Inkjet printhead performance enhancement by feedforward input

Inkjet printhead performance enhancement by feedforward input
Inkjet printhead performance
enhancement by feedforward input
design based on two-port modeling
Inkjet printhead performance
enhancement by feedforward input
design based on two-port modeling
PROEFSCHRIFT
ter verkrijging van de graad van doctor
aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus Prof. dr. ir. J.T. Fokkema,
voorzitter van het College voor Promoties,
in het openbaar te verdedigen op
maandag 12 februari 2007 om 12.30 uur
door
Matthijs Benno GROOT WASSINK
werktuigkundig ingenieur
geboren te Leiden
Dit proefschrift is goedgekeurd door de promotoren:
Prof. ir. O.H. Bosgra
Prof. dr. ir. D.J. Rixen
Samenstelling promotiecommissie:
Rector Magnificus
Prof. ir. O.H. Bosgra
Prof. dr. ir. D.J. Rixen
Prof. dr. ir. J. van Eijk
Prof. dr. ir. M. Steinbuch
Dr. ir. J.F. Dijksman
Prof. dr. D. Lohse
Dr. ir. S.H. Koekebakker
Prof. ir. R.H. Munnig Schmidt
voorzitter
Technische Universiteit Delft, promotor
Technische Universiteit Delft, promotor
Technische Universiteit Delft
Technische Universiteit Eindhoven
Philips Applied Technologies Eindhoven
Technische Universiteit Twente
Océ-Technologies B.V.
Technische Universiteit Delft, reservelid
This research is supported by Océ-Technologies B.V. in Venlo, The Netherlands.
The research reported in this thesis is part of the research program of the Dutch
Institute of Systems and Control (DISC). The author has successfully completed
the educational program of the graduate school DISC.
ISBN 978-90-9021484-9
c 2007 by M.B. Groot Wassink
Copyright All rights reserved. No part of the material protected by this copyright notice may
be reproduced or utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and retrieval
system, without the prior permission of the author.
Voorwoord
Degene die het promoveren associëren met vier jaar lang zwoegen achter een computer in een hokje op de universiteit, kan ik direct een illusie armer maken: de
afgelopen vier jaar hebben mij in ieder geval het tegendeel bewezen. Zo heb ik
de enorme vrijheid in het onderzoek, het verdiepen en verbreden van kennis en
vaardigheden, het samenwerken met Océ en het deelnemen aan internationale conferenties ervaren als een combinatie die uniek is bij een eerste ’baan’. Toegegeven,
het zwoegen klopt wel af en toe, maar ja, bij welke baan heb je dat nou niet?
Kortom: het is een prachtige tijd geweest. Maar wat het vooral mooi heeft
gemaakt is de samenwerking met een (flink) aantal mensen. In dat kader gaat mijn
grootste dank uit naar Okko. Hij heeft mij zowel de vrijheid als steun gegeven
bij het opzetten en uitvoeren van dit onderzoek: zijn onovertroffen kennis en
inzicht is van enorm belang geweest bij de totstandkoming van dit proefschrift.
Ook Daniel ben ik veel dank verschuldigd. De inhoudelijke discussies vanuit zijn
expertise heb ik enorm gewaardeerd en hebben het behaalde resultaat aanzienlijk
verbeterd.
Daarnaast heb ik het geluk gehad om tijdens de promotie vier goede afstudeerders
te hebben kunnen begeleiden: Anton, Niels, Ferry en Pieter. Het pressiemiddel
’de exponentiele functie’ heeft zeer zeker effect gehad: jullie resultaten zijn dan
ook terug te vinden in dit boekje, waarvoor dank! Ook dank aan Océ voor het
mogelijk maken van dit onderzoek en de ondersteuning die ik vanuit Venlo heb
gekregen: Sjirk, Herman, Rob, Marc en vele anderen dank!
En wat zou promoveren zijn zonder mede-lotgenoten? Met veel plezier denk ik
terug aan de vele humorvolle en relativerende gesprekken tijdens de talloze koffieen lunchpauzes. Dank daarvoor aan alle oud-collega’s van de vroegere vakgroep
Systeem- en Regeltechniek en het huidige Delft Center for Systems and Control.
Het zijn er teveel om op te noemen... Tot slot: Elske, ouders, familie en vrienden,
ook jullie dank voor jullie begrip en luisterende oren de afgelopen jaren!
Matthijs Groot Wassink,
Den Haag, December 2006.
i
Contents
Voorwoord
i
1 Introduction
1
1.1
Inkjet technology . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.1 A historical overview . . . . . . . . . . . . . . . . . . . . . .
1.1.2 A generic manufacturing technology . . . . . . . . . . . . .
1.2
System description . . . . . . . . . . . . .
1.2.1 An archetypal PIJ printhead . . .
1.2.2 Limitations of current designs . . .
1.2.3 Towards a controlled environment
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2 Problem formulation
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The research objective . . . . . . . . . . . . . . . . . . . . . . . . .
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2.2
A decomposition in research questions . . . . . . . . . . . . . . . .
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2.3
The structure of this thesis . . . . . . . . . . . . . . . . . . . . . .
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3 Experimental exploration
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3.1
Description of the experimental setup
3.1.1 Piezo sensor signal . . . . . . .
3.1.2 CCD camera . . . . . . . . . .
3.1.3 Laser-Doppler interferometry .
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3.2
Description of the experimental printheads
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Identification method . . . . . . . . . . . . . . . . . . . . . . . . . .
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Piezo-based experimental identification . . . . . . . . . . . . . . . .
3.4.1 With bridge-structure . . . . . . . . . . . . . . . . . . . . .
3.4.2 Without bridge-structure . . . . . . . . . . . . . . . . . . .
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3.5
Laser-vibrometer based experimental identification . . . . . . . . .
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3.6
Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . .
47
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iv
CONTENTS
4 Modeling of the ink channel dynamics
4.1
4.2
4.3
4.4
PIJ printhead model survey . . . . . . .
The two-port model . . . . . . . . . . .
4.2.1 The acoustic path . . . . . . . .
4.2.2 The fluidic path: the nozzle . . .
4.2.3 The fluidic path: drop formation
4.2.4 The fluidic path: a review . . . .
4.2.5 The actuation path . . . . . . . .
The bilateral coupling . . . . . . . . . .
Concluding remarks . . . . . . . . . . .
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5 Model validation
5.1
5.2
5.3
5.4
5.5
Introduction . . . . . . . . . . . . .
Piezo-based validation . . . . . . .
Laser-vibrometer based validation
Discussion . . . . . . . . . . . . . .
Concluding remarks . . . . . . . .
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6 The control framework
6.1
6.2
6.3
6.4
6.5
Introduction . . . . . . . . . . .
The lifted ILC control structure
The control goals . . . . . . . .
ILC design . . . . . . . . . . .
6.4.1 LQ-optimal control . . .
6.4.2 Constrained ILC . . . .
Concluding remarks . . . . . .
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7 Application of feedforward control
7.1
7.2
7.3
7.4
7.5
Introduction . . . . . . . . . . . . . . . . . . . .
Piezo-based ILC . . . . . . . . . . . . . . . . .
7.2.1 SISO ILC: reducing residual vibrations .
7.2.2 MIMO ILC: minimizing cross-talk . . .
7.2.3 Constrained MIMO ILC . . . . . . . . .
Laser-vibrometer based ILC . . . . . . . . . . .
Discussion . . . . . . . . . . . . . . . . . . . . .
Concluding remarks . . . . . . . . . . . . . . .
8 Conclusions and recommendations
8.1
8.2
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143
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
CONTENTS
v
A Hamiltonian ILC design
149
Bibliography
153
Glossary of symbols
163
Summary
167
Samenvatting
169
Curriculum Vitae
171
Chapter 1
Introduction
The importance of inkjet technology as key-technology for today’s industry has been
and still is the driving force behind the major improvements that this technology
has undergone over the last decades. This thesis contributes to that development
of inkjet technology. As justification of our particular research approach, an inventory of the current state of the art of this technology is essential. To that
purpose, this chapter presents a characterization of inkjet technology. As a result, the limitations of current designs will emerge, based on which several possible
research directions are identified.
1.1 Inkjet technology
In this section, a historical overview is presented of inkjet technology. Simultaneously, the unique capabilities of piezoelectric inkjet technology compared to
other forms of inkjet technology are addressed as well. Next, an inventory of
the applications of piezoelectric inkjet technology is given illustrating its versatile
functionality.
1.1.1 A historical overview
The rapid development of inkjet technology started off around the late fifties.
Since then, literally countless inkjet devices have seen the light of day. In this
overview, the attention is mainly restricted to the development towards the two
most important inkjet concepts of today, namely piezoelectric and thermal inkjet,
see Fig 1.1. At the end of this section, both concepts are discussed vis-à-vis. For
a more extensive overview of the history of inkjet technology, one is referred to
[Pon00].
The foundation of inkjet technology is attributed to the Belgian physicist Plateau
1
2
1.1
INTRODUCTION
inkjet
technology
binary deflection
multiple deflection
...
continuous
(CIJ)
drop
on
demand
(DOD)
thermal (TIJ)
piezoelectric (PIJ)
electrostatic
squeeze
bend
push
shear
...
Figure 1.1: Classification of inkjet technology
and English physicist Lord Rayleigh. Though Plateau was the very first to publish
on this field with his article ’On the recent theories of the constitution of jets of
liquid issuing from circular orifices’ in 1856 ([Pla56]), most of the credit belongs
to Lord Rayleigh. He published a series of founding papers including ’Instability
of jets’ in 1878 ([Ray78]), ’On the instability of cylindrical fluid surfaces’ in 1892
([Ray92b]), and ’Investigations of capillarity’ in 1899 ([Ray92a]). Still, it took
several decades before application of these physical principles took place in working devices. The first pioneering work in that direction was performed in the late
1940s by an employee of the Radio Corporation of America (RCA), who invented
the first drop-on-demand device. By means of a piezoelectric disc, pressure waves
could be generated that caused a spray of ink drops, see Fig. 1.2. However, this
invention was never developed into a commercial product.
Figure 1.2: The first drop-on-demand inkjet device (US Patent 2,512,743)
The honor of the first commercial inkjet apparatus is considered to go to the
1.1
INKJET TECHNOLOGY
3
Minograf of the Siemens-Elema company released in 1952. Instead of being an
inkjet printer, it was merely a voltage recorder quite similar to current seismic
apparatus.
The early work of Plateau and Lord Rayleigh and the two jet-writing concepts can
be regarded as first steps towards inkjet printing. The rapid growth of electronic
information systems in the late sixties induced a renewed scientific interest and
started research into the two major directions of inkjet technology: continuous
inkjet (CIJ) and drop-on-demand (DOD), see Fig. 1.1. During the sixties, progress
was established in three important regions:
• DOD thermal inkjet. With sudden steam printing, a researcher from the
Sperry Rand Company basically invented thermal inkjet printing, see Fig. 1.3.
By boiling aqueous ink at certain time instances, a drop of ink could be generated. The strength of this design clearly was not acknowledged, since the
company did not elaborate this idea into a commercial product. The idea
was abandoned until the late seventies when Canon and Hewlett Packard
(HP) picked it up.
Figure 1.3: Sudden steam printing (US Patent 3,179,042)
• DOD electrostatic pull inkjet. The basic working principle comprises the
following. Conductive ink is held in a nozzle by negative pressure. By
application of a high voltage pulse to an electrode located outside the nozzle,
a charged droplet of ink is pulled out. By application of the appropriate
deflection field, the droplet can be located on the substrate. Companies
developing electrostatic pull inkjet devices were the Casio, Teletype, and
Paillard company. With the model 500 Typuter, the Casio company released
4
1.1
INTRODUCTION
in 1971 a printer of this type. The Inktronic Teletype machine in the late
1960s was marketed by the Teletype company.
• Continuous inkjet. The major achievement in CIJ was the synchronization of the jet breakup. By adding periodic (acoustic) actuation, the random drop formation process becomes synchronized to that period as was
predicted by Lord Rayleigh. Consequently, the resulting droplets can be
charged and deflected to the desired position. Main players in the field were
Sweet of the Stanford University who came up with the Inkjet Oscillograph.
This device was elaborated for use by the Stanford Research Institute (SRI)
for inkjet bar coder work for Recognition Equipment Incorporated (REI).
The A.B. Dick Company elaborated Sweet’s invention to be used for character printing. With their Videojet 9600 in 1968, it was the first CIJ printing
product ever.
Despite these developments in inkjet technology, the products that came to the
market can be characterized as unreliable and having a poor print quality. In
the seventies, the DOD electrostatic pull principle was abandoned due to poor
printing quality and reliability. The development of DOD thermal principle was
put on hold. Of the principles in development, only CIJ remained and was developed further. In addition, the seventies are marked with the emergence of
the DOD piezo-electrical inkjet, abbreviated as PIJ, principle. More specifically,
these developments comprised the following:
• CIJ with binary drop deflection. This approach is depicted in Fig. 1.4. The
charged droplets are deflected to the paper or to the gutter where it is
recycled. This track of research and development continued the work that
was started in the sixties. Main players are the A.B. Dick Company, REI,
the Mead Company, and IBM. The A.B. Dick company and REI continued
their work in bar code printing. The Mead company introduced DIJIT
in 1973 used for advertising purposes. The huge research efforts of IBM
resulted in one product only, the IBM 6640.
high voltage
deflection plate
paper
HV
drop
generator
charge
electrode
gutter
Figure 1.4: CIJ with binary drop deflection
1.1
INKJET TECHNOLOGY
5
• CIJ with multiple drop deflection. This approach is illustrated in Fig. 1.5.
Two companies that were involved in this branch of CIJ were the Sharp and
Applicon company. The former released their Jetpoint in 1973, the latter
their color image printer in 1977.
high voltage
deflection plate
paper
HV
drop
generator
charge
electrode
gutter
Figure 1.5: CIJ with multiple drop deflection
• DOD piezo-electrical inkjet. Generally, the basis of piezo-electrical inkjet
(PIJ) printers is attributed to three patents. The first one is that of Zoltan
of the Clevite company (US Patent 3,683,212), proposing a squeeze mode of
operation. The second one of Stemme of the Chalmer University (US Patent
3,747,120) utilizes the bend mode of piezoelectric operation. Finally, Kyser
and Sears of the Silonics company (US Patent 3,946,398) used a diaphragm
mode of operation. Common denominator of these three patents is the
use of a piezoelectrical unit to convert a pulse of electrical energy into a
mechanical pressure to overcome the surface tension forces holding the ink
at a nozzle. Drops are only created when an actuation pulse is provided,
hence drop-on-demand. Obviously, the main discriminator between these
patents is the used dominating deformation mode of the piezoelectric material together with the geometry of the ink channels. The patents of Howkins
(US Patent 4,459,601) describing the push mode version and Fischbeck (US
Patent 4,584,590) proposing the shear mode, completed the now commonly
adapted categorization of printhead configurations. In general, four types of
PIJ printheads can be distinguished, namely the squeeze, push, bend, and
shear mode, see Fig. 1.6.
Major advantages of PIJ over CIJ printers include the fact that there is
no need for break-off synchronization, charging electrodes, deflection electrodes, guttering and recirculation systems, high pressure ink-supplies and
complex electronic circuitry. The first piezoelectric DOD inkjet printer to
reach the market was in 1977 with the Siemens PT-80. Silonics was the
second company to introduce a piezoelectric DOD printer, namely the Quietype in 1978.
6
1.1
INTRODUCTION
squeeze
bend
shear
push
Figure 1.6: Classification of piezoelectrically driven inkjet printheads
All the inkjet printers that had been introduced so far had failed to be commercially successful. It proved to be extremely difficult to combine print quality,
throughput, cost, and reliability all into one single inkjet printing device with either CIJ or PIJ. Though CIJ is capable of attaining high throughput, it required
high costs to achieve the required high print quality in addition to reliability.
With PIJ, it turned out to be problematic to achieve both excellent print quality and reasonable throughput simultaneously. The realization of high density of
piezoelectric actuators was difficult. Consequently, it was impossible to miniaturize the design to an acceptable format.
