number of studies were performed to develop the empirical correlations for the
estimation of the engine power output and the engine efficiency.
West [125] proposed to use a dimensionless parameter called Beale number to predict
the engine power (named after Prof. William Beale).
The Beale number is defined in terms of a Stirling engine's operating parameters:
(2.6)
where
is the Beale number [-],
mean average gas pressure [bar],
is the power output of the engine [W],
is the engine cycle frequency [Hz] and
is the
is swept
3
volume of the power piston [cm ].
The Beale number with value of 0.015 was suggested to use for a wide range of Stirling
engines with different configurations. To take into account the effect of the temperature
range in the working cycle, West proposed to introduce into the Beale number the
empirical factor F of 0.035 to more accurately calculate the engine’s power output.
Senft in [126] recommended that for LTD Stirling engines the greater value of the
factor F is required to improve the accuracy of predictions.
To find an appropriate correlation for calculating the power output of the LTD Stirling
engine of the gamma configuration, Kongtragool and Wongwises in [127] investigated
the experimental performance of the engine. They came to the conclusion that the
Beale’s formula was not valid for conditions in which there were the high temperature
ratio. However, they recommended that the original Beale formula and the mean
pressure based power output formula with a certain magnitude of the factor F would
make it possible to accurately predict the engine’s power output.
Other example of development of an empirical correlation for prediction power out of
Stirling engines include works of Prieto et al. [128] and Prieto et al. [129]. They
analysed a wide range of experimental results and applied methods of the dynamic
similarity and quasi-static simulations for various LTD and HTD Stirling engines. To
improve the accuracy of predictions of the indicated power of Stirling engines, Prieto
and Stefanovskiy [128] used dimensionless analysis to characterise the gas leakage and
40
Chapter 2
mechanical power losses. The dimensionless mechanical power losses were determined
using the empirical data. Several Stirling engines were modelled using this
dimensionless analysis. In [129] Prieto et al. found that such the correlation based on
dimensionless analysis of the engine’s working process and different losses in the cycle
provided more accurate prediction of the performance of three LTD Stirling engines
compared to that obtained using the modified Kolin's model.
2.6.2
Thermodynamic modelling
Numerous mathematical models based on thermodynamic theory have been developed
for studying different designs of Stirling engines and their cycles. These models were
classified in [130-132] by splitting them in a big number of groups using various
criteria. In these investigations, however, the models are categorised into onedimensional and multi-dimensional model. In turn, one-dimensional model of Stirling
engines is subdivided into three subgroups called the first, the second and third-order
models.

One-dimensional first-order models: the first-order models are based on algebraic
equations of mass and energy conservation equations applied to describe the
working process of Stirling engines and such approach allows us to obtain analytical
equations to estimate power output and efficiency of machines.
Schmidt model described in [56] was the first in the series of the first-order models
and this model made it possible to derive analytical equation [57] to describe
pressure variations in the cycle assuming that an engine’s space could be presented
as three main spaces, namely the compression space, the expansion space and the
dead volume. It was assumed that processes in each of these spaces take place at a
corresponding constant temperature levels.
Toda et al. [58] presented a model which takes into account the effectiveness of the
drive mechanism in the Schmidt theory to analyse the engine performance of the
LTD Stirling engine of a gamma configuration. The results of experiments on a 10
kW LTD Stirling engine demonstrated that the mechanical effectiveness of the
engine has a strong influence on the shaft power of engine.

One-dimensional second-order models: the second-order models are superior to the
first-order thermodynamic models. Such the models may generally be based on
Schmidt analysis. Usually, an engine’s internal gas volume is separated into several
spaces. For each space a set of ordinary differential equations of the mass and
Stirling engines
41
energy conservation are used to describe heat and mass transfer processes. Various
heat and the pressure losses occurring in real engines are taken into account in the
model for a more accurate prediction of the engine’s performance.
Martini [132] presented such the isothermal model based on Schmidt theory and
taking into account heat losses and pressure losses. The temperatures of hot and cold
gases were obtained by taking into account temperature drops in the heater and
cooler. Numerical predictions for the GPU-3 Stirling engine had the 20 % error
compared to experimental results.
The first second-order model using an adiabatic analysis was developed by
Finkelstein in 1960 and described in [79]. The model defined as adiabatic
compression and expansion spaces whilst all heat exchangers were treated as perfect
heat exchangers with isothermal processes. Urieli and Berchowitz [138] described
the further development of the adiabatic model by Finkelstein. The model was
implemented as a computer code and corresponding differential equations were
solved using the fourth-order Runge-Kutta method. Additionally, to improve
numerical predictions, Urieli and Berchowitz [138] developed a so-called Simple
analysis which in fact was the adiabatic model with incorporated non-ideal heat
exchangers. Isothermal processes were still assumed in all heat exchangers but the
heater’s and cooler’s gas temperatures were then recalculated applying the heat
transfer equations.
There have been some Stirling engine models developed based on the Simple
analysis approach. Abbas et al. [139] developed method based on the Simple model
taking into account a presence of a non-ideal regenerator. The regenerator space was
divided into two parts in this work and the model also included calculation of
various losses such as the shuttle loss, regenerator loss, regenerator wall heat
conduction and pumping loss. The work loss due to a pressure drop was subtracted
from the indicated work. Another model using the Simple analysis was developed
by Strauss and Dobson [140] in which the pumping work loss and regenerator heat
loss described in Urieli [141] were evaluated. The original method to include into
calculations the regenerator wall heat conduction was described in this paper. Heat
losses and the pumping work loss were included into the value of the rejected heat
in the cooler. This caused the reduction of the pressure inside the engine and of the
power output. When compared to experimental data, the model demonstrated a good
accuracy in predicting performance of the General Motor GPU-3 Stirling engine.
42
Chapter 2
In addition to the Simple analysis method, the quasi steady flow model of Stirling
engines was published by Urieli and Berchowitz in [138]. The concept of non-ideal
heat exchanger was realised and introduced into the calculation procedure. The heat
exchanger wall temperature was defined as a constant during operation of the
machine. The model required the use of heat transfer correlations. The fourth-order
Runge-Kutta technique was used to solve a set of ordinary differential equations.
The pressure drop in heat exchangers was determined and it rectified pressure
values in the expansion and compression spaces. Urieli and Berchowitz [138]
validated the model with experimental data for the General Motor GPU-3 Stirling
engine and reported an improved accuracy of predictions compared to the Simple
analysis method.
Several mathematical models for Stirling engines were developed based on the quasi
steady flow approach. Thus such models were developed for analysis of the SOLO
161 Stirling engine with the alpha configuration to be used in the Eurodish 10 kWe
dish/Stirling system by Granodos et al. [142] and Nepveu et al. [143]. Granodos et
al. [142] developed their model which is combined with calculations of the cavity
type receiver in MATLAB. In this study the engine’s volume was split into 19
control volumes with ten control volumes being allocated to the regenerator. The
developed quasi steady flow model took into account the pressure drop in the
channels of heat exchangers. The model developed by Nepveu et al. [143]
considered 32 control volumes in the engine’s gas circuit with the regenerator being
split into 8 control volumes. As in the previous case, the model described in [143]
also coupled to calculations of the heat receiver. This model’s validation was carried
out using experimental results obtained by Reinalter et al. [144] on the Eurodish 10
kWe dish/Stirling unit installed at the CNRS-PROME laboratory, France. Parlak et
al. [145] analysed the performance of a gamma-type Stirling engine using the
developed model based on the quasi steady flow approach. The procedure for
determination of the pressure in each working space using the values of the
calculated pressure drop was main feature of the model. For calculations of medium
temperature difference engines, Tlili et al. [146] developed the second-order
adiabatic mathematical model based on the quasi steady flow approach described by
Urieli and Berchowitz [148]. They called this the Dynamic model and in the
modelling procedure took into account various thermal losses and pressure drops in
each component of a MTD Stirling engine. The losses included the energy
dissipation due to the pressure drop in all heat exchanges, internal and external
Stirling engines
43
conduction losses in the regenerator and shuttle losses caused by reciprocating
motion of a displacer. The gas spring hysteresis losses in the compression and
expansion spaces were taken into account to improve predictions of the engine
performance by Timoumi et al. [147]. In this paper authors also used for calibration
experimental data obtained from the General Motor GPU-3 Stirling engine and
reported that there was a good correlation between the numerical and experimental
results.
Schulz and Schwendig [148] described a mathematical model for simulation of
Stirling engines. The pressure drop and heat transfer in the oscillating flow of gas
through the heat exchanger were considered at turbulent flow conditions. The
ordinary differential equations for each control volume of a Stirling engine were
presented and the real gas state equation was used in the model. The absolute value
of Nusselt number for laminar and turbulent flow conditions were calculated in the
modelling process. It was found that there was a good agreement when theoretical
results were compared to experimental data for a Vuilleumier engine.
Mahkamov and Ingham [149] analysed the working process and influence of
mechanical losses in a 1-kW solar Stirling engine of an alpha configuration. The
second-order mathematical model which considered the hydraulic losses and heat
transfer in all control volumes. For the determination of the mechanical losses, cycle
variations in the frictional force between the sealing ring and the cylinder (single
and double sealing rings), in the sealing of the shaft, the frictional torque in all
rolling bearings and the aerodynamic resistance of a flywheel were calculated. As a
result of simulations, the predicted indicated power of 3190 W and the brake power
of 1585 W were obtained. It was shown that total mechanical losses in the Stirling
engine were approximately 50 % of the indicated power.
Ataer and Karabulut in [150] presented a mathematical model for analysis of a Vtype Stirling-cycle refrigerator which does not take into account the pressure drop in
the cycle. The machine was split into fourteen control volumes and the mass and
energy conservation equations were written for each volume. The constant heat
transfer coefficient was used to calculate heat transfer in the cooler and heater.
Karabulut et al. [151] further developed the model by adding a set of kinematic
equations to analyse a Stirling engine with a concentric piston and displacer.
However, the constant heat transfer coefficient was still used when considering heat
transfer on heater and cooler surfaces.
44
Chapter 2
Anderson et al. [152] developed a Stirling engine model with 13 control volumes
with ordinary differential energy and mass conservation equations written for each
volume. The model validation was carried out using a 9 kW Stirling engine and
there was an acceptable agreement in numerical predictions.
Several mathematical models of LTD Stirling engines were developed which took
into account differences in their design compared to conventional Stirling engines.
Martaj et al. [153] carried out thermodynamic study of a LTD Stirling engine using
the steady state energy, entropy and exergy analysis. The engine was divided into
three isothermal cells, namely the heating, cooling and regenerator cells and these
were analysed using energy, mass, entropy and exergy balance equations. It was
found that the thermal and exergy efficiencies for the whole machine were inversely
proportional to the regenerator's dead volume. They found that there was an
acceptable agreement between calculated and experimental work outputs for the
engine operating with a rotational speed of 199 rpm. Robson et al. [154] developed
a mathematical model of a LTD Ringbom Stirling engine which describes
parameters of the gas in expansion and compression spaces and in a regenerator.
The model takes into account a power piston that is connected to the flywheel with a
displacer being a free piston. The mass and energy conservation differential
equations for gas in each space were applied. Kinematical and kinetic equations of
the transmission mechanism were used to describe the characteristics of piston
motions and to determine the engine’s power output. The pressure variation
description obtained using the developed mathematical model was in a good
correlation with experimental results obtained on the LTD Stirling engine designed
by Senft. However, it was concluded that in this model a very small time step is
necessary during integration of equations to obtain convergence in the simulation of
the unsteady behaviour of the engine.

One-dimensional third-order models: the third-order model is more complex
compared to the second-order one and needs much more intensive numerical
calculations. In the third-order model mass, volume and momentum equations are
written for several control volumes of the gas circuit of the engine in the form of
partial differential equations.
Examples of such models are reviewed by Dyson et al. [155] and Anderson [156].
GLIMPS is the simulation code developed by Gedeon in 1986 and based on a thirdorder model. The 2D finite difference grid in terms of time and space was used to
solve the mass, energy and momentum partial differential equations. The governing
Stirling engines
45
equations were applied for both the working gas and the structure of engine. Tew et
al. [157] reported that the simulation of Stirling engines using Sage code was
performed in a NASA’s project.
H-FAST that is Stirling engine analysis code using the harmonic analysis method
was presented by Huang cited in [155]. The mass, energy and momentum equations
were formed in simplify assumption. Furthermore, the result was deducted with
several losses to obtain the correct result.
Anderson in [156] described in details the PROSA software (version 3.0) developed
by Thomas. Anderson et al. in [158] presented a one-dimensional modelling
approach for a Stirling engine which takes into account the compressibility in the
unsteady gas flow. The losses due to the finite temperature heat transfer, the flow
friction were described using the empirical correlations. When compared with
experimental data, simulation results were in a good agreement with experimental
data. It was pointed out that using correct empirical correlations for calculating the
heat transfer and friction in the regenerator and heat transfer in the displacer
clearance is very important for the accurate prediction of the efficiency and power
output. Furthermore, the optimisation study with the use of the shooting method was
performed in this work.

Multi-dimensional modelling: in addition to using various one-dimensional models
for analysis of a working process of Stirling engines, there have been some
investigations performed on application of two-dimensional (2D) and threedimensional (3D) modelling in order to better predict the working performance of
each component of Stirling engines.
Mahkamov and Ingham [159] demonstrated the application of 2D computational
fluid dynamic (CFD) modelling for analysis of the working process of a V-type
solar Stirling engine. An axisymmetric CFD model with the standard k-ε turbulence
model for compressible flow was chosen to numerically simulate the operational
cycle. Preliminary to that, the second-order mathematical model which includes heat
and hydraulic losses was used and its results were compared with the results of the
CFD model. The better accuracy in the prediction of the engine performance with
the CFD model was reported.
Ibrahim et al. [160] reported on the development of a 2D CFD model for analysing
the components of Stirling engines. The so-called CAST model was used to solve
the governing equations with the k-ε turbulence model. The predicted gas hysteresis
loss from CAST was compared to results obtained from simulations using the
46
Chapter 2
commercial CFD software, namely CFD-ACE+. The acceptable agreement between
numerical results was found but these significantly differed from experimental
results obtained on the test rig.
Further to analysing the working process with application of the 2D CFD modelling,
the 3D CFD modelling has been used for the design improvement of a Stirling
engine. Thus, Mahkamov and Djumanov [161] analysed the whole Stirling engine
operation using 3D CFD modelling with commercial software, namely ANSYS. The
modelling deployed the k-ε turbulence model for numerical investigations of the
working process inside a gamma-type Stirling engine. Obtained numerical data
included temperature, pressure and the velocity contours in the Stirling engine and
the indicated power of the Stirling engine was calculated using obtained pressurevolume diagrams in the expansion and compression spaces. Tan et al. [162]
presented, using as a tool the 3D CFD (ANSYS) modelling, working process details
of a small beta-type kinematical Stirling engine coupled to a parabolic dish
concentrator. Two improved design configurations were suggested to increase the
indicated power of the engine. Moreover, Mahkamov [163] carried out 3D CFD
modelling for the design improvement of a biomass engine. In this paper the
working process of a gamma-type Stirling engine run on biomass (heat source) was
first investigated by using the developed second-order mathematical model and then
the 3D CFD model was applied for calculations. It was demonstrated that the
accuracy of the second-order and CFD models was 30 and 12-18 %, respectively,
when compared to experimental data. The commercial CFD software, ANSYS, was
used for simulating the gas flow and heat transfer in each component of the Stirling
engine and the standard k-ε turbulence model for compressible flow was deployed in
modelling the cycle. The design improvements were proposed which included a new
regenerator with the increased porosity and the change of engine’s configuration to
the alpha-type. A similar work was performed by Mahkamov and Eid [164] for a
Stirling engine heated by synthetic oil. The investigation of the heat transfer
phenomena from synthetic oil to the heater of a V-type Stirling engine was
performed. The engine was designed for application as a part of a medium
temperature solar power plant. The 3D CFD simulations using ANSYS were run to
design the heater of the engine.
Dyson et al. [165] reported the computational simulations using 3D CFD modelling
based on ANSYS for a part of opposed Stirling convertors. The temperature
contours in the cooler, heater and regenerator were presented and also pressure
Stirling engines
47
variations in the expansion and compression spaces were determined. The numerical
results obtained using the 3D CFD model were compared to results obtained with
the use of the one dimensional model software, namely Sage.
Some of the above described mathematical models of Stirling engines were used as
tools for optimisation of design of Stirling engines. Such investigations are
presented in the next section.
2.7
Optimisation of the design of Stirling engines
This section reviews a number of papers related to the optimisation of Stirling engines
using various mathematical models and optimisation methods.
Blank and Wu [166] presented results of investigations of an extra-terrestrial Stirling
engine operating with solar heat source using the irreversible thermodynamic cycle
analysis. The optimum power and the efficiency at the optimum power condition were
calculated using the finite time approach.
Erbay and Yavuz [167] also analysed the performance of the Stirling heat engine at the
maximum power conditions. The power and the efficiency were determined by using
the model of the thermodynamic cycle of a Stirling engine with a realistic regenerative
heat exchanger. The maximum power density technique was used for analysing the
engine performance.
Costea et al. [168] optimised the operating temperature for a solar Stirling engine using
the mathematical model based on the Stirling thermodynamic cycle. This model took
into account fluid friction and mechanical losses in the engine using empirical
correlations. Results were obtained by solving a system of nonlinear equations using the
method of Lagrangian undetermined multipliers.
Hsu et al. [169] investigated the performance of a free piston Stirling engine operated
using the heat from an incinerator. In order to determine the optimal design of an
engine, the efficiency and optimal power output at variable heat source and heat sink
temperatures were investigated by using a cycle-averaged heat transfer coefficient
model. It was found that the efficiency and optimal power output were proportional to
the heat source temperature. Additionally, Hsieh et al. [170] studied the feasibility of
the power production using the waste heat of an incinerator and a free piston Stirling
engine. The performance of a Stirling engine was predicted by using the heat transfer
model with irreversible heat transfer processes and the optimisation of the work output
was investigated using the Lagrange multipliers method. The maximum power was used
as an objective function in the optimisation procedure.
48
Chapter 2
Martaj et al. [171] presented the energy, entropy and exergy analyses of the hot-end and
cold-end heat exchangers and the regenerator of a Stirling engine. The mathematical
model of the Stirling engine with the finite dimension thermodynamics processes was
used in this paper. The engine was analysed using the entropy and exergy balance
equations under the steady state condition. In order to optimise the performance of the
Stirling engine, this research defined the criteria parameters in terms of the maximum
power, the maximum thermal efficiency, the minimum entropy generation and the
maximum exergy efficiency. A MATLAB computer program was used to run
optimisation calculations.
Boer presented in [172] the one dimensional model for analysing the regenerator of
Stirling engines. The model was analytically derived from a simplified model and
included viscous and thermal losses. The optimisation performed provided the optimal
condition for the regenerator conductance and the piston phase angle for the maximum
power output
Yaqi et al. [173] investigated the optimisation of the solar HTD Dish-Stirling engine.
The finite heat transfer and heat losses were considered in the model. The optimisation
performed in terms of the heat absorber temperature, the concentrating ratio of the
system, the effectiveness of the regenerator and the heat leak coefficients resulted in the
maximum efficiency of the Dish-Stirling power unit.
Optimisation of Stirling engines based on deploying the Schmidt model is of a
particular interest in this subject field. Senft [174] developed the model for optimisation
using the principle of the forced work integrated to the classical Schmidt theory. The
optimisation calculations provided the values of the optimal swept volume ratio and
phase angle to achieve the maximum brake work. The geometry of a gamma-type
Stirling engine was optimised and discussed in this work. Schmidt model was also used
by Cullen et al. [175] to obtain preliminary results of modelling an Otto/Stirling cycle
hybrid engine. The simulations with respect to the engine speed using the Schmidt
model were compared to the simulation results obtained using the direct method. The
mathematical model based on the Schmidt model was described by Formosa and
Despesse in [176]. The regenerator efficiency, the heat exchanger effectiveness and a
number of heat loss sources were taken into account in this model. Comparison of the
calculated power output with published results from other models for the GPU-3
Stirling engine [177] was carried out to validate the developed model. From
optimisation calculations, the effect on the brake power output of the fluid mass, the
frequency and the cooler efficiency were determined. However, the author suggested
Stirling engines
49
that the proposed model could be used only for the preliminary designing of the
engines.
For better accuracy, higher order models of Stirling engines should be used for analysis
and optimisation. Altman [177] presented a commercial program for analysing the
working process of Stirling engines and obtaining the optimal design parameters. Thus,
a numerical analysis program named SNAPpro was first presented in 1999/2000 for
designing the various types of Stirling engine. The isothermal second-order model
described by Martini [132] was the design tool for simulating the working process. This
program written as an Excel program calculated the performance of the Stirling engines
and also presented graphs such as P-V diagrams and heat losses. Moreover, it was
coupled to Genetic algorithm code and could optimise up to 20 engine design
parameters.
Orunov et al. [178] presented the design of the tri-generation power unit on the basis of
an alpha Stirling-Stirling cycle for generating electricity and heat/cold production. The
first-order model of a Stirling cycle with taking into account hydraulic losses was used
at the first stage of the design process. At the second stage, the second-order model
taking into hydraulic losses in the all heat exchangers was used for analysing the
working gas process. In the optimisation procedure, the dimensionless work parameter
and dimensionless length of the heater, cooler and regenerator were used as the
optimisation criteria.
Timoumi et al. [146] described the optimisation procedure for the engine parameters of
the GPU-3 Stirling engine. The developed quasi steady flow model called Dynamic
model was used for the parametric analysis. The effects of the thermal conductivity and
heat capacity of materials, the porosity of the regenerator, the temperature of the gas
flowing into the regenerator, the regenerator volume, the mass of the working fluid and
also the expansion volume on the engine performance were investigated.
Zarinchang and Yarmahmoudi in [179] demonstrated the optimisation of a 20 kW
Stirling engine. The optimisation was performed by using so-called OPTIMUM code
based on the second-order Stirling engine model. The main heat exchangers such as the
heater, cooler and regenerator were redesigned by using the sensitivity analysis. Then
the results were recalculated by using the third-order program called STRENG for
better predicting the engine performance.
Some studies, though being on the research of conventional Stirling engines, are related
to the optimisation of designs of LTD Stirling engines. Some of such papers are
reviewed.
50
Chapter 2
Rochelle [180] proposed the mathematical model based on the finite dimension
thermodynamics processes and used for simulation and optimisation of a LTD Stirling
engine with a gamma configuration. The model uses several operating parameters in the
dimensionless form. The ratio between the displacer swept volume and the working
volume was optimised so to achieve the maximum efficiency and the maximum work
output in the LTD Stirling engine without and with a perfect regenerator. The results of
the optimisation provided optimal dimensionless values for both above cases.
Abdullah et al. in [181] analysed the design of the engine by using the first and third
order mathematical models for a double-acting LTD Stirling engine operating at the
temperature difference of 50 °C with a heat source being a thermo-syphon solar water
heater producing the hot water with the temperature of 70 °C. The preliminary design
parameters were obtained using the Schmidt analysis and then the third-order model
based on Martini’s model [181] was deployed to provide the optimal design parameters.
It was demonstrated that there were considerable heat and frictional losses inside all
heat exchangers of the Stirling engine. The parametric study of the diameter of the
cooler and heater tubes were performed.
From review of published paper carried out in this section, it is obvious that the
optimisation methods used for designing engines required the mathematical models
providing the closed form solution and furthermore the parametric analysis was
commonly used for obtaining the optimal engine parameter. Only few published papers
describe application of Genetic algorithm method for the optimisation of Stirling
engines.
Chapter 3
Linear generators
A linear machine can be defined as being the results of a cylindrical rotary machine
which has been mentally split long a radial plane and unrolled, so that instead of
produce a torque (as in traditional rotating machines) it produce a linear force along its
length.
3.1
Linear generator typologies:
3.1.1
Induction machines
The induction machine is the workhorse of the electric motor industry. Its robustness
arises from the simple topology of its rotor, particularly in a squirrel cage motor where
there is no requirement for brushes or slip rings. There is a slight preference for
induction machines to be used as motors, in particular linear induction machines have
previously been of primary interest for transportation actuators. In a linear induction
machine a three phase alternating current is used to excite the stator windings, which, in
combination with the winding configuration, sets up a synchronous speed travelling
magnetic field in the air-gap region. Consequently, if the translator is not moving at
synchronous speed, i.e. for all non zero-slip speeds, an emf is induced within the
translator, this induce an opposing current in the mover windings that, in turn, create a
magnetic field that react again the stator field. The mover accelerates until the force,
provided by the magnitude of induced mover current, balance the applied mechanical
load. The transition of machine operation from motor to generator occurs as the mover
speed overtakes the field speed and goes super synchronous. By feeding the stator coils
through a controllable inverter any slip speed can be achieved at any mover velocity.
Controlling the output power is hence relatively simple for a variable speed input and as
such the rotary version is commonly used in wind turbines. The moving part can have a
52
Chapter 3
very simple and light construction, consisting of either conductors embedded in slots of
a back iron structure, or possibly just a single conducting sheet. The main drawback of
such kind of machine is the low efficiency. In fact, for a linear machine, where there is
likely to be a larger air-gap than its rotary counterpart, the inductance and hence
reactance of the excitation coil is low. Consequently a large excitation current is
required and the overall efficiency of the machine is reduced.
3.1.2
Synchronous machines
Synchronous machines are the primary devices used for high speed rotary power
generation in the world's electric power systems today [182]. The rotor is fed with a
direct current via brushes and slip rings and so sets up a magnetic field which follows
its movement. The stator surrounds the rotor with a set of stationary coils, such that they
are cut by the rotor flux pattern. Controlling the excitation currents in these machines,
which are generally more efficient than induction machines, provides a way of
regulating the output voltage which would otherwise be dependent on speed.
Operating as a motor for high speed transportation devices, linear synchronous
machines have been shown to be more favourable than their induction equivalents, with
an efficiency of 90 % when compared to 82 % and a power factor of 0.8 as opposed to
0.52 [183]. At low velocities, however, the small pole pitches necessary for rapid flux
change limits the current / size of the excitation coils.
There are wear issues associated with field-wound synchronous machines when
compared to induction machines. The necessity of physical contact with the translator in
order to transfer magnetising current will enforce routine maintenance for inspection or
replacement of brushes. The variable speed bi-directional oscillatory motion considered
here will only serve to increase wear on the brushes when compared to their rotary
counterpart.
3.1.3
Permanent magnet machines
In a machine excited by PMs there is no need for the provision of field excitation, as
magnetised material is used to supply pole flux as opposed to current carrying coils. A
simple example of such a machine consists of surface mounted ferrous magnets
mounted on an iron translator which oscillates within a cylindrical distributed three
phase winding, held in place by slotted iron structure.
Linear generators
53
As the translator moves, so too does the flux pattern resulting from the presence of
magnets. An emf is hence induced in the stationary coils as the flux cutting them is
changed. The flow of magnetic flux can be simplified by assuming infinitely permeable
iron, in which case the total magneto-motive force (mmf) produced by a magnet is
dropped across the magnetic gap length (i.e. in the air-gap and within the magnet itself).
The flux density in the air-gap
[T] is given by:
(3.1)
where
is remnant flux density in the magnet [T],
is the magnet thickness [mm],
is the permanent magnet relative permeability [-] and
is the air-gap length [mm],
respectively.
By use of Lorenz’s law, the machine thrust force
[N] on the electrical windings
results:
(3.2)
where is the winding current [A] and
is winding length [m], respectively.
The use of PMs allows achieving greater efficiency compared to synchronous machines,
due to the lack of mover windings, cause of losses. Furthermore, PMs allow obtaining
higher specific power, lighter mover, higher strength and less maintenance.
3.2
Variable reluctance permanent magnet machines
One of the main problems of linear machine is the strength connection between the
primary moving source and the mover of linear generator. Accordingly, the output
electric frequency is the same of mechanical frequency. In the slow speed applications,
this could be a problem, because low frequency means large size of the machine in
order to convert the same quantity of energy.
A slow physical velocity may be converted to a high speed flux change by a process
known as magnetic gearing. Figure 3.1b shows a single stator pole, split into 3 teeth,
interacting with five individual mover poles, each a surface mounted permanent magnet.
If the mover moves by one mover pole, the direction of flux flow through the pole can
be seen to reverse. In non-toothed designs (Figure 3.1a), the mover would have to move
a distance five times this value. This toothed design has therefore increased the rate of
change of flux fivefold. However, the mover is only interacting with three stator poles
at any one time, reducing the maximum value of flux by a factor of 3/5. The net
magnetic gearing of the split mover and stator poles is hence threefold.
54
Chapter 3
(a)
(b)
Figure 3.1. Magnetic gearing: non-toothed (a) and toothed (b) linear generator designs.
Recent developments have resulted in a new class of machine, known as Variable
Reluctance Permanent Magnet machines (VRPM). There are many topologies of these
machines [184-189], all bound by the same principles of operation. Alternative return
paths are offered to the magnetic flux through the interaction of rare earth PMs and
small pitch iron teeth. Force is developed as the path of minimum reluctance is sought.
These machines are capable of shear stresses unmatched by other electrical machines,
which has led to a general reduction in the size of machine. These advantages are
further amplified if rare earth magnets are used.
A common problem to all VRPMs is that they tend to operate at low power factors
under load, values in the range of 0.35-0.55 are typical [190]. The highly effective
magnetic circuit of the machine, which is key to its high shear stresses, presents this
inherent disadvantage. The problem is compounded by the desire for many turns on the
coil to further enhance the change in flux linkage from the slow physical velocity. The
result of these factors is that current flowing within the coils produces a strong flux
pattern. Any change in this current flow is· hence resisted by a large change in flux
flow, producing a back emf. In practice this phenomena manifests itself as a large
inductance in series with the electrical machine resisting any change in current. The
subsequent phase lag between emf and current results in the terminal voltage collapsing
when current flows. The low power factor of VRPMs will hamper their useful power
output. The requirements of this topology and the utilisation of magnetic gearing make
it an inherent feature which must be overcome by electronic means.
In order to increase the power factor and bring the current and voltage of the generator
in phase, reactive power must be provided. This can be done by the addition of a
parallel capacitor across the load or by means of an active rectifier that force the output
current to be in phase with internal emf, and then bringing the power factor up to unity.
Linear generators
3.2.1
55
Transversal flux machines
In a transversal flux machine (TFM) (Figure 3.2) the main flux flow is in a direction
perpendicular to the direction of travel. The translator consists of surface mounted PMs
in a configuration that forces the flux to vary both axially and circumferentially. There
are stator coils mounted either side of the translator with a series of iron yokes
channelling the flux in such a way that alternate mover poles excite the same coil [184185]. The result is an inherently three dimensional flux path, and flux from all the
magnets contributes to linking one or other of the coils at all times. As such the TFM is
the ultimate VRPM machine, producing very high shear stresses values approaching
200 kN/m2 have been reported [184].
Moving
direction
Flux
path
Figure 3.2. Transversal flux permanent magnet linear machine (TFPMLM).
The mounting of the stator yokes in combination with the three dimensional flux path
present various problems with the support structure of this machine. The typical three
dimensional structure does not allow the ferromagnetic elements to be laminated and
the necessary small pitch often demands a large number of individual parts, or the use
of SMC materials. Furthermore, inherent in this machine is the presence of cogging
torque: the tendency of the magnets to align themselves with the path of least
reluctance. It has been reported as 30 kN/m2 in one machine [185].
If the mover magnets are replaced with an equivalent current sheet and the magnets and
teeth are misaligned, the force
[N] resulting in the interaction of the current sheet and
the flux due to the coil current is given by:
(3.3)
(
where
(
)
) is the flux density in the air-gap [T],
is permanent magnet equivalent current [A] and L is the permanent
magnet axial length [m].
56
Chapter 3
3.2.2
Vernier hybrid machines
The Vernier hybrid machine (VHM) (Figure 3.3) has a significantly more conventional
machine structure, and a two dimensional flux pattern [186, 189]. The stator mounted
PMs interact with the purely iron toothed mover, in order to produce the flux path
shown. This two dimensional path links all four coils, four magnets and four air-gaps. A
similar topology has been proposed, [187], which had the magnets on the moving part
and the stator face toothed. If the mover moved one magnet pitch, equal to half a mover
pitch, in either direction, it can be seen that the teeth would fully align with the opposite
poles to those shown, and hence the flow of magnetic flux would be completely
reversed.
Moving
direction
Figure 3.3. Vernier hybrid machine.
Because of the slots in the mover the flux density is not constant throughout the width
of the airgap. The flux pattern with a higher density is observed under the tooth region
than under the slot region. Due to the magnets being alternatively magnetised, the
greatest force is reacted when the magnets and teeth are exactly misaligned and hence
the equivalent current carrying coils are in the area of highest flux density. The flux
under the slot region causes a force which opposes that under the teeth.
The peak resultant force per tooth pitch Fn [N] will be the difference of these opposing
forces, given by:
(
where
)
is the flux density under the tooth [T] and
region [T].
(3.4)
is the flux density under slot
Linear generators
3.3
57
Principle of operation
While rotary motion electric generators are multiphase (generally 3-phase) machines, in
general, the linear motion generators (LGs) tend to be single-phase machines, due to
linear oscillating motion that forces a change of phase sequence for a change in the
motion direction.
Referring to a harmonic motion, the mover position [m] can be described as follows:
( )
̂
(
)
(3.5)
where ̂ is the maximum value of the mover semi-stroke [m] and
is the oscilating
frequency [rad/s].
The electromagnetic force (emf) is, in general:
( )
where
[m/s].
( )
is the PM flux linkage in the phase coil [Wb] and
(3.6)
is the mover speed
Putting equation (3.5) in equation (3.6), results:
( )
̂
To obtain a sinusoidal emf waveform,
(
)
(3.7)
has to be constant with mover position.
This means that the PM flux linkage in the phase coil has to vary linearly with mover
position. The linear variation of PM flux linkage with mover position is met only
approximately in practice, due to iron core magnetic saturation. Generally, the
flattens toward excursion ends, which leads to the presence of third, fifth and seventh
harmonics in ( ).
Ideally, with
position-independent, emf varies as the speed does.
The thrust force that acts on mover is as follows:
( )
Ideally, with
( ) ( )
( )
(3.8)
position-invariant, thrust force varies as the current.
From equation (3.8), can be noted that the highest thrust force per given current occurs
when the ( ) and ( ) are in phase with each other. Supposing
position-invariant,
from equation (3.7), it follows that ( ) is in phase with mover speed; thus, to achieve
the highest thrust force, the current has to be in phase with the speed as well.
58
Chapter 3
The above relation is valid only if
that the phase inductance
can be considered position-invariant, this means
[H] is independent of mover position, that is, if the
reluctance force is null.
In the hypothesis of ideal LG (
position-invariant), the complex variables can be
used to describe the machine operation, the output voltage ̅ [V] results:
̅
̅
) ̅
(
̅
is the coil resistance [Ω] and ̇
The delivered apparent power ̅ is as follows:
where
̅
The delivered active power
where
̇ ̅
(3.9)
is the coil impedance.
̅̌
(3.10)
( ̅ ̌)
(3.11)
is:
is the phase shift between output voltage and current.
To deliver electrical power to a grid the LG output voltage has to be larger than grid
voltage
. A series of capacitors may be used to reduce the phase shift
inductance
3.4
, due to coil
.
Dynamic equations
The motion of LG is regulated by the following force [N] balance:
(3.12)
where
is the force provided by primary engine,
force of linear generator,
force (where
is the electromagnetic
is the cogging force,
is the spring constant [N/m]) and
is the spring
is the total moving mass [kg].
In steady-state harmonic motion, the primary engine force has to completely balance by
only the electromagnetic force of linear generator. Then, from equation (3.12), has to
be:
(
as
̂
(
) , it follow that:
[(
then, finally:
(3.13)
)
)
]̂
(
)
(3.14)
Linear generators
59
(
√
)
(3.15)
Equation (3.15) spells out the mechanical resonance condition. So the electrical
frequency (equals to the mechanical one) results equals to the spring-mass proper
frequency.
In this case, the prime mover has to provide only the useful electromagnetic power,
while the mechanical spring has to provide the cogging and inertia forces, does the
conversion of electrical to kinetic (and back).
To obtain the best condition of energy conversion the spring has to design in order to
achieve the resonant frequency for that particular moving mass and working frequency:
(3.16)
3.5
Structures of linear machine
Firstly, the linear machines can be classified based on their planar geometry:
 flat (planar) linear machines, shown in Figure 3.4a, that is obtain making an
ideally cut and straight a conventional rotating machine. They, in turn, can be
classified in single-sided and double-sided if they are assembled in opposite;
 tubular (cylindrical) linear machines (Figure 3.4b and Figure 3.5) that is obtain
starting from a flat linear generator and then re-rolling the machine in the
moving direction.
(a)
(b)
Figure 3.4. Process to obtain a linear machine: single-sided flat (a) and tubular (b) structures.
The linear machines can be classified according to the stator length respect to mover
length:
 long stator (or short mover): the length of the supply part is longer than the
excitation way, in most cases the excitation part is mobile;
 short stator (or long mover): the supply part is shorter (or equal) than the
excitation way. The supply part is, in most cases, mobile.
60
Chapter 3
Figure 3.5. Cross-section of a tubular linear machine.
The linear machines can be sort by the thrust (propulsion) force is generated, or rather
the excitation typology:
 coil: the traveling magnetic field is produced by the current that flow through in
a winding (coil exited machines);
 PM: the traveling magnetic field is produced by permanent magnets (PM exited
machines).
3.6
Control of permanent magnet linear generators
The control of the PMLGs depends on the prime mover and load characteristics.
Typical prime movers are as follow:
 Free piston engines (Stirling engines) [191];
 Spark-ignited linear internal combustion engines (ICEs) [192];
 Very low-speed reciprocating wave energy converters (average speed from 0.5
to 2.0 m/s) [191].
The control of stroke be instrumented through the prime mover, while the adaptation of
alternator voltage to the load – independent or at power grid – is to be performed by
power electronics. The motion frequency may be maintained constant and equal to
power grid frequency, securing operation at the mechanical resonance frequency. If the
frequency of motion varies within some limits, then the complexity of power electronics
has to increase, if the load demands constant frequency and voltage output.
The control solution can refer to constant or variable motion frequency solutions and, as
concerning the load, a stand-alone non-demanding load or a strong power grid.
In the case of constant motion frequency, the control is decoupled as shown in Figure
3.6. Particularly, the motion amplitude control is provided by prime-mover governor in
Linear generators
61
order to deliver more or less mechanical power while voltage adaption is required in the
case of operation at the power grid or at a certain voltage.
Figure 3.6. Constant motion frequency solution (power grid connection).
The power error is the input of a power regulator follower by the stroke length
,
regulator. Finally, the fuel injection rate is modified to produce the required motion
amplitude
, according to electric power requirements.
As the frequency is constant and connection to the grid is required, the motion phase
may be modified slowly to provide the synchronization to the grid connections. The
voltage adapter may be an autotransformer with variable ratio via a small servomotor.
Alternatively, a soft-starter may be used to reduce the transients during connection to
the grid. An additional resistance may be added before synchronization, in series with
the alternator, and then short-circuited a few second later, as is done with induction
generators.
When the motion frequency is allowed to vary, a front-and rectifier plus inverter may be
applied, as shown in Figure 3.7. In this case, the series capacitor is no longer required,
as the DC link capacitor does the job. The DC link voltage may now be larger than the
LMA emf.
62
Chapter 3
Figure 3.7. Constant motion frequency solution: front-and rectifier plus inverter.
A bidirectional converter makes the interface between the variable voltage and
frequency of the PMLG and the load requirements. The control of the load-side
converter is different for independent load in contrast to power grid operation; it should
be similar for the single-phase alternators. The bidirectional converter also allows for
motoring and thus, could help in starting or stabilizing the prime mover’s motion when
the operation takes place at the power grid or with a backup battery. In this case, the
smooth connection to the power grid is implicit.
Note that for ocean-wave linear translator alternators, three-phase configurations were
proposed [191]. A three-phase bidirectional converter is required in this case. The
control of such a system is similar to the case of rotary PM generators.
3.7
History and applications of linear generator
A brief review of the history, types and applications of linear electric actuators and
generators was given by Boldea et al. [193].
Amara et al. [194] investigated a tubular linear permanent magnet machine that might
offer the highest efficiency and power/force density. The machine studied had a nineslot, ten pole configuration, with a fractional number of slots per pole per phase. Since
the Halbach magnetic array was used for the mover, the flux at the inner bore is quite
small, permitting the use of nonmagnetic supporting tube for the magnets. The problem
Linear generators
63
of eddy current reduction is the main design issue for this type of machine
configuration.
A typical application of the linear permanent-magnet synchronous generator (LPMSG)
is in a wave-energy to electric energy conversion system. The perpetual vertical motion
of the sea waves is exploited to drive the mover of a linear generator for producing
electricity. A pilot plant using this principle, namely the Archimedes Wave Swing
(AWS), was designed for 4-MW peak power [195]. A three-phase linear permanentmagnet generator was designed to extract electric energy from the motion of the floater
[196].
Polinder et al. [197] showed that the linear PM generator was cheaper than linear
induction generators for such applications. Conventional LPMSG had magnets mounted
on the mover, but the authors also proposed a transverse-flux PM generator (TFPMG)
which had flux concentrators, magnets and conductors all on the stator, while the
translator only consisted of iron. Such a machine configuration was more difficult to
build and hence further investigations would be needed.
Wang et al. [198] describe the design and experimental characterization of a
reciprocating linear permanent magnet generator. Their paper describes a design
methodology which embraces both the electromagnetic and mechanical design. The
electromagnetic design synthesis is based around analytical field solutions which allow
the optimization of the generator parameters, and it is further verified by non-linear
electromagnetic finite element analysis. The design methodology utility is illustrated by
a design study and extensive experimental characterisation of a prototype device.
In another paper [199] Wang at al. discuss the issues about the design of a linear
permanent magnet generator for application in a free-piston energy converter. To
achieve the required high power density, high efficiency and low moving mass, a
tubular machine equipped with modular windings and quasi-Halbach magnetization is
chosen. Analytical expressions for predicting the open-circuit magnetic field
distribution, emf, thrust force and armature reaction, as well as the self and mutual
inductances of the machine, are established and verified by means of finite element
analysis. It is shown that the machine design can be optimized with respect to three
dimensional ratios while satisfying other performance specifications.
Fraiz at al. [200] discuss how to reduce the cogging force in a linear permanent magnet
generator. To reduce this force they have design and simulated a tree-phase linear
permanent magnet generator for direct wave energy conversion and predicted its
performance using the finite element method. They studied the influence of different
64
Chapter 3
design parameters (such as PMs length, magnetization orientation and stator slot
typologies) on the cogging force and minimized this force by varying the proposed
parameters. The results obtained confirm a large reduction in the cogging force and an
enhancement in the generator performances.
Delli Colli et al. [201] illustrate the operation of a tubular-machine drive as a linear
generator for a heave-buoy wave energy conversion. The paper summarized the
principle of marine wave buoy interaction and reports the design analysis and control of
a permanent magnet synchronous tubular linear machine based on a scaled generator
prototype and on a rotating simulation test bench.
Soheil at al. [202] describe an improved topology of a linear permanent magnet
generator for sea wave energy harvesting and presents an electrical model for analysis
of this generator. The electrical model is established based on the magnetic equivalent
circuit of the machine. The model can be used to design a power electronic interface for
capturing maximum power at the resonant frequency of a sea wave. Validity of the
model is verified via comparing the calculated inductance of the machine with that of
obtained from numerical analysis based on finite element method.
Francois at al. [203] set forth a 3D analytical model of a tubular linear induction
generator. The article describes the electromagnetic model of the linear generator and
draw comparison with a finite element model, in addiction to identifying the elements
of the equivalent electrical diagram and displaying results from the multi-objective
optimization study performed using a genetic algorithm.
Chapter 4
Magnetic materials utilized in
linear electric machines
4.1
Introduction
The magnetic materials can be classified into three categories: diamagnetic,
paramagnetic and ferromagnetic materials. The classification depends on the magnetic
dipole moment of atoms of the material and on the interactions among the atoms.
When the different magnetic materials are placed in a uniform magnetic field, the field
lines are changed as shown in Figure 4.1.
(a)
(b)
(c)
Figure 4.1. Field lines distribution of diamagnetic materials (a), paramagnetic materials (b) and
ferromagnetic materials (c) placed in a uniform magnetic field H.
66
Chapter 4

Diamagnetism
When a diamagnetic material is exposed to a strong magnetic field, electrons in
this material rearrange their orbits (orbital motion of electrons) and creating
small persistent currents which oppose the external magnetic field. Hence,
diamagnetic materials have a very weak and negative susceptibility to external
magnetic fields, and also slightly repelled by the magnetic field. In addition,
diamagnetic material does not retain the magnetic properties when the external
field is removed and does not have permanent net magnetic moment per atom
since its orbital are fully-filled.
Examples of diamagnetic material are bismuth, copper, water, alcohol, mercury
etc.
The permeability of diamagnetic materials is slightly less than or equals to the
free space permeability (
). The field lines distribution of a diamagnetic
material placed in a uniform magnetic field are reported in Figure 4.1a.
The properties of these materials are independent of temperature.
Some materials behave like superconductor at very low temperature and they are
perfect diamagnetic materials which have µr = 0 and B = 0 (no magnetic field
could be established inside superconductor materials since they expel all the
magnetic fields that applied on it). The most popular application of diamagnetic
materials is magnetic levitation, where an object can be made to float in the air
above a strong magnet.

Paramagnetism
Paramagnetic materials are weakly attracted to magnets and have a small
positive susceptibility to magnetic fields. Paramagnetic properties are due to the
presence of some unpaired electrons that produce the net spin magnetic
moments which tend to align themselves in the direction of the external
magnetic field. They do not retain the magnetic properties when the external
field is removed.
Paramagnetic materials are sensitive to the temperature: they become more
magnetic when theirs temperature reduces. Paramagnetic materials are typically
considered nonmagnetic since they have very small positive susceptibility (of
the order 10-5) as compared to ferromagnetic materials.
Magnetic materials utilized in linear electric machines
67
The permeability of paramagnetic materials is slightly more than or equals to the
free space permeability (
). The field lines distribution of a paramagnetic
material placed in a uniform magnetic field are reported in Figure 4.1b.