The invention of thermal inkjet (TIJ) in the early eighties fundamentally changed
inkjet research. By the replacement of the piezoelectric by a thermal transducer,
the main bottleneck of PIJ concerning miniaturization was resolved. Not only
the size of the thermal transducer was favorable being a simple resistor, but also
the low cost of manufacturing. TIJ can be manufactured using mass-production
based on IC-manufacturing technology making the cost per nozzle much lower
than the cost per nozzle of a PIJ printhead. Typically, a TIJ nozzle costs around
several euro-cents whereas a PIJ nozzle cost lies around ten euro-cents. Both the
fact that inkjet printers now could be miniaturized and its low cost of manufacturing made TIJ to the superior inkjet technology at that time. Canon was the
first company to bring TIJ to the market in 1981. Their lead in the TIJ devel-
1.1
INKJET TECHNOLOGY
7
opment was translated in a great number of patents, practically giving Canon
the means to control the TIJ market. Of the companies that Canon licensed its
patents to, HP was the only company that could keep up the pace with Canon.
The milestones in TIJ printing are so extensive that a list is omitted.
After the introduction and immense success of TIJ, PIJ research efforts were
largely diminished. Only a few companies continued their research into PIJ. In the
nineties, only a few companies that conducted research in PIJ were left, among
which Spectra, Xaar, Seiko-Epson, Trident, and Lexmark. CIJ-based printers
and research practically disappeared, except for some sporadic publications (e.g.
[Die98], [Sch99], [Hei00]). An important impulse to PIJ research was provided by
ongoing developments in the manufacturing of multilayer piezoelectric actuators.
One of the major barriers now had been lifted: that of miniaturization. Epson’s
advances in piezoelectric transducer fabrication have allowed it to remain competitive.
Despite the eminent success of TIJ printing, there are some fundamental advantages of PIJ over TIJ:
• Ink properties. TIJ only works with aqueous inks whereas PIJ can work with
a broad latitude of ink properties, including hotmelt ink. This is favorable
in two ways. First, certain applications require a special type of material
to be deposited such that PIJ is the only technology capable of doing so.
Second, the types of ink that can be used with PIJ results in general in a
higher print quality.
• Durability. PIJ printers have a higher durability than their TIJ equivalents.
Typically, a PIJ nozzle is capable of jetting around 10 billion drops per
lifetime whereas a TIJ nozzle is only capable of around 200 million droplets.
The reason for that is the harm that is posed to the heater element of a TIJ
printer. Each time a droplet is jetted, it is heated and cooled quite quickly
successively. This affects the life-time considerably.
• Attainable jetting frequency. PIJ printers can achieve higher jetting frequencies than TIJ printers.
• Drop-size modulation. Since control of the bubble collapse is not possible
with TIJ, drop-size modulation is fundamentally not possible with TIJ.
With PIJ, the necking of the drop-formation process can be controlled and
therefore gives an opportunity for drop-size modulation. This can be used
to further increase the resolution and thus print-quality.
At present, both TIJ and PIJ printing have evolved into the two most important
technologies when it comes to printing. The initial advantages of TIJ over PIJ
have been levelled over the years by further development of the PIJ technology.
8
INTRODUCTION
1.1
Also, current applications of inkjet technology simply require the sketched unique
capabilities of PIJ that the TIJ technology is unable to provide.
1.1.2 A generic manufacturing technology
A fundamental strength of the PIJ technology is its ability to deposit a wide
variety of materials on various substrates in certain patterns. Next to this characteristic, several additional advantages can be mentioned that apply to inkjet
technology in general. To start with, its on-demand character makes it a very
flexible manufacturing technology. Furthermore, when used for manufacturing,
the use of PIJ printing usually reduces the number of manufacturing steps necessary. Additionally, due to its additive character, there is a reduction of the use of
possibly expensive materials or equivalently a reduction in waste as well. Finally,
it is a non-contact and non-contaminating process which can be very favorable in
a manufacturing process. Altogether, these characteristics make inkjet technology
a very versatile manufacturing technology.
The importance of PIJ printing for the industry is best illustrated by the large
range of applications. Due to this wide variety of applications it is practically
impossible to present a complete overview. Also, each categorization of the applications remains artificial to some extent. Nevertheless, the following division
is adopted:
• Graphics. Most likely, inkjet technology is first associated with this field of
applications. This is hardly surprisingly given the huge amount of (desktop)
printers present in offices and the like. Accordingly, the amount of printer
types is also large. A subdivision can be made based on for example the
type of ink used (e.g. aqueous, hotmelt, UV-curable), substrate (e.g. paper,
textile, food, canvas), and format (e.g. narrow or wide format printing).
Some of these fields are dominated by TIJ printing, others by the PIJ printing. In general, PIJ printers are utilized in case the ink cannot be deposited
by TIJ printers or the required quality is high.
• Displays. In the display market, PIJ technology is used to manufacture Flat
Panel Displays (FPD), Liquid Crystal Displays (LCD), color filters (a part
of LCDs), Polymer Light Emitting Diodes (PLED), and flexible displays.
The accompanying performance criteria are one of the major driving forces
behind much research and development efforts concerning PIJ. Examples
can be found in e.g. [Has02; Ben03].
• Electronics. Within this market, PIJ printheads are used to create functional electrical traces using conductive fluids on both rigid and flexible
substrates. One of the first applications of inkjet technology within this
field was that for the production of Printed Circuit Boards (PCB). Other
applications include the fabrication of electric components and circuits such
1.2
SYSTEM DESCRIPTION
9
as Radio Frequency Identification (RFID) tags, wearable electronics, solar
cells, fuel cells, and batteries. Challenges for the PIJ technology within this
field include the spreading of the ink and the required guarantees of continuity of the jetted lines. Examples of the manufacturing of electronics with
PIJ technology can be found in e.g. [Hei05; Szc05; Kno05].
• Life science. This market is rapidly expanding with new requirements for
precise dispensing of DNA and protein substances. The high costs of these
fluids make PIJ technology with its precision placement and tight flow control an excellent dispensing tool. Applications include the use for DNA research, various medical purposes such as dosing of drugs, and food science.
A quite futuristic application is the use of inkjet printing for the fabrication
of living tissue. Examples can be found in [Che96; Jam98; Coo01; Rad05].
• Chemical. Within this market, the PIJ technology is mainly used as tool
for research purposes. Again, the unique capacity of the technology for
dispensing small doses of liquids specifically makes it useful for this market.
Applications include material and substrate development as well as coating
purposes. Examples can be found in [Oht05; Nak05]
• Optical. Jetting of UV-curable optical polymers is a key technology for
the cost-effective production of micro-lenses. These tiny lenses are used
in devices from fiber optic collimators to medical systems. The ability of
PIJ technology to precisely jet spheres in variable but consistent drop sizes
provide opportunities for the cost reduction of existing optical components
and innovative new designs, see e.g. [Cox96; Che02; Bie04].
• Three-dimensional mechanical printing. This category claims the PIJ technology as tool for rapid prototyping, small volume production, and the production of small sensors. Examples can be found in [Wal02; Voi03; Yeo04].
As discussed, performance requirements imposed by various applications are quite
strict. In light of future applications, it is expected that these requirements will
become even tighter. In combination with current limitations, this motivates
ongoing research, as will be discussed in Section 1.2.3.
1.2 System description
1.2.1 An archetypal PIJ printhead
The variety encountered in PIJ printhead design is enormous. Apart from the
variation in actuation principle (see Fig. 1.6), the possibilities in geometry are
seemingly endless. Despite the differences between the various designs, some
common denominators can be distinguished:
10
1.2
INTRODUCTION
1. Basic working principle. Though the operation of a PIJ printhead involves
many fields of science, a major role is assigned to that of acoustics.
2. The ink channel design. Despite the sketched diversity in printhead designs,
four basic components keep returning. These include the channel itself, the
nozzle, the ink supply, and the piezo-unit.
3. The operation of printheads. Typically, actuation pulses are manually shaped
input pulses based on physical insight of the design.
The work presented in this thesis focusses on the common principles of PIJ printheads, among which the ones listed above, yet will be elaborated on one particular
PIJ printhead design. Since the fundamental characteristics of this design does
not differ from most other PIJ printhead designs, the results presented throughout this thesis will be still generally applicable. So to speak, the employed PIJ
printhead design is truly an archetypical one.
In this section, a description of the working principle of the used PIJ printhead
design is given. At the same time, it provides a perfect example of the three
sketched characteristics above. Here, the focus lies on the basics rather than the
details of the design. Those will be discussed in Chapter 3 and 4. Note that
various experimental curves shown in the remainder of this chapter have been
measured with one of the experimental printheads, see Chapter 3.
x=0
[reservoir]
piezo unit
x=L
[nozzle]
ink channel
Vpuls
1
2
3
4
t
5
Figure 1.7: A schematic side view of an inkjet channel and its working principle
In Fig. 1.7, a schematic side view of a channel of the PIJ printhead subject in
this thesis is depicted. A schematic front view of an array of channels is depicted
in Fig. 1.8. As can be seen, all piezo-units are connected to the same substrate.
The channel has a length of several millimeters. The reservoir is connected to the
channel as an open end. As explained in [Gro03], the piezo-unit is concurrently
1.2
SYSTEM DESCRIPTION
11
substrate
piezo
unit
piezo
unit
ink
channel
ink
channel
Figure 1.8: A schematic view of an a cross-section of a PIJ printhead
used as actuator and sensor. Physically, it senses the force that results from the
pressure distribution in the channel acting on the piezo’s surface that borders
the channel. This force creates a charge on the piezo-unit. Since only changes
in charge are measured, in fact the time derivative of the instantaneous present
force is sensed. Furthermore, since the resulting voltage drop of this current over
a resistance is measured, we have that a voltage is the resulting sensor signal.
For the trapezoidal pulse used for actuation, a typical sensor signal is depicted in
Fig. 1.9, p. 13. Typically, around 75 nozzles per inch are integrated in an array
that forms a printhead.
To fire a droplet, a trapezoidal pulse is provided to the piezo actuator, see Fig. 1.7.
Then, ideally, the following occurs, see e.g. [Bog84; Ant02]. To start with, a negative pressure wave is generated in the channel by enlarging the volume in the
channel (step 1). This pressure wave splits up and propagates in both directions
(step 2). These pressure waves are reflected at the reservoir that acts as an open
end and at the nozzle that acts as a closed end (step 3). Note that the negative
pressure wave reflecting at the nozzle causes the meniscus to retract. Next, by
decreasing the channel’s volume to its original value a positive pressure wave is
superimposed on the reflected waves exactly when they are located in the middle
of the channel (step 4). Consequently, the wave traveling towards the reservoir is
canceled whereas the wave traveling towards the nozzle is amplified such that it
is large enough to result in a droplet (step 5).
Another common denominator is the operation of an PIJ printhead. For most
designs, an input wave form is manually shaped based on physical insight in the
working of a printhead. Clearly, for the design presented here, the actuation
pulse is tuned to the first eigenfrequency of the ink channel. Additionally, somewhat more complex waveforms are designed for purposes like smaller droplets and
damping of the residual vibrations. Details will be discussed in Chapter 4.
1.2.2 Limitations of current designs
The applications discussed in Section 1.1.2 require certain performance criteria to
be met. For a PIJ printhead, an important set of requirements is related to the
resulting drop properties, namely:
12
INTRODUCTION
1.2
• Drop-speed. The resulting droplets are required to have a certain speed,
typically around several m/s.
• Drop-volume. Depending on the application under consideration, the performance requirement concerning volume typically varies from 5 to 15 picoliter.
Smaller drop-volumes are for example required with the manufacturing of
PolyLEDs. The smallest drop-volumes are around 2 to 3 picoliter. For some
applications, it is required that the drop-size can be varied during operation. For example, for large areas that need to be covered large drops are
desired, whereas for high resolution printing small drops are desirable. This
is referred to as drop-size modulation.
• Drop-speed and -volume consistency. The variations in drop-volume and
drop-speed between successive drops and between the nozzles must stay
within a certain percentage band, typically ranging from 2 to 15 percent.
This is to avoid irregularities in the printed object. In this thesis, only
drop-to-drop consistency is considered.
• Drop-shape. The drop-shape is influenced negatively by the formation of
tails or satellite drops. These are highly undesirable for the quality of the
print. For example, for the production of PolyLEDs, tails or satellites induce
cross-contamination.
• Jet straightness. The droplets have to be deposed in a straight line to the
substrate, typically within 5 to 14 mrad accuracy. Note that as the dropvolume decreases, this requirement becomes even more important.
These requirements are only explicitly concerned with the drop itself. The following important two requirements are more related to the jetting process:
• Productivity. The productivity of a PIJ printhead is mainly determined by
the jetting frequency, defined as the number of drops that a channel jets
within a certain time, and the amount of nozzles per inch (npi-ratio), see
[Bru05] for details. Though these two parameters are highly dependent on
the specific design of printhead, typically it is around 10-20 kHz at 50-100
npi to guarantee acceptable productivity.
• Stability. Stability of the jetting process is one of the most important performance requirements for PIJ printheads. In this context, stability is defined
as the absence of nozzle failure per a certain amount of jetted drops, e.g.
one failure per one million jetted drops.
In addition to these requirements, more general requirements are imposed, including the lifespan of the printhead (typically more than ten billion actuations
per channel), the materials compatibility (a wide variety of inks must be deposable), the maintainability, and the cost of production and manufacturability of
1.2
13
SYSTEM DESCRIPTION
the printhead. In this thesis, we restrict ourselves to the requirements posed for
the drop itself plus the two requirements concerning the jetting process itself.
Meeting these performance requirements is severely hampered by the following
operational issues that are associated with the design and operation of printheads
as discussed in Section 1.2.1. Major issues that are generally encountered are the
following:
• Residual vibrations. After a drop has been jetted, the fluid-mechanics within
an ink channel are not at rest immediately: apparently traveling pressure
waves are still present. These are referred to as residual vibrations. In
Fig. 1.9, the system’s response to a standard actuation pulse is depicted.
Also, the time instant of drop-ejection is indicated (around 17 µs in Fig. 1.9).
Usually, the fixed actuation pulse is designed under the assumption that a
channel is at rest. To guarantee consistent drop properties, one has to
wait for these residual vibrations to be sufficiently damped out to fulfill
this assumption. Since this takes about 100 to 150 µs, it limits the maximally attainable jetting frequency with all the consequences concerning the
productivity and drop-consistency of a printhead. If the presence of residual vibrations is ignored and the jetting frequency is increased nonetheless,
drop-properties start varying. As example, the so called Drop-on-Demand
(DOD) speed curve is depicted in Fig. 1.9, showing the dependency of the
drop-speed on the jetting or DOD frequency. As can be seen, considerable
speed fluctuations result.
−6
4
x 10
4.5
3
2
Droplet speed [m/s]
Integrated sensor signal [Vs]
4
1
3.5
3
0
2.5
−1
−2
0
10
20
30
40
50
Time [µs]
60
70
80
90
100
2
2
4
6
8
10
12
DOD frequency [kHz]
14
16
18
20
Figure 1.9: Residual vibrations (left, measured response (black) and the corresponding actuation pulse (gray, scaled)) and its effect on the DOD-speed curve
(right)
• Cross-talk. Cross-talk is the phenomenon that one ink channel cannot be actuated without affecting the fluid-mechanics in neighboring channels. Crosstalk occurs in various ways:
14
1.2
INTRODUCTION
1. Electrical cross-talk. This form of cross-talk usually does not play a
significant role. It occurs at the level of electrical circuits that are
present in any printhead to operate the channels, for example in the
form of leakage currents.
2. Acoustic cross-talk. The phenomenon that pressure waves within one
channel influence other channels is called acoustic cross-talk. It can
occur via the ink reservoir. Though it is a more important effect than
electrical cross-talk, the overall influence can generally be considered
small.
3. Structural cross-talk. Structural cross-talk can occur in many ways.
For example, as can be seen in Fig. 1.8, all piezo-fingers are connected
to a substrate. As a result, deformation of one piezo-unit induces a
deformation of the neighboring units. Another path is via the deformation of a channel itself. As a result, the volume of the neighboring
channels changes also which induces pressure waves in those channels.
The deformation of the printhead structure can originate from two
sources. The first one is the result of a channel being actuated and is
referred to as direct voltage cross-talk. The second one is the result
of the occurring pressure wave that causes deformation of the channel
and is called indirect or pressure cross-talk.
−6
3
x 10
5
4.9
2
4.8
1.5
4.7
Droplet speed [m/s]
Integrated sensor signal [Vs]
2.5
1
0.5
4.6
4.5
0
4.4
−0.5
4.3
−1
0
10
20
30
40
50
Time [µs]
60
70
80
90
100
4.2
−10
−8
−6
−4
−2
0
2
Channel number [−]
4
6
8
10
Figure 1.10: Cross-talk (left, measured response of an actuated channel (gray) and
a neighboring channel (black)) and the consequences on the drop-speed (right)
In Fig. 1.10, the effect of cross-talk on the fluid-mechanics of a neighboring channel is shown. Also, its effect on the drop-speed of simultaneous
actuation of neighboring channels is depicted. In this figure, the resulting
drop-speed of channel zero is depicted when in turn neighboring channels
are actuated. For example, when the neighboring channel at the right of
channel zero is actuated, the drop-speed of channel zero drops from 4.9 m/s
1.2
SYSTEM DESCRIPTION
15
to 4.2 m/s. As can be seen, the effect of cross-talk on the drop-speed in
particular is substantial. Though this figure only shows the drop-speed,
cross-talk influences other drop-properties as well.
To minimize the effects of cross-talk, a number of measures have been taken.
First, operation of ink channels is designed such that two neighboring channels are not actuated simultaneously. However, this limits the possibilities
considerably. Also, ink channels are actuated with a small delay to allow
the worst effects to be damped out. Another measure to minimize the effect
of cross-talk involves the printhead design itself. As can be seen in Fig. 1.11,
the amount of piezo-units is twice that of the design depicted in Fig. 1.8.
The redundant piezo-units B bordering the piezo-unit A form a so called
bridge structure that provide additional stiffness to the design. If piezo-unit
A is actuated to jet a droplet, the piezo-units B (short circuited) reduce the
effects of structural cross-talk. However, this reduces the variations in drop
speed only slightly. Furthermore, it is a costly solution, since the number
of required piezo-unit for an array doubles. Also, it limits the attainable
npi-ratio.
substrate
piezo
unit
B
piezo
unit
A
piezo
unit
B
ink
channel
Figure 1.11: A schematic view of an a cross-section of a PIJ printhead with a
bridge structure
• Changing/varying dynamics. There are various phenomena that account
for changing or varying dynamics. First, some materials suffer from aging
and their properties change over time. For example, piezo-material has a
notorious reputation when it comes to aging. Second, due to the extreme
sensitivity of an ink channel’s behavior for small changes in material properties, ink channel dynamics vary even within a range of a couple of channels.
Changing or varying dynamics in combination with fixed actuation pulses
affect the performance negatively. Conventional measures to minimize these
effects, such as enforcing strict material properties during production, are
usually very expensive and boost the cost of production considerably.
• Robustness against disturbances. There are a number of disturbances possibly affecting the performance. To start with, air-bubbles or dirt particles
may cause a channel stop functioning. Also, the various structural modes
16
INTRODUCTION
1.2
of a PIJ printhead itself influences the performance negatively. Using one
fixed actuation pulse simply cannot handle these issues effectively.
These operational issues form boundaries for the attainable performance and
hence are a major drive behind the research and development conducted into
inkjet technology. An inventory of solution strategies for these limitations is presented in the following section.
1.2.3 Towards a controlled environment
Applications of inkjet technology as presented in Section 1.1.2 impose tight performance criteria on the printheads. In the near future, these performance requirements become tighter. For some of these applications, even today’s performance
already is insufficient. Given these facts, several operational issues have been
identified in Section 1.2.2 that exactly limit the attainable performance. These
observations provide a clear motivation for ongoing research in the field of inkjet
technology.
The objective of this section is to identify suitable research directions that can
improve the performance of PIJ printheads in face of the operational issues. To
obtain such an inventory of possible solution strategies, it is necessary to first
distance oneself from the specific PIJ printhead and focus on the various disciplines involved in printhead engineering. In this section, after having obtained an
overview of these disciplines and their individual contributions, the focus again
shifts to the printhead design itself and it is discussed how the various disciplines
can offer solution strategies to the issues at hand.
Research and development of the PIJ technology require a wide variety of disciplines to be involved, see Fig. 1.12. After all, due to the complexity of inkjet
systems, it is impossible to attribute all the necessary specialist knowledge to
one engineering domain. While restricting to the design and development of a
PIJ printhead only, already the following disciplines are typically represented in
printhead engineering:
• Applied physics. The role of applied physics consists mainly of gaining
fundamental understanding of the relevant phenomena that form the basis
of a PIJ printhead. This is of great importance during practically every
phase of printhead development. Typical examples include studies into the
drop-formation process, the inclusion of air-bubbles, and the assessment of
print quality.
• Mechatronics. The field of mechatronic engineering can be regarded as the
combination of mechanical, electronic, software, and systems and control engineering. To assess the role of mechatronics within printhead engineering,
the field is split up according to its origins:
1.2
SYSTEM DESCRIPTION
17
(a) Mechanical engineering. A mechanical engineer applies physical principles to (re)design a certain device, in this case a PIJ printhead. Their
expertise mainly aims at the application of the concepts of for example
(fluid) dynamics, strength of materials, and applied thermodynamics.
(b) Electrical engineering. Electrical engineering is a discipline that deals
with the application of electricity. Their contribution covers a wide
range from the selection of suitable actuators and sensors, designing
and testing electrical networks that support the functioning of a PIJ
printhead, to the digital signal processing to manipulate the relevant
signals.
(c) Software engineering. This computer science discipline is concerned
with developing large software applications. Their involvement with
printhead engineering usually comes at a later stage, when the printhead is mounted in the complete printing system. Therefore, their role
is somewhat limited during the design of a PIJ printhead itself.
(d) Systems and control. Engineers specialized in systems and control deal
with both the design and operation of a printhead, though main emphasis is given to the control part. For example, based on knowledge
of the system optimal input pulses can be designed.
Mechatronic engineers form the core of the printhead engineering team.
After all, a printhead truly is a mechatronic device. An important remark
concerns the role of systems and control that is so often associated with
mechatronics. Though the systems and control discipline is acknowledged as
being an important aspect, the application thereof lags considerably behind,
especially in the field of printhead engineering.
• Materials engineering. Materials engineering is a multidisciplinary field focusing on functional solids, whether the function served is structural, electronic, thermal, or some combination of these. Their work within the printhead engineering focuses on the choice for materials, keeping an eye on
issues such as manufacturability, cost, and function. This discipline plays
an important role in printhead engineering, since the consequences of these
choices have high impact for example on the cost per nozzle.
• Chemical engineering. The involvement of chemical engineering in the design of a PIJ printhead confines itself mainly to the ink. The influence of ink
properties on the functioning of a PIJ printhead however is large. Though
the development of ink can be performed quite independently from that
of the printhead, it is important that some critical parameters of ink, e.g.
viscosity, are established in mutual consult.
Based on this overview, an inventory of research directions can be drawn up.
Not surprisingly, each of these disciplines solves the operational issues from their
18
INTRODUCTION
1.2
particular perspective. A categorization of the various research directions can be
given as follows:
• Mechanical (re)design. This solution approach to the operational issues
comprises a mechanical redesign of the printhead, either by starting from
scratch, applying only minor changes, or anything in between. Some possible
(combinations of) directions include:
(a) Geometry. The geometry of an ink channel or an entire printhead
influences the performance considerably. A few examples thereof are
the following. A reduction of the channel-length induces the creation of
smaller droplets. The way the ink is supplied to an ink channel largely
determines the boundary condition of a channel and thus the operation
of a printhead. An investigation in the geometry in all its details
therefore is a suitable research direction in face of the operational issues
encountered.
(b) Actuation. Actuation is one of the key-issues in printhead design. For
example, the specific implementation of the piezo-electrical actuation
not only determines the amount of cross-talk, but also the controllability and observability of the jetting process. Even the choice for a
piezo-electrical actuator could be subject of discussion.
(c) Material. The choice of materials also has its influence on the operation
of a printhead. An example is the wetting of the nozzleplate that
might be solved by using a different type of coating. Also, the cost of
manufacturing is largely dependent on the choices regarding material
as well.
• Ink properties. Apart from the printhead itself, the ink plays an extremely
important role in the jetting process. Rather than focussing on the printhead design itself, the ink is an important research direction as well. As
illustration, recall that the drop-formation is largely dependent on the ink
properties.
• Control. The application of the principles of system- and control to a truly
mechatronic device such as the printhead is a promising research direction.
As argued, printhead engineering lacks a systems and control up to this
point while there are a lot of possibilities for this research direction.
The first two research directions characterize current research efforts quite well.
These efforts are mainly centered around on the mechanical (re-)design of PIJ
printheads. Input for the (re-)design of the printheads originates from among
others applied physicists. Related, but relatively autonomous tracks comprise the
chemical and material engineers performing further research on their particular
area of interest. Apparently, the fact that a printhead is a mechatronic device
1.2
SYSTEM DESCRIPTION
19
does not automatically result in adopting a truly mechatronic approach, i.e. with
the proper attention to systems and control, to solve the performance limiting issues. In Fig. 1.12, the sketched characterization of current printhead engineering
is schematically depicted in the figure on the left. Here, the systems and control
approach plays only a modest role. In our view, however, a more prominent part
for systems and control in general, and the application of control to PIJ printheads in particular, is indispensable to lift current performance limitations of PIJ
printheads. In Fig. 1.12, the importance of systems and control within printhead
engineering is depicted in the figure on the right. To get a better understanding
for systems and control as solution strategy for the operational issues, the major
benefits of this approach are inventoried.
systems
&
control
applied
physics
mechatronics
applied
physics
mechatronics
systems
&
control
materials
engineering
chemical
engineering
materials
engineering
chemical
engineering
Figure 1.12: Characterization of printhead engineering: current situation (left)
and with the proposed direction (right)
Basically, systems and control can play a crucial role in two ways. To start with,
its systematic approach to the functioning of complex systems in the aggregate
offers structuring of the research and focus on the major performance determining mechanisms. Second, it provides an additional degree of freedom to enhance
the performance of PIJ printheads by means of control. These added values of
systems and control within printhead engineering are advantageous for the improvement of existing printhead designs as well as for the development of new
ones. For example, the use of control is a very cost-beneficial option to enhance
the performance of existing PIJ printheads without having to perform a redesign.
Also, during the design of new printhead, both the systematic approach and the
additional degree of freedom in the form of control provide tools to tune the design such that optimal performance can be achieved.
Having introduced the relevance of systems and control for PIJ printheads in general, let us elaborate a bit more on the role of control in particular. In most cases,
the term control is associated with feedback control. Feedback control aims primarily on stabilization and disturbance rejection. However, for a PIJ printhead,
being a stable system by its nature, usage of feedback control is not of direct im-
20
INTRODUCTION
1.2
portance. Next to feedback control, feedforward control can be considered. For
systems that act predictable based on their physical design, feedforward control is
a suitable option. A PIJ printhead fulfills this requirement perfectly. Here, feedforward is considered as tool for the design of actuation signals for PIJ printheads.
To the best of our knowledge, the use of feedforward for this purpose has been
virtually unexplored. The related field of input shaping has been investigated at
least at one occasion, see [Jon97].
In the next chapter, the systems and control approach as solution strategy to lift
the performance limitations of current PIJ printheads is further elaborated to a
research objective and several research questions.
Chapter 2
Problem formulation
In this chapter, the discussion in the introduction of this thesis is formalized in a
research objective. This objective is then divided in three main research questions.
Finally, the structure of this thesis is outlined.
2.1 The research objective
In the previous chapter, the main performance limiting operational issues that
are commonly encountered in PIJ technology have been discussed. Given the
fact that performance criteria for PIJ printhead applications become increasingly
tight, these boundaries must be lifted to be able to meet future requirements.
Based on an inventory of solution strategies that can resolve these operational
issues, a systems and control approach has been chosen to be explored in this
thesis. To the best of our knowledge, this research direction has been formerly
unprecedented within the printhead engineering community, at least in the open
literature. Therefore, only few work is available that can serve as starting point
for the research conducted here. In this light, the research objective to fully explore the possibilities of systems and control for PIJ printheads is formulated as:
Develop a unifying modeling and control framework for a PIJ printhead to investigate the possibilities and limitations of current designs in face of the commonly
encountered operational issues.
Let us clarify the various elements present in this objective. To start with, ’a
unifying modeling and control framework ’ relates first and foremost to the two
basic ingredients of a systems and control approach, namely modeling and control. To possess unifying properties in light of the research presented here, a model
should describe the functioning of a printhead on a system level, incorporating
all performance relevant dynamics. The input as well as a firm theoretical back21
22
PROBLEM FORMULATION
2.2
ground for these dynamics is often provided by the various disciplines involved in
printhead engineering. The resulting model therefore is able to relate the overall
performance of a PIJ printhead on a system level to the various detail studies
performed by the various research groups within printhead engineering. Hence,
the classification ’unifying’ is adopted. In addition, a solid control framework
enables the systematic exploration of the to be introduced feedforward control
option together with the obtained insight to come up with practical solutions to
the operational issues at hand. Together, such a unifying modeling and control
framework provide a solid basis to systematically ’investigate the possibilities and
limitations of current designs in face of the commonly encountered operational
issues’. The word ’possibilities’ reflects the utilization of the resulting framework
to lift current boundaries posed by the ’commonly encountered operational issues’
to enhance the attainable performance of PIJ printheads. At the same time, new
boundaries are expected to emerge. These more fundamental ’limitations’ of current printhead designs can however offer valuable insight to be used in the design
process of future PIJ printheads. The generality of the research conducted in this
thesis and the various results is emphasized by the use of the phrases ’current designs’ and ’commonly encountered ’. The results obtained throughout this thesis
apply to more PIJ printheads than the ones considered here.
2.2 A decomposition in research questions
In this section, the research objective is decomposed in three main research questions. Together, the solutions to these questions provide an overall solution to
the research objective of this thesis.
Question 1: How should a PIJ printhead be modeled given its intended use for
the proposed systems and control approach?
Basically, this research question is closely related to the suitability of the PIJ
model for the purposes in mind. Within the advocated systems and control approach, the role of the model is versatile. For one, the model should provide
insight in the working of a PIJ printhead, both for the implementation of control
and the use for (re-)design purposes. Also, it should facilitate the implementation
itself of (feedforward) control. Several additional requirements could be formulated. Now, the better the model fulfills these and other requirements, the more
beneficiating the systems and control approach can become.
An important aspect throughout the discussions regarding the modeling (and control) concerns the linearity of the jetting process. Though the jetting of a drop
each time a channel is actuated induces nonlinear behavior, it remains to be seen
how this affects the overall behavior of an ink channel from a systems and control
point of view.
2.3
A DECOMPOSITION IN RESEARCH QUESTIONS
23
Question 2: Can we design actuation wave forms which will be implemented
as feedforward control such that the performance of current PIJ printheads is
improved?
As discussed previously, the introduction of control provides an additional degree of freedom to a PIJ printhead. Without having to perform a redesign of
an existing printhead, its performance can be optimized by a simple tuning of a
controller. For new designs, the performance can be increased by taking the presence of the control into account. Now, for both existing and new PIJ printhead
designs, the question arises how the incorporation of control can help to overcome
the operational issues and thereby enhancing the performance of PIJ printheads.
A related, but certainly equally important question concerns to what extent the
attainable performance can be increased.
In this thesis, feedforward is investigated. Given the fact that an PIJ printhead
acts predictably based on its physical design and is inherently stable, feedforward
control is the most suitable choice. More specifically, given the highly repetitive
character of the jetting process, Iterative Learning Control (ILC) is a logical choice
as control strategy. Though ILC has proven its value for high-precision motion
systems, it has not been used in the field of inkjet technology yet. A systematic
exploration of the possibilities of ILC given the operational issues is therefore a
fitting approach to this research question. Additionally, the generic character of
the proposed framework renders it generally applicable to a broad range of PIJ
printheads.
For the implementation of control, an important issue concerns the choice for the
controlled and manipulated variable. Two options are considered in this thesis,
namely piezo-based and laser-vibrometer based ILC. Though this choice is highly
dependent on the particular PIJ design at hand, there is no loss of generality.
Question 3: Can we improve current PIJ printheads such that some basic limitations with respect to the attainable performance are lifted?
The utilization of a systems and control approach to the operational issues at hand
will lift some of the present boundaries concerning the attainable performance.