Ferromagnetism
Ferromagnetic materials have a large and positive susceptibility to an external
magnetic field. They exhibit a strong attraction to magnetic fields and they are
able to retain their magnetic properties after the external field has been removed.
Ferromagnetic materials have some unpaired electrons so their atoms have a net
magnetic moment. They get their strong magnetic properties due to the presence
of magnetic domains within which the magnetic moments of all its atoms (1019)
are aligned parallel to each other.
Examples of ferromagnetic materials are iron, nickel and cobalt, and their alloys
such as alnico.
The permeability of ferromagnetic materials is much greater than the free space
permeability (
). The field lines distribution of a ferromagnetic material
placed in a uniform magnetic field are reported in Figure 4.1c.
When a ferromagnetic material is in the un-magnetised state, the domains are
nearly randomly organised as shown in Figure 4.2a and the net magnetic field
for the part as a whole is zero. When a magnetising force is applied, the domains
become aligned as shown in Figure 4.2b to produce a strong magnetic field
within the part. The boundaries between the adjacent domains consist of thin
transition regions called as domain walls.
(a)
(b)
Figure 4.2. Domain comparison in a un-magnetised (a) and magnetised (b) ferromagnetic
material.
68
Chapter 4
In addition, strong magnetic ferromagnetic materials lose all their magnetic
properties if they are heated to a critical temperature (Curie temperature). This is
because the magnetised domains will organise themselves randomly after theirs
atoms are being heated.
The main characteristics of magnetic materials are summarized in Table 4.1.
Table 4.1. Characteristics of magnetic materials.

Hysteresis
The magnetisation behaviour of the ferromagnetic materials is described by the
B-H magnetisation curve (hysteresis loop) as shown in Figure 4.3b.
The loop is generated by measuring the magnetic flux B of a ferromagnetic
material while the magnetising force H is changed. A ferromagnetic material
that has never been previously magnetised or has been thoroughly demagnetised
will follow the dashed line (initial magnetization curve) as H is increased, as
shown in Figure 4.3a. As the line demonstrates, the greater the amount of
current applied, the stronger the magnetic field in the component. At point “a”
almost all of the magnetic domains are aligned and an additional increase in the
magnetising force will produce very little increase in magnetic flux. The
material has reached the point of magnetic saturation.
When H is reduced down to zero, the curve will move from point "a" to point
“b”. At this point, it can be seen that some magnetic flux remains in the material
even though the magnetising force is zero. This is referred to as the point of
Magnetic materials utilized in linear electric machines
69
retentivity on the graph and indicates the remanence or level of residual
magnetism in the material, where some of the magnetic domains remain aligned
but some have lost their alignment. As the magnetising force is reversed, the
curve moves to point “c”, where the flux has been reduced to zero. This is called
the point of coercivity on the curve: the reversed magnetising force has flipped
enough of the domains so that the net flux within the material is zero. The force
required to remove the residual magnetism from the material, is called the
coercive force or coercivity of the material.
B
H
(a)
(b)
Figure 4.3. Hysteresis loop: initial magnetization curve (a) and hysteresis loop profile (b).
As the magnetising force is increased in the negative direction, the material will
again become magnetically saturated but in the opposite direction (point “d”).
Reducing H to zero brings the curve to point “e”. It will have a level of residual
magnetism equal to that achieved in the other direction. Increasing H back in the
70
Chapter 4
positive direction will return B to zero. Notice that the curve did not return to the
origin of the graph because some force is required to remove the residual
magnetism. The curve will take a different path from point “f” back to the
saturation point where it completes the loop. The complete close loop “abcdefa”
is called as a hysteresis loop.
The saturation field strength Hsat [A/m] is the minimum field strength that has to
expend to reach the region of magnetic saturation. For field strength H ≥ Hsat the
ferromagnetic material behaviour is like to free space, thus:
(4.1)
The remanence induction or residual flux density Br [T] is the induction
remaining in a closed magnetic circuit if after achieve the saturation condition
(H ≥ Hsat), the field strength return to zero.
The coercive force or coercivity Hc [A/m] is the magnetic field strength at which
the induction of a magnetic material previously magnetised up to saturation
becomes zero.
The maximum energy product (BH)max [GOs (Gauss-Oersted) = 8·103 Ws/m3] is
the maximum product of the flux density and field strength attained on the
demagnetization curve (Figure 4.4).
The values of B and H at (BH)max are designated as Ba and Ha.
Figure 4.4. Energy product of a magnetic material, the dot represents the maxim value (BH) max.
The maximum energy product occurs at the point on the demagnetisation curve
where a contour line would just touch it. Such feature is a very important
characteristic of permanent magnet materials, because it is a measurement of the
maximum amount of magnetic energy stored in a magnet.
In general, magnetic materials can be classified as magnetically "soft" and
"hard" materials. The first ones have a smaller hysteresis loop (Fig. a), while the
second ones have a bigger hysteresis loop (Fig. b).
Magnetic materials utilized in linear electric machines
71
(b)
(a)
Figure 4.5 Comparison between hysteresis loop of a hard material (a) and of a soft material (b).
The characteristics of soft magnetic materials are: the facility to magnetize and
demagnetize, they have low hysteresis loss, high permeability and low magnetic
stored energy. On the other hand, hard magnetic materials retain their
magnetism and are difficult to demagnetize even after the removal of applied
magnetic field. They have large hysteresis loss and a high magnetic stored
energy.
For such reason, soft magnetic materials are used to build iron cores of electrical
machines, whereas hard magnetic materials are used for to build permanent
magnets.
4.2
Hard ferromagnetic materials - permanent magnets
The hard ferromagnetic materials are used to build permanent magnets.
The most popular permanent magnet materials are: ferrite, alnico, SmCo and NdFeB.
While there are several processing options, magnets can be divided into fully dense or
bonded, leading to isotropic or anisotropic magnets.
The historical trend of permanent magnets is reported in Figure 4.6, and in Figure 4.7
shows the magnetization curve.
72
Chapter 4
Figure 4.6. Historical trend of permanent magnets.
Figure 4.7. Magnetization curves of hard ferromagnetic materials.
The main properties, in terms of residual flux density, coercivity, intrinsic coercivity
and maximum energy product are reported in Table 4.2.
Table 4.2. Comparison of the main properties of PM materials in terms of: residual flux density,
coercivity, intrinsic coercivity and maximum energy product.
Material
Grade
Br
[kG]
Hc
[kOe]
Hci
[kOe]
BHmax
[MG Oe]
NdFeB
NdFeB
SmCo
Alnico
Ceramic
39H
26
B10N
5
8
12.8
6.8
10.5
12.5
3.9
12.3
5.8
9.2
0.6
3.2
21.0
10.3
10.0
0.6
3.3
40.0
26.0
10.0
5.5
3.5
Magnetic materials utilized in linear electric machines

73
Ferrite
Ferrite magnets, also called ceramics, are made of a composite of iron oxide and
barium carbonate (BaCO3) or strontium carbonate (SrCO3). Widely available
since the 1950’s, this materials is commonly available and at lower cost than
other types of materials used in permanent magnets.
They are made using pressing powder in a die and then fusing the powder into a
solid material with heat. While the sintered magnets are solid, their physical
properties are more similar to a ceramic and are easily broken and chipped.
Ferrite magnets are brittle and hard, they generally require diamond wheels to
grid and shape. These magnets come in a number of different grades: they can
be isotropic, namely with equal magnet properties in all directions; or they can
be anisotropic, that are magnetised in the pressing direction. The anisotropic
method delivers the highest energy product among ceramic magnets at values up
to 3.5 MGOe.
Ferrite magnets have a good balance of magnetic strength, resistance to
demagnetizing and economy. Nowadays, they are the most widely used magnets
in cheap applications.
Beneficial characteristics of ferrite magnets include their low cost, high coercive
force, resistance to corrosion, and high heat tolerance. Drawbacks include their
low energy product or “strength”, low mechanical strength, and the ferrite
powder on the surface of the material which can rub of and cause soiling.

Alnico
Alnico magnets are made up of an alloy of aluminium (Al), nickel (Ni) and
cobalt (Co), with small amounts of other elements added to enhance the
properties of the magnet. They have good temperature stability, good resistance
to corrosion, but are prone to demagnetization.
Alnico magnets are produced by two typical methods: casting or sintering.
Sintering offers superior mechanical characteristics, whereas casting delivers
higher energy products (up to 5.5 MGOe) and allow for the design of intricate
shapes. There are anisotropic grades that provide for a preferred direction of
magnetic orientation.
Beneficial characteristics of Alnico magnets include their high corrosion
resistance, high mechanical strength and very high working temperatures.
Drawbacks include their higher cost, low coercive force, easy of
demagnetization and low energy product.
74
Chapter 4

Samarium cobalt (SmCo)
Samarium cobalt is a type of rare earth magnet material that is highty resistant to
oxidation, has a higher magnetic strength than ferrite and alnico and better
temperature resistance than neodymium materials. They offer the best
temperature characteristics of all rare earth magnets and can withstand
temperatures up to 350 °C.
Sintered SmCo magnets are brittle and prone to chipping and cracking and may
fracture when exposed to thermal shock.
Due to the high cost of samarium, SmCo magnets are used only in application
where high temperature and corrosion resistance is critical.
Beneficial characteristics of SmCo magnets include their high corrosion
resistance, high energy product and high temperature stability. Drawbacks
include their high cost and very low mechanical strength.

Neodymium iron boron (NdFeB)
Neodymium iron boron is another type of rare earth magnetic material. NdFeB
is the most advanced commercialized permanent magnet material available
nowadays. This magnet has similar properties as the SmCo magnets except that
it is more easily oxidized and generally does not have the same temperature
resistance. However, NdFeB magnets have the highest energy products
approaching 50 MGOe and they are mechanical stronger than SmCo magnets.
NdFeB material is more costly by weight than ferrite or alnico, but produces the
highest amount of flux per unit of volume or mass making it very economical
for many applications. Their high energy products lend themselves to compact
designs that result in innovative applications and lower manufacturing costs.
NdFeB magnets have to protect again the corrosion. Surface treatments have
been developed that allow them to be used in most applications. These
treatments include copper, silver, gold, nikel, zinc and tin plating and epoxy
resin coating.
Beneficial characteristics of NdFeB magnets include their very high energy
product, very high coercive force and moderate temperature stability.
Drawbacks include lower mechanical strength and low corrosion resistance
when not properly coated or plated.
Magnetic materials utilized in linear electric machines

75
Bonded magnets
All of the above mentioned materials are available as bonded grades by either
extrusion, compression, calendaring or injection molding processes. The
magnetic properties are lower because they sometimes lose their anisotropy and
they are not fully dense due to the introduction of resins and epoxies. The main
advantage to this group is that they can be made in complex shapes and can be
insert, over-molded and co-molded with other materials.
The main properties of above mentioned PM materials, with advantages and drawbacks,
are listed in Table 4.3. In Table 4.4 are reported the Curie
and the maximum
temperatures of main PM materials.
Table 4.3. Comparison of the main characteristics (advantages and drawbacks) of the main PM
materials.
Table 4.4. Curie temperature and maximum temperature of main PM materials.
Material
NdFeB
SmCo
Alnico
Ceramic
[°C]
310
750
860
460
[°C]
150
300
540
300
76
4.3
Chapter 4
The effect of temperature variation
All permanent magnet materials are temperature sensitive and it is important to take this
into consideration when designing magnetic circuits where a high degree of temperature
stability is necessary or where high maximum temperatures are likely to be reached in
operation.
Temperature changes have an effect on the demagnetisation curves with different
coefficients of both remanence and coercivity. As can be seen from Figure 4.8, the
general shape of the curve remains the same but has varying values of remanence and
coercivity with temperature changes.
Figure 4.8. Hysteresis loop changing as a function of temperature (material: MQP-14-12).
To minimise the effect of temperature on the field produced by a permanent magnet, it
is advisable to operate above the ‘knee’ in the demagnetisation curves.
However the temperature effects are of two types. The reversible changes where the
initial flux is restored when the magnet returns to the original temperature; and the
irreversible changes where the losses are permanent, but can be restored by remagnetising the ferromagnetic material.
The reversible changes are usually only dependent upon the material composition,
whereas the irreversible losses are largely dependent on the working point and intrinsic
coercivity of the material grade.
The manufacturer data-sheet usually report two type of temperature:
• TCurie: is the Curie temperature at which the elementary magnetic moments are
randomized and the material is demagnetized, the magnetism can only be restored by
renewed magnetization below this temperature;
Magnetic materials utilized in linear electric machines
77
• Tmax: is the maximum practical operating temperature in air, for general classes of
major materials.
Table 4.4 reports these temperatures for the mentioned PM materials.
Changes in temperature affect both flux output, which is proportional to Br, and
resistance to demagnetization, which is proportional to Hc (Fig. ). The amount to which
these change is called the reversible temperature coefficient of induction ( ) or the
reversible temperature coefficient of coercivity ( ). The change in Br and Hc as a
function of temperature is generally not linear.
The induction and coercivity as function of the temperature variation is given by:
where
and
( )
[
(
)]
(4.2)
( )
[
(
)]
(4.3)
are the induction and coercivity at temperature
, respectively.
Table 4.5 reports the reversible temperature coefficients of induction and coercivity for
different PM materials. Note also, that ferrite has a large positive reversible coefficient
of coercivity. That means as temperature is reduced, intrinsic coercivity decreases
resulting in a knee developing in the normal curve.
Table 4.5. Reversible temperature coefficients of induction ( ) and coercivity ( ).
Material
NdFeB
SmCo
Alnico
Ceramic
4.4
[°C-1]
-0.12
-0.04
-0.02
-0.20
[°C-1]
-0.60
-0.30
0.01
0.30
Manufacturing and magnetization process
Depending on the type of material, the following processes are used to manufacture
permanent magnets:

Sintering: the sintering process involves compacting fine powders at high
pressures in an aligning magnetic field, then sintering it into a solid shape. After
sintering, the “ingot” is rough and must be machined to achieve close tolerances.
The complexity of shapes that can be pressed using this process is limited.
Neodymium and Samarium cobalt powders are compressed to form magnets by:
 Isostatic pressing: the powder is compacted with the same force from all
directions. These magnets have the highest possible magnetic values
because of the higher density achieved using this technique.
78
Chapter 4
 Compressing in tools:
a. Transverse Field Pressing. The powder is compacted at right angles
to the magnetic flux. These magnets have a weaker field compared to
the isostatic pressed ones, but stronger compared to the parallel
pressed magnets.
b. Parallel Compressing. The powder is compacted parallel to the
magnetic field.

Compression Molding: this method is commonly used to make NdFeB magnets
using melt spun Nd powders that are epoxy coated. The powders are compacted
and then heat cured for the epoxy to perform the binding function. These
magnets are typically isotropic.
Injection Molding Neodymium, Samarium Cobalt, and Ferrite Materials can be
manufactured by injection molding. Common binders for injection molding are
polyamides. The advantage of using this method is the possibility to get a better
tolerance directly from the tool with no required heat treatment, and the magnets
can be produced in complex shapes. They can be combined with other materials
for over-molding or insert molding.

Casting: this process is used for manufacturing Alnico. Process is similar to the
casting of other metals. Parts are formed in sand casts which can be complex.

Calendering and Extruding: flexible NdFeB and Ferrite magnets with Nitrile
rubber binders are made using this method. The process is similar to that for
vinyl sheets. Parts are later die cut or stamped from sheets of various
thicknesses.
As regard the magnetization process, magnets can be delivered magnetized and unmagnetized. Many automated production methods rely on the megnets being
incorporated into assemblies in a un-magnetized state and later magnetized once the
assembly is completed. A magnet can be magnetized in a variety of directions (Figure
4.9).
Anisotropic material must be magnetically orientated during the production of the
material. They therefore are inclined to a fixed orientation. The radial orientation is
quite difficult to produce and it is only available in some isotropic materials.
Magnetic materials utilized in linear electric machines
79
Figure 4.9. Magnetizing patterns.
Permanent magnet materials are believed to be composed of small regions or “domains”
each of which exhibit a net magnetic moment. An un-magnetized magnet will possess
domains which are randomly oriented with respect to each other, providing a net
magnetic moment of zero. Thus a magnet when demagnetized is only demagnetized
from the observer’s point of view. Magnetizing fields serve to align randomly oriented
domains to give a net, externally observable field.
The objective of magnetization is initially to magnetize a magnet to saturation, even if it
will later be slightly demagnetized for stabilization purposes. Saturating the magnet and
then demagnetizing it in a controlled manner ensures that the domains with the least
commitment to orientation will be the first to lose their orientation, thereby leading to a
more stable magnet. Not achieving saturation, on the other hand, leads to orientation of
only the most weakly committed domains, hence leading to a less stable magnet.
Anisotropic magnets must be magnetized parallel to the direction of orientation to
achieve optimum magnetic properties. Isotropic magnets can be magnetized through
any direction with little or no loss of magnetic properties. Slightly higher magnetic
properties are obtained in the pressing direction.
80
Chapter 4
4.5
Soft ferromagnetic materials - iron core materials
Soft magnetic materials are used to build the iron cores of the electric machines, thanks
to their low hysteresis loss.

Hysteresis losses
The hysteresis losses can be calculated considering a magnetic circuit of length
L and section S, supply by a sinusoidal current with I RMS value. The supply
power results:
(4.4)
Thus, the energy supply by the source in going from point “g” to point “b” (Fig. a),
results:
∫
∫
(4.5)
This integral corresponds to the shaded area, shown in Figure 4.10a; it is equal
to the energy supplied per unit volume of the magnetic core.
When the current is in the same direction but is decreasing (from point “b” to
point “c” of Figure 4.10b), the polarity of the induced e.m.f. is reversed,
according to Lenz’s law, the result that the energy return to the source is:
∫
(4.6)
(a)
(b)
Figure 4.10. Hysteresis losses: energy supply by the source (a) and energy return to the source
(b).
Magnetic materials utilized in linear electric machines
81
Finally, considering a complete loop “gbcdefg”, the energy supply by the source
[J], results:
∮
(4.7)
The area of the hysteresis loop in tesla-ampere turns/meter or in weber-ampere
turns/m3 is therefore the number of joules dissipated per cubic meter and per
cycle in the core (energy dissipated per unit volume per cycle). The energy
dissipated per unit volume per cycle is named as the hysteresis loss (Figure
4.11). This energy loss is due to the friction encountered during domain-wall
motion and rotation.
Figure 4.11. Hysteresis losses.
Hysteresis loss can be minimised only by selecting a soft magnetic material with
a narrow hysteresis loop. This is because the small loop area corresponds to a
low hysteresis loss. Thus, soft magnetic materials are suitable for armature and
transformer core. On the other hand, hard magnetic materials have larger loop
areas which correspond to higher hysteresis loss. Due to this reason, hard
magnetic materials are not suitable for alternating magnetisation but they are
suitable for permanent magnet.
The specific power of the hysteresis loss
[W/m3] is empirically given by
Steinmetz hysteresis law:
̂
where
(4.8)
is the coefficient of hysteresis losses (it depends on the nature of the
ferromagnetic material),
density [T] and
is the frequency [Hz], ̂ is the peak value of the flux
is a coefficient included between 1.5 and 2.5
82
Chapter 4

Eddy current losses
When a changing magnetic field cuts through a sample of metal or magnetic
materials that is not connected to a circuit, by Faraday’s law, a circulating
current is induced. This current is known as eddy current, it is localised within
that material and has a flow pattern as shown in Figure 4.12.
Figure 4.12. Eddy current (black arrows) induced within the metal or magnetic material and the
current induced by the external magnetic field (red arrows).
Eddy current creates losses through Joule heating, and it reduces the efficiency
of device that operates under alternating magnetic field condition such as iron
core of transformers and alternating current motors. This power loss is known as
eddy current loss due to the induced eddy current in the metal or magnetic
materials.
In general, the resistance
[Ω] of a material is given by:
(4.9)
where
is the resistivity of the material [Ω m], is the length of the piece of
material [m], and
is the cross-sectional area of the piece of material [m2].
In order to reduce the resistance the material resistivity can be increased by
adding silicon. Another effective way to achieve low eddy current loss is by
using lamination of electrical metal sheets, this means to reduce the surface
available for the eddy current. These metal sheets are coated with insulator
which breaks the eddy currents path.
As for the hysteric losses, the eddy current losses
analytically way:
̂
where
[W/kg]
can be expressed by
(4.10)
is the eddy current losses coefficient (it depends on the nature of the
ferromagnetic material).
The total eddy current caused by the local eddy currents is generally higher than
that predicted by the formulation under the global eddy current assumption. The
different is known as the excess loss. Since it is very difficult to calculate the
Magnetic materials utilized in linear electric machines
83
total average eddy current loss analytically, by statistical analysis, it was
postulated that for most soft magnetic materials under a sinusoidal magnetic
field excitation, the excess loss
[W/kg] can be predicted by:
⁄
where
̂
⁄
(4.11)
is the excess losses coefficient (it depends on the nature of the
ferromagnetic material).
The total loss can be calculated by summing equations (4.9), (4.10) and (4.11)
The Figure 4.13 illustrates the separation of alternating core loss of Lycore-140,
0.35 mm non-oriented sheet steel at 1 T. By equations (4.9), (4.10) and (4.11)
the coefficients of different loss components can be obtained by fitting the total
core loss curves, as shown in Figure 4.13.
Figure 4.13. Separation of alternating core loss of Lycore-140 at B=1 T.
Although eddy current reduces the efficiency of an electromagnetic device, but
in proper circumstances, it can be used as a non-destructive tool to measure
material thickness and coating thickness. It also can be used in the crack
detection, material identification, heat damage detection, and heat treatment
monitoring.
4.6
Soft magnetic composite materials
The ideal soft magnetic composites (SMC) material would be an isotropic electric
insulator with very high magnetic permeability, low coercivity and high saturation
induction. In addition the material should be easy to shape into 3-dimensional structures
in order to fully take advantage of the material’s isotropic nature.
84
Chapter 4
The most common soft magnetic material used today is steel sheet, either as pure iron or
alloyed with elements like Si in order to enhance performance. The steel sheets (with an
electrically insulating coating) are punched into 2-dimensional sheets and stacked to
form 2-dimensional laminated structures. Steel laminate is clearly the dominant material
for applications up to several hundred Hz. There are many reasons for this, good
magnetic properties (high induction, low losses at lower frequencies) and normally low
cost. However, this material also has drawbacks; it is by its laminated nature essentially
a 2-dimensional material. For higher frequencies, very thin and often alloyed sheets
must be used in order to minimize the eddy current losses. In these cases, the cost for
both rolling the steel sheets and punching those increases dramatically. Saturation
induction will also decrease due to the increased air-gaps between the sheets. The
punching operation also results in plastic deformation (distortion of the crystal lattices),
which causes deterioration of the magnetic properties.
From this we can conclude that none of the traditional materials fully meet the
requirements for an ideal soft magnetic material as outlined above.
Powdered Soft Magnetic Composites (SMCs) do however in many respects resemble
the ideal material, as they are isotropic materials consisting of small insulating iron
particles. Intricate 3D-shapes could be obtained by the well-established, cost-effective
P/M-compaction process. Saturation induction is close to the laminates and eddy current
losses are significantly lower due to the smaller size of the particles ( typically 5200μm) compared to the thickness of the steel sheets (normally 200-1000μm). However
SMCs have a distributed air-gap, leading to lower permeability and further, due to the
plastic deformation of the particles that takes place during the compaction step, higher
hysteresis losses. A heat treatment operation after the compaction will partly relieve the
stresses.
The benefits of replacing the conventional laminated core in an electric machine with
the powdered iron composites are considerable, and include:
 Essentially unity iron stacking factor, due to the fact that a SMC core is of
compact structure.
 Potential for reduced air gap length as a result of the tight tolerances maintained
in manufacturing SMC material.
 Increased copper fill factor (up to 66% vs. the typical value of 33%); reduced
copper volume as a result of increased fill factor and reduced end winding
length.
 Reduced copper loss as a result of the reduced copper volume;
Magnetic materials utilized in linear electric machines
85
 Reduced high frequency tooth ripple losses since the SMC has essentially no
eddy current losses;
 The above two items suggest a potential increase in the overall efficiency;
 Modular construction allows the possibility of easy removal of an individual
modular unit for quick repair or replacement;
 Possibility of producing three dimensional flux patterns in the SMC material and
thus, to improve the electric machine performance.
 Reduced axial length–over–end–winding dimension as a result of the more
compact end winding.
 Absence of the phase insulation as a result of using non–overlapping windings;
also, there is an additional potential elimination of the ground wall insulation
since the SMC core (stator) itself acts as an insulator.
 There is no need to stress relief, as in the case of the stator lamination after
punching and assembling the stack; this is a relatively costly and time
consuming task, (stress relief is, however, included in the process of
manufacturing the SMC parts).
 Reduced bearing currents in the presence of PWM waveforms, because of the
use of SMC which acts as insulation against this type of current flow.
 Stator is easily recyclable since the stator can again be compressed back into
powered form with pressure and the copper windings readily removed.
These advantages are, of course, accompanied by a number of drawbacks. The most
significant of these is the relatively low relative permeability of roughly 500 for the
most commonly used SOMALOY 500. Other concerns include relatively low rupture
strength, as well as an important portion of iron losses at lower frequency, due to the
increased hysteresis loss, although there is almost no eddy–current loss.
The SMC concept is very flexible as the final properties of the component will depend
on the properties of the iron powder, coating, lubricant / binder, compaction and heat
treatment. A schematic of the factor that affect the SMCs properties is show in Figure
4.14.
86
Chapter 4
Figure 4.14. Factor affecting SMCs properties.
4.6.1
Powder characteristics
Particle shape
In principle an SMC material could be built up by particles having different shapes. The
widely used particle shape to build SMC materials are: fibres, flakes and spheres
(Figure 4.15).
(a)
(b)
(c)
Figure 4.15. An example of tree possible particle shape of SMC materials: fibre (a), flake (b) and sphere
(c).

Fibre: The individual fibre is essentially a 1-dimensional element and its ability
to carry a magnetic flux is much better in the direction along the fibre than
across it. If fibres are filled into a die and compacted, the fibres will, due to the
prevailing pressure situation in the die, essentially be aligned in a plane
perpendicular to the compaction direction. This means a 2-dimensional material,
but because the distributed air-gap will be large in all directions in this material
it makes no sense.
The most effective packing of the fibres will occur if they, in the plane
perpendicular to the compaction direction, all were aligned in the same
direction. Although difficult to obtain in practice, an essentially 1- dimensional
Magnetic materials utilized in linear electric machines
87
material would be created. The distributed air-gap along the fibres would in this
direction be relatively small and eddy currents limited to the thickness of the
fibres. The fibres would probably be effectively packed, thus the material could
be expected to have very good magnetic properties in this direction. In the
transverse directions the situation would however be the opposite; a very large
distributed air-gap and thus a poor permeability.

Flake: Flaky particles will upon compaction, for the same reasons as discussed
above, essentially be aligned perpendicular to the compaction direction in order
to create an efficient 2-dimensional material. The distributed air-gap in this
plane will be small and eddy currents limited to the thickness of the flakes.
Therefore good properties in this plane could be expected, while the opposite
situation occurs in the plane perpendicular to the flake surfaces. Nevertheless,
this would be the superior 2-dimensional SMC-material.
Another factor influencing the anisotropy for both the fibre and the flake is the
demagnetisation factor, Nd.
The demagnetisation factor determines the strength of the demagnetisation field
Hd opposed to the magnetization field M, in each particle.
Hd = Nd M
(4.12)
where Nd is influenced by the shape of the particles and the direction of
magnetization in such a way that it will be small in directions parallel to the
fibre length and flake surface, while the opposite will occur in transverse
directions. This fact makes the anisotropy even more pronounced for these
materials while the demagnetizing factor is constant for isotropic shapes i.e.
spheres.

Sphere: This shape, is by its nature, not possible to align in any direction and a
true 3-dimensional, isotropic structure is created upon compaction. As the shape
is isotropic, the distributed air-gap is larger compared to the optimal directions
for the fibre and flake materials, but much smaller in other transverse directions.
Even if the isotropic material can in some respects be considered as a
compromise, this is the only alternative that will clearly benefit from the unique
shaping opportunities of the compaction process. The sphere makes it possible to
manufacture parts with 3-dimensional shapes that can carry a 3-dimensional
flux. This combination opens up new avenues to improve existing designs for
electrical motors, linear actuators and ignition coils etc.
88
Chapter 4
Purity
For steel sheets, purity of the iron is known to be of considerable importance to the
magnetic properties. The same will in principle apply to iron powders. It is well known
from traditional P/M that impurities affect compressibility negatively. A lower density
means a larger distance between particles, i.e. a larger distributed air-gap and poorer
magnetic properties. Interstitials such as O, N & C are perhaps the most poisonous and
are known to have a negative effect on both compressibility, which indirectly affects the
properties, and also directly on the base material, causing a significant drop in
permeability and increase of coercivity. Secondary effects like so called magnetic
ageing might also occur after long term operation whereupon interstitials are diffused to
certain areas in the material causing pinning of magnetic domains. The effect of other
alloying elements will depend from case to case. Normally the small trace elements of
other metals in iron powder will not significantly affect the properties. Nowadays iron
powders used for SMC materials are very pure with an extremely low content of
interstitials and harmful alloying elements and hence very suitable as soft magnetic
material.
The size of the particles will significantly affect the characteristics of the material.
Decreasing the particle size will increase the distributed air-gap, leading to lower
permeability and induction. Eddy currents will, however, decrease due to the smaller
particle size. Smaller eddy currents mean a larger skin depth and as a consequence
improved frequency stability and lower losses at higher frequencies. These facts are
illustrated in Figure 4.16, in which two high frequency materials, PermiteTM 55K(< 45
micron) and PermiteTM 75K(< 150 micron) with different particle sizes are compared.
Depending on the demands of the application, the appropriate particle size should be
selected.
(a)
(b)
Figure 4.16. Effect of SMC particle size on permeability (a) and core loss (b).
Magnetic materials utilized in linear electric machines
89
Insulating Coating
The demands on the insulating coatings of SMC materials are extremely high. Firstly
they must not be too thick, as that would increase the distributed air gap. Secondly, the
coating must be chemically stable in order not to be destroyed by chemical reactions
during processing of the material or service in the application. Thirdly, the coating must
be both hard and ductile in order to survive the pressures and friction during
compaction.
In this study the influence of insulating coating on the magnetic performance was
investigated. Two 100-mesh powders, uncoated and coated, were mixed with 0.5%
KenolubeTM, compacted at 800 MPa and heat treated at 500°C for 30min in air. The
coated material resembles SomaloyTM 500. The results are presented in Figure 4.17.
(a)
(b)
Figure 4.17 Effect of SMC insulation coating on induction (a) and permeability (b).
As shown, the coating will hardly affect the induction and permeability compared to the
uncoated material. In fact, the inorganic coating on the particles will facilitate the
compaction, as smearing in the compaction tool will be less of a problem. From this we
can conclude that the coating typically used is extremely thin and that a very small
additional air-gap is being introduced.
Looking at the losses, the very positive effect of the coating is obvious. The coating
significantly improves the frequency stability. This is a result of increased resistivity
and means also that the coating will significantly increase the skin depth for the
materials.
4.6.2
Manufacturing process
Compaction
SMC-materials could take direct advantage from the well-established compaction
technology for traditional PM-materials.
90
Chapter 4
The most critical compaction parameter influencing material performance is the
compaction pressure. It is always beneficial to compact at as high compaction pressure
as possible, see B-H curves in Figure 4.18.
Higher density means higher induction for a certain applied field, leading to benefits
such as higher torque in rotating machine applications.
At the same time, increasing the compaction pressure makes compaction more difficult.
The tool wear increases as well as the risk for tool failure. However, as the general trend
for powder metallurgy is towards higher density, knowledge on how to successfully
compact at high pressures is continuously improving and many companies already
today are using pressures up to 800 MPa or higher.
Figure 4.18. Effect of the compaction pressure on induction.
Heat Treatment
During compaction, plastic deformation in the particles occurs which leads to distortion
of the crystal lattices. This distortion acts as an energy barrier which the domain wall
motion and rotation have to overcome. As a consequence, the coercivity of the material
will increase and the response to an applied field will not be so strong, i.e. a flatter and
wider B-H curve. Therefore a stress-relieving operation of the compacted SMC-part,
allowing for a recovery of the crystal lattice will be very beneficial for the soft magnetic
properties. The inorganic coating used for the SomaloyTM-type of materials has been
optimised in order to withstand heat treatment to about 500°C and at this temperature
the stress relaxation of the material is significant. Lowering the temperature leads to less
Magnetic materials utilized in linear electric machines
91
efficient stress relaxation while higher temperatures destroy the inorganic coating, as
Initial Permeability
seen in Figure 4.19.
Figure 4.19. Effect of heat treatment temperature on frequency stability.
4.7
Analytical analysis of magnetic circuits
4.7.1
Equivalent electric circuit
Consider a simple structure consisting of a current carrying coil of N turns and a
magnetic core of mean length
[m] and a cross sectional area
[m2] as shown in the
Figure 4.20. The permeability of the core material is µc [H/m]. Assume that the size of
the device and the operation frequency are such that the displacement current in
Maxwell’s equations are negligible, and that the permeability of the core material is
very high so that all magnetic flux will be confined within the core.
Figure 4.20. Structure of a simple magnetic circuit.
92
Chapter 4
By Ampere’s law:
∮
(4.13)
∮
results:
(4.14)
where Hc is the magnetic field strength [A/m] in the core and
the
magnetomotive force [A].
The magnetic flux
[Wb] through the cross section of the core can express as:
(4.15)
where
the flux density in the core.
Therefore, results:
(4.16)
where
⁄(
) is the reluctance of magnetic circuit.
It is worth to point out, that if we take the magnetic flux
magnetomotive force
as the “emf of a voltage source” and
as the “current”, the
as the “resistance”, we
have an analogy with Ohm’s law of an electrical circuit theory.
Some corresponding quantities in electric and magnetic circuit are listed in Table 4.6.
Table 4.6. Corresponding quantities in electric and magnetic circuit.
Electric quantities
Magnetic quantities
Current
Magnetic flux
Current density
Magnetic flux density
Conductivity
Permeability
Resistance
⁄(
Conductance
)
⁄
Electromotive force
Electric field intensity
Reluctance
⁄(
Permeance
)
⁄
Magnetomotive force
Magnetic field intensity
For nonlinear magnetic circuits, where the nonlinear magnetization curves need to be
considered, the magnetic reluctance is a function of magnetic flux since the
permeability is a function of the magnetic field strength or flux density (Figure 4.17b).
Numerical or graphical methods are generally used to solve this kind of problems.
The equivalent electric circuit of a magnetic circuit is shown in Figure 4.21.
Magnetic materials utilized in linear electric machines
93
𝝓𝒄
𝑀
𝓡𝒄
Figure 4.21. Equivalent electric circuit of a simple magnetic circuit.
The differences between electric and magnetic circuits are as below:

The path of the magnetic flux flows is perpendicular to the current flows in the
circuit. In other words, the directions of B and J are perpendicular.

For a given temperature, electric resistance is constant and does not depend on
current density. However, the magnetic reluctance depends on magnetic field
and flux intensity since the permeability is not constant.

Current flowing in an electric circuit involves dissipation of energy, but for
magnetic circuit, energy is needed to generate magnetic flux.
Now, consider a magnetic circuit with an air gap of length
cut in the middle of a leg,
as shown in Figure 4.22a. As they cross the air gap, the magnetic flux lines bulge
outward somewhat, as illustrated in Figure 4.22b. The effect of the fringing field is to
increase the effective cross sectional area
(a)
of the air gap.
(b)
Figure 4.22. Structure of a magnetic circuit with an air gap (a) and flux line distribution in the air gap
(b).
By Ampere’s law, results:
(4.17)
where
(4.18)
94
Chapter 4
and
(4.19)
According to Gauss’ law in magnetics:
(4.20)
∮
Results:
(4.21)
Therefore:
(
)
(4.22)
That is, the above magnetic circuit with an air gap is analogous to a series electric
circuit. Further, if we regard Hclc and Hglg as the “voltage drops” across the reluctance of
the core and the air gap respectively, the above equation from Ampere’s law can be
interpreted as an analogy to the Kirchhoff’s voltage law (KVL) in electric circuit theory,
that is:
∑
∑
(4.23)
The Kirchhoff’s current law (KCL) can be derived from equation (4.20), that is:
∑
(4.23)
Where N is the total number of branches with currents flowing towards or away from
the node.
The equivalent electric circuit of the magnetic circuit with an air gap is shown in Figure
4.23.
𝝓𝒄
𝓡𝒄
𝑀
𝓡𝒈
Figure 4.23. Equivalent electric circuit of a magnetic circuit with an air gap.
Magnetic materials utilized in linear electric machines
95
Consider a piece of permanent magnet of a uniform cross sectional area of
length
and a
, as shown in Figure 4.24a. The demagnetization curve of the magnet is a
straight line with a coercive force of Hc and a remanent flux density of Br, as shown in
Figure 4.24b. The demagnetization curve can be expressed analytically as:
(
⁄
where
)
(4.25)
is the permeability of the permanent magnet, which is very close
to o, the permeability of free space (for a NdFeB magnet, PM = 1.05 0).
The magnetic “voltage drop” across the magnet can be expressed as:
(
where
is the reluctance and
)
(4.26)
is the magnetomotive force (“voltage source”) of
the magnet. It should be noted that in the magnet, BPM and HPM are in opposite directs.
The equivalent electric circuit of a permanent magnet is reported in Figure 4.24c.
Figure 4.24. Structure (a), demagnetization curve (b) and equivalent electric circuit (c) of a permanent
magnet.
4.7.2
Limitations of the magnetic circuit approach
The number of problems in practical magnetic circuits that can be solved by the
magnetic circuit approach is limited, despite the similarity of this approach to simple dc
electric circuit theory. The purpose of introducing magnetic circuits is to state some
very fundamental principles and definitions that are necessary in order to understand
electromagnetic systems. The limitations of magnetic circuit theory rest primarily in the
nature of magnetic materials as contrasted with conductors, insulators, and dielectric
materials. Most of these limitations have already been introduced as "assumptions" in
the discussion of magnetic circuits. Let us assess the significance of these assumptions.

Homogeneous
magnetic
material:
most
materials
used
in
practical
electromagnetic systems can be considered homogeneous over finite regions of
96
Chapter 4
space, allowing the use of the integral forms of Maxwell's equations and
calculations of reluctances and permeances.

Isotropic magnetic materials: many sheet steels and ferrites are oriented by
means of the metallurgical process during their production. Oriented materials
have a "favored" direction in their grain structure, giving superior magnetic
properties when magnetized along this direction.

Non-linearity: this is an inherent property of all ferro- and ferrimagnetic
materials.
However, there are many ways of treating this class of nonlinearity analytically:
 as can be seen from the B-H curve, a considerable portion of the curve for
most materials can be approximated as a straight line, and many
electromagnetic devices operate in this region;
 numerous analytical and numerical techniques have been developed for
describing the B-H and other nonlinear magnetic characteristics;
 the nonlinear B-H characteristic of magnetic materials manifests itself in
their relationship between flux and exciting current in electromagnetic
systems; the relationship between flux and induced voltage is a linear one, as
given by Faraday's law. It is possible to treat this nonlinear excitation
characteristics separately in many systems, such as is done in the equivalent
circuit approach to transformers and induction motors;
 an inductance whose magnetic circuit is composed of a magnetic material is
a nonlinear electric circuit element, such as a coil wound on a magnetic
toroid;
 with an air-gap in the magnetic toroid, however, the effect of the nonlinear
magnetic material on the inductance is lessened. Actuators have air-gaps in
their magnetic circuits, permitting the basic theory of these devices to be
described by means of linear equations;

Saturation: all engineering materials and devices exhibit a type of saturation
when output fails to increase with input, for instance, in the saturation of an
electronic amplifier. Saturation is very useful in many electromagnetic devices,
such as magnetic amplifiers and saturable reactors.

Leakage and Fringing Flux: this is a property of all magnetic circuits. It is best
treated as a part of the generalized solution of magnetic field distribution in
space, often called a boundary value problem. In many actuator magnetic
circuits, boundaries between regions of space containing different types of
Magnetic materials utilized in linear electric machines
97
magnetic materials (usually a boundary between a ferromagnetic material and
air) are often planes or cylindrical surfaces that, in a two-dimensional cross
section, become straight lines or circles. Leakage inductances can frequently be
determined in such regions by calculating the reluctance or permeance of the
region using fairly simple integral formulations. The spatial or geometric
coefficients so obtained are known as permeance coefficients.
98
Chapter 4
Chapter 5
Semi-analytical model of a linear
machine
5.1
Introduction
The previous chapters described the working principle of Stirling engines and the
different typology of linear machine, utilized in the cogeneration system and renewable
energy converter. The application is referred to a micro co-generation system − based
also on renewable energy sources – to supply domestic or isolated low rated loads.
Mechanical energy is provided by a free-piston energy converter (FPEC), which
converts thermal energy produced by a thermal source into mechanical energy;
specifically, an external combustion engine, based on Stirling cycle, was considered
[204]. Mechanical energy is then converted into electrical energy by means of linear
generator (LG). The FPEC systems can be connected to the distribution grid by a backto-back converter according the schemes adopted for variable speed generators; as for
the LG operation, this scheme allows optimal starting and synchronization by switching
the generator to motoring operation, motion regularization and optimization of the
energy output by a proper current control strategy [205]. The general scheme of such
system is reported in Figure 5.1.
Figure 5.1. Scheme of a free-piston energy converter integrating a Stirling engine and a single phase
TLPMG.
100
Chapter 5
The main advantages of FPECs with respect to conventional crank–rod based on rotary
machine systems are:

high specific power thanks to their greater compactness;

high reliability thanks to the reduction of the moving parts;

high efficiency;

cheapness thanks to their constructive simplicity;

low maintenance;

low noise pollution.
The short stroke permanent magnet linear generator (TPMLG) can be convenient as it
represents a good compromise between the production costs and the output
performances.
The main advantages of tubular configuration are:

higher specific power respect to flat configuration;

no normal component of the force;

high modularity;

high manufacturing simplicity;

possibility to adopt ring winding.
The core lamination of tubular configuration results quite difficult because the
lamination, in order to reduce the eddy current loss, should be arranged in the axial
direction and this involves a hard manufacturing production. The axial lamination of a
tubular linear machine is reported in Figure 5.2 [206].
(a)
(b)
Figure 5.2. Stator core fabricated from modules formed from I-shaped laminations (a), schematic of
laminated linear generator [206].
Semi-analytical model of a linear machine
101
For such reason, the use of SMC materials is practically binding to obtain complex core
shapes by suitable molding process. In addition, the SMC materials present a lower
eddy current loss but a lower permeability than conventional lamination.
Finally, the use of PMs involves the following features:

higher efficiency respect to electrically excited machines thanks to absence of
copper loss in the mover;

lighter mover, this feature allows achieving a greater dynamic performance;

higher costs due to the rare earth utilization to build the PMs;
Magnetization pattern is a relevant issue for the electromagnetic design of the machine,
since it can heavily affect the TPMLG performances. Several papers [207,208] suggest
Halbach or quasi-Halbach arrangement as the more appropriate magnetization pattern
for such kind of applications. The main features given by the Halbach and quasiHalbach magnetization arrangement are:

the axially magnetized magnets essentially (or the axial component of
magnetization in the case of Halbach arrangement) provide a return path for the
radial airgap flux, and, hence, the flux in the mover yoke is relatively small;

the permanent magnets may be mounted on a non-magnetic support tube without
significantly compromising the thrust force capability [208];

the mass of the moving piston and permanent magnet armature assembly can be
greatly reduced, which is essential for achieving a high output power and
dynamic performances of the cogeneration system;