At the same time, however, several new boundaries will inevitably emerge. The
source of these boundaries can be attributed to the design itself: they cannot be
lifted any other way than changing the design itself. Only by the application of the
proposed systems and control approach, these new boundaries become apparent.
Having identified these fundamental limitations, the question arises how these
boundaries can be dealt with. Several indications will be provided throughout
this thesis.
24
PROBLEM FORMULATION
2.3
2.3 The structure of this thesis
This thesis is organized as follows. In Chapter 3, the experimental setup and
the various PIJ printheads are introduced. Among other things, the sensor functionalities are addressed, the various properties of PIJ printheads are reviewed,
and experimental identification is treated. The findings of this experimental exploration form the starting point for both the theoretical and experimental work
that is presented throughout this thesis. By using the results of Chapter 3 as
starting point, the subjects treated in this thesis are directly related to actual
verifiable data rather than being somewhat artificial. After having performed our
experimental exploration, the theoretical modeling of an ink channel is treated in
Chapter 4. To start with, the need for a new model is thoroughly motivated. This
model is constructed as a series of bilaterally coupled multiports and is based on
first principles only. Special attention is paid to the choices made, e.g. concerning
white box modeling and the use of a two-port approach. The resulting two-port
model is validated in Chapter 5. The results are discussed in detail. Also, directions for future research concerning the two-port modeling of an ink channel are
given. Together, Chapter 4 and 5 provide an answer to research question Q1. At
the end of Chapter 5, research question Q3 will be addressed based on the results
obtained so far. Note that at the end of Chapter 7, some of these findings are
revisited to provide conclusive answers to research question Q3. In Chapter 6,
the feedforward control framework is introduced. Details concerning this framework are treated, such as for example the formulation of a suitable control goal,
the ILC controller synthesis, and the incorporation of constraints in the actuation
signal. Next, the implementation of ILC to the experimental setup is addressed
in Chapter 7. Both the results of the so-called piezo- and laser-vibrometer based
approaches are presented. It is shown that the obtained learned actuation pulses
provide solutions to the two most prominent performance limiting operational
issues: residual vibrations and cross-talk. Consequently, the productivity and
drop-consistency is improved. For several other operational issues, it is indicated
how they can be solved by the proposed control strategy. Research question Q2
is addressed both Chapter 6 and 7. Research question Q3, that has already
been discussed to some extent in Chapter 5, is revisited based on the results obtained in Chapter 7. At the same time, a conclusive answer to this third research
question can then be provided. Finally, Chapter 8 presents the conclusions and
recommendations of this research.
Chapter 3
Experimental exploration
In this chapter, the experimental setup used to investigate PIJ printheads is discussed in detail. Special attention is given to the sensor functionalities present.
Then, the various PIJ printheads that are used during the research are introduced.
The relevant printhead dynamics of these PIJ printheads are identified and the results are presented. Here, these experimental results are not shown for validation
or control purposes. For that, one is referred to Chapter 5 and 7. Instead, the
data are used to be able to from this point relate the main topics covered in this
thesis directly to actual verifiable data. In our view, such an approach contributes
to the verifiability of our research according to [Buc95].
3.1 Description of the experimental setup
A schematic overview of the experimental setup is depicted in Fig. 3.1. The experimental setup itself is depicted in Fig. 3.2. With this setup, PIJ printheads can
be investigated in various ways. The only actuator is the piezo-unit of the inkjet
printhead. Three sensors are available in this setup. First, the piezo-unit not only
can be used as actuator but also as sensor. Second, the meniscus (ink-air interface in the nozzle) movements can be captured by the laser-vibrometer. Third,
properties of the resulting droplet can be monitored by a CCD camera. These
sensor functionalities will be discussed in detail in the subsequent subsections.
The PIJ printheads under investigation use a hotmelt type of ink that require
heating of the printhead. The required reference temperature is reached by a PID
controller (Eurotherm 2408), which measures the printhead’s temperature with
thermocouples and controls the input voltages by means of heating elements.
Next, to monitor the ink level inside the reservoir, a level sensor is incorporated
in the printhead. Furthermore, a printhead is mounted in vertical direction with
the nozzles faced down, similar to its position in an inkjet printer. To avoid that
25
26
3.1
EXPERIMENTAL EXPLORATION
actuation signal
waveform
generator
switch
board
amplifier
scope
pc
piezo sensor
signal
meniscus
velocity
printhe
temperature
control unit
ad
air pressure
unit
image
ink level
indicator
strobe
light
CCD camera
+ microscope
laser-vibrometer
+ detector
mirror
(45 deg.)
Figure 3.1: A schematic overview of the experimental setup
the ink simply flows out of the nozzles under the influence of gravity, an air pressure unit (TS 9150G) makes sure that the pressure in the ink reservoir remains
below the ambient pressure.
As depicted in Fig. 3.1, the setup is connected to a personal computer that is
equipped with National Instruments IMAQ PCI 1409 and PCI GPIB cards for
image processing and communication, respectively. On the computer, the desired actuation signals can be programmed and relevant data can be stored and
processed. After defining the actuation signal, it is sent to an arbitrary waveform
generator (Philips PM 5150/Fluke 195). The waveform generator sends the signal to an amplifier unit (Krohn-Hite 7602), which has a certain gain. From the
amplifier unit, the signal is fed to a so-called switch-board. The switch-board is
controlled by the personal computer and determines which channels are provided
with the appropriate actuation signals. For the tracing of both the actuation
and various sensor signals, an oscilloscope (Tektronix TDS 420/TDS 3034B) is
used. This oscilloscope is connected to the computer and displayed data can be
downloaded to the personal computer.
3.1
DESCRIPTION OF THE EXPERIMENTAL SETUP
27
Figure 3.2: The experimental setup
3.1.1 Piezo sensor signal
The first and most important sensor functionality discussed is that of the piezounit. For a detailed discussion on the piezo-unit, one is referred to Chapter 4.
Here, the fundamentals are treated, required for the explanation of the simultaneous use of the piezo-unit as actuator and sensor.
As generally known, a piezo can be used as actuator or sensor, see e.g. [Waa91].
For that, one uses the piezo’s indirect (actuator) and direct (sensor) piezo-electric
effect. The former comprises the following. If an electrical potential V is applied
to the piezo-unit, a deformation of the piezo-unit u results. The latter refers to
the following phenomenon. If a force F is applied to a piezo’s surface, an electric
charge q results. Together, this behavior can be described as:
u
d 1/k
V
=
(3.1)
q
C
d
F
with C the piezo’s capacity, d the piezoelectric charge constant, and k the stiffness
of the piezo. Schematically, (3.1) can be represented as two-port, as depicted in
Fig. 3.3. The piezo-unit is bilaterally coupled with an impedance Zc representing
the remainder of the ink channel. The use here of the two-port concept anticipates the derivation of the two-port model of an ink channel to be presented in
28
3.1
EXPERIMENTAL EXPLORATION
Chapter 4. For an introduction of the two-port modeling approach, one is referred
to the next chapter.
V
d
+
u
+
1
k
C
+
q
+
d
Zc
F
Figure 3.3: The piezo-block: the ink-channel as impedance
Now, rather than using the piezo-unit as either actuator or sensor, during the
research presented in this thesis it is used as actuator and sensor simultaneously.
This is accomplished as follows. The measured signal q is made up of two contributions. The first is that of the applied actuation voltage V via the piezo’s
capacity C and is referred to as the direct-path. The second contribution originates from the force F exerted by the ink in the channel via the piezoelectric
charge constant k and is referred to as the indirect-path. Since only this second
contribution is the required sensor signal, it has to be extracted from the measured signal q. However, the contribution of the direct-path is considerably larger
than that of the indirect-path, being typically 10-20 mA and 50-100 µA, respectively. Consequently, it is difficult to measure the sensor signal (indirect-path)
simultaneously while using the piezo as actuator. Basically, there are two options
to do so still:
1. Using software-compensation. Given knowledge of the applied electrical
field V and the availability of an accurate model of the piezo’s capacity
C, the contribution of the direct-path can be computed. By subtracting
this contribution from the measured signal q, the required sensor signal can
be established, see e.g. [Dos92; And94]. Note that for the discrimination
between the direct and indirect-path in our case, a rather accurate model
has to be available. The model inaccuracies should be at least significantly
smaller than the sensor signal that one is trying to obtain.
2. Using hardware-compensation. Rather than modeling the piezo’s capacity
C, an actual piezo is used to predict the contribution of the direct-path. In
Fig. 3.4 and 3.5, this is schematically depicted. The measured signal q of a
full ink channel comprises both the direct- and indirect-path. The measured
signal q of an empty ink channel only consists of the contribution of the
3.1
DESCRIPTION OF THE EXPERIMENTAL SETUP
29
direct-path. Again, by subtracting both measured signals, the indirect-path
or sensor signal can be obtained.
direct-path
'piezo'
indirect-path
'ink'
d
V
+
u
+
1
k
C
+
'piezo'
+
+
q
Zc
d
F
'ink'
+
Figure 3.4: Division into a piezo- and ink-block diagram
A drawback of software compensation relates to the required accuracy of the
piezo model. Since modeling of the piezo’s capacity C is extremely difficult given
its nonlinear behavior, this method is hard to implement. On the other hand,
hardware compensation requires that both piezo-units are exactly the same. Small
differences, e.g. due to drift or production tolerances, are always present. This
influences the accuracy of the resulting sensor signal negatively. Of both methods,
hardware compensation is the only feasible method to simultaneously use the piezo
as actuator and sensor in case of a PIJ printhead. To minimize the effects of piezo
capacity differences, the following measures are taken:
'piezo'
+
'ink'
-
'piezo'
=
'ink'
+
full channel
empty channel
Figure 3.5: The basic principle to obtain the actuation and sensor signal simultaneously as used in the piezo-sensing device
• Temperature differences. Differences in piezo capacity occur due to temperature differences of both piezo-units. By isolating the PIJ printhead these
differences are satisfactorily minimized.
• Differences in piezo capacity. Matching the impedance of various piezo-units
usually results in a satisfactory pair.
30
3.1
EXPERIMENTAL EXPLORATION
• Influence of structural effects on the sensor measurement. Even though the
ink channel is empty, a small contribution due to the deformation of the
structure may be present in the indirect-path. This effect can be neglected
though.
For details, one is referred to [Gro03]. The measured frequency response of the
electronic conditioning of the piezo-sensing device, i.e. the subtraction as shown
in Fig. 3.5, is depicted in Fig. 3.6. Note that modeling of the piezo-unit itself, i.e.
the piezo-block as depicted in Fig. 3.3, is postponed until Section 4.2.5. Apparently, as can be seen in Fig. 3.6, the magnitude as well as the phase are distorted
for the low and high frequency range. However, for the frequency range of interest, roughly from 20 kHz up to 250 kHz, the resulting sensor signals are minimally
affected by the piezo-sensing device.
5
Magnitude [dB]
0
−5
−10
−15
−20
−25
−30
−35
1
10
2
10
3
4
10
10
5
10
6
10
Frequency [Hz]
100
Phase [deg]
50
0
−50
−100
−150
1
10
2
10
3
10
4
10
5
10
6
10
Figure 3.6: Measured FR of the piezo-sensing device
Having discussed the technical implementation of the simultaneous use of the
piezo-unit as actuator and sensor, the question arises what the sensor signal represents. Physically, it senses the force that results from the pressure distribution
in the channel acting on the piezo’s surface that borders the channel. This force
creates the discussed electric charge on the piezo-unit (the indirect-path). Since
only changes in electric charge are measured, in fact the time derivative of the
instantaneous present force is sensed. Furthermore, since the resulting voltage
drop of this current over a resistance is measured, we have that a voltage is the
3.1
DESCRIPTION OF THE EXPERIMENTAL SETUP
31
resulting sensor signal. A typical sensor signal as result of a standard trapezoidal
actuation pulse is depicted in Fig. 3.14, p. 43.
The following remarks are in order. First, the piezo sensor is located in the
channel whereas the droplet formation takes place in the nozzle. Second, due to
the integrating character of the sensor the resulting signal is an average of the
pressure that is present in a channel. Finally, since all the piezo’s are connected
to the same substrate, the actuation as well as sensing is influenced by structural
cross-talk. Despite all these facts, the current sensor signal can be regarded as
representative for the jetting process.
3.1.2 CCD camera
A second sensor functionality is provided by the Charge-Couple Device (CCD)
camera equipped with a microscope, that can observe the generated droplets. A
stroboscope provides a short light flash at a defined instant after the droplet is
ejected and an image is obtained on which the droplet seems to be fixed in the air.
A necessary requirement for this approach to succeed is that the repeatability of
the drop formation is high. Then, since both the time duration and the distance
that the droplet has traveled are known, an estimate of the droplet speed can
easily be obtained. Moreover, it is possible to estimate the volume of the droplet,
because the droplet diameter can be determined. Other information which can be
obtained concern the droplet’s angle, the formation of satellites and the stability of
the jetting process. A great advantage of the CCD camera is that direct information about a droplet is obtained. Unfortunately, this information is only available
at discrete time instants. In case the drop formation is not repeatable, a more
expensive high-speed camera could be used to obtain the required drop properties.
Note that the resulting droplet properties are the result of image processing. By
altering some of the parameters of this process, e.g. the threshold used for the
black-white conversion, the outcome may change. This affects the quality of the
measurements negatively.
3.1.3 Laser-Doppler interferometry
The third and last sensor functionality is the laser-vibrometer. The principle of
Laser-Doppler interferometry consists of the splitting of a laser beam in two different paths and, finally, combine the beams again. One beam travels over a fixed
path and the path of the other beam is varied. In case a beam is reflected against
a moving object, a Doppler shift takes place. When the object is moving towards
the beam, the frequency of the signal increases and when the object is moving
away from the beam, the frequency decreases. This way, the combined signal contains information about the phase difference and the frequency shift between the
two signals. This information is measured by a detector. With a Laser-Doppler
32
EXPERIMENTAL EXPLORATION
3.1
interferometer or laser-vibrometer it is possible to measure the velocity of the
meniscus inside a nozzle. Here, the meniscus surface is the moving object which
reflects the beam. Unfortunately, this type of measurement can only be applied
to a small range of the dynamics. It is namely not possible to jet during this
measurement, without taking special measures.
In the experimental setup, the laser-vibrometer is used to measure the meniscus
velocity. It consists of a Polytec OFV-5000 vibrometer controller containing a
Polytec VD-02 velocity decoder. Furthermore, a Polytec OFV-512 fiber interferometer and a Polytec OFV-130-3 micro-spot sensor head complete the setup.
The resulting laser beam of approximately 3 µm in diameter is aligned via a
mirror in the center of a nozzle that has a diameter of 32 µm. It is assumed
that a Poiseuille velocity profile occurs in the nozzle during operation, such that
the laser-vibrometer setup measures the maximum velocity. Due to the use of
a laser-vibrometer via a mirror that is situated directly in front of the nozzle
exit, the experiments are restricted to the non-jetting situation. Practically, this
means that only experiments at a lower voltage can be performed. However, if it
is assumed that the ink channel behaves linearly, the resulting learned actuation
pulses at a lower voltage can be scaled up to a jetting voltage and implemented.
This important linearity assumption will be discussed in detail in the subsequent
chapters. The following remarks are in order. First, the impossibility to use
meniscus-based ILC in a jetting situation does not conflict with its intended use
as design tool for wave forms. Second, a sensor that is integrated in the printhead
as replacement of the laser-vibrometer is currently being developed, see [Gro06a].
Then, limitations with respect to the used voltage are removed.
The following remarks are in order with respect to the use of the laser-vibrometer:
• Laser alignment. Due to the reflective property of the nozzleplate, alignment of the laser beam is quite difficult. Initial alignment is performed
based on a camera image of the laser-spot on the nozzleplate. Since the
wetting is clearly visible, the jetting channel can easily be established. The
final alignment takes place by observing the resulting sensor signal on the
scope. The expected amplitude of the response is known from calibration
experiments conducted earlier.
• Sensor output. A remaining issue concerns the physical interpretation of
the resulting sensor signal. If the laser is not aligned in the center of the
nozzle, it is not known what velocity is measured. This might still be the
maximum component of the meniscus.
• Limited measurement capabilities. To start with, only in a non-jetting situation the measurements can be carried out. Second, a considerable phase-lag
is introduced by the velocity decoder of the Polytec equipment. Since this
phase-lag is known, it can be compensated for.