sinusoidal distribution of air-gap flux density.
The Halbach magnetization enhances the quasi-Halbach advantages, but with the
drawback of more complicated manufacturing and uncertain resulting profile.
The aim of the research was to develop a model that allows analysing, optimising and
thus designing a TPMLG; so the goal is achieving the results, in terms of thrust force,
output power and efficiency, varying the main electromagnetic quantities that affect the
TPMLG performances.
One of the most important issues to investigate, from electromagnetic design point of
view, is the magnetization pattern, since it can heavily affect PM amount and
consequently the mover mass and, in turn, dynamic performances. For this purpose,
analytical and numerical methods are proposed to study tubular configurations
[209,210]. Due to the asymmetrical magnetic configuration and to the remarkable
fringing end effects, analytical methods – generally relying on the assumption of
periodic boundary conditions – can be hardly applied.
102
Chapter 5
On the other side, finite element analyses (FEAs) carried out on the whole configuration
for different mover positions can be excessively time-consuming for a preliminary
design, when the influence of several PM magnetizations must be investigated.
In order to reduce the computational effort, an approach combining harmonic
decomposition and automated magnetostatic FEA sequences is proposed, to simulate
the PM motion and different magnetization profiles by analysing fixed magnetization
space harmonic distributions.
Another important issue is the current supply strategy definition. Such matter is
essential in order to determine the best solution to maximize the power conversion.
The method is based on the superposition of the effects of each magnetization space
harmonic and the effect of the supply current, while the geometry is being unvaried. In
order to apply the superposition principle the magnetic linearity of iron cores (stator and
mover) materials is assumed, that means to consider a constant magnetic permeability
for the ferromagnetic parts.
Actually, the mover yoke saturation is generally not negligible, due to high dynamic
requirements that constrain the mover mass reduction as much as possible. Thus, the
demagnetizing effect due to mover yoke saturation is taken into account by a suitable
current sheet distribution on the mover surface, determined by an iterative procedure.
The proposed semi-analytical method allows overcoming the inherent limitations of
analytical methods (asymmetrical magnetic configuration, remarkable fringing end
effects and mover local magnetic saturation) that can be simply managed only in some
simplified cases or by adopting a complex formulation and appropriate corrections
[211]. On the other hand less computational effort is required than normal FEAs in the
case of high number of electromagnetic parameter variation, as it usually occurs for a
machine design and optimization (i.e. several mover positions, different magnetization
and current supply solutions).
In fact, the proposed method, after a suitable number of FEAs, allows investigating of
several magnetization patterns and supplying strategies, without requiring any other
FEAs.
The method allows taking into account several SMC materials for the ferromagnetic
parts in the hypothesis of unvaried linear magnetic permeability (the permeability value
before the saturation effect becomes remarkable). In this condition is possible utilize the
same space harmonic FEAs, adjusting only the mover current sheet distribution
according to BH curve of specific SMC material.
Semi-analytical model of a linear machine
5.2
103
Preliminary design
The proposed procedure is applied to the analysis of a short-stroke 3-poles TPMLG
(Figure 5.3), using a commercial software package (Ansys Maxwell®).
Figure 5.3. TPMLG prototype; (a): stator assembly with twelve SMC modular cores; (b): mover
consisting of bonded PM ring mounted on a steel shaft.
The main TLPMG ratings and material magnetic properties are shown in Table 5.1. In
particular the maximum PM magnetization MPM, is typical of a bonded rare-earth PM
material, and it is kept unvaried for all the examined magnetization schemes, including
pure radial, Halbach and quasi-Halbach magnetization patterns.
Table 5.1. Main ratings design data and magnetic properties.
m/s
kA⁄m
=
A⁄
1.32 A
0
The TPMLG is preliminarily sized according to the maximum flux density allowable in
the SMC cores (
1.3 T) [212] and assuming simple harmonic motion and sinusoidal
current waveform. The asymmetrical pole shoe profile is devised to increase the side
flux linkage and to reduce both the cogging effects
parametric analysis, varying the pole shoes height (
(
16, 18, 20 mm) and internal (
Halbach magnetization and current
5.2.
and the coil inductance . A
3, 4 mm) and the external
12, 14, 16 mm) width, was carried out for
3 A. The parametric data are reported in Table
104
Chapter 5
Table 5.2. Parametric analysis per preliminary design of pole show profile (Halbach
magnetization,
3 A).
N.
[mm]
[mm]
[mm]
[N]
[mH]
3
4
3
4
3
4
3
4
3
4
3
4
3
4
3
4
3
4
16
16
18
18
20
20
16
16
18
18
20
20
16
16
18
18
20
20
12
12
12
12
12
12
14
14
14
14
14
14
16
16
16
16
16
16
79.3
87.6
22.8
27.7
32.9
34.3
22.7
18.8
58.4
57.9
93.7
90.8
64.6
55.7
117.3
123.5
159.9
152.3
441.7
448.8
453.2
460.4
465.2
470.1
464.6
474.9
476.7
486.3
489.1
493.1
499.9
512.8
511.8
525.7
525.6
534.2
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
From Table 5.2 results that the minimum value of cogging force and inductance is
achieved for the analysis number 3 and number 7. Fur such reason it is chosen
mm,
17 mm and
3
13 mm. Main geometrical data of the reference and
simplified models are reported in Table 5.3.
Table 5.3. Main geometrical model data (see Figure 5.4; sizes in mm).
44.5
6
29.4
52
300
40
8
10
= 6.4
= 14
17
1
13
200
=3
90
Being the coil section is 448 mm2 and assuming a 50% filling factor, the coil resistance
at operating temperature is estimated at RG = 20.5Ω.
In order to limit the mover mass, an adequate shaft stiffness being maintained, mover
yoke thickness
must be limited as much as possible, provided an excessive
magnetic saturation is avoided.
Semi-analytical model of a linear machine
5.3
105
Simplified FEA model
The simplified FEA model (called HFEA: harmonic finite element analysis) aims to
improve the performance and optimize the linear generator after the preliminary design
with a significant reduction of the computational effort [213-214].
The semi-analytical model is based on the superposing the effect of different order of
magnetization space harmonics, which are fixed with respect to the stator reference. The
magnetization space harmonics are obtained by means of the Fourier analysis of PMs
magnetization profile. The PMs magnetization distribution can be reproduced
recombining, through the relative Fourier coefficients, every magnetization space
harmonic order. For such purpose, the PMs are replaced with a PM band with the same
permeance of PMs and with a sinusoidal magnetization with different harmonic orders.
The geometrical model utilized for simplified FEA derives from real LG geometry
(reference model, shown in Figure 5.4a). Thank to symmetry (in this case the mover is
kept fixed respect to stator) is possible to use a simplified model, considering only half
geometry (Figure 5.4b).
(a)
(b)
Figure 5.4. FEAs geometrical models: reference model (a), simplified model (b). E1, E2: symmetry
boundaries; : sample air-gap line for the flux density; C: coil section.
106
Chapter 5
In such model the stator ferromagnetic parts are unchanged and modelled with a
constant permeability value. Differently, the PM band length
with respect to PM length
must be extended
to cover the maximum value of the stroke
, in order to
enable the reproduction of the spatial magnetization function, that is:
(5.1)
The best solution is obtained with infinite-length PM band, in fact this solution allows
considering only the odd harmonics and neglecting the even ones. In order to obtain
such condition, even/odd condition is applied to the boundary edge. In this case the
mover yoke length has to extend. However, the mover yoke length cannot reach the
maximum value (continuous mover yoke between adjacent TPMLGs) in order to avoid
fictitious flux leakage on the E2 boundary. Figure 5.5 shows the cogging force profile
for different mover yoke configuration. As can be seen the complete mover
configuration (red line) does not reproduce the force profile in the correct way due to
the flux path between adjacent models (real and fictitious models).
Figure 5.5. Cogging force profile for different mover yoke configuration.
In order to deal with mover yoke saturation an iterative procedure was developed,
introducing in the former linear model a set of 2Z fictitious current sheet coils wound
around suitable zones of the mover yoke – where saturation is more remarkable – to
simulate the demagnetizing effect due to saturation, the permeability value
at
linearity being unvaried (Figure 5.6). These coils are fixed with respect to the stator
reference frame and symmetrical with respect to r-axis.
Semi-analytical model of a linear machine
107
𝒘𝒎 𝒉
Figure 5.6. Scheme of the current sheet distribution on the mover surface in the generic hth zone.
In order to choose the number and the width of the zones a preliminary FEA was carried
out, in order to identify the zones where the flux density remain approximately constant
for the different mover positions. Alternatively, the zones widths can be choose
analytically. Particularly, has been chosen Z = 2 zones of width equals to
⁄
.
In the hypothesis of magnetic linearity is possible to apply the superposition principle,
analysing separately the effect of each magnetization harmonic, current supply, and
mover current sheet.
In order to study any magnetization patterns, two different magnetizations are
considered for PM band: radial (indicated with letter “r”) and axial (indicated with letter
“z”). The PM band magnetization is set a space harmonic at a time, with radial and axial
magnetization separately. In the case that the software does not allow setting the
magnetization profile, is possible to reproduce it through suitable current densities (for
the radial magnetization) and current sheets (for the axial magnetization).
To reconstruct the PMs magnetization profile by means of Fourier principle, we have to
impose two different magnetization profile that results with a 90° phase difference
between them. These two magnetizations have the magnetic axes along the d and q axes
respectively, as shown in Figure 5.7 that report the decomposition of the fundamental
harmonic (
(
and
) of a square wave radial magnetization profile (
) for a generic stroke s.
) in two components
108
Chapter 5
Figure 5.7. Components
harmonic
and
in the d - q and stator reference frame of the fundamental
related to a pure radial magnetization
;
: maximum PM magnetization.
To summarise, for each harmonic order we carried out four FEAs, with the follow
magnetizations:
,
,
e
. Four different FE models were developed,
with the same geometrical model but different magnetization profile, as shown in Figure
5.8.
Figure 5.8. FE geometrical model with current density and current sheet utilised to reproduce the
magnetization.
In the follow are reported the magnetization (just for simplicity with unitary amplitude)
for the four different models for the generic harmonic :
a) Radial component, d-axix:
results:
. The current density in the PM band
(
The boundary conditions on the edges are:
E1: odd;
E2: even.
)
[A/m2]
(5.2)
Semi-analytical model of a linear machine
b) Radial component, q-axix:
results:
109
. The current density in the PM band
(
)
[A/m2]
(5.3)
The boundary conditions on the edges are:
E1: even;
E2: odd.
c) Axial component, d-axix:
results:
. The current on the PM band surface
(
)
[A/m]
(5.4)
The boundary conditions on the edges are:
E1: odd;
E2: even.
d) Axial component, q-axix:
results:
. The current on the PM band surface
(
)
[A/m]
(5.5)
The boundary conditions on the edges are:
E1: even;
E2: odd.
As can be noted, the magnetization is independent to mover position and results fixed
with respect to stator reference. For such reason is possible to consider only odd
harmonics because the even harmonics give a null contribution.
The only models that give a non-null contribution to the flux linkage are the “b” and
“c”.
As regards the contribution due to current supply, a unitary value is set in “b” or “c”
model, after setting to zero the magnetization.
As regards the current sheet distributions on the mover yoke, utilised to reproduce the
demagnetising effect due to magnetic saturation, they are reproduced in the “c” or “d”
model. In fact, they are the unique models that give an axial component of the flux
density and thus can represent the demagnetizing effect of the saturation. Also in this
case, just for simplicity, the values are set, one at a time, to unitary value, while the PM
band magnetization is virtually switched off.
110
Chapter 5
To summarise, the carried out FEAs are (let N number of harmonics considered to
reconstruct the PMs magnetization and Z number of zones of mover division):

2N analyses by only applying the PM current distributions;

one analysis by applying only the coil current;

Z analyses by only applying the current sheet on the mover surface.
As can be seen, the most computational effort of the HFEA method is due to the
simulations for the space harmonics magnetization. This suggests choosing carefully the
number of the harmonics N, in order to reconstruct correctly the PMs magnetization
without an excessive computational time.
Another important issue which affects the computational effort of the HFEA is the
length
of the model.
To choose the number of harmonic to consider, the cumulative contribution of the
harmonic is compared to the FEA carries out for the reference model. This process has
been done for some different mover positions, Figure 5.9 shows the cogging force trend
for no-load condition and s=10mm.
Figure 5.9. Harmonic contribution on cogging force (no-load, s=10mm).
It can be concluded that, after about 50 harmonics, the force value is reconstructed well
enough. Moreover this solution does not involve an excessive effort in terms of time
consuming, in fact it require about 25 simulations for each model.
In conclusion, considering
harmonics, HFEA method requires little more than
100 simulations. Considering that, during the analysing stage of the machine, we should
consider different mover positions (at least 30 positions) and different current supply
Semi-analytical model of a linear machine
111
values (at least 10 values), that means at least 300 simulations; it is clear the advantage
to use the simplified method. Besides, the simplified model considers only half
geometry of the TPMLG, and this involves a further time consuming reduction. In the
case of machine optimization the HFEA results even more convenient, since different
magnetization patterns and different current strategy have been considered.
The HFEA execution to obtain the complete model on an ordinary PC requires about 25
minutes. Even if preliminary calculation time for each harmonic increases with
the
subsequent solution takes only some tenth of a second per configuration, including nonlinear iterative procedure. On the contrary, a full model direct FEA takes about 1 min
per solution, the same HFEA energy error threshold being maintained. Accordingly,
direct FEA is likely more convenient to analyse a single magnetization pattern and a
limited number of mover positions; instead, HFEA seems to be more profitable for
taking into account multiple magnetization schemes, finer position samplings and
variation of the supply strategy.
Another item to take into account is the meshing process of the FEA. One of the main
problems is the meshing the object near the airgap, because a not thick enough mashing
may result in a not correct reconstruction of the solution fields. On the other hand, if the
mesh is thick enough near the airgap, this may involve an excessive mesh in the other
part of the model, and this result in excessive time consumption.
This problem can be solved by subdividing the airgap between the stator core and PMs
into several smaller parts. In this way is more simple to control the number of the mesh
elements in the object, and thus find the right balance between the solution accuracy and
the computational effort require for the solution.
5.4
Working principle of simplified model
In the previous paragraph the geometrical models utilised for FEAs have been
described.
The HFEA procedure aims at the calculation of the following quantities:

radial and axial components of the airgap flux density:
( ) ( )
( )
(the
superscripts indicate: magnetization direction (radial or axial), axis direction (daxis or q-axis); the subscripts indicate: component direction (radial or axial),
harmonic order).
Such quantities are sampled on suitable points
in air-gap lines (see
in
Figure 5.3 as an example). The sampling should be dense enough to reconstruct
112
Chapter 5
the flux density profile in the correct way; particularly, the flux density was
sampled on 1500 points. The flux density samples are utilised to calculate the
force value by means of the Maxwell stress tensor;

radial and axial components of the flux density in the stator
( ) ( )
( ) ( )
( )
and mover
cores. To the purpose, a grid inside the cores is defined to sample with
enough accuracy the flux density distribution. The samples are utilised to
estimate the core loss, by means of the hysteresis, eddy current and excess
coefficients;

the coil flux linkage (
,
). Only d-axis magnetization component gives a
non-null contribution to the flux linkage. It is utilised to evaluate the total flux
linkage and thus to calculate the electromotive force (emf).
As mentioned in the previous paragraphs, the HFEA method is based on the
superimposing the effect of the different harmonics. The process that has been used is
reported in the following, firstly referring to a non-saturated condition:
1) the PMs magnetization profile has been decomposed in Fourier series in order to
determine the Fourier coefficients with reference to simplified model PM band
width
. The Fourier coefficients depend on mover position s and geometrical
quantities. In the case of two magnetization components (radial and axial), they
have to be considered separately, calculating the Fourier coefficients for each
one;
2) the HFEA is carried out to obtain the TPMLG electromagnetic model. In
particular it consists of the calculation of the flux density components, in the
airgap and in the stator and mover cores, and of the coil flux linkage for each
magnetization harmonic and coil current. Such data are then stored for
subsequent elaborations;
3) thank to linearity condition, the total quantity values can be reconstructed (those
that are related to the real model), in terms of airgap flux density and flux
linkage, multiplying the electromagnetic quantity of a specific harmonic order
by the related Fourier coefficient and after summing the contribution for each
harmonic order. As regards the current supply contribution, the electromagnetic
quantities related to unitary current supply are multiplied for the real current
supply value. Thus, for a generic load condition, the total quantity values are
given by the sum of the harmonic contributions and current supply contribution.
Semi-analytical model of a linear machine
113
By such approach, mover position dependence is taken into account by the
Fourier coefficients, without requiring additional FEAs.
4) starting from airgap flux density and from flux linkage is possible to calculate
the thrust force and electromotive force values respectively. These values are
affected by the number of the harmonics considered.
5) the demagnetizing effect on the mover yoke, due to magnetic saturation, is to
take into account by a suitable procedure described in the next paragraph;
6) the TPMLG performances are evaluated with the estimation of the core losses
and efficiency.
The first point regards the decomposition of PMs magnetization profile in Fourier series
to obtain the Fourier coefficients. For a general function ( ) with period , from the
definition of the Fourier series results:
( )
where
∑(
(
)
is the order of harmonic the coefficients
(
and
))
(5.6)
are calculated according to
the specific magnetization distribution.
The more utilized magnetization patters are: radial, quasi-Halbach and Halbach. All the
considered magnetization profiles are symmetrical with respect to r-axis (or rather qaxis) for s = 0. The Fourier coefficients relative to these magnetization outlines are:

Radial: the radial magnetization corresponds to a square-wave profile with only
the radial component
(Figure 5.10).
r
z
Figure 5.10. Radial magnetization scheme.
For such kind of magnetization, the PMs could not be contiguous but can be an
air gap between them (quasi-square wave distribution):
⁄ .
where
114
Chapter 5
The Fourier coefficients results:
∫
(
)
(
)
(
)
(5.7)
∫
(
)
(
)
(
)
(5.8)
where:
( )
∫
(
)
(5.9)
[
(
)
(
)
(
)]
with:
(5.10)

quasi-Halbach: the quasi-Halbach magnetization corresponds to a square wave
distribution for both
and
(Figure 5.11).
r
z
Figure 5.11. Quasi-Halbach magnetization scheme.
For such kind of magnetization profile, the PMs are alternately magnetized in
radial and axial directions in order to the axially magnetized magnets essentially
provide a return path for the radial air-gap flux, and, hence, the flux in the inner
bore of the armature is relatively small.
For the radial component of magnetization the Fourier coefficients results:
∫
(
)
(
)
(
)
(5.11)
∫
(
)
(
)
(
)
(5.12)
Semi-analytical model of a linear machine
115
where
( )
∫
(
)
[
(
(
))
(5.13)
(
(
))
(
)]
For the axial component of magnetization the Fourier coefficients results:
∫
(
)
(
)
(
)
(5.14)
∫
(
)
(
)
(
)
(5.15)
(
(
Where
[
))
(
(
))
(5.16)
(

(
))
(
(
))]
Halbach: the Halbach magnetization corresponds to sinusoidal spatial
distributions for
and
(Figure 5.12).
z
Figure 5.12. Halbach magnetization scheme.
For such kind of magnetization the axially magnetization provides a return path
for the radial air-gap flux giving the same benefit provided from quasi-Halbach
magnetization, furthermore the air-gap flux density distribution approximate in
a better way the sinusoidal profile, and thus the emf results with a lower
harmonic content.
116
Chapter 5
The radial (axial) component of magnetization the Fourier coefficients results:
( )
∫
( )(
)
(
)
( )
(
)
(5.17)
( )
∫
( )(
)
(
)
( )
(
)
(5.18)
(
)[
(
(
)(
where:
)]
(5.19)
)
(5.20)
with:
(5.21)
After executing the FEAs for the four models (step 2), is possible to reconstruct the total
values of the electromagnetic quantities as radial and axial component of airgap flux
density and total coil flux linkage.
The no-load radial (axial) component of the flux density
( )
(
)
∑[
( )
( )
( )
( )
( )
( )
( )
( )
( )
results:
( )
( )
( )
( )]
This is a general formulation; it can be used to calculate the airgap flux density
as well as the flux density inside the mover
( )
(5.22)
and stator
( )
( )
cores.
According to Figure 5.4a reference frame, only d-axis magnetization components
contribute to coil flux, so the no-load flux linkage results:
( )
∑
( )
( )
(5.23)
By the no-load airgap flux density formulation and according to Figure 5.4a, the noload (cogging) force
and is determined by Maxwell stress tensor:
( )
∫
(
)
(
)
(5.24)
Semi-analytical model of a linear machine
By (23), the no-load emf
117
results:
( )
( )
∑(
where
( )
(5.25)
)
is the number of coil turns. It’s worth pointing out
is the mover speed and
that the flux linkage derivative obtained from the analytical expressions of the
coefficients
and
is much more robust from the numerical point of view than
other numerical procedures.
A separate FEA is executed with unitary total coil current and de-energized PMs. The
flux density components
and
as well as the flux linkage
are then
determined as at no-load.
The radial and axial flux density components
,
and flux linkage
at load are
calculated as:
( )
(
)
( )
where
(
)
( )
( )
( )
( )
and the emf
( )
(5.27)
and
at load are obtained as in equation (5.24) and (5.25), where
are replaced by
,
formulation including the dependence of
written as (assuming the flux linkage
( )
and
and
, respectively. An alternative
on the variation of
with
can be
nearly independent on ):
( )
( )
( )
5.5
(5.26)
[A] is the total coil supply ampere-turns.
The force
,
( )
(5.28)
(5.29)
Analysis take into account the mover saturation
In order to limit the mover mass, an adequate shaft stiffness being maintained, mover
yoke thickness
must be limited as much as possible, provided an excessive magnetic
saturation is avoided. In principle mover yoke saturation cannot be nevertheless
neglected (as shown in Figure 5.13), so, at a first glance, the above described procedure,
relying on the superposition principle, is invalidated.
118
Chapter 5
Figure 5.13. Flux density distribution for radial magnetization, s = 20mm, Mc = 1000 A.
Therefore an iterative procedure is developed to include saturation, introducing in the
former linear model a set of 2Z fictitious current sheet coils wound around suitable
zones of the mover yoke – where saturation is more remarkable – to simulate the
demagnetizing effect due to saturation, the permeability value
at linearity being
unvaried. These coils are fixed with respect to the stator reference frame and
symmetrical with respect to r-axis.
Let
the linear current density of each h-th coil (with
), the effect of each
pair of symmetrical coils (i.e., h-th and (2Z+1-h)-th coils) (Figure 5.14a) can be
obtained by summing the effect of two even-symmetrical current sheets
5.14b) and two odd-symmetrical current sheets
(Figure
(Figure 5.14c), given by:
(5.30)
(5.31)
As a result, the analysis can be restricted to a halved FE model for both the cases,
changing only boundary conditions on E1 and E2. This model is analysed by a set of 2Z
FEAs, supplying each k-th coil one by one with
energized PMs. As a result, the flux density values
, with
points in the h-th zone) along the line
the mover yoke are obtained.
1 A/m, setting
( )
( )
in the points
, de(
of Figure 5.14a in the middle of
Semi-analytical model of a linear machine
119
(a)
(b)
(c)
Figure 5.14. Fictitious current sheets introduced to allow for saturation ( : sample line for
( )
evaluation of
) (a) and decomposition into even
(b) and odd
(c) current sheets.
The current density
is determined according to the mismatch
between the
magnetic field value H at saturation and the corresponding value
at linearity
(permeability
), the same flux density being maintained (Figure 5.15).
Figure 5.15. Definition of the magnetic field mismatch
to take into account magnetic saturation.
120
Chapter 5
Applying the method of images, the magnetic field due to the current sheet outside (in
the PMs)
and inside the mover
is given by, respectively:
(532.)
(5.33)
and consequently the boundary continuity conditions for the magnetic field yield (
:
magnetic field at linearity):
(
(
with
(
)
)
(5.34)
)
(
)
permeability variation with respect to
expressed as a function of
by:
(5.35)
where
is the total (the resultant effect of magnetization, coil current and mover
current sheet densities) flux density in the points
Substituting (5.35) in (5.34) and solving for
.
gives finally:
(5.36)
Since the field generated by the fictitious coils affects in turn the flux density inside the
mover, the mismatch ΔH is iteratively re-adjusted, according to the flow diagram of
Figure 5.16. Starting from an initial set of the estimated current density values
), the even/odd components
can be obtained by (5.30) and (5.31).
( )
(
Multiplying the corresponding unit-current flux density values
the estimated flux density contributions
sample point
the Z
(
)
(with
(
)
)
by
( )
yields
to the flux density variation on each
due to each k-th current sheet. Summing such contributions for all
( )
pairs to the linear flux density value
magnetization and coil current, the updated flux density values
corresponding permeability
(
) and magnetic energy
according to the mover magnetic characteristic.
(
( )
(
due to
)
and the
) are obtained,
Semi-analytical model of a linear machine
121
Figure 5.16. Conceptual scheme of the iterative procedure to define the h-th zone current density.
Let the h-th zone equivalent permeability
, the weighted average of
taking
as
weighting function, is calculated as:
∑
(
)
∑
The corresponding flux density value
mismatch
(
(
)
(5.37)
)
, the linear
values of the magnetic field are obtained.
, non-linear
and
value is updated putting
in (5.36) and multiplying by the due sign.
This process is repeated for each zone to obtain a new current density set for a further
step of the iterative procedure, stopped according to a suitable convergence test. It is
worth remarking that, even if the above described procedure must be repeated for each
mover position, i.e. s value, the re-evaluation of the unit-current flux density
(
)
is
not required, since fictitious coils are fixed with respect to the stator. These values are
122
Chapter 5
therefore evaluated and stored not only in the mover sample points
air-gap point set
but also in the
, as well as the corresponding per-turn coil flux linkage
current density
due to
A/m (q-component does not contribute to coil flux linkage).
Evidencing the dependence on s and adopting the above notations, the expressions
(5.26) and (5.27) of the airgap flux density and the coil flux linkage at linearity (denoted
by
subscript) are adjusted taking into account the current sheets contributions as
follows:
( )
(
)
( )
(
( )
(
( )
)
)
∑(
( )
( )
( )
( )
( )
( )
( )
∑(
(
)
( ))
(5.38)
)
(5.39)
)
(5.40)
Correspondingly, force and emf are obtained, respectively:
( )
( )
where the derivative
⁄
∫
(
( )
)
(
∑
(5.41)
is now determined by numeric interpolation.
The influence of the sheet coil pairs number Z on the HFEA accuracy with magnetic
saturation was investigated. With reference to a pure Halbach magnetization and a
constant current supply iG = 2 IG, the results related to a linear HFEA, HFEAs with Z =
1 and Z = 2 and non-linear FEA were compared. The values of the axial force, reported
in Figure 5.17, shows that the HFEA with Z = 2 and non-linear FEA results match very
well, so Z = 2 is adopted for the further analyses.
Semi-analytical model of a linear machine
Figure 5.17. Mover axial force
123
with saturated mover yoke (
on the accuracy.
): influence of the zone number
In Table 5.4 are reported the FEAs stored quantities, such as the flux density in the
airgap and in the mover and stator cores, and the flux linkage) due to magnetization,
coil current and mover sheet current. Inside the square brackets are reported the
dimension of the stored matrix.
Table 5.4. HFEA storage quantities: in the bracket array dimension.
Magnetization
,
,
Airgap flux density
,
,
,
,
Coil current
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
[
[
Mover flux density
Mover sheet current
(
,
(
,
)]
[
)]
,
[
(
,
,
[
)]
,
,
(
(
,
[
)]
,
)]
,
(
)]
,
,
,
,
,
,
(
)]
,
Stator flux density
,
[
Flux linkage
,
(
,
[
(
)]
(
[
,
,
[
)]
)]
[ ]
[ ]
,
124
5.6
Chapter 5
Core losses and efficiency estimation
The SMC core losses, affecting the TLPMG electromagnetic efficiency, can be
estimated by evaluating the radial and axial component of flux density as a function of
time on a suitable grid of sampling points
. The density of spatial and time sampling is
a compromise between the requirements of limited computational effort and accurate
prediction of the loss distribution. Therefore, the samples along
used for both
inside the mover are
calculation (for equivalent mover yoke saturation) and core loss
calculation. As for the stator core, three distinct sampling subsets are defined (Figure
5.18): S1 in the horizontal yoke, where only Bz is evaluated, S2 in the vertical yoke,
where only Br is evaluated, and S3 in the pole shoe, where both the components must be
taken into account, since flux density direction varies with time.
Figure 5.18. Stator zones division for core loss estimation: S1: horizontal yoke, S2: vertical yoke and S3:
pole shoe.
Assuming a steady-state condition, the flux density time dependence can be expressed
by truncated Fourier series expansions, harmonic coefficients being derived by
interpolating the magnetostatic FEA results. This approach smooths the numerical
errors due to FE solution and consequently reduces noise in time derivative evaluation,
provided that an adequate sample number is considered. The average value of the core
loss density
[W/m3] is determined elaborating the formulation suggested in [213] as:
| {(
∫ (
〈
|
|
〉)
(
〈
|
| )
where 〈 〉 brackets stands for time averaged values and
absolute value of flux density vector,
and
〉) }
(5.42)
| |
√
⁄ is the fundamental frequency,
is the
,
are the hysteresis, eddy current and excess loss coefficients, respectively.
Such coefficients are obtained by the elaboration of the specific loss data as functions of
the maximum flux density at different frequencies [215-217]. The total core losses
Semi-analytical model of a linear machine
125
are finally obtained by the integration of
on the volume of the predefined stator and
mover regions.
Preliminary elaborations evidence that the hysteresis loss contribution is largely
predominant in the frequency range below 1 kHz. Consequently, a current-controlled
supply strategy could be profitably adopted to limit current peaks, the same rms value
being maintained, to improve efficiency by reducing the maximum flux density
magnitude at load.
The instantaneous input electromagnetic power
( ) and the output power
( ) are
given by:
( )
( )
where
( ) and
( )
( ) ( )
( )
[
( )
(5.43)
( )]
( )
(5.44)
are the TPMLG instantaneous output voltage and coil resistance,
respectively.
Neglecting friction and windage losses, the mean values of the input mechanical power
and the output electric power
can be therefore calculated as:
( )
∫
∫
(5.45)
( )
(5.46)
Using the above quantities, further indexes are considered, mainly related to the
electrical system performance and requirements. In particular, the following
calculations are useful for design purpose:

the efficiency, which accounts for the contribution of electromagnetic losses,
results:
(5.47)

the total power factor, which is related to the grid converter sizing, results:
(5.48)
where

∫
( )
is the output voltage rms value;
the power ripple, which is involved in the sizing of appropriate filters or storage
systems at the converter output terminals, results:
{
( )}
{
( )}
(5.49)
126
5.7
Chapter 5
Model validation
The main HFEA parameters (number of harmonics nm and of sheet coil pairs Z) should
assure a good accuracy without an excessive calculation time. To this purpose, a quasiHalbach
configuration
(
,
)
was
preliminary
analysed, comparing the Br air-gap profiles obtained by HFEA and by a direct FEA of
the full model to check the influence of saturation on HFEA accuracy. The very good
agreement between the two methods for both unsaturated and saturated mover core
(linear and non-linear HFEA, respectively) is shown in Figure 5.19 (results related to N
= 55 and Z = 2 for s = /2 and iG = IG). It’s worth noticing the mover saturation is quite
relevant, causing about 13% reduction of the maximum flux density.
Figure 5.19. Radial component of the flux density calculated on the air-gap line lg by HFEA procedure
and direct FEA (N = 55 and Z = 2 for s = /2 and iG = IG).
As can be seen, the HFEA method reproduces correctly the airgap flux density. This
means that the thrust force, given by the product between radial and axial components
of airgap flux density, is reproduced in the correct way as well.
In order to validate the proposed model, a condition of significant mover magnetic
saturation is chosen. In Figure 5.20 is reported the no-load flux linkage for pattern III in
the case of linear and non-linear magnetic characteristics of SMC material. The flux
linkage reduction of non-liner with respect to linear characteristic is remarkable (
12%).
Semi-analytical model of a linear machine
127
Figure 5.20. Comparison of no-load flux linkage between linear and non-linear magnetic characteristics
(pattern III).
Figure 5.21 shows the no-load force and flux linkage as functions of the mover position
for both the Halbach magnetizations. With regard to such figures, the following aspects
can be evidenced:

the agreement with the actual FEM model is very good for both the
magnetization schemes;

the ideal Halbach scheme presents very low cogging force and higher flux
linkage (+11% with respect to quasi Halbach), the latter being practically
sinusoidal;

the cogging force related to quasi Halbach scheme has higher harmonic content,
which can harmfully affect the mover dynamics.
Figure 5.21. No-load force for ideal Halbach and quasi Halbach magnetizations (wmr=w'mz= 20 mm,
w''mz= 10 mm).
128
Chapter 5
To check the effectiveness of the calculation procedure, the core losses at both no-load
and
are evaluated by HFEA and by direct FEA, with a very good agreement
(Table 5.5). It is worth to point out that transient package required by the latter analysis
is considerably more time consuming than a magnetostatic FEA. Losses are mainly
confined in the stator core (about 85% of the total value) and particularly in the zone S 3;
at load, core losses further increase due to the magnetizing effect of the coil mmf.
Table 5.5. Core losses in W calculated by HFEA and direct FEA.
Operating
condition
No-load
5.8
HFEA
Zone S1
Zone S2
Zone S3
Mover
Total
Direct
FEA
3.6
4.6
1.8
2.1
3.3
4.2
1.4
2.3
10.1
13.2
10.2
12.8
Example of application
The HFEA can be utilised to study and analyse the performance of different
magnetization profile and different current supply strategy [214]. Through a parametric
analysis, an optimization of TPMLG can be carried out.
For the example of application, a simple harmonic motion and a load current source
with a predefined number nG of odd harmonics are considered:
( )
( )
∑
̂
(
)
(
(5.50)
)
(5.51)
To such purpose, three different PM magnetization patterns are compared by HFEA,
namely:

pattern I: pure Halbach magnetization, corresponding to sinusoidal spatial
distributions for mr and mz;

pattern II: radial magnetization corresponding to a square-wave mr distribution,
with

⁄ ;
pattern III: quasi-Halbach magnetization corresponding to square wave
distributions for both mr and mz, the PM width ratio being
with
to maximize the no-load flux at
⁄ .
⁄
,
Semi-analytical model of a linear machine
129
As regards the current supply, two possible strategies are considered:

strategy A: sinusoidal current supply (nG = 1, current in phase with no-load
√
e.m.f. eG0) with

, wher e is rated value of the coil current;
strategy B: current supply by suitable harmonic components in order to
maximize the mean converted power, the total current rms value being equal to
.
The condition is fulfilled by choosing the harmonic amplitudes according to the
following equations:
(5.52)
∑
{
with
harmonic amplitudes and
rms value of the no-load emf calculated by the
magnetostatic analyses.
As regards the pattern III, the calculation of the no-load flux linkage by (5.23) in the
range
by a parametric analysis defines the optimal value
(Figure 5.22): for lower and higher values the flux decreases due to the lower Br
contribution and to the higher mover saturation, respectively.
Figure 5.22. No-load flux linkage
⁄ at as a function of the PM width ratio related to pattern III.
An example of parametric analysis can be also carried out in order to optimize the noload flux, by changing the PMs width wPM. Such analysis can be straightforwardly
accomplished by HFEA, since it requires only the analytical elaboration of the results
obtained by the preliminary FEA sequence, according to (5.23).
130
Chapter 5
An example of this calculation is shown in Figure 5.23: an almost perfectly sinusoidal
waveform is achieved by a very limited number of iterations for wPM=5/8
= 25 mm.
This case allows achieving a sinusoidal waveform for the emf as well. A low harmonic
content in emf reduces the core losses and thus results in an efficiency improving.
Figure 5.23. No-load flux linkage as a function of position for different wPM values.
In the same figure, the curve related to wPM=3/4
= 30 mm is also reported, as it
matches different requirements of limitation of PM amount, flux maximization and
reduction of the harmonic content. This condition maximize the flux/PM amount ratio,
that means, at the same time, the maximization of the flux/cost ratio, since the PM cost
is the highest one in relation to the other TPMLG active parts.
Another important issue to investigate is the cogging force
, because it affects the
starting of linear generator and in general its dynamic behaviour. The cogging force
results from the interaction between the iron-structure stator teeth and the permanent
magnets mounted on the translator part.
For such purpose four other different magnetization patterns have been considered:

pattern I: pure Halbach magnetization;

pattern II: quasi Halbach magnetization with

pattern III: quasi Halbach magnetization with
⁄ ;
⁄ ;

pattern IV: radial magnetization with
⁄ .
⁄
and
Semi-analytical model of a linear machine
131
Figure 5.24 shows the influence of the magnetization pattern on the cogging force
.
The agreement with nonlinear FEAs was confirmed for all the configurations. The
lowest cogging force is achieved in the pure Halbach scheme, thank to more smoothed
magnetization profile and thus lower interaction between the magnet and the stator core.
The worst condition is represented by pattern III.
Figure 5.24. Cogging force
for different magnetization patterns obtained by HFEA with Z=2.
As for the comparison at load condition, the calculation of the current harmonic content
for strategy B, carried out by (5.52) with nG=5, evidences the presence of an appreciable
3rd harmonic, mainly for pattern II (
); on the other side, pattern III is
characterized by the higher fundamental and 5th harmonics
,
). The
main performances for both the supply strategy A and B are compared in Table 5.6:
mean power and efficiency values are close and satisfactory for both patterns I and III.
It’s worth pointing out that strategy B generally leads to lower flux density peak values
inside cores and consequently to lower hysteresis losses. Though benefitting by PM
amount saving and an easy-to-manufacture magnetization profile, pattern II, yielding
about 17% less mean power, has poorer efficiency and power factor.
Table 5.6. Performance comparison for different magnetization schemes and current supply strategy.
Pattern
I
II
III
(r=0.75)
Supply
strategy
PG
[W]
PG
[-]
PFe
[W]
[%]
[-]
A
B
A
B
A
B
452.0
488.0
368.0
387.0
439.0
456.0
1.57
2.09
1.49
1.99
1.41
1.80
15.1
14.1
10.8
10.1
14.0
13.0
89.9
90.7
88.8
89.4
89.8
90.3
0.856
0.831
0.852
0.787
0.881
0.812
132
Chapter 5
The most remarkable PG improvement (
) is obtained by applying strategy B to
pattern I, with a slight efficiency increase as well, with the drawback of a higher power
fluctuation, as shown by Figure 5.25. Generally, it can be seen that the greater is the
deviation of the eG0 waveform from sinusoid (due to non-uniform speed) the more
profitable is the pulsed current supply in comparison with other control strategies.
Figure 5.25. Instantaneous output power and no-load emf related to pattern I.
HFEA flexibility and computational effectiveness enables also a fast parametric
analysis of the TLPMG performances as functions of the current load ratio
⁄ , as
shown in Figure 5.26, related to strategy B. Efficiency is satisfactory in a pretty wide
load range, and power rate is only a little less than linear, due to saturation.
Figure 5.26. TPMLG performances as functions of load (pattern I - strategy B).
Semi-analytical model of a linear machine
133
Since a wide SMC commercial variety is available, differing from each other by
magnetic properties as well as cost, it is convenient to investigate how the former affect
both the TLPMG flux linkage and core losses, especially in presence of distorted flux
density time variation. Such parametric analysis, useful to identify the most convenient
material for the ferromagnetic cores, is applied to the materials reported in Table 5.7
[216-218]. Even if the linearized magnetic permeability values are close, these materials
differ by both BH curve saturation and specific core loss values.
Table 5.7. Magnetic properties (magnetic field and core losses) of SMC materials.
N.
1
2
3
4
Material type
GKN 304913
Ancorlam®
Ancorlam® HR
Somaloy® Prototyping
H [kA/m]
@1T
H [kA/m]
@ 1.5 T
1.5
2.2
2.7
2.5
4.9
7.9
8.5
12.0
400
400
300
400
pFE [W/kg] @ 1 T
60 Hz
400 Hz
9.0
7.6
5.9
71.7
67.0
53.0
47.0
With reference to pattern I (the most effective as for the output characteristics) and to
supply strategy B, the results of comparison are reported in Table 5.8: the highest PG
value is obtained by the N.1 material with less saturated BH curve. As for efficiency,
the material N.4 could be preferable due to its lower core losses. However, the relatively
low operating frequency is not worth using expensive materials with high bulk
resistivity, making preferable the achievement of other benefits in terms of
manufacturing features and power performance.
Table 5.8. Performance comparison for different SMC materials.
N.
1
2
3
4
5.9
Material type
GKN 304913
Ancorlam®
Ancorlam® HR
Somaloy® Prototyping
UG
[V]
PG
[W]
PFe
[W]
[%]
[-]
443
434
439
428
488
474
483
465
14.1
15.9
15.8
10.7
90.7
90.2
90.4
90.9
0.831
0.824
0.829
0.819
Conclusion
In this chapter an analytical procedure based on magnetization space harmonic was
developed. The proposed HFEA method combines the results of linear FEAs related to
the space magnetization harmonics with an iterative procedure to define suitable surface
current sheets, effectively taking into account the heavy mover saturation. The method
134
Chapter 5
is based on the superposing of the effects due to the different magnetization harmonic
orders, the effect of coil current and the effect of the fictitious current sheets on the
mover surface. By this procedure is possible to reconstruct some interesting quantities
such as the airgap flux density, utilised to estimate the thrust force value by means of
the Maxwell stress tensor; the flux density inside the stator and mover cores, utilised to
estimate the core losses.
After the preliminary elaborations, any magnetization pattern can be analysed and
therefore HFEA can be effectively adopted for parametric analyses and design
optimization.
The HFEA procedure enables a considerable time saving with respect to direct FEAs,
since it operates on fixed geometric configurations with predefined sinusoidal
magnetization patterns. It is particularly convenient to improve the TPMLG
performance by comparing different magnetization schemes and supply strategies and
can apply to parametric variations and optimization analyses as well. Its effectiveness is
confirmed by the good agreement with the corresponding direct FEA results even with
heavy mover saturation.
The example of application shows that, assuming a simple harmonic motion, the best
performances as for output power and efficiency are obtained by Halbach-type
magnetization and non-sinusoidal current-controlled supply. The procedure enabled also
to investigate on the convenience of adoption of high permeable SMC material for the
cores.
Chapter 6
Characterization of
electromagnetic parameters
6.1
Introduction
The tubular permanent magnet (TPM) LG results the more indicated electromagnetic
configuration for the free-piston co-generation system, as the mover can be easily
integrated with the engine piston, resulting in a very compact arrangement.
Furthermore, it is possible to scale up the power rating simply by lining up more stator
and PM elements.
Remarkable benefits can derive from the use of molded powder materials, such as soft
magnetic composites (SMC) for the stator core and bonded rare-earths for the PM ring,
which can be magnetized after the installation on the mover shaft. Among them, the
high manufacturing automation of the TPMLG active parts, the compact design, the
isotropic magnetic properties of the stator SMC core (no complicated laminated
structures are requested) [218] and the option to adopt different PM magnetization
patterns to achieve particular design objectives make the TPMLG a very promising
solution.
To check the effectiveness of the procedure used to analyse and design a TPMLG,
presented in the previous chapter, a prototype of TPMLG was built. The stator and
mover of the three-pole single-phase TPMLG prototype is reported in Figure 6.1.
The stator assembly (Figure 6.1a) consists of twelve SMC modular core segments, held
together by the insulation box including a ring coil. The PM ring (Figure 6.1b) is
divided into six identical pieces mounted on a solid iron hollow cylinder, suitably sized
to reduce the moving mass, without appreciably affecting the magnetic saturation. This
structure is joined to the inner shaft of the linear motion transmission. To optimize the
output power/PM mass ratio, the coil flux linkage at the mover stroke ends should be
136
Chapter 6
maximized. For this purpose, the moving PMs should be always aligned to the stator
pole axis at the stroke end.
Insulation
box
PM
SMC
piece
Coil
(a)
(b)
Figure 6.1. 3-poles single phase TPMLG prototype; (a): stator with SMC modules and ring coil; (b):
mover with surface PM ring.
Though general simplified assumptions can be used for a preliminary analytical design
[210], a reliable assessment of the TPMLG output performances requires an accurate
definition of the material properties, to be identified by suitable numerical procedures.
Uncertainties related to coarse supplier specifications, process variability and heavy
machining may lead to unreliable results and then to unsatisfactory set-up of the
generator system, as for the design optimization as well as the control strategy
assessment [219].
This chapter describes a procedure elaborating the results of both measurements and
finite element analyses (FEAs) aiming to the determination of the main magnetic
properties, such as the magnetization curve and the core loss coefficients of the SMC
material as well as the actual PM magnetization function. The parameters defining the
magnetic properties, obtained by the proposed procedure, are then implemented in the
FE model to evaluate the expected performances in terms of voltage, force and core loss
values, enabling both possible design improvements and the definition of the TPMLG
mathematical model for the dynamic operation analysis [220].
The TPMLG prototype allows to check the procedure elaborated in the chapter 5, as
well as to obtain an effective characterization of the magnetic material properties. Such
quantities can be remarkably affected by the manufacturing process, leading to
unexpected performance worsening. Then an appropriate procedure should be set up to
evaluate possible mismatch with respect to the provider’s specifications and to enable
the definition of reliable models for design optimization and control strategy
assessment.
Characterization of electromagnetic parameters
137
Generally, the uncertainties are related to:

the SMC magnetization curves and core loss coefficients, heavily affected by the
raw sample manufacturing, the machining process, possibly deteriorating the
particle insulation of the magnetic powder, and by the number of core segments;

the magnetization distribution along the PM ring depending on the magnetizer
arrangement, on the back-iron conductivity and on the material magnetic
characteristic.
The preceding issues can be crucial for the performance prediction of the prototype
under consideration. For such purpose, a procedure is developed to derive the main
magnetic properties of the SMC material (linear and saturated permeability values, core
loss coefficients) from the analytical elaboration of suitable experimental tests.
The experimental setup consists of Ne = 4 regularly spaced identical coils (excitation
coils) with Nt = 180 turns each, placed around one half part of the stator core to obtain a
quasi-toroidal field distribution (Figure 6.2), with low leakage fluxes and almost
uniform magnetic flux on every transversal section of the stator core. Taking into
account the isotropic magnetic characteristics of the SMC core, such arrangement
should reproduce similar magnetic properties of the actual TPMLG. The adopted
arrangement enables to achieve higher saturation than the actual configuration, since
flux lines have to cross narrow stray air-gaps, provided that the SMC modules are
assembled as tightly as possible.
Figure 6.2. Experimental setup for the magnetic characterization of the SMC core with the sketch of a
radial section; 1,…,4: subdivisions of the core section used for the determination of the magnetic
parameters.
138
Chapter 6
6.2
Determination of the permanent magnet parameters
As for the PM characteristics, the spatial flux density distribution along the PM surface
is measured, the mover being extracted from the stator. The flux density is measured by
a gaussmeter with a regular sampling step Δ = 2 mm, enabling to define the coercivity
and the magnetization profile to be implemented in the FE model.
Three different magnetization spatial distributions are considered (dashed lines of
Figure 6.3), compatible with measures and datasheet from PM manufacturer:

type 1: a pure radial scheme, being

type 2: an equivalent quasi Halbach scheme, being
the PM coercivity;
the PM coercivity and
the ratio between the axial and the radial magnetization amplitudes (light
blue line);