3.2
DESCRIPTION OF THE EXPERIMENTAL PRINTHEADS
33
• Heating of the ink. The heating of the ink by the laser can be neglected due
to the low power intensity of the laser beam.
3.2 Description of the experimental printheads
A schematic representation and nomenclature of the PIJ printheads used in the
research presented in this thesis are depicted in Fig. 3.7. Specific details concerning the geometry and physical properties of these printheads are listed in Table 3.1
and 3.2, respectively. All printheads used in this thesis are similar, except for one
point. This concerns the presence of the so-called bridge structure, see Fig. 3.8.
As discussed in Chapter 1, this bridge structure is used for the minimization of
structural cross-talk effects. Some printheads have the bridge structure (233e02
and 293e02) while others have not (DG074). During the discussions throughout
this thesis, it is clearly indicated which printhead has been used.
nozzle
substrate
piezo-finger
reservoir
channel
connection
Figure 3.7: Nomenclature of an ink channel
substrate
substrate
piezo
unit
piezo
unit
channel
A
channel
B
piezo
unit
B
channel
B
piezo
unit
A
piezo
unit
B
channel
A
Figure 3.8: Cross-section of a PIJ printhead without(left) and with (right) bridge
structure
As can be seen in Fig. 3.7 and Table 3.1, the channel and connection have a different cross-section. Normally, a change in cross-section gives rise to an impedance
change with corresponding transmission and reflection conditions. However, due
to the flexibility of the (actuated) channel wall, the impedances of both the channel and connection match. Hence, effectively, there is no impedance change and
34
3.2
EXPERIMENTAL EXPLORATION
Channel (actuated)
Channel (not actuated)
Connection
Nozzle
length
height
width
length
height
width
length
height
width
length
diam. (start)
diam. (end)
7.61
106
266
0.40
106
266
1.06
230
230
100
100
32
mm
µm
µm
mm
µm
µm
mm
µm
µm
µm
µm
µm
Table 3.1: Data of the printhead geometry
Density
Dynamic viscosity
Surface tension
Speed of sound
Effective speed of sound
ρ
µ
ν
c
ceff
1090
0.011
0.028
1250
900
kg/m3
Pa s
N/m
m/s
m/s
Table 3.2: Overview of the physical properties of ink
the effect of the changing cross-section can be neglected.
In Table 3.2, a distinction is made between the speed of sound and the effective
speed of sound. The former applies for the non-actuated parts of the ink channel.
The latter is used for the actuated channel. Due to the fluid-structure interaction,
the effective speed of sound is lower. By using these different values for various
parts of an ink channel, this effect is accounted for.
Throughout this thesis, it is assumed that all channels are identical. The validity
of this assumption as well as the consequences if not, are discussed in Chapter 6
and 7. In Fig. 3.9, an overview of the nomenclature of the various transfer functions is provided that is adopted in this thesis. The direct transfer functions
are denoted by Ha and Hb , the indirect or cross transfer functions by Hab and
Hba . The identification is performed using two of the three sensor functionalities.
First, the piezo is used as actuator and sensor. This is referred to as piezo-based
identification. Second, the laser-vibrometer instead of the piezo-unit is used as
sensor. This is referred to as laser-vibrometer based identification.
The printhead’s main eigenmodes can be determined using modal analysis. In
general, resonance frequencies can be computed according to:
3.2
DESCRIPTION OF THE EXPERIMENTAL PRINTHEADS
uA
35
uB
Hab
ink
channel
Ha
channel
B
channel
A
Hb
ink
channel
Hba
yA
yB
Figure 3.9: Nomenclature of two neighboring channels
ceff
(3.2)
λ
with ceff the effective speed of sound and λ the wave length of the appropriate
standing wave in an ink channel. In principle, the ink channel’s basic resonance
frequency is the 1/4 λ mode, given the fact that one open (reservoir) and one
closed (nozzle) end is present. Note that λ equals in our case Lch + Lco + Ln .
However, for frequencies up to approximately 100 kHz, the nozzle acts as an
open rather than a completely closed end. Therefore, for low frequencies the ink
channel acts more as a 1/2 λ resonator, see [Ant02]. This phenomenon can be
explained as follows. Suppose that the nozzle dynamics can be described by an
equivalent mass-spring-damper system, where the mass represents the ink in the
nozzle. For low frequencies, the mass-spring-damper system oscillates whereas
for high frequencies it does not. Thus, the mass-spring-damper system, i.e. our
nozzle, acts as a low-pass filter. This phenomenon is discussed in more detail in
Section 4.2.2.
fr =
Now, in Table 3.3 and 3.4, the theoretical resonance frequencies of an ink channel
are listed, accounting for the occurring switch in resonating behavior at approximately 100 kHz. Anticipating on the identification of the frequency responses
in Section 3.4 and 3.5, the corresponding measured resonance frequencies for the
piezo-based (293e02) and laser-vibrometer based (233e01) transfer functions are
listed also. These values have been determined based on Fig. 3.13 and 3.17. When
comparing the theoretical and measured values of Table 3.3 and 3.4, the following
remarks are noteworthy. In the laser-vibrometer based case, the second mode
considerably deviates from the theoretical predicted frequency. This will be addressed in Chapter 5. Furthermore, the remaining (small) differences result from
the particular differences of the 233e01 and 293e02 printhead.
36
3.2
EXPERIMENTAL EXPLORATION
theoretical mode
1/2 λ (0.50)
2 · 1/2 λ (1.00)
5 · 1/4 λ (1.25)
7 · 1/4 λ (1.75)
frtheoretical
50 kHz
100 kHz
125 kHz
175 kHz
measured mode
0.48 (≈ 1/2 λ)
0.91 (≈ 2 · 1/2 λ)
1.19 (≈ 5 · 1/4 λ)
1.83 (≈ 7 · 1/4 λ)
frmeasured
48 kHz
90 kHz
118 kHz
182 kHz
∆ fr
2 kHz
10 kHz
7 kHz
7 kHz
Table 3.3: Overview of the theoretical and measured (293e02, see Fig. 3.13, p. 41)
resonance frequencies in the piezo-based approach
theoretical mode
1/2 λ (0.50)
2 · 1/2 λ (1.00)
5 · 1/4 λ (1.25)
7 · 1/4 λ (1.75)
frtheoretical
50 kHz
100 kHz
125 kHz
175 kHz
measured mode
0.43 (≈ 1/2 λ)
0.76 (≈ 3 · 1/4 λ)
1.30 (≈ 5 · 1/4 λ)
1.71 (≈ 7 · 1/4 λ)
frmeasured
43 kHz
76 kHz
129 kHz
170 kHz
∆ fr
7 kHz
24 kHz
4 kHz
5 kHz
3
3
2
2
1
0
−1
−2
Scaled pressure [Pa]
Scaled pressure [Pa]
4
0
2
4
6
8
Position [mm]
10
1
0
−1
−2
12
2
2
1
1
Scaled pressure [Pa]
Scaled pressure [Pa]
Table 3.4: Overview of the theoretical and measured (233e01, see Fig. 3.17, p. 46)
resonance frequencies in the laser-vibrometer approach
0
−1
−2
−3
0
2
4
6
8
Position [mm]
10
12
0
2
4
6
8
Position [mm]
10
12
0
2
4
6
8
Position [mm]
10
12
0
−1
−2
−3
−4
Figure 3.10: Pressure waves in an ink channel at 107 kHz sinusoidal actuation at
four time instances; right traveling wave (gray), left traveling wave (gray dotted),
resulting standing wave (black), and piezo-unit actuation (black dotted)
3.3
DESCRIPTION OF THE EXPERIMENTAL PRINTHEADS
37
Finally, one last phenomenon is to be addressed. If the piezo-unit is actuated
with a sinusoid at 107 kHz (or a multiple thereof), the ink in the channel below
the piezo-unit’s surface oscillates with the same frequency whereas the ink in the
remainder of the ink channel, connection and nozzle is almost completely at rest,
see Fig. 3.10. Fig. 3.10 is obtained using a finite volume model of the ink channel
dynamics, see [Wij04]. Note that this effect is also clearly visible in Fig. 3.17, p. 46.
One possible explanation for this phenomenon is the occurrence of destructive
interference below the piezo-unit’s surface, and comprises the following. Suppose
that the piezo-unit can be modeled as a finite set of point sources each emitting
traveling waves in both directions of an ink channel, see Fig. 3.11. If it is assumed
that the piezo-unit deforms uniformly over its length (see Section 4.2.5), these
point sources oscillate uniformly for every frequency. If the piezo-unit is actuated
with a frequency whose wavelength corresponds to the length of the piezo-unit (λ
= l), destructive interference occurs. Now, the generated pressure waves for two
point sources spaced at exactly d = 1/2 λ are illustrated in Fig. 3.11. As can be
seen, the waves of this set of sources are amplified below the piezo-units surface,
yet are canceled at any other location. Since the piezo-unit can in principle be
represented by an infinite set of point sources spaced at 1/2 λ apart, this effect
only is increased if more point sources are taken into account, see Fig. 3.11. As a
result, the ink below the piezo-unit is oscillating whereas the fluid-mechanics in
the remainder of the ink channel, connection, and nozzle are almost at rest. The
frequency at which this phenomenon occurs can be computed as:
d
substrate
piezo unit
Figure 3.11: Illustration of destructive interference phenomenon
fr =
ceff
900
=
= 118 kHz
λ
7.61 10−3 s
(3.3)
This theoretically computed value corresponds nicely to the measured anti-resonance
at 107 kHz, see Fig. 3.17, p. 46.
38
EXPERIMENTAL EXPLORATION
3.3
3.3 Identification method
In this section, the identification method of the Frequency Response (FR) from
the piezo actuator to either the piezo sensor or laser-vibrometer is discussed. Note
that the construction of the accompanying Frequency Response Functions (FRF)
based on these established FRs is discussed in Chapter 5: it additionally requires
the choice of a model structure and subsequent determination of its parameters.
For now, the focus lies on the determination of a non-parametric model to be used
for validation of the theoretical model to be constructed.
To start with, one has to select a particular type of input signal. The following
input signals have been considered:
• Sinusoids. Sinusoids are employed as input signals during a (pseudo) sinesweep identification procedure. A finite number of sinusoidal input signals
are then provided to a system. Important properties of the (pseudo) sinesweep are the following. First, the energy content is the same for each
frequency. For systems having a lot of noise, this is very advantageous. The
signal-to-noise ratio then remains large. Second, the transient effects can be
minimized by increasing the time spent per frequency. At the same time,
this also relates to a first drawback of the sine-sweep measurement. Since
the experiments take relatively much time, the effect of drift affects the
outcome. This is particularly true for the piezo, known for its drift. One
major cause for piezo drift is formed by the temperature fluctuations of
the piezo, according to the pyroelectric effect [Waa91]. Another drawback
concerns the resolution of the sine-sweep. Since only a finite number of
sinusoids are used, some frequencies are not excited at all.
• Step. Identification procedures applying a step response can be performed
fast. In face of the piezo drift, this can be very advantageous. However,
a number of disadvantages are present. First, a step remains band limited
such that high frequencies are often not excited. Second, due to the short
measurement, transient effects affect the result. Since lower frequencies are
particularly vulnerable for this effect, the quality of the identification of the
lower frequency range is influenced negatively.
• White noise. Another option is the use of white noise for system identification. One important property of white noise is its flat frequency spectrum:
the energy is equally distributed over the frequencies within the bandwidth
of the white noise signal. At the same time, this property may cause the
signal-to-noise ratio to deteriorate. This should be taken into account when
applying white noise as input signal for identification.
For the identification of the inkjet channel FR, the sinusoids were selected as input signal. The application of a sinusoids as input for the piezo, superposed on
3.3
IDENTIFICATION METHOD
39
the bias voltage, is schematically illustrated in Figure 4.16. The choice for the
amplitude of the sinusoids is discussed in subsequent sections. Next to the choice
of the type of input signal, the selection of the sample frequency is of importance.
To avoid aliasing, the signal that is being sampled should not contain frequencies
beyond the Nyquist-frequency. The Nyquist frequency fN is defined as half the
sampling frequency fs . Given the fact that there are no significant inkjet channel
dynamics present beyond 4 MHz, the sample frequency of 10 MHz suffices. For
most experiments presented and discussed in this thesis, we are only interested
in frequencies up to 500 kHz. Therefore, a Krohn-Hite 7206 low-pass filter with
a cut-off frequency of 500 kHz is employed. Its FR is depicted in Fig. 3.12.
10
Magnitude [dB]
0
−10
−20
−30
−40
4
10
5
10
Frequency [Hz]
6
10
100
Phase [Deg.]
0
−100
−200
−300
−400
4
10
5
10
6
10
Figure 3.12: Measured FR of the Krohn-Hite 7206 low-pass filter with a cut-off
frequency of 500 kHz
Based on the traced output signals and knowledge of the provided inputs, the
following procedure is applied to construct a non parametric model, see [Pei96].
For each frequency point, the Fourier components of the input and output are
determined using a Discrete Fourier Transform. At the same time, possible trends
present in the data are eliminated in the procedure of [Pei96]. The FR is then
obtained by a simple division per frequency point. Note that though related
to an Empirical Transfer Function Estimate (ETFE), there are several critical
differences. First, the trend is removed in the procedure of [Pei96]. Second, since
the exact frequency points are known, the computations are more accurate. The
noise present can be filtered out more accurately. Since the inkjet channel has
40
EXPERIMENTAL EXPLORATION
3.4
rather much noise, this effect is considerable.
3.4 Piezo-based experimental identification
The naming of the experimental identification presented depends on the sensor
functionality employed. If the piezo is used as sensor, the resulting identification
is referred to as piezo-based. In this section, the results thereof are presented.
Given our interest in an array of channels, two neighboring channels are selected
for identification. As discussed in Section 3.2, the presence of a bridge structure
influences the transfer functions considerably due to its effect on the structural
cross-talk. Therefore, the piezo-based identification is carried out for two different
printhead geometries: with and without the bridge structure.
3.4.1 With bridge-structure
In Fig. 3.13, the direct and cross FR are depicted. During identification, the amplitude of the sinusoids was chosen such that the inkjet channel was not jetting.
Once the nonlinearity as a result of droplet ejection is eliminated, the system
behaves linearly. This has been verified by various superposition experiments. Of
course, validity of the resulting model in the jetting situation remains to be seen
and is addressed below.
At first glance, the +1 slope in the direct FR seems surprising. As discussed in
Section 3.1.1, this is caused by the differentiating character of the piezo as sensor:
it senses changes in electric charge (i.e. current) rather than the electric charge
itself. Stated otherwise, the changes in channel pressure are measured instead of
the pressure itself. From a physical point of view, it makes more sense to control
the channel pressure rather than the changes thereof. For example, if there is no
change in pressure, the channel is not necessarily in rest. Therefore, an integrator
is added. This has several important consequences. One is the importance of the
various resonance frequencies that are visible in Fig. 3.13. The first resonance
frequency at 45 kHz (corresponding to the theoretically computed one) seems less
important compared to the other resonance frequencies due to its limited magnitude. However, adding an integrator renders the first resonance frequency the
most important one. The apparent physical importance of the first resonance
frequency is confirmed by the actuation pulse that is used, see Chapter 1. This
pulse is namely completely tuned to the first eigenfrequency of the inkjet channel.
As can be seen in Fig. 3.13, the cross FR does not have a large magnitude.
Also, the resonance frequencies are barely recognizable. Apparently, the bridgestructure is quite effective in the reduction of the cross-talk. The conclusion that
the cross-talk is eliminated is not correct, however. Though with limited mag-
3.4
PIEZO-BASED EXPERIMENTAL IDENTIFICATION
41
Magnitude [dB]
0
−20
−40
−60
−80
3
10
4
5
10
6
10
10
Frequency [Hz]
200
Phase [Deg.]
0
−200
−400
−600
3
10
4
5
10
6
10
10
−20
Magnitude [dB]
−30
−40
−50
−60
−70
−80
−90
4
5
10
10
Frequency [Hz]
0
Phase [Deg.]