type 3: a pure radial scheme with a 3rd harmonic injection, being
and
the fundamental and 3rd harmonic coefficients, respectively.
The parameters
,
,
,
,
and
are determined by minimizing the
deviation between measured and simulated data, the latter obtained by corresponding
2D rz FEA. In Figure 6.3, the flux density profiles with
for the type
1,
,
,
for the type 2 and
,
for the type 3 are compared. The FEA results show that the flux density
profile is not adequately reproduced by a pure radial magnetization. Even if the
presence of asymmetries between the lateral and middle pole is fairly reproduced by a
type 1 pattern, only the magnetization type 2 satisfactorily accounts for the spikes in the
middle pole region, and is more consistent with the PM segmentation and the actual
magnetizer assembly. An appreciable reduction of the magnetic coercivity with respect
to the declared design value (-400 kA/m) is evidenced for both the cases, and a
consequent TPMLG performance derating is therefore to be expected.
Characterization of electromagnetic parameters
139
Figure 6.3. Comparison between flux density data related to manufacturer sheet, laboratory
measurements and FE simulations.
6.3
Determination of SMC magnetization curve
In order to reproduce the material magnetization curve
( ), a general analytical
representation is considered. Among the different proposed formulations, the following
function for the relative permeability
( )
( ) is selected [223]:
{
(6.1)
(
with
)
relative permeability of the linear unsaturated characteristic,
density value delimits the linear region and
,
[T] flux
-2
[T ] coefficients representing the
decrease. The function appropriateness is evidenced by the very good agreement, even
for high saturation, between the input material data and the interpolation obtained by the
best-fit coefficients in (6.1) (Figure 6.4).
The coefficients
and
related to the prototype B(H) curve are determined
evaluating the core magnetic flux as function of the excitation m.m.f. To such purpose,
the excitation coils are supplied at low frequency (f = 30 Hz, sinusoidal voltage
waveform) and the instantaneous values of the current supply and the induced e.m.f. on
a probe coil (Np = 50 turns), tightly coupled to the core, are recorded. Then, the flux
linkage is straightforwardly determinated by integrating the probe coil per turn e.m.f.,
no correction for winding leakage fluxes and heating being required.
140
Chapter 6
Figure 6.4. Interpolation of the Somaloy® 700 3P relative permeability as a function of the flux density B
(
;
and
).
The current peak value ̂ in the excitation coils related to the j-th test (
with
test number, * superscript identifies measured quantities) goes from 0.31 A to
8.05 A, the latter one corresponding to a remarkably high saturation. Figure 6.5 shows
the dependence of the probe coil per turn flux linkage on the supply current. The
material hysteresis loops centred on the magnetization characteristic ̂ ( ̂ ), clearly
recognizable in the figure, are obtained by the Fourier interpolations of both current and
flux time samples. Such characteristic is related to the mean value of the flux density
inside the whole core section; nevertheless, it takes by no means into account the nonuniform flux density distribution in the section itself due to the dependence of H on r.
Figure 6.5. Probe coil per turn flux linkage as a function of the supply current for different maximum
current values ( ̂
0.31, 0.64, 1.23, 2.25, 3.34, 6.27 A).
Characterization of electromagnetic parameters
141
Therefore, an iterative procedure is set up, based on the determination of the core flux
contributions of Ns = 4 subdivisions of the stator cross section (Figure 6.2) a uniform
flux density being assumed in each zone, by virtue of its limited radial extent. This
assumption is confirmed by a preliminary 3D magnetostatic FEA, using the maximum
current supply (I = 5 A). The corresponding flux density map is shown in Figure 6.6. In
particular the analytical prediction of the flux density mean value for each region is only
2.5% lower than the FE simulation, enabling a reasonable evaluation of the actual
permeability values.
Figure 6.6. Flux density map corresponding to I* = 5 A
Known the excitation m.m.f. and given the coefficients
and
, the corresponding
permeability values and the core fluxes in each region, can be calculated analytically.
Based on such considerations, an iterative process varies
and
to minimize the
deviations between the measured data and the analytical evaluations.
A detailed flow diagram of the procedure is represented in Figure 6.7. The following
calculation steps can be identified:
a) with reference to j-th experimental test, the maximum instantaneous value of the
magnetic field ̂
in the k-th region (
̂
with ̅ mean radius of the k-th region;
) is determined as:
̂
̅
(6.2)
142
Chapter 6
b) given a set of the {
permeability
( )
( )
( )
} coefficients (h: iteration index), the relative
in the k-th region can be determined by (1);
( )
c) the corresponding magnetic reluctance
obtained by the expressions:
(
and flux peak value ̂
)
can be
( )
(6.3)
( )
̂(
with
,
and
̂
)
(6.4)
( )
axial width, outer and inner radius of the k-th region,
respectively;
(
d) the j-th total flux linkage ̂
)
is then obtained by summing the flux contribution
of each k-th core region;
e) by repeating the steps a) to d), the sum of the square flux relative deviations
( )
( )
with
with respect to the experimental values is calculated; if the variation of
respect to the previous step value
(
)
is lower than a predefined threshold ,
the iteration is stopped, otherwise a new set {
( )
( )
} is imposed by the
minimization algorithm.
Table 6.1 reports the main coefficients related to the SMC magnetic properties
corresponding to the datasheets (model A) and to the elaboration of the flux-current
measurements according to the flow diagram of Figure 6.7, assuming uniform bulk
properties (model B) or including the influence of parasitic air-gaps between the core
modules (model C). Such condition is simulated by introducing a constant magnetic
reluctance
, to account for the difference between the datasheet and the
experimental m.m.fs for a given flux value in the linear region.
Table 6.1. SMC magnetic parameter comparison between manufacturer data sheet values (model A) and
experimental data (model B and C).
Source of elaboration
(model)
Data sheet (model A)
Experimental data ( model B)
Experimental data (model C)
10-3
[ ]
0.64
0.64
0.64
711
400
711
1.60
3.32
3.50
[
1.69
2.64
2.58
]
Characterization of electromagnetic parameters
143
Measured
value
Analytical
calculation
Figure 6.7. Flow diagram of the procedure to evaluate the SMC magnetization parameters
In order to obtain the same datasheet
and
.
, a reluctance value
has to be assumed, equivalent to an air-gap width
,
consistent with the core assembly allowance. However, Figure 6.8 evidences a
remarkably higher saturation for
with respect to datasheets, whatever the
model adopted. For instance, at the design working point of the stator core (
), the permeability is about 1/3 (from 326 to 110).
144
Chapter 6
Figure 6.8. Comparison between the magnetization functions obtained from the SMC datasheet (model
A) and the measurement elaboration (models B and C).
Both models A and B are checked by 3D FEAs, agreeing with measurements. Taking
advantage of symmetry conditions, the FEA can be restricted to only one-quarter model
(Figure 6.9).
Excitation coil
SMC core
Figure 6.9. One fourth of 3D FEA geometric model for the calculation of the magnetic flux.
Figure 6.10 shows the magnetic flux as a function of the total excitation ampere-turns.
In particular, it evidences a very good concordance between the FEA applying model B
(FEA-B) and the measurement elaboration. With respect to the results related to
Characterization of electromagnetic parameters
145
datasheets (FEA-A), flux values are lower by about −20% for ampere-turns higher than
2 kA.
Figure 6.10. SMC magnetic flux as a function of total coil ampere-turns, obtained by measurement and
FEA data elaboration.
6.4
Determination of SMC core loss coefficients
Due to the high remanence of the adopted NdFeB PMs, an appropriate utilization of the
SMC material calls for a careful evaluation of the core losses. Despite of the limited
operating frequency, saturation effects and distorted flux density waveform can
appreciably affect the TLPMG electromagnetic efficiency. Furthermore, in this
particular case, the prototype machining process could exalt possible eddy current
generation especially on the stator module surfaces, due to the deterioration of the
powder insulation coating [224].
Because of the test bench excitation symmetry, only the dependence on time of flux
density circumferential component
the core loss density
eddy current
must be determined. Thus the average value of
[W/m3] can be calculated by the sum of the hysteresis
and the excess loss
̂
where
, the
terms, according to the formulation [224]:
∫ (
|
|
|
⁄ is the fundamental frequency, ̂ is the peak value of
| )
,
are the hysteresis, eddy current and excess loss coefficients, respectively.
(6.5)
,
and
146
Chapter 6
The total core losses
on the volume
are finally obtained by integrating the average core loss density
of each k-th stator subdivision:
∑∫
where
is the width of the k-th zone,
zone respectively,
∑
and
(6.6)
∫
are the inner and outer radius of k-th
is the radius coordinate referred to k-th zone. It is worth pointing
out the equation (6.6) is numerically evaluated, elaborating the flux density values
sampled on a uniform grid.
Likewise for
and
, the coefficients
,
and
are determined by minimizing
the deviation between the data obtained by the elaboration of the measurements on the
prototype and the results of the above described analytical formulation.
The procedure is based on the elaboration of data related to measurements with different
and current peak values ̂ (with
frequencies
index). Let
and
and current and
, with
test
the corresponding recorded instantaneous values of supply voltage
the excitation winding resistance, the total core losses
derived
from measurements are determined as the difference between the input power
copper loss power
:
∫ (
where
⁄
and
)
(6.7)
is the period.
The frequency ranges from 20 Hz to 200 Hz, the latter value to the maximum allowable
for the supply equipment. Because of voltage limitations, the maximum supply current
has to be decreased for the highest frequency values, thus limiting to some extent the
saturation conditions. Actually, some smoothing and regularization procedure has to be
applied, to mitigate the influence of error sources (influence of temperature on
,
disturbances, sample discretization, etc.). Corresponding flux linkage peak values ̂
are evaluated by integrating the e.m.f.:
̂
The values
⁄
∫(
)
(6.8)
of the core energy loss per cycle are then fitted by a function of
frequency and flux linkage peak value given by
̂
̂
. The first term,
related to hysteresis losses, can be extrapolated from energy losses for
hysteresis losses can be estimated by the function:
, and
Characterization of electromagnetic parameters
̃ (̂
Then, the residual losses ̃
147
)
̂
(6.9)
are obtained, including the excess and the eddy current
loss contributions:
̃
̃
The core loss model (i.e.,
̃
,
̃ (̂
and
̃
)
(6.10)
coefficients) is determined by a fitting
procedure, aiming to match the values obtained according to ̃
and ̃
,
respectively.
The main calculations can be summarized as follows:
a) the instantaneous magnetic field in the m-th sampling point of the core section
(
with
number of sample points in the core section) and
related to the j-th experimental test, is determined as (
: sample radius):
(6.11)
b) the corresponding flux density tangential component
(
) is obtained by
the magnetization curve; the Fourier analysis (13 odd harmonics are considered)
of the instantaneous values yields the maximum value ̂
derivative function
and the time
;
c) the total core losses are obtained integrating the core loss densities by equation
(6.6);
d) by repeating the steps a to c for all
deviations
( )
tests, the sum of the relative square flux
with respect to the experimental values is calculated (h: iteration
index); if the variation of
( )
with respect to the value
(
)
at the previous
(h-1)-th step is lower than a predefined threshold , the iteration is stopped,
otherwise the core loss coefficients are updated and the process is repeated.
The flow diagram of the procedure utilized to determine the core loss coefficients is
reported in Figure 6.11.
148
Chapter 6
Measured
value
Analytical
calculation
Figure 6.11. Flow diagram of the procedure to evaluate the SMC core loss coefficients.
As for the core loss model, Table 6.2 reports the coefficients derived from material
datasheet elaboration and the application of the procedure described in Figure 6.11,
applied to the measured losses. A good correspondence is obtained between the
hysteresis coefficients
(about 12% difference). On the contrary, in the datasheet
elaboration, the excess losses are prevalent [9], while the residual losses evaluated by
experimental elaboration depend on
, suggesting to adopt
.
Table 6.2. Core loss coefficients comparison between data sheet and experimental values.
Source of elaboration
Datasheet
Experimental data
[
615
688
]
[
0.14
12.2
]
[
]
9.64
-
Such result confirms the possible core loss increase due to the machining process. The
hysteresis and residual core losses obtained from the measurement elaboration and
Characterization of electromagnetic parameters
evaluated by the coefficients
and
149
related to datasheet elaboration are compared
in Figure 6.12: a satisfying agreement is attained in the low and medium frequency
range, whatever the flux value. On the other side, the discrepancy is appreciable for the
highest frequency value (150 Hz), which can however regard minor flux density
harmonic contributions with respect to the TPMLG rated frequency.
(a)
(b)
Figure 6.12. Hysteresis losses (a) and residual losses (b) vs flux peak value ̂ and frequency f by
measurement elaboration and by analytical evaluation.
150
Chapter 6
From Figure 6.12 it's worth noticing the relevant values of the eddy current losses for
the experimental elaboration which become even higher than the hysteresis losses as
frequency
, whatever the flux value. Such effect is not recognizable in the
⁄
measurement elaboration data, where the ratio
is always lower than 25% even at
high frequency values. Even if core machining could have introduced a significant loss
increase with respect to the SMC pressure molding process adopted for mass
production, as a precautionary approach, the calculated core loss model is adopted to
evaluate the TPMLG performances.
Furthermore,
{
}
a
sample
{
transient
3D
FEA
related
to
the
condition
}is executed to evaluate the core losses by the code built-in
routine. The FEA adopting the same coefficients as for experimental elaboration yields
, 10.2% lower than the measured value (
approach for experimental elaboration provides
). Analytical
(-6.8% than
measurement, +3.8% than FEA). Even in this case, the accuracy of the procedure can be
therefore considered satisfactory.
6.5
Evaluation of the actual TPMLG performances
Because of the appreciable mismatch between design and experimental PM and SMC
magnetic properties, an update of the analysis of the TPMLG prototype performances is
required. To such purpose, the material magnetic models related to the datasheet and
calculated parameters are considered.
Carbon steel magnetic properties are assigned to the mover shaft, suitably downgraded
to take into account eight lightening holes (
). For the model related to
datasheet elaborations, a PM radial magnetization is assumed with coercivity
, while type 3 magnetization (quasi Halbach scheme with
and
) is assumed for experimental model.
In order to compare the results of the experimental model with the results explained in
the previous chapter (in which the electromagnetic properties has been provided by the
manufacturer datasheets), the same working conditions have been considered:

simple harmonic motion: ( )

sinusoidal load current:
( )
̂
√
(
);
(
).
The instantaneous voltage and power (related to both datasheet and experimental
models) are shown in Figure 6.13: in spite of the sinusoidal dependence of flux linkage
on s, no-load voltage
is non-sinusoidal due to the speed variation; voltage
at load
Characterization of electromagnetic parameters
151
is even more distorted, due to the armature reaction. The high third to fundamental
harmonic amplitude ratio for no-load voltage (
and
for datasheet and
experimental model, respectively) confirms this configuration is more suitable for a
pulsed rather than sinusoidal current supply [212].
The heavy second harmonic component of output power
(greater than mean value) is
inherent in single-phase operation, and likely could require some suitable storage device
to interface with the electrical network. Correspondingly, high input shaft thrust
oscillations result, to be taken into account in the engine design.
(a)
(b)
Figure 6.13. Simulated TPMLG instantaneous quantities of voltage and power calculated by the
datasheet and experimental models. (a): no-load
and load voltage ; (b): output power
with
corresponding mean value.
152
Chapter 6
Table 6.3 shows the remarkable influence on the electromagnetic quantities of the
mismatch between the datasheet and the measured magnetic properties. In particular,
both no-load voltage
and mean output power decrease
(−1 . % and −1 . %,
respectively), while stator core losses remarkably increase (+21.4%) due to higher core
loss coefficients. Nevertheless, the detriment of efficiency is not so dramatic (about
1.7%).
Furthermore, both cogging force peak ̂
〈
value and winding inductance mean value
〉 remarkably decrease, enhancing dynamic performance and power factor,
respectively.
Table 6.3. Comparison of TPMLG performances and output parameters for datasheet and experimental
models.
Model
Datasheet
Experimental data
6.6
[V]
̂
[N]
〈 〉
[H]
[W]
[W]
[W]
293
242
48.4
25.6
0.405
0.330
557
464
10.6
8.1
1.1
6.1
Conclusion
In this chapter, an analytical technique to estimate the electromagnetic parameters of a
TPMLG by suitable experimental tests is developed. Measurement with variable supply
voltage frequency and amplitude are elaborated to define the SMC magnetic properties,
resulting in a remarkable mismatch with respect to datasheet, likely due to the heavy
impact of the manufacturing processes. In detail, an analytical procedure to characterize
the magnetization curve (BH curve) and the core losses of the SMC material utilized in
stator core. The determination of the magnetization curve is based on the minimization
of the flux linkage obtained by measurement and by analytical way. The procedure to
estimate the core losses is based on the estimation of the core loss coefficients due to
hysteresis, eddy current and excess losses. The characterization of core loss coefficients
is deduced from minimization between the core losses evaluated by measurement and
by analytical way.
The procedure’s effectiveness is nevertheless confirmed by the agreement of FEA
results with measurements.
The actual PM magnetization pattern is identified by measurement as well. The flux
density on the PMs surface of the removed mover was measured by means of a gaussmeter. The measured values are then compared with the results obtained from FEAs for
Characterization of electromagnetic parameters
153
3 different possible magnetization profiles. It was found that the magnetization profile
that fit the measured values in the best way is the Halbach magnetization profile.
The characterization of the TPMLG parameters enabling to analyse the main
performances of the machine with the updated material characteristics: a power output
decrease results, anyway not invalidating the initial design as to the application
effectiveness.
154
Chapter 6
Chapter 7
Dynamic model of Stirling
Free-piston cogeneration system
7.1
Introduction
Linear generators find application in cogeneration systems powered by free-piston
engines. Such kinds of systems present several advantages, such as the reduction of
moving parts that means an increasing of reliability, a reduction of maintenance and
then an improving in terms of efficiency.
The Stirling engine represents one of the most promising solution as regard the freepiston engine since it allows to achieve very high efficiency (theoretically the Carnot
efficiency).
One of the most favourable features for their application is the possibility to integrate
the LG mover with the machine member driven by energy source, with no additional
mechanical couplings. As a result of the simplification of the kinematic system, both
dynamic response and conversion efficiency are improved.
Particularly, the linear generator analysed and optimised in the previous chapters, cause
of its low power ratings, it is designed to fit in a small domestic cogeneration system. In
this case, the Stirling engine utilizes the waste heater of a boiler, which would otherwise
be lost, to move a linear generator for electric energy production. Therefore, the Stirling
engine converts the thermal energy obtained from the gas combustion process into
mechanical energy that, in turn, is converted in electrical energy by linear generator.
The linear generator is connected to the grid by means of an electric converter that
allows the power exchange with the grid. It can be integrated with an energy storage
system, such as a batteries or ultracapacitors bank, to store the surplus of electric energy
that is not required to the load. The scheme of a free-piston cogeneration system is
reported in Figure 7.1 where is also shown the major component of the system.
156
Chapter 7
Figure 7.1. Scheme of a free-piston energy converter integrating a Stirling engine and a single phase
TLPMG.
The operation of the whole co-generation system can be simulated only by developing a
dynamic model of the power conversion devices (Stirling engine, linear generator and
static converter) taking into account the electromechanical interaction among each
other. To the purpose a Matlab Simulink® code is set-up aiming to the solution of both
the mechanical and the electrical equations.
In particular, such code allows analysing the cogeneration system for the different load
condition, as well as for different heat supply. Moreover, it allows finding the optimum
solution in term of mechanical oscillating frequency
[rad/s]. In fact, on one hand, is
advisable to work at high frequency in order to obtain high e.m.f. and then high
converted power values, but, on the other hand, this means to have a large spring
constant
[N/m], in order to comply with the resonance condition:
√
where
(7.1)
[kg] is the mass of the moving part.
Furthermore, at high speed, the thermal exchange between the heat exchangers and the
working fluid considerably worsens.
It has been shown [225] that, due to the low-pass-filter effect of the mechanical massspring-damper subsystem, the piston velocity
[m/s] and the mover displacement
[m]
according to the expressions:
(
∫
where
and
)
(
(7.2)
)
(7.3)
⁄ are the maximum stroke and the maximum speed that
can be achieved by the linear generator.
Dynamic model of Stirling Free-piston cogeneration system
157
Another important issue that should take into account is the non-ideality of Stirling
cycle, since the relatively high speed value. In this condition, the Stirling cycle deviates
considerably from the ideal one, because of the non-ideality of energy transformations,
which can be reproduced by adiabatic rather than isothermal curves in the pressurevolume (pV) diagram (Figure 7.2).
p
V
Figure 7.2. pV diagram of Stirling cycle: isothermal transformations (blue lines) and adiabatic
transformations (red lines).
The above mentioned phenomenon is due to the extreme rapidity in which the fluid
evolves inside the cylinder, which does not allow at working fluid to exchange heat with
heater and cooler. For such reason, the temperature of working fluid cannot keep
unvaried. This phenomenon involves a remarkably efficiency reduction that can be
achieved the 50% of the ideal Carnot efficiency.
The simplest way to control the cogeneration system consists in controlling the reaction
force of linear generator. Such condition is obtained by adjusting the load resistance in
order to control the load current and then the linear generator force. The dynamic model
is therefore convenient also to set up the control strategy in order to optimize the energy
yield.
The model of the Stirling engine is developed starting from the ideal gas equation, and
then it was elaborated, taking into account the non-ideality. For such purpose a real
spatial thermal profile of the heat exchangers (heater and the cooler) was considered.
As for the model of the linear generator, it was developed by elaborating suitable FEAs,
where the force and the flux linkage were calculated for different position and current
values, storing the values in suitable lookup tables.
To connect the output of linear generator to the grid an electric converter arranged by a
rectifier and an inverter has been considered. The output frequency of linear generator
158
Chapter 7
depends on the spring constant and the moving mass, as shown in equation (7.1).
However, this value can vary in the neighbourhood of the mechanical resonant
frequency in function of the electric load. In any case, generally, the mechanical
resonant frequency can be lower than the grid one, requiring controlled interface
converters to adapt both frequency and voltage to the grid values. The converter action
is simulated by a variable resistive load which value is adjusted according to the signal
elaborated by a PI regulator. Therefore it is possible to correct the position error with
respect to the reference value established by resonant condition changing the LG current
and then the thrust. The real position value can be provided by simply elaborating an
output voltage measurement. In fact, measuring the output voltage and subtracting the
internal voltage drop of the linear generator, is possible to achieve the speed value, and
finally the mover position by integrating the speed estimation.
7.2
Dynamic model of cogeneration system
In a dynamic point of view, the resulting motion of the Stirling engine piston (Figure
7.3) is described by the Newton’s second law:
( )
(
)
(7.4)
{
where:
is the resultant force;
of the mover position;
is the force provided by free-piston engine, function
is the linear generator electromagnetic force, function of the
mover position and of the load current ;
[N/m] constant of the spring;
and bearing friction, with
is the force provided by the spring, with
is the frictional force due to the working fluid viscosity
fluid friction coefficient [
⁄ ].
𝐹𝑠
𝐹𝑟
𝐹𝑙𝑔
𝐹𝑆𝑡
𝐹𝑓
Figure 7.3. Force balance on the Stirling engine piston.
Equation (7.4) denotes the system nonlinearity therefore requiring a numerical solution.
To the purpose, a Matlab Simulink code is developed schematically reported in Figure
7.4.
Dynamic model of Stirling Free-piston cogeneration system
159
𝐹𝑓
𝐹𝑆𝑡
𝐹𝑟
𝐹𝑠
𝐹𝑙𝑔
𝐹𝑙𝑔
𝑘𝑠
Figure 7.4. Scheme of the dynamic model implemented with Matlab Simulink®.
The model integrates the mechanical and the electrical systems. The mechanical system
reproduces the force balance expressed by Equation (7.4). The electric model
implements the voltage equation including the LG parameters and the equivalent load
which simulates the converter action. In particular, the TPMLG represents the
electromagnetic coupling between the mechanical and the electrical system, namely it
converts the input mechanical power into the output electrical power at net losses. For
TPMLG the following equations can be applied:

for the electrical system (Faraday-Neumann-Lenz):
|
|
|
where the flux linkage
( )
(7.5)
is only due to PMs magnetization and
( ) is defined
as the back-emf coefficient of the TPMLG, is dependent on the displacement of
the permanent magnet.

For the mechanical system:
( )
where
( ) is defined as the thrust force coefficient of TPMLG, it is
dependent on the mover displacement as well and
The coefficient
TPMLG.
(7.6)
( )
( )
is the cogging force.
( ) is the electromagnetic coupling coefficient of
160
Chapter 7
In order to simplify the model it is convenient to report all the quantities in the same
system. If we compare the differential equation of a mass-spring-damper mechanical
system (Figure 7.5a):
( )
∫ ( )
(7.7)
with the differential equation of a series RLC electric circuit (Figure 7.5b):
( )
( )
( )
∫ ( )
(7.8)
the relationship between the two systems results as following:
( )
( )
( )
( )
𝑣(𝑡)
𝐹(𝑡)
(a)
(b)
Figure 7.5. Mass-spring-damper mechanical system (a) and equivalent RLC series electric circuit (b).
This analogy allows converting the mechanical elements into equivalent electrical
elements and in this way a pure-electric system is obtained (Figure 7.6).
Figure 7.6. Electric circuit utilized for the dynamic study of the system. The mechanical elements are
converted into equivalent electrical elements.
Dynamic model of Stirling Free-piston cogeneration system
7.3
161
Stirling engine model
There is an increasing interest in the use of Stirling engine based cogeneration systems
for residential and commercial cogeneration because of their prospect for high
efficiency, good performance at partial load, fuel flexibility, low emission level, low
vibration and noise level.
The model of a Stirling engine results quite troublesome to develop since there are
many causes to take into account. Particularly, the main challenge is to describe the heat
exchange between the working fluid and the heater/cooler, since it depends on the
configuration (geometry, materials, etc.) of heat exchangers (geometry, materials, etc.).
To such purpose, firstly an ideal model (assuming ideal gas) was developed. Then, the
model was improved by taking into account the non-ideal phenomena, such as dead
volumes, non-isothermal expansion and compression, thermal and mechanical losses
and imperfect regeneration. These non-ideal phenomena result in efficiency decay with
respect to the ideal case, and strongly affect the engine operating condition.

Ideal model
In the case of ideal gas (viz. ideal Stirling engine), the instantaneous value of the force
( ) [N] provided by the Stirling engine can be obtained by rearranging the ideal-gas
equation
, where
3
the piston volume [m ],
] and
[ ⁄
is the gas mass [g],
( )
is
is the universal gas constant
( ) is the gas temperature [K], respectively:
( )
where
( )
is the pressure of the gas [Pa],
( )
( )
( )
is the piston surface [m2] and
( )
( )
( )
( )
(7.9)
is the piston position, with
the mean oscillating position of the piston.
During the isothermal expansion, the piston speed is positive and the gas temperature is
maintained at the heater temperature value ( ). Whereas, during the isothermal
compression, the piston speed result negative and the gas temperature is maintained at
the cooler temperature ( ). Then, the equation (7.9) can be arranged as:
( )
( )
{
( )
( )
(
)
(7.10)
(
( )
)
162
Chapter 7
The Stirling engine force can be expressed by a general formulation:
̅
( ( ))
( )
where ̅
(
(7.11)
( )
)⁄ is the mean temperature of the working fluid [K] and
is the temperature gap of the working fluid [K].
By defining ̅
̅ and
, the equation (7.11) can be rearranged
as:
̅
( )
( ( ))
( )
(7.12)
Another important issue to investigate in order to describe the operating process of the
system is mean position of the spring
[m]. In fact, the mean position of the spring
affects the mean position of the piston and consequently the dynamic of the system. For
such purpose some energetic considerations were carried out.
The potential energy provided by the spring
[J] during a complete cycle is given by
the difference between the maximum stored energy that corresponding to the maximum
⁄
compression condition (
), and the minimum stored energy, that
⁄
corresponding to the minimum compression condition (
[(
with
)
(
) ]
(
):
)
(7.13)
the stroke length [m] (Figure 7.7).
The work provided by the Stirling engine during the compression phase, is given by the
area under the lower isothermal (
) curve of the of the pV diagram. This area can
be evaluated by analytical way by means of the integral:
(̅
∫
)
(
)
(7.14)
The energy stored by the spring should be equals to the energy of Stirling engine during
the compression phase. Therefore, by equating (7.13) and (7.14), it results:
̅
(
)
(7.15)
Dynamic model of Stirling Free-piston cogeneration system
163
Figure 7.7. Spring and the piston position reference.

Simplified model
The ideal model does not take into account the non-ideal phenomena that adversely
affect the TPMLG efficiency. Such phenomena can be summarized as:
 dead volumes: dead volumes are defined as the total void volume in a Stirling
engine. In general, the dead volume refers to the volume of working fluid
contained in the total dead space in the engine, including the regenerator and
transfer port. It is evidenced that a real Stirling engine must have some
unavoidable dead volume. In normal Stirling engine design practice, the total
dead volume is approximately 58% of the total volume [228].
Dead volumes alter the efficiency, in fact the output work as well as the
adsorbed heat is proportional to:
(7.16)
 non isothermal expansion and compression: due to high working frequency of
the machine that does not allow at working fluid a heat exchange at constant
temperature;
 thermal losses: due to conduction, convection and radiation losses;
 mechanical losses: due to friction between the operating parts of the system;
 imperfect regeneration: due to finite thermal capacitance of the regenerator and
the not null thermal capacitance of the working fluid.
164
Chapter 7
For such purpose equation (7.12) does not describe in the correct way the operation of
the system. However, equation (7.12) can be generalized by:
( )
( )
( )
(7.17)
where ( ) is a suitable non-linear function given by:
( )
with
( ) and
( )(
∫(
( ))
( )(
Figure 7.8.
and
( ) and
( ) is reported in
are two coefficients proportional to heater
temperature, respectively.
Figure 7.8. Typical spatial profile of the heater
Function
(7.18)
( ) functions that represent the spatial profile of the heater and cooler
heat exchange, respectively. A typical spatial profile of
and cooler
( )))
( ) and
( ) and cooler
( ) heat exchange functions.
( ) can be expressed analytically as:
̅
̅
( )
(7.19)
{
̅
( )
{
(7.20)
̅
Figure 7.9 reports the simulation comparison of the an ideal and a practical Stirling
cycle. It’s worth pointing out that the area of the real Stirling cycle results lower than
the ideal one. This is due to the working fluid temperature that is not able to achieve the
heater and the cooler temperature. This results in an output power decreasing (
) with respect to the ideal case (
).
Dynamic model of Stirling Free-piston cogeneration system
Figure 7.9. Comparison between ideal and real Stirling engine (
̅
, ̅
).
165
,
,
,
In order to validate the proposed model, the experimental results of a free-piston engine
proposed in the literature are chosen as reference [226]. They are related to the model of
the RE-1000 studied by the NASA during the 80’s, having the characteristics reported
in Table 7.1.
Table 7.1. Most important characteristics of RE-1000.
Output power [W]
955
Overall efficiency [%]
23.7
Swept volume [cm ]
74.9
Operating frequency [Hz]
30.1
Piston stroke [cm]
4
Mean pressure [MPa]
7.034
Working fluid
Helium
Total mass of fluid [kg]
4·10-4
600
Cooler temperature
25
0.25
Piston diameter [mm]
3
Heater temperature
Piston weight [kg]
[°C]
[°C]
35
By comparison between Figure 7.9 and Figure 7.10, it can be observed the good
agreement between the manufacturer data and the values obtained by means of
mathematical model can be therefore considered satisfactory.
166
Chapter 7
Figure 7.10. Clapeyron diagrams for compression and expansion spaces [227].
7.4
TPMLG model
The model of linear generator is based on two lookup tables obtained by means of a
suitable sequence of FEAs. Through FEAs, the thrust force
[Wb] for different values of supply current
carried out, thus lookup tables
The e.mf.
(
) and (
[N] and the flux linkage
[A] and mover position
[m] were
) are developed.
[V] provided by linear generator to the load is given by:
(7.21)
And then the load current [A] is given by:
(7.22)
where
is the internal resistance of linear generator [Ω] and
(with
the
output voltage [V]) is the equivalent resistance of the load [Ω], which value is
controlled adjusting the parameters of the electronic converter.
The scheme of the liner generator implemented in the Simulink code of the linear
generator is reported in Figure 7.11.
Dynamic model of Stirling Free-piston cogeneration system
(
(
167
)
)
Figure 7.11. Linear generator model.
7.5
Electronic power converter model
The supply voltage of linear generator is usually at low frequency (a typical value is
around at 30 Hz) since the mechanical constraint (mover mass and spring constant)
limits the dynamic performances of the system. Moreover, the amplitude of the output
voltage is related to the speed, which can vary during the operation of the machine. For
such reason, an electronic converter is needed to connect the linear generator to the grid
or to the electric load regulating the frequency and the amplitude of the output voltage.
Such device generally consists of an AC/DC converter (i.e. a buck/boost or a simple
rectifier) and an inverter.
A convenient control strategy of electronic converter is achieved by keeping the output
current in phase with respect to the output voltage in order to maximize the highest
converted power.
7.6
Control system
A cogeneration system produces simultaneously electricity and useful heat. Thanks to
multi-fuel operation of the Stirling engine, the primary energy source can be derived
from different heat sources such as fossil fuels or green fuels (biomass, geothermal and
thermodynamic solar). Some renewable energy sources presents discontinuous and
often unpredictable energy yield, since they are related to the climatic conditions. In
addition, also the electric and the heat loads may present large variations especially in
case of weaken. In particular, a sudden variation of the heating temperature in the
Stirling engine system (due to fuel injection or heat load of the heater temperature
involves a variation of the force provided by Stirling engine that, in turns, induces a
variation of the speed of the system. In case of electric load variations, the TPMLG
168
Chapter 7
force may be affected altering the system speed. For example, if the load is sudden
disconnected, the output current and then the reaction force of linear generator drops
instantly.
For such reason, a control system has to be developed, in order to maintain the speed
within the design limits. In fact, if the speed exceeds the maximum value means that the
mover stroke would be greater than the mechanical constraints on the mover position.
The control the input fuel, where possible, is generally not convenient, since the thermal
dynamics results too slow due to the high thermal capacitance of the heater. The most
convenient solution is represented by the control of the load current, in order to comply
with the reference speed profile.
The control of the load current is based on the load compensation, namely to adjust the
load resistance by means of a fictitious resistance. The fictitious resistance is varied by
changing the parameter of the electric converter in order to the linear generator supply
to the load the required current. The electric converter works as a variable resistance
and adjusts the resistance to the required value. In the case the output power of linear
generator is greater than the power required to the actual load, the excess power may be
provided to a storage system.
The scheme of the current control is reported in the Figure 7.12.
(a)
Measured speed
(from voltage mesurement
)
| |
(b)
Figure 7.12. Model of current control based on load compensation: current control (a) and detail of load
compensation (b).
Dynamic model of Stirling Free-piston cogeneration system
169
The fictitious resistance of the load compensation is regulated by a PI controller from
the error es between the reference value sref of the maximum stroke smax and the
measured value. The maximum value of the stroke is obtained from the following
procedure:
 calculation of the absolute value (| |);
 integrating by synchronization with the zero crossing, in order to achieve the
profile of the stroke;
 sampling on the maximum value (sample on falling edge of the speed).
It’s worth pointing out that the speed
can be obtained by a simple voltage and current
measurement. The speed can be obtained from the following procedure:
(7.23)
( )
where
[V] and
(7.24)
[A] are the measured output voltage and current values,
respectively.
The advantage of this technique is that no mechanical measurement is required
(seemingly sensor-less control with consequent improvement of the system reliability
and maintenance and costs reduction).
The parameters of the PI controller are chosen in order to achieve a prompt dynamic
response without an excessive overshoot. The saturation block after the PI controller
limits the load compensation so that the load current does not excess the rated value due
to thermal constraints.
7.7
Dynamic analysis
The Stirling engine considered in the proposed model is the RE-1000, since it fits the
requirements of the cogeneration system and it is consistent with linear generator sizing.
The most important features of the RE-1000 engine are reported in Table 7.1.
The dynamic model is utilized to analyse the electromechanical performances of the
free-piston cogeneration system. In the free-piston arrangements, develop a model of
the whole system is very important as, unlike to crank-shaft based arrangement, the
working condition is heavily affected by the working point of the system. Since freepiston arrangements are not equipped of gearbox, the speed of the system is the same
for all the parts that compose the system. For such reason, it is very important to
170
Chapter 7
determine the best solution from a speed point of view. The speed of the system is
strongly linked to the mechanical resonant frequency
.
From a performance point of view, the mechanical resonant frequency should be as
large as possible, since the no load e.m.f.
is proportional to the speed , as shown by
(7.5). On the other hand, a high mechanical resonant frequency means having a high
value of the spring constant and then a large spring that, inevitably, increase the size and
the costs of the system. Furthermore, increasing the working fluid speed, it has not been
able to achieve the temperature of the heater and cooler (no isothermal transformations).
The real cycle of the Stirling engine deviates significantly from the ideal one, and thus
results in an efficiency decreasing.
Figure 7.13 reports the simulated (real) Stirling cycle for different mechanical resonant
frequency.
7.6
ideal cycle
real cycle f = 10 Hz
real cycle f = 30 Hz
real cycle f = 50 Hz
7.4
pressure [MPa]
7.2
7.0
6.8
6.6
6.4
6.2
6.0
20
40
60
80
100
120
total volume [cm3]
140
160
180
Figure 7.13. Comparison of simulated (real) Stirling cycles for different mechanical resonant frequency.
Therefore, the speed of the system can be regulated by changing the constant value of
the spring. It is possible only by an adjustable gas spring, for example by means of
pressure regulation. It is clear that such kind of technique cannot be employed as control
strategy, since it would require a compressor that would involve an increasing of losses,
costs and dimensions. In any case, it has not been able to fulfil the dynamic
requirements of the system, such as the constraint regarding the maximum stroke of the
mover, as it is not fast enough.
The other parameter that affects the dynamic system is the friction due to the viscosity
of the working fluid. It is represented by the frictional coefficient
and it is
Dynamic model of Stirling Free-piston cogeneration system
171
proportional to the speed. The friction can be characterised as an electrical load and
particularly it behaves as a resistor. It is worth to point out that increasing the speed of
the system the friction losses increase as well.
Another important issue to investigate is the heater and cooler temperature. From
equation (7.12), it can be noted that the mean temperature affects the main value of the
thrust force, whereas amplitude of the force is proportional to the temperature rise
between the heat exchanger (heater and cooler). It is therefore convenient to have a
temperature rise the largest possible.
The temperature of the heater depends on heater behaviour. When the heat is provided
by fuel (it can be green fuel such as biomass or biogas or even traditional fossil fuel),
the temperature depends on the amount of injected fuel. In the cogeneration system it
depends on the thermal consumption as well. If the heat is provided by a
thermodynamic solar panel, the temperature is linked with solar radiation. Whereas, the
cooler temperature depends on the temperature of the exchanger fluid, generally water.
Of course such approach cannot be adopted to prompt speed regulations because the
large time constants characterizing the thermal phenomena.
As previously mentioned, the linear generator model is based on two lookup tables,
(
) and
(
), obtained by suitable FEAs where the experimental values of the
magnetic material characteristics are used.
The thrust force
and the flux linkage
were evaluated for current
and for position
(Table 7.2), resulting in 1681 different conditions and as many simulations. The
current and position limits were chosen in order to take into account any working
condition. Particularly, the maximum current value is about 3 times the rated value.
This allows, for brief periods, to supply the load with higher current values with respect
to the nominal one, in order to improve the dynamic performances of the system.
Table 7.2. Initial, final and step values of the current and the position utilized to build the lookup tables.
Quantity
Initial value
Current [A]
√
Final value
√
Step value
√
Position [mm]
The thrust force and flux linkage profiles, utilised in the lookup tables, evaluated for
different position and output current values are reported in Figure 7.14.
172
Chapter 7
600
I=4A
400
I=2A
force [N]
200
I=0
0
-200
I = -2 A
-400
I = -4 A
-600
-20
-15
-10
-5
0
5
mover position [mm]
(a)
10
15
20
10
15
20
6
I=4A
4
flux linkage [mWb]
I=2A
2
I=0
0
I = -2 A
-2
I = -4 A
-4
-6
-20
-15
-10
-5
0
5
mover position [mm]
(b)
Figure 7.14. Interpolated FEAs of the force (a) and the flux linkage (b) as function of current supply and
mover position values.
From Figure 7.14a it can be noted that the cogging force can be neglected with respect
to the load force. From Figure 7.14b can be noted the ferromagnetic cores results rather
saturated when the current supply achieve the maximum value, the saturation effect
involves a reduction of the flux linkage of
25.2%. It worth pointing out that the effect
due to current supply is more pronounced than the effect due to PMs, involving a
demagnetizing effect.
Another important issue that should be investigated is the control strategy of the
cogeneration system. Particularly, the aim of the control system is firstly to regulate the
Dynamic model of Stirling Free-piston cogeneration system
173
speed of the co-generator in order to maximize the converted power, and then to
constrain the mover position within the mechanical limits.
Rearranging the equation (7.7) results:
∫(
∫
)
(7.25)
whereby, the unique quantities that can be varied in order to control the speed of the
system are:
 the Stirling engine force
temperatures gap
;
 the linear generator force
 the spring force
: it proportional to the heater and cooler
: it is proportional to the output current ;
: it is proportional to the spring constant , that affects the
mechanical resonant frequency
as well.
Some numerical analyses were carried out in order to check the effectiveness of the
control system. To such purpose an instantaneous variation of the heater temperature
was considered. The thermal dynamics was neglected in order to focus the attention
only on the mechanical and electrical systems that have a faster dynamic than thermal
system. To the purpose, it is assumed the application of two step variations to the heater
temperature. In particular, after 5 and 10 seconds two steps of
°C with
respect to the steady state temperature ̅
°C, then after 15 seconds the
temperature is reset to the initial value ̅
; such condition can, for example,
refers to a sudden variation of the solar irradiation in the case of solar thermal panel.
The reference value of the maximum stroke was chosen in order to take into account
any overshooting due to PI controller. Thus, a safety coefficient was considered in order
to prevent that the mover position exceeds the limit due to mechanical constraint, thus
34.8 mm.
The instantaneous values of the heater temperature
, the maximum output current
7.15.
, the equivalent load resistance
and the mover position
are reported in Figure
Chapter 7
maximum stroke [mm]
maximum load current [A]
equivalent resistance [Ω]
heater temperature [K]
174
800
750
700
650
600
550
0
2
4
6
8
10
time [s]
(a)
12
14
16
18
20
0
2
4
6
8
10
time [s]
(b)
12
14
16
18
20
0
2
4
6
8
10
time [s]
(c)
12
14
16
18
20
400
350
300
250
200
150
100
2.5
2.0
1.5
1.0
0.5
36.5
35.5
34.5
measured value 𝑠𝑚𝑎𝑥
reference value 𝑠𝑟𝑒𝑓
33.5
32.5
0
2
4
6
8
10
time [s]
(d)
12
14
16
18
20
Figure 7.15. Instantaneous values of the heater temperature (a), equivalent load resistance
maximum value of the output current (c) and maximum mover position (d).
(b),
As can be noted from the Figure 7.15, the control compensates satisfactorily the
variations of maximum mover stroke. The maximum stroke value remains always
within the mechanical constraints.
From Figure 7.15 is also possible note that an increasing of the heater temperature
involves an increasing of the Stirling engine force that in turn tends to increase the
speed and maximum mover stroke. This involves an increasing of the position error
,
Dynamic model of Stirling Free-piston cogeneration system
175
so the PI controller decrease the compensation resistance (
), thus the current and
consequently the linear generator reaction force increase. As results, the linear generator
force increases the speed and consequently the maximum stroke value decrease. In the
case the heater temperature decreases, the process is the same but with the signs
reversed.
load current [A]
The instantaneous values of load current and the speed are reported in Figure 7.16.
2
1
0
-1
-2
9.8
9.9
10.0
10.1
time [s]
(a)
10.2
10.3
10.4
9.9
10.0
10.1
time [s]
(b)
10.2
10.3
10.4
speed [m/s]
4
2
0
-2
-4
9.8
Figure 7.16. Instantaneous values of the load current (a) and the speed (b).
7.8
Conclusion
In this chapter the dynamic model of a free-piston cogeneration system was developed.
The dynamic model allows analysing and tuning the parameters of the cogeneration
system. Since in the FPEC the alternating motion is obtained through the interaction
between the piston force and a spring reaction, it is important to tune the system
operation at the mechanical resonance frequency. In this condition, the best
performance can be obtained in terms of output power and efficiency.
The dynamic operation is studied by the development of a numerical code in
Matlab/Simulink environment, integrating the Stirling engine, the linear generator and
the electronic power converter. The analytical model of the Stirling engine was firstly
developed assuming ideal gas and ideal Stirling cycle and then elaborated to introduce
the non-ideality due to non-conservative phenomena. The model parameters are chosen
with reference to an existent Stirling engine, the RE-1000, developed by NASA in
176
Chapter 7
1986. The output quantities calculated by means of analytical model are in agreement
with the quantities provided by the datasheet of RE-1000.
The linear generator model is based on two lookup tables obtained by means a sequence
of FEAs, using the magnetic parameters derived from the measurement elaborations on
a scale prototype.
The electronic converter model is developed by simulating an equivalent load connected
to the LG terminals. Such approach is adopted mainly to set up a convenient control
strategy to be applied for the speed regulation. In particular, the control system is based
on the load current adjustment in order to vary the linear generator reaction force. The
control is also necessary to keep the mover position within the limit due to mechanical
constraints for any working condition.
The results show a good dynamic response and accuracy, even in the presence of a rapid
variation of the heater temperature. This situation, used especially to test the model,
may be linked to variations in the quantity of fuel or solar radiation which can occur in
typical applications of free-piston Stirling engine.
Chapter 8
Conclusions and future work
In this thesis a micro-cogeneration system, consisting of a free-piston Stirling engine
(FPSE) and a tubular permanent magnet linear generator (TPMLG), was investigated.
Particularly, the following issues have been addressed:

development of a semi-analytical model (HFEA), based on simplified FEA, that
allows to analyse and optimize the TPMLG in terms of magnetization profile
and supply current strategy;

development of a technique to estimate the electromagnetic parameters of the
TPMLG by the elaboration of suitable experimental tests, in order to predict the
actual performance of the machine;

development of a dynamic model of the whole cogeneration system, in order to
study and to predict the output performance with different thermal, mechanical
and electrical conditions.
Firstly, a brief description with the state-of-the-art, the history, the applications, the
configurations and the working principle of Stirling engines were illustrated. As regards
the linear generators, their different typologies as well as their possible structures and
the working principle were presented.
Different hard ferromagnetic materials utilised as permanent magnets, such as NdFeB,
SmCo, Alnico and ceramic were compared, highlighting strengths and weaknesses. The
soft magnetic composites (SMC) materials, utilised in the ferromagnetic stator and
mover cores, were described.
From the preliminary literature review, it was found that a single-phase, tubular,
permanent magnet linear generator (TPMLG) represents the best choice from the
technical-economic point of view.
Particularly, the tubular configuration allows achieving a higher power density and the
axial component of the force is eliminated. As regards the magnetization profile, the
Halbach magnetization results in the more appropriate for such kind of applications.
178
Chapter 8
The most important advantages of Halbach magnetization are the reduction of the
harmonic content in the emf and the reduction of the mover saturation. This feature
allows reducing the mover thickness, involving a mover mass reduction and thus a
greater performance from the dynamic point of view.
The SMC materials utilized to build the stator core allow achieving complicated
geometry profile and a core losses reduction, thanks to their lower eddy current loss
with respect to the traditional laminated machines.
An innovative calculation procedure, called HFEA, based on simplified FEAs was
described and developed. An iterative procedure to define suitable current sheets on the
mover surface was developed in order to take into account the mover yoke saturation,
generally not negligible in such kinds of applications. From such analysis the flux
density and the flux linkage can be reconstructed by superposing the effects of the
different magnetization harmonic order, the effect of the coil current and the effect of
the mover sheet current. By means of such quantities the force, the emf, the core losses
and thus the machine efficiency were estimated.
After the preliminary elaborations, any magnetization pattern can be analysed and
therefore HFEA can be effectively adopted for parametric analyses and design
optimization.
The HFEA procedure enables a considerable time saving with respect to direct FEAs,
since it operates on fixed geometric configurations with predefined sinusoidal
magnetization patterns. It is particularly convenient to improve the TPMLG
performance by comparing different magnetization schemes and supply strategies and
can apply to parametric variations and optimization analyses as well. Its effectiveness is
confirmed by the good agreement with the corresponding direct FEA results even with
heavy mover saturation.
The example of application shows that, assuming a simple harmonic motion, the best
performances as for output power and efficiency are obtained by Halbach-type
magnetization and non-sinusoidal current-controlled supply. The procedure enabled also
to investigate on the convenience of adoption of high permeable SMC material for the
cores.
An analytical technique to estimate the electromagnetic parameters of a TPMLG by
suitable experimental tests was developed. Measurements with variable supply voltage
frequency and amplitude are elaborated to define the SMC stator core magnetic
properties, resulting in a remarkable mismatch with respect to the datasheets, likely due
to the heavy impact of the manufacturing processes and stray air-gaps between the
Conclusions and future work
179
modules. The procedure effectiveness is nevertheless confirmed by the agreement of
FEA results with measurements. The actual PM magnetization pattern is identified by
measurement as well, enabling to analyse the TPMLG main performances with the
updated material characteristics: an output power decrease is appreciated, anyway not
invalidating the initial design as to the application effectiveness.
A dynamic model of the cogeneration system was developed. Such a model allows
analysing the whole cogeneration system, including the Stirling engine, the TPMLG and
the electronic converter. The model of the Stirling engine reproduces the effect on the
Stirling cycle of the spatial profile of the heater and cooler temperatures. The model of
the linear generator is obtained from lookup tables provided from suitable FEAs,
representing dependence of both the flux linkage and the thrust on the mover position
and on the coil current.
Due to the low-pass-filter effect of the mechanical mass-spring-damper subsystem, the
piston velocity and the displacement of the compressor vary essentially sinusoidally
with time, and higher order harmonics can be neglected.
As regards the further works, a simplified dynamic model, which considers only the
fundamental component of the forces provided by Stirling and linear generator, should
be developed. This kind of model allows to study the system with lower time
consumption and to optimize the system in order to achieve the highest power
conversion. Furthermore, a dynamic experimental setup should be arranged in order to
check the effectiveness of the dynamic model.
180
Chapter 8
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