−100
−200
−300
−400
−500
−600
−700
4
10
5
10
Figure 3.13: Measured FR from the piezo-actuator to the piezo-sensor; direct
(above) and cross (below) (293e02)
42
3.4
EXPERIMENTAL EXPLORATION
nitude, the cross-talk effect is still large enough to affect the droplet properties
negatively, as will be discussed in subsequent chapters. In addition, the bridge
structure limits the attainable nozzles per inch and thus productivity.
A substantial phase lag is present in both measured FRs, that is only partially
resulting from the various (anti-)resonances. Another part originates from various
devices in the hardware loop (see Fig. 3.1):
• The amplifier. The amplifier for the pulses generated by the waveform generator introduces a phase lag due to its limited bandwidth. For frequencies
below its bandwidth, the phase lag can be approximated by a linear phase
delay of 0.08 degrees per kHz, resulting in 80 degrees delay at 1 MHz.
• The waveform generator and scope. The internal clock of the waveform
generator and that of the scope may cause a delay. Both clocks are sampling
at 10 MHz, but are not coupled. In the worst case, this results in a delay
of 0.1 µs or a phase delay of 36 degrees at 1 MHz.
• ZOH sampling. The signal sampling in combination with a Zero-OrderHold (ZOH) also introduces a delay of half a sample interval. For a sample
frequency of 10 MHz this results in 9 degrees delay at 1 MHz.
• The low-pass filter. As discussed, during some experiments, a Krohn-Hite
low-pass filter is used, see Fig. 3.12. This results in approximately 400
degrees additional phase lag at 1 MHz. Note that this filter was not applied
during the identifications as presented in Fig. 3.13.
• The piezo-sensing device. The piezo-sensing device not only introduces
phase lag for frequencies beyond approximately 100 kHz, but also phase
lead for the low-frequency range, see Fig. 3.6. The phase lag at 1 MHz
equals 140 degrees.
• The laser-vibrometer. The phase introduced by the laser-vibrometer can be
computed according to the following formula ([Pol00]):
∆φ(fr) = −100
fr
− 0.00038fr
frc
(3.4)
where fr is the frequency in Hz at which the phase lag is to be computed.
frc is the cut-off frequency of the low-pass filter of the laser-vibrometer
and equals 1.5 MHz. Though this effect is not relevant for the piezo-based
identification, it is for the laser-vibrometer based identification. This will
be discussed in the next section.
For the piezo-based FRs, the total phase lag amounts to 274 degrees at 1 MHz if
the Krohn-Hite low-pass filter is not used.
3.4
43
PIEZO-BASED EXPERIMENTAL IDENTIFICATION
A final remark concerns the following. There is a peculiarity in the amplitude in
the low-frequency range for the direct FR as depicted in Fig. 3.13. One should
expect that the magnitude of FRF goes to −∞ for low frequencies. After all,
for those frequencies the ink channel has two open ends and the ink can oscillate
freely. In practice, however, the measured FR goes to a certain small, but constant, value. It is assumed that this mismatch is caused by electronic conditioning
of the piezo-sensing device used during the measurement, see Fig. 3.6.
0.08
1
0.8
0.06
0.6
0.04
Sensor signal [V]
Sensor signal [V]
0.4
0.2
0
0.02
0
−0.2
−0.4
−0.02
−0.6
−0.8
0
0.1
0.2
0.3
0.4
0.5
Time [s]
0.6
0.7
0.8
0.9
1
−4
x 10
−0.04
0
0.1
0.2
0.3
0.4
0.5
Time [s]
0.6
0.7
0.8
0.9
1
−4
x 10
Figure 3.14: Measured response in the jetting mode from the piezo input to the
piezo output; direct (left) and cross (right) (293e02)
The measured response of an actuated and neighboring channel to a trapezoidal
pulse are depicted in Fig. 3.14. A frequency spectrum of the response of the actuated ink channel as depicted in Fig. 3.14 reveals that the dominating frequency
of the response equals that of the first eigenfrequency of the ink channel. Apparently, despite the limited magnitude around 45 kHz, the standard actuation pulse
is designed such that this mode is excited the most. Linearity will be discussed
in Chapter 5.
3.4.2 Without bridge-structure
In Fig. 3.15, the direct and cross FRs for a PIJ printhead without bridge structure is depicted. Except for several small printhead specific differences, the direct
FR is similar to that of a printhead with bridge structure. However, this does
not hold for the measured cross FR. In the absence of a bridge structure, the
cross FR is far more evident. Though less smooth, several important resonance
frequencies can be detected. These correspond fairly good to those of the direct
FR. Note that for the phase, the same arguments hold as the FRs measured with
a printhead having a bridge structure.
44
EXPERIMENTAL EXPLORATION
3.4
−15
Magnitude [dB]
−20
−25
−30
−35
−40
−45
−50
5
10
Frequency [Hz]
0
Phase [Deg.]
−100
−200
−300
−400
−500
−600
5
10
−20
Magnitude [dB]
−25
−30
−35
−40
−45
−50
−55
5
10
Frequency [Hz]
200
Phase [Deg.]
100
0
−100
−200
−300
−400
5
10
Figure 3.15: Measured FR from the piezo-actuator to the piezo-sensor; direct
(above) and cross (below) (DG074)
3.5
0.8
0.15
0.6
0.1
0.05
0.4
0
Sensor signal [V]
Sensor signal [V]
45
LASER-VIBROMETER BASED EXPERIMENTAL IDENTIFICATION
0.2
0
−0.05
−0.1
−0.2
−0.15
−0.4
−0.6
−0.2
0
0.1
0.2
0.3
0.4
0.5
Time [s]
0.6
0.7
0.8
0.9
1
−4
x 10
−0.25
0
0.1
0.2
0.3
0.4
0.5
Time [s]
0.6
0.7
0.8
0.9
1
−4
x 10
Figure 3.16: Measured response in the jetting mode from the piezo input to the
piezo output; direct (left) and cross (right) (DG074)
In Fig. 3.16, the responses of the actuated as well as a neighboring inkjet channel
to a standard trapezoidal actuation pulse are depicted. The response of the
actuated channel seems to be in anti-phase with that of a non-actuated channel.
This corresponds to the physical behavior of the actuator block. By decreasing the
volume of the actuated channel, the actuators of both neighboring channels are
lifted via the substrate. Consequently, the volume of the non-actuated channels
is enlarged, leading to the exact opposite response in pressure than the actuated
channel.
3.5 Laser-vibrometer based experimental identification
As indicated in the previous section, the naming of the experimental identification
depends on the sensor functionality used. Here, the laser-vibrometer is used as
sensor functionality. Correspondingly, the resulting identification is referred to as
laser-vibrometer based. Note that the measurements presented here are obtained
from a printhead without a bridge structure.
In Fig. 3.17, the FR from the piezo-actuator to the meniscus velocity is depicted.
The following remarks are noteworthy. First, the magnitude of the first resonance
frequency of the direct FR turns out to be dependent on the used excitation voltage. This effect is not present in the cross FR. To visualize this effect, the direct
FR has been measured using three different excitation voltages. For the cross
FR, one excitation voltage sufficed. This nonlinear behavior can be explained
as follows. Even at these relatively low excitation voltages, the beginning of the
drop formation process can be observed. At this point, the viscous forces become
larger than the surface tension forces at the free-surface. As a result, the outward
meniscus velocity detected by the laser-vibrometer is larger than the inward ve-
46
3.5
EXPERIMENTAL EXPLORATION
0
Magnitude [dB]
−10
−20
−30
−40
−50
−60
−70
4
10
5
10
Frequency [Hz]
6
10
Phase [Deg.]
0
−500
−1000
sweep amplitude 1V
sweep amplitude 2.5V
sweep amplitude 4V
−1500
4
10
5
10
6
10
−20
Magnitude [dB]
−30
−40
−50
−60
−70
−80
4
10
5
10
Frequentie (Hz)
6
10
0
Phase [Deg.]
−1000
−2000
−3000
−4000
−5000
4
10
5
10
6
10
Figure 3.17: Measured FR from the piezo-actuator to the meniscus velocity; direct
(above) and cross (below) (233e01)
3.6
47
CONCLUDING REMARKS
locity due to the inertia of the newly formed droplet beginning. These effects are
confirmed by simulations of a finite volume model programmed in Flow3D, see
[Wij04]. The larger the excitation voltage, the more distorted the sine-response
becomes and the smaller the magnitude of the identified FR. This effect only
occurs at the first resonance frequency and cannot be detected at higher frequencies. Second, considerable phase lag can be seen in Fig. 3.17. This originates
from substantial time-delays present in the system as discussed with piezo-based
identification. In addition, considerable phase lag is introduced by the Polytec
laser-vibrometer.
The measured FR at 2.5 V is selected for use in the sequel of this thesis. Note
that the choice for the 2.5 V FR has been rather arbitrary, the 1.0 V FR could
have been used equally well instead. However, since the first resonance frequency
is hardly present in the 4.0 V FR, this last FR would not have been a proper
choice. In Fig. 3.18, the measured response to a standard trapezoidal actuation
pulse at a jetting frequency of 10 kHz at 2.5 V is depicted. In the sequel of this
thesis, further attention is paid to this nonlinearity as well as the limitation of
the laser-vibrometer setup.
1
0.3
0.8
0.25
0.2
0.15
Meniscus velocity [m/s]
Meniscus velocity [m/s]
0.6
0.4
0.2
0
0.1
0.05
0
−0.2
−0.05
−0.4
−0.6
−0.1
0
0.2
0.4
0.6
0.8
1
Time [s]
1.2
1.4
1.6
1.8
2
−4
x 10
−0.15
0
0.2
0.4
0.6
0.8
1
Time [s]
1.2
1.4
1.6
1.8
2
−4
x 10
Figure 3.18: Measured response in the jetting mode from the piezo input to the
meniscus velocity; direct (left) and cross (right) (233e01)
3.6 Concluding remarks
In this chapter, a comprehensive experimental exploration of PIJ printheads has
been performed. To start with, the experimental setup itself has been introduced.
The various sensor functionalities have been discussed, in particular the use of the
piezo-unit as actuator and sensor simultaneously. Next, the PIJ printheads have
been introduced. The geometry and FRs of various PIJ printheads have been
presented. Also, the various FRs were clarified using the physical background of
48
EXPERIMENTAL EXPLORATION
3.6
the printheads. With the results discussed in this chapter, a solid basis for the
upcoming issues in this thesis has been obtained.
The following two assumptions, introduced in this chapter, are of particular importance in the sequel of this thesis:
• Identical channels. The validity of the assumption that all channels are
uniform still is an open issue. Validity of this assumption would simplify
the identification and control of an PIJ printhead considerably. In case small
differences in channel dynamics turn out to be present, the approach can
possibly be made robust against the corresponding model uncertainties. In
the next chapters, the uniformness of the ink channels will be investigated
further.
• Linearity of the jetting process. Another important assumption concerns the
linearity of the jetting process. Based on the jetting process itself, validity of
this assumption is certainly not trivial. After all, the jetting of a drop each
time a channel is actuated induces nonlinear behavior. The actual impact
of this effect on the operation of a PIJ printhead from a systems and control
point of view is a subject that will be investigated further in the subsequent
chapters of this thesis.
In this chapter, it has been assumed that the PIJ printhead behaves linearly from a systems and control perspective. The identification has been
performed while keeping the excitation voltages low such that the channels
were not jetting. Now, several linearity related questions emerge. For one,
the usefulness of the identified FRs in the jetting case must be reviewed.
This is a particular relevant issue for the laser-vibrometer based approach
with the apparent nonlinear behavior with respect to the channel’s first
eigenfrequency. In the remainder of this thesis, these issues are given the
appropriate attention.
In the next chapter, modeling of a PIJ printhead is discussed. Having gained insight in various physical properties of PIJ printheads in general, and our ’archetypal’
experimental printheads in particular, the modeling can start with the appropriate prior knowledge. Next to this physical background, a preview was given of
the adopted approach to the modeling of an ink channel. The use of two-ports,
employed here to explain the functioning of the piezo-unit, forms namely a key
feature of our modeling approach. At the end of the next chapter, the insight in
the working of a PIJ printhead obtained in this chapter will be further extended.
Chapter 4
Modeling of the ink channel
dynamics
This chapter starts with a survey of known mathematical PIJ printhead models.
Given our modeling purposes, the need for an alternative model will become apparent. In this chapter, therefore, a new theoretical model is derived. To that purpose,
an ink channel is divided into a number of functional blocks each representing a
part of that channel. During the derivation, all the assumptions necessary are
discussed in detail. It will not only turn out that the unique characteristics of the
model result in a model that breaks the trade-off between accuracy and model complexity, but simultaneously form a suitable framework for control and redesign.
The derivation of the model is concluded by pointing out future research directions concerning this model. Validation of the resulting so called two-port model
are postponed until the next chapter.
4.1 PIJ printhead model survey
Given the research objectives as formulated in Chapter 2, let us start this chapter
by formulating the associated model requirements:
• Accuracy. Though accuracy is an obvious requirement, it still is of importance to state our exact objectives in this matter. As depicted in Fig. 4.1,
the resulting model is to be used for the control and redesign purposes as
formulated in Chapter 2. The objective with respect to accuracy is therefore formulated in light of these intended model applications. To start with,
the behavior of an ink channel on an input-output level is to be predicted
accurately. Consequently, the model can be used for the application of
control. Also, the major performance determining mechanisms of an ink
49
50
MODELING OF THE INK CHANNEL DYNAMICS
4.1
channel should be predicted accurately. This provides the required insight
for redesign.
• Model complexity. The resulting model’s complexity is to be kept as low as
possible. First, for the use of the model for the redesign purpose in mind,
focussing on the performance determining processes in the first place leads
to a simple model that provides the insight needed. Second, linked to the
model complexity is the computational complexity. It is favorable to keep
the computational load as small as possible since it facilitates the use for
control.
Modeling
(Re-)design
Control
Figure 4.1: Modeling for control and (re-)design
Since modeling usually comprises a trade-off between accuracy and model complexity, the requirements posed above form no exception. This is particularly true
for a PIJ printhead. Modeling a PIJ printhead or even one ink channel is considered a complex issue. Gaining insight into the origins of this supposed complexity
is of importance when discussing PIJ printhead modeling. This complexity is
mainly caused by the following interconnected issues:
• Multiscale and multiphysics modeling. PIJ printhead dynamics cover a wide
range in applied mechanics. To start with, the piezo-unit dynamics can be
described by the governing equations of solid mechanics. The ink dynamics
on the other hand require the relationships of fluid-mechanics. Even within
the fluid-mechanics, various rather diverse topics are represented in a PIJ
printhead. A good example is the droplet formation, where free-surface flow
is the center of attention. Another example concerns the simplification of the
governing equations towards acoustics. Often, this simplification is highly
desirable to reduce the often high computational load. Finally, the modeling
of the electrical circuitry has not even been considered yet. Altogether,
the described presence of diverse topics in continuum mechanics requires
the knowledge of all those fields in detail to successful modeling the PIJ
printhead. Also, it gives rive to several other difficulties when modeling a
PIJ printhead.
• Fluid-structure interaction. Fluid-structure interaction (FSI) occurs when
a solid interacts with a fluid. In case of a PIJ printhead, the piezo-unit
4.1
PIJ PRINTHEAD MODEL SURVEY
51
interacts with the ink in a channel. A major cause of the associated difficulty
is the moving boundary. Stated otherwise, the domain occupied by the
medium is one of the unknowns in the problem. In fact, this is related to
the free-surface problem that occurs during droplet formation. There also
the boundary is moving and is not known prior to the computation. Another
associated difficulty with FSI is the coupling, see below.
• Coupling. More generally speaking, next to the FSI, coupling is another
issue. Given the usage of several fields in continuum mechanics for various
parts of an ink channel, the coupling of these parts to one model is not trivial.
This is especially true in case many parts are present, such as with a PIJ
printhead. Additionally, the admissible time step can be severely limited
due to this method of coupling, leading to large computational times.
• Geometry. The particular geometry of a PIJ printhead often forms an issue.
Due to its complexity, e.g. the geometry of a nozzle, it is difficult to generate
a proper mesh when using a Computational Fluid Dynamics (CFD) package.
analytical
numerical
combination
FV
FE
FD
complete ink
channel dynamics
[Bel98] [Dij84] [Dij99]
[Ten88a]
[Ten88b]
[Sak00] [Kol02b] [Ber03]
[Wij04] [Lio02]
[Wij04] [Bad98; Bad01]
[Pan02] [Che99] [Sch86]
[Sch87] [Asa92] [Yu03]
[Sei04]
nozzle dynamics
and drop formation
[Fro84] [Shi05] [Mar06]
[Kol02a]
drop formation
only
[Egg95]
[Wil99] [Yeh01]
[Yu05]
[Wu05]
Table 4.1: Overview of available piezo-electric ink channel models
Given the model requirements and the complexity of PIJ printhead modeling, the
question arises whether a suitable model already is available in open literature.
As can be seen in Table 4.1, a great number of printhead models can be found.
The categorization of these models presented here is based on the fact whether
the governing equations are solved using analytical or numerical techniques:
1. Numerical. Numerical means not continuous thus discrete. To solve the
governing equations, being partial differential equations, numerically, one
has to discretize those in place and time. Based on the usual discretization
methods in place of the common CFD packages, the following subdivision
is adopted:
• Finite volume. The finite volume method discretizes a volume into a
number of cells of arbitrary shape. Subsequently, the governing equations are solved on these discrete control volumes by ensuring conser-
52
MODELING OF THE INK CHANNEL DYNAMICS
4.1
vation of mass, momentum, and energy in the fluids between finite
volumes.
• Finite element. With the finite element method, the governing differential equations are solved in terms of minimum residuals over an
element. The unknowns inside an element are approximated by shape
functions which amplitudes are controlled by nodal displacements.
• Finite difference. At each intersection of the lines of the finite difference mesh used, the governing differential equation is replaced by
a finite difference approximation. The main disadvantage is that it
requires structured meshes, and coordinate transformations for complicated geometries.
For the time discretization, numerous options are available. Examples include the explicit and implicit Euler method, the midpoint rule, and the
trapezoid rule. In explicit time integration schemes, the admissible time
step for the solver during integration is limited. This limit is determined
by the Courant number, defined as the ratio between the time step and the
wave propagation time within an element:
C=
∆t
∆x/c
(4.1)
where ∆x is a characteristic length of the cell and c the speed of sound. To
ensure correct computations in explicit methods, the Courant number may
not exceed 1.
For the modeling of free surface, e.g. for the modeling of droplet formation,
additional methods are employed within a CFD package. These are used to
track the moving boundary and can be categorized into surface or volume
methods. Examples of the former is the Marker and Cell method (MAC).
An example of the latter is the Volume of Fluid (VOF) method.
2. Analytical. In some cases, the governing equations can be solved analytically.
Usually, a number of (simplifying) assumptions are then required. Lumped
parameter approaches are considered to fall into this category.
3. Combination. Models that solve the governing equations analytically in
some direction and numerically in others belong to this third group.
As can be seen in Table 4.1, the models are further divided according to the part
of an ink channel that is modeled. Not all models incorporate the complete ink
channel. Based on this overview, the question arises whether or not these are
useful for the research presented in this thesis.
4.2
THE TWO-PORT MODEL
53
In general, the numerical models of Table 4.1 are very accurate. Rather than
being constructed to assess the overall performance of a PIJ printhead, their goal
is often to describe certain phenomena. For example, the working of the nozzle is
to be predicted. Also, it is employed to establish the cause of arising problems,
such as clogging of the nozzle. For these purposes, the level of detail and the
accuracy of the models render them extremely suitable. At the same time, the
high accuracy is frequently accompanied by large model complexity. Numerical
models are often programmed in CFD packages using complex meshes and solving techniques to get an answer. This makes it hard to obtain the insight that is
required for the application of control or redesign.
Though the model complexity of analytical models is usually low, the accuracy
is on average less than that of the numerical models. For the intended purposes,
the decrease in accuracy is acceptable. However, sometimes over-simplification or
under-simplification is performed such that the resulting model provides too few
or just too much information, respectively. For the models presented in Table 4.1,
this is the case. Especially with respect to the insight the models are supposed
to provide the models are not adequate.
The models that combine numerical and analytical models usually combine the
drawbacks of both previous mentioned categories. They only take care of a small
reduction in computational time. However, this is not exactly an issue.
Altogether, there still is a need for a model that really breaks the trade-off between
accuracy and model complexity. Such a model that is suitable for the control and
redesign purposes in mind cannot be found in the open literature. Therefore,
a new PIJ printhead model will be derived in the subsequent sections of this
chapter. A thorough discussion of the resulting model as well as the validation of
the derived theoretical model is provided in the following chapter.
4.2 The two-port model
Given the objectives as formulated in the previous section, it is chosen to employ
the concept of bilaterally coupled systems (BCS) for the modeling of an ink channel. The notion of BCS in connection with the modeling of dynamical systems has
been first introduced in [Pay61]. The related energy port and multiport systems
have been developed in the work of [Ros72] and [Kar77]. In our view, several
properties of this concept can play a crucial role in achieving these aims. The
most important properties of a BCS are the following. To start with, it enforces
a causal internal model structure for the system to be modeled. Among other
things, such a structure guarantees the possibility for physical interpretation of
a system at all times. Next, the interaction of a BCS with its surroundings is
taken into account explicitly by means of so called impedance and admittance
54
4.2
MODELING OF THE INK CHANNEL DYNAMICS
relationships. These relationships can be viewed as extension of the system’s
boundary to represent a part of the behavior of its surroundings. As a result,
the role and effect of the boundary conditions, the input and output impedances,
that are imposed to the system becomes clear. Considered at a more abstract
level, the concept of BCS forms the ideal combination of the main focus of a
systems and control approach (directing its attention towards the input-output
relations of a system, the boundary of a system) and an applied physics or fluidmechanics approach (that is more concerned with the structure itself of a system).
The successful application of this concept to a PIJ printhead is subject of the
current chapter. A PIJ printhead typically consists of an array of ink channels.
Here, one such an ink channel is considered as the system to be modeled. To obtain
a model of a complete printhead, several channel-models can be integrated to form
an array of ink channels. For the application of BCS, the overall boundaries of the
system under investigation are to be selected first. At the nozzle of an ink channel,
the exact place and time instant a drop splits from the ink present in the nozzle
represents the first boundary. The reservoir and the electrical circuitry form the
remaining boundary conditions for an ink channel. Second, having established
these boundaries, the system itself can be divided into several subsystems that
together make up a complete ink channel. In Fig. 4.2, a schematic overview
is given of the two-port model of the inkjet channel depicted in Fig. 1.7. This
partition into subsystems is based on the specific design of the inkjet channel under
consideration. To start with, the segment of the channel that is actuated by the
piezo-actuator is called the channel block. It differs only from the connection
block by the fact that the latter is not actuated. The reservoir forms the physical
boundary of an inkjet channel and also forms the boundary of the model. As all
these three blocks can be modeled using acoustics, they are referred to as acoustic
path. The following two blocks, those of the nozzle and droplet formation, are
modeled using the basic equations in fluid-mechanics. Together, they make up
the fluidic path. The last block is that of the piezo-actuator. In the various
upcoming subsections, each of these functional blocks will be discussed. Finally,
the coupling of these subsystems making up the complete two-port system is
treated in Section 4.3.
Piezo
Piezo
actuator
actuator
Reservoir
Channel
Connection
Nozzle
Drop
formation
Figure 4.2: A schematic overview of the two-port model of an inkjet channel
4.2
THE TWO-PORT MODEL
55
4.2.1 The acoustic path
The acoustic path consists of the channel, connection, and reservoir. The following
assumptions are done:
• To start with, it is assumed that for these blocks a one dimensional approach
can be used. This implies that only plane waves occur during operation of
an inkjet channel. Flow3D simulations confirm the validity of this approach.
• Second, it is assumed that there is no mean flow and that only small perturbations occur. This is a valid assumption, since the volume that is jetted
is so small that this is hardly noticeable as mean flow in the channel.
• The reservoir is assumed to act as open end. In practice, this is almost true.
• Finally, the dissipation also is assumed to be negligible. Dissipation by the
ink in the channel is determined by the oscillation frequency of the ink itself
as a result of the actuation, see [And84]. These oscillations are relatively
small compared to the velocity in the nozzle.
The application of BCS to the modeling of fluid transmission lines has been introduced in [Bro62] and [Bro65]. Friction can also be accounted for in this approach,
see [Bro69a] and [Bro69b]. A recent elaboration of BCS to the modeling of a fluid
transmission line without friction can be found in [Bos02]. In this section, the
approach is extended to account for the presence of an actuator. This translates
into adjusting the governing equations for a variable cross-section A(x, t).
The modeling of the channel is treated first. To that purpose, we start with the
conservation of mass and momentum for a channel with variable cross-section
A(x, t):
∂A(x, t)ρ(x, t) ∂A(x, t)ρ(x, t)v(x, t)
+
=0
∂t
∂x
∂A(x, t)ρ(x, t)v(x, t) ∂A(x, t)v 2 (x, t)ρ(x, t) ∂A(x, t)p(x, t)
+
+
=0
∂t
∂x
∂x
(4.2)
(4.3)
Here, v(x, t), p(x, t), A(x, t), and ρ(x, t) are the velocity, pressure, channel crosssection, and density, respectively. (4.2) can be written as:
∂A(x, t)ρ(x, t)
∂v(x, t)
∂A(x, t)ρ(x, t)
+ A(x, t)ρ(x, t)
+ v(x, t)
=0
∂t
∂x
∂x
and (4.3) as:
(4.4)
56
MODELING OF THE INK CHANNEL DYNAMICS
v(x, t)
4.2
∂A(x, t)ρ(x, t)
∂v(x, t)
∂v(x, t)
+ A(x, t)ρ(x, t)
+ 2A(x, t)ρ(x, t)v(x, t)
∂t
∂t
∂x
(4.5)
+ v 2 (x, t)
∂A(x, t)ρ(x, t) ∂A(x, t)p(x, t)
+
=0
∂x
∂x
Using the mass balance (4.4), (4.5) can be written as:
A(x, t)ρ(x, t)
∂v(x, t)
∂v(x, t)
∂A(x, t)p(x, t)
+ A(x, t)ρ(x, t)v(x, t)
+
= 0 (4.6)
∂t
∂x
∂x
If A(x, t)v(x, t) is replaced by the flow φ(x, t), (4.2) becomes:
∂A(x, t)ρ(x, t) ∂ρ(x, t)φ(x, t)
+
=0
∂t
∂x
Elaborating the partial derivatives in (4.7) leads to:
A(x, t)
∂ρ(x, t)
∂A(x, t)
∂φ(x, t)
∂ρ(x, t)
+ ρ(x, t)
+ ρ(x, t)
+ φ(x, t)
=0
∂t
∂t
∂x
∂x
(4.7)
(4.8)
Furthermore, it is assumed that the variations in density and pressure under
adiabatic conditions are related through:
∂ρ
1 ∂p
∂ρ
1 ∂p
dp = c2w →
= 2
and
= 2
(4.9)
dρ adiabatic
∂t
cw ∂t
∂x
cw ∂x
where cw is the wave propagation velocity. (4.8) can be written as:
A(x, t) ∂p(x, t)
∂A(x, t)
∂φ(x, t) φ(x, t) ∂p(x, t)
+ ρ(x, t)
+ ρ(x, t)
+
= 0 (4.10)
c2w
∂t
∂t
∂x
c2w
∂x
or equivalently as:
∂p(x, t) c2w ρ(x, t) ∂A(x, t) c2w ρ(x, t) ∂φ(x, t)
∂p(x, t)
+
+
+ v(x, t)
=0
∂t
A(x, t)
∂t
A(x, t)
∂x
∂x
(4.11)
For the elaboration of (4.6), we make use of the following relations:
∂v(x, t)
∂φ(x, t)
∂A(x, t)
=
− v(x, t)
∂t
∂t
∂t
∂v(x, t)
∂φ(x, t)
∂A(x, t)
A(x, t)
=
− v(x, t)
∂x
∂x
∂x
A(x, t)
(4.12)
4.2
THE TWO-PORT MODEL
57
Using the relations of (4.12), (4.6) can be written as:
∂φ(x, t)
∂A(x, t)
∂φ(x, t)
− v(x, t)ρ(x, t)
+ ρ(x, t)v(x, t)
∂t
∂t
∂x
∂A(x,
t)
∂A(x,
t)p(x,
t)
+
=0
− ρ(x, t)v(x, t)2
∂x
∂x
ρ(x, t)
(4.13)
Elaborating the partial derivatives of (4.13) results in:
∂A(x, t)
∂φ(x, t)
∂φ(x, t)
− v(x, t)ρ(x, t)
+ ρ(x, t)v(x, t)
∂t
∂t
∂x
∂A(x,
t)
∂p(x,
t)
∂A(x,
t)
− ρ(x, t)v(x, t)2
+ A(x, t)
+ p(x, t)
=0
∂x
∂x
∂x
ρ(x, t)
(4.14)
or equivalently:
∂φ(x, t)
∂A(x, t)
∂φ(x, t)
− v(x, t)
+ v(x, t)
∂t
∂t
∂x
p(x, t)
∂A(x,
t)
A(x,
t) ∂p(x, t)
+
− v(x, t)2
+
=0
ρ(x, t)
∂x
ρ(x, t) ∂x
(4.15)
Now, both equations (4.11) and (4.15) are linearized. Suppose that all variables
are derived by a constant plus a small perturbation:
φ(x, t) = φ0 + φ̃(x, t)
v(x, t) = v0 + ṽ(x, t)
A(x, t) = A0 + Ã(x, t)
(4.16)
ρ(x, t) = ρ0 + ρ̃(x, t)
p(x, t) = p0 + p̃(x, t)
Here, it is assumed that p0 , φ0 , and A0 are not a function of x. If (4.16) is substituted in (4.11) and (4.15) and the higher order terms are dropped, the linearized
equations are obtained. The tilde is omitted for denoting a perturbation, renaming v0 , A0 , and ρ0 in v, A, and ρ, respectively, the set of conservation laws can
be written in vector form as:
∂
∂t
"
−
p(x, t)
φ(x, t)
#
2
cw ρ
A
v
+
"
v
A
ρ
∂
A(x, t) +
∂t
c2w ρ
A
v
#
0
v2 −
∂
∂x
p
ρ
p(x, t)
φ(x, t)
∂
A(x, t)
∂x
=
(4.17)
58
4.2
MODELING OF THE INK CHANNEL DYNAMICS
"
v
Note that we assume v 6= 0 for now. The eigenvalues of matrix
A
ρ
c2
wρ
A
v
the values λ1,2 = v ± cw . Its corresponding right eigenvectors are:
cw ρ cw ρ − A
A
m2 =
m1 =
1
1
#
have
(4.18)
If we now define the following state transformation:
z1 (x, t)
z2 (x, t)
=
m1
m2
−1
p(x, t)
φ(x, t)
=
"
A
2cw ρ
− 2cAw ρ
1
2
1
2
#
p(x, t)
φ(x, t)
(4.19)
then (4.17) can be brought to the form:
∂
∂
z1 (x, t)
v + cw
0
z1 (x, t)
+
=
0
v − cw ∂x z2 (x, t)
∂t z2 (x, t)
#
"
v−cw ρv 2 −p
∂
∂
2ρ
2
A(x,
t)
+
A(x, t)
v+cw
ρv 2 −p
∂t
∂x
2
2ρ
(4.20)
Note that z1 (x, t) and z2 (x, t) have the physical dimension of flow. After application of the Laplace transform while assuming zero initial conditions and some
reshuffling we obtain:
∂
∂x
z1 (x, s)
z2 (x, s)
=
+
"
− cws+v
0
s
cw −v
−s(cw −v)
2(cw +v)
−s(cw +v)
2(cw −v)
#
0
z1 (x, s)
z2 (x, s)
" ρv2 −p
A(x, s) +
2ρ(cw +v)
p−ρv 2
2ρ(cw −v)
(4.21)
#
∂
A(x, s)
∂x
This renders the partial differential equation to an ordinary one that can be solved
straightforwardly. Prior to that, the forcing function A(x, s) is defined to be the
product of A(x) and A(s). A(x) represents the shape of the piezo-actuator when
actuated. It is assumed that the piezo creates a uniform cross-sectional variation
K over its complete length. The amplitude of this mode as well as the trajectory
in time is determined by A(s), though being Laplace transformed. The solution
to (4.21) can be computed straightforwardly. If A(x, s) is replaced by KA(s), the
first ordinary differential equation (ODE) of (4.21) reads as:
∂
s
−s(cw − v)
z1 (x, s) +
z1 (x, s) =
KA(s)
∂x
cw + v
2(cw + v)
(4.22)
Using the solution at x = 0, z1 (0, s), as boundary condition, the solution to (4.22)
at x = L can be written as:
4.2
THE TWO-PORT MODEL
−sL
z1 (L, s) = z1 (0, s)e cw +v −
59
−sL
KA(s)(cw − v) 1 − e cw +v
2
(4.23)
−sL
KA(s)(cw + v) 1 − e cw −v
2
(4.24)
A similar computation reveals the solution for the second ODE of (4.21):
−sL
z2 (0, s) = z2 (L, s)e cw −v +
The solution to (4.21) can be written in vector form as:
z1 (L, s)
z2 (0, s)
=
"
−sL
e cw +v
0
0
−sL
#
z1 (0, s)
z2 (L, s)

−sL
(4.25)
e cw −v

− (cw2−v) 1 − e cw +v
 KA(s)
+  (c +v) −sL
w
cw −v
1
−
e
2
Now, (4.25) represents a two-port system as depicted as in Fig. 4.3 (block 1).
Here, Lch , Lco , Aco , Sp represent the length of the channel, the length of the
connection, its cross-section, and the surface of the piezo bordering the channel,
respectively. Note that v = 0 since we assumed that there is no mean flow. As
can be seen, the solution admits a nice interpretation as travelling waves. To
obtain the original physical states p(x, s) and φ(x, s), the inverse transformation
of (4.19) can be applied to the states z1 (x, s) and z2 (x, s) (block 3).
For the connection, a similar approach can be used, except that the cross-section
remains constant and can be left out of the mass and momentum equations. The
solution is depicted in Fig. 4.3 (block 2).
The last subsystem of the acoustic path is the reservoir. For the waves that come
from the channel, the reservoir acts as open end, p(0, t) = 0, since the reservoir
contains a large amount of ink compared to the channel. Using (4.18) and (4.19),
this boundary condition can be written as:
p(0, t)
φ(0, t)
=
cw ρ
A
1
ρ
− cw
A
1
z1 (0, t)
z2 (0, t)
=
0
φ(0, t)
(4.26)
(4.26) can only be satisfied when z1 (0, t) = z2 (0, t). In Fig. 4.3, this behavior of
the reservoir is taken into account (block 4). In the actual system, regarding the
reservoir as open end is not completely true. The coupling between the channel
and the reservoir also takes place via a connection and so a more gradual transition
to an open end is obtained. The error made however is so small that this is allowed
without introducing a large error.
60
4.2
MODELING OF THE INK CHANNEL DYNAMICS
F
A(s)
Kcw
(1 − e
2
Sp
−sLch
cw
)
-
e
z1 (0, s)
1
−sLch
cw +v
2cw ρ
Aco
−sLco
+z
1 (L, s)
e cw +v
1
2
−1
3
4
+
−cw ρ
Aco
+
+
z2 (0, s)
p
+
+
e
−sLch
cw −v
z2 (L, s)
e
−sLco
cw −v
+
1
Figure 4.3: Block diagram of the acoustic path
4.2.2 The fluidic path: the nozzle
The fluidic path consists of the nozzle and droplet formation. In this section, the
fluid-mechanics in the nozzle are modeled. In light of the model requirements
posed in Section 4.1, various options for the modeling of the nozzle dynamics are
presented and discussed.
The starting point for the discussion forms the governing equation: the NavierStokes equation. In addition, the fluid dynamics in the nozzle can be considered
incompressible, as proven in [Wij04] and [Mar06]. The trade-off between accuracy and model complexity boils down to the number of dimensions considered
when solving the Navier-Stokes equation for the nozzle at hand. In this section,
four options are considered: two one- and two two-dimensional approaches. The
derivation of the first one-dimensional model comprises a simple elaboration of
Newton’s second law. The second one-dimensional approach is based on solving
the governing equations for a one-dimensional variable control volume, see e.g.
[Han67]. The application thereof to the specific nozzle at hand for a Poiseuille
flow profile can be found in [Hei98]. Here, [Hei98] is extended to allow for the
consideration of more complex flow profiles, based on the work of [Mar04]. The
first two-dimensional approach is based on the so called stream-function vorticity
method for solving the governing equations, see e.g. [Poz97]. This approach has
been introduced for modeling nozzle dynamics in [Fro84]. For the nozzle geometry under investigation, this has been elaborated in [Mar04] and [Mar06]. The
second two-dimensional model uses a CFD-package and is discussed in [Wij04].
φ
4.2
THE TWO-PORT MODEL
61
Both two-dimensional models will only be shortly discussed in this thesis. A threedimensional model is deliberately not considered here. Such a model is only useful
in the following cases. First, if the nozzle geometry is not axis-symmetric, e.g. in
case of a square or elliptic nozzle, a three-dimensional model forms an added value.
Second, in case the nozzle geometry can be regarded axis-symmetric, modeling
of the inclusion of air-bubbles or dirt particles, requires a full three-dimensional
model also. Note that even in the latter case, a two-dimensional model could
suffice as well provided that several assumptions are made.
After the introduction of these four nozzle models, the drop formation is discussed
in the next section. At the end of that section, all four models of the complete
fluidic path are critically evaluated. Based on this evaluation, a decision is made
regarding the nozzle model to be used in the sequel of this chapter.
A 1D nozzle model: an impedance
p
1
Z(s)
v
An
φ
5
Figure 4.4: Block diagram of the fluidic path
Our first approach to the modeling of the nozzle dynamics excels in its simplicity.
Apart from the fact that a one-dimensional approach is adopted, it is also assumed
that the nozzle is filled with ink at all times. This allows us to model the nozzle
as one fixed impedance. The starting point for the derivation is Newton’s second
law, which reads for the nozzle, stated in terms of p(s) and v(s), as follows:
p(s)An = ρAn Ln sv(s) + 8πµLn v(s)
(4.27)
with An , Ln , and µ being the nozzle’s cross-section, length, and viscosity, respectively. The viscous friction due to the pressure gradient across the nozzle is
accounted for in the second term, assuming a Poiseuille flow profile, see [Han67].
According to the definition, the nozzle impedance can be written as:
Z(s) =
ρLn An s + 8πµLn
p(s)
=
v(s)
An
(4.28)
62
4.2
MODELING OF THE INK CHANNEL DYNAMICS
To compute output φ(s) given the input p(s), we get:
φ(s) = An v(s) = An
p(s)
A2n
=
p(s)
Z(s)
ρLn An s + 8πµLn
(4.29)
In Fig. 4.4, the fluidic path is depicted (block 5).
Using the parameters listed in Table 3.1 and 3.2, the corresponding frequency
response of (4.29) can be obtained, see Fig. 4.5. As discussed in Section 3.2, the
nozzle acts as open end for low frequencies and switches to a closed end for higher
frequencies. This behavior corresponds to the frequency response of Fig. 4.5,
showing a corner frequency of the nozzle dynamics of approximately 100 kHz of
the first order system (4.29).
−270
Magnitude [dB]
−275
−280
−285
−290
−295
−300
−305
4
10
5
6
10
10
7
10
Frequency [Hz]
0
Phase [Deg.]
−20
−40
−60
−80
−100
4
10
5
10
6
10
7
10
Figure 4.5: Theoretical frequency response of the nozzle block from input p to
output φ
A 1D nozzle model: a deformable control-volume
The Navier-Stokes equation is solved for a deformable control volume representing
the nozzle. With this approach, it is assumed that the flow can be described using
one dimension only. Furthermore, the surface tension at the meniscus is neglected.
The conservation laws for mass and energy in integral form are invoked to describe
the flow inside the printhead’s nozzle. Using the integral form implies the use of
4.2
THE TWO-PORT MODEL
63
deformable control volumes. A deformable CV is a CV that may change in time,
which basically means that a certain volume V and its surrounding area A has
some or all boundaries moving at a certain velocity b(t). Suppose that the fluid
velocity itself is denoted by v(t), an observer fixed to the CV sees a relative
velocity v r (t) of the fluid crossing the control surface (CS) defined by:
v r (t) = v(t) − b(t)
(4.30)
Given the incompressibility of the nozzle flow, the conservation of mass of the
deformable CV can be written as:
Z
Z
∂
ρdV −
ρ(v r · n)dA = 0
(4.31)
∂t
CV
CS
Here, n represents the outward normal at the CS. (4.31) simply states that the
rate of change of mass within the CV equals the rate of flow of mass into the CV
minus the rate of flow out of the CV. The equation of conservation of energy can be
derived by forming the scalar product of the local velocity v(t) with the equation
of motion. If one assumes that the nozzle operates adiabatically and gravitational
forces are neglected, the equation of mechanical energy of a deformable CV can
be written as (see [Han67]):
∂
∂t
Z
CV
1
ρ|v|2 dV +
2
Z
CS
1
ρ|v|2 (v r ·n)dA = −
2
Z
p(n·v)dA+
CS
Z
v ·(∇·σ)dV (4.32)
CV
Here, ρ is the density, p the pressure, and σ the viscous stress tensor. The first
term of (4.32) at the left indicates the change in kinetic energy of the control volume. The second term represents the fluxes of kinetic energy in and out through
the (moving) boundaries. The first term on the right side of (4.32) stands for
the power of the surface forces, in this case pressure. The second term on the
right accounts for the shear work due to viscous stresses. (4.32) is a scalar equation of energy and uses the three components of the velocity. Using cylindrical
coordinates the velocity equals:
v(t) = vz (r, θ, z, t)
(4.33)
In Fig. 4.6, the nomenclature and the geometry of the nozzle is depicted. To
start with, the velocity profile is regarded 2D axis-symmetric and (4.33) can be
simplified to vz (r, z, t). Only in case an air-bubble or dirt particles are present,
validity of this assumption is questionable. Note that even in these cases the
assumption of axis-symmetry can be used without introducing major inaccuracies.
Furthermore, instead of considering vz (r, z, t), an average velocity vz,av (z, t) over
the cross-section is assumed in the one-dimensional approach. Using (4.31), this
means that the balance equations can be written in terms of volume flow φ(t)
only. The volume flow is:
64
4.2
MODELING OF THE INK CHANNEL DYNAMICS
r
z
R(z)
p(z0 , t)
φ(t)
zk (t) Ln
z0
Figure 4.6: Geometry of the nozzle
φ(t) = vz,av (z, t)A(z) = vz,av (z0 , t)A(z0 ) = vz,av (zk , t)A(zk )
(4.34)
The average velocity vz,av depends on the actual occurring velocity profile and
can be computed according to:
1
vz,av (z, t) =
A(z)
Z
vz (r, z, t)dA
(4.35)
However, by considering only an average velocity vz,av (z, t) the computations of
energies over the cross-section are in error. To compensate for the error, correction factors are used: the kinetic energy and the momentum-flux correction
factor, α and β, respectively. These factors greatly depend on the actual flowprofile occurring in the nozzle.
For the one-dimensional case, (4.32) can be written as:
∂
∂t
Z
CV
1 2
ρv dV +
2 z
Z
1 2
ρv (vz,rel ·ez )dA = −
2 z
CS
Z
p(ez ·vz )dA+
CS
Z
v·(∇·σ)dV (4.36)
CV
using the fact that the velocity is only in one direction (normal to the surface).
The first term of (4.36) can be rewritten using the fact that the nozzle is axissymmetric:
∂
∂t
Z
CV
1 2
∂
ρvz dV =
2
∂t
Z
CV
1
∂
ρvz (r, z, t)2 dV = 2π
2
∂t
Z
zk (t)
z0
Z
0
R(z)
1
ρvz (r, z, t)2 rdrdz
2
(4.37)
4.2
THE TWO-PORT MODEL
65
To further simplify (4.37) from the two-dimensional to the one-dimensional case,
(4.37) is written in terms of the average velocity and an additional correction
factor. The following equation then must hold:
2π
Z
R(z)
vz (r, z, t)2 rdr = βvz,av (z, t)2 πR(z)2
(4.38)
0
since:
2π
Z
R(z)
2
2
vz,av (z, t) rdr = 2πvz,av (z, t)
0
1 2
r
2
R(z)
= vz,av (z, t)2 πR(z)2
(4.39)
0
Stated alternatively, (4.38) states that the flux in kinetic energy of a slice dz of the
CV computed using the actual flow-profile vz (r, z, t) must equal the outcome when
using the average velocity vz,av (z, t). Consequently, correction factor β equals:
β(z, t) =
2
vz,av (z, t)2 R(z)2
Z
R(z)
vz (r, z, t)2 rdr
(4.40)
0
β is known as the momentum-flux correction factor and is dependent on the actual occurring flow-profile vz (r, z, t).
Similar to these computations, the kinetic correction factor α can be computed.
To that purpose, the second term of (4.36) is written as:
Z
1 2
ρv (vz,rel · ez )dA = 2π
2 z
CS
Z
0
R(z)
1
ρvz (r, z, t)2 (vz,rel (r, z, t) · ez )rdr
2
(4.41)
If (4.41) is written in terms of the average velocity and the kinetic correction
factor:
2π
Z
R(z)
vz (r, z, t)3 rdr = αvz,av (z, t)3 πR(z)2
(4.42)
0
(4.42) simply states that the kinetic energy taken at a point over the cross-section
computed using the actual flow-profile must equal that computed with an average
velocity vz,av (z, t).
The kinetic correction factor α equals:
2
α(z, t) =
vz,av (z, t)3 R(z)2
Z
R(z)
vz (r, z, t)3 rdr
0
Now, using the correction factors α and β, (4.32) can be written as:
(4.43)
66
∂
∂t
4.2
MODELING OF THE INK CHANNEL DYNAMICS
Z
zk (t)
z0
1
1
ρβ(z, t)vz,av (z, t)2 πR(z)2 dz + ρ α(z, t)vz,av (z, t)2 (vz,av,rel (z, t) · ez )πR(z)2
2
2
Z
=−
k (t)
z0
(4.44)
Z
p(ez · vz )dA +
CS
z
v · (∇ · σ)dV
CV
For further elaboration of (4.44), an assumption regarding the occurring flow
profile is required. Given the pulsating nature of the traveling pressure waves
within an ink channel, a Womersley velocity profile seems a logical choice, see
[Hal55]. A characterization of pulsating flow is provided by the Womersley number
Wo. Wo is defined as:
R
Wo =
2
r
ω
ν
(4.45)
with R the characteristic diameter of a tube, ω the pulsating frequency of the flow,
and ν the kinematic viscosity of the fluid. However, actuation of an ink channel
results in pressure fluctuations that are quite highly irregularly rather than purely
sinusoidally. For that reason, a Womersley profile may not be suitable. On the
other hand, the conditions for a Poiseuille profile are not fulfilled either. In
Section 4.2.4, it is shown that the accuracy of the nozzle model derived here is
not influenced by adopting a Womersley profile instead of a Poiseuille profile.
Therefore, for the further elaboration of (4.44) a Poiseuille profile is adopted.
After all, this greatly reduces the complexity of the resulting nozzle model. A
Poiseuille flow profile can be described as:
vz (r, z) =
−
∂p R(z)2
r2
1−
∂z
4µ
R(z)2
(4.46)
where p is the driving pressure drop and µ the dynamic fluid viscosity. Note that
this profile is not dependent on time t. According to (4.35), the average velocity
vz,av (z) equals:
vz,av (z) =
2π
πR(z)2
Z
R(z)
0
vz (r, z)rdr = −
∂p R(z)2
∂z 8µ
(4.47)
The momentum-flux correction factor is computed as:
β(z) =
8
R(z)2
Z
0
R(z) 1−
r2
R(z)2
2
rdr =
and the kinetic correction factor as:
R(z)
8
1 2 1 r4
1 r6
4
r
−
+
=
R(z)2 2
2 R(z)2 6 R(z)4 0
3
(4.48)
4.2
THE TWO-PORT MODEL
3
Z R(z) 16π
r2
α(z) =
1−
rdr
πR(z)2 0
R(z)2
R(z)
16
1 2 3 r4
1 r6
1 r8
=
r
−
+
−
=2
R(z)2 2
4 R(z)2
2 R(z)4
8 R(z)6 0
67
(4.49)
Given both correction terms, (4.44) will be now elaborated termwise. The subscript av is omitted.
1. Since
vz (z, t) =
φ(t)
φ(t)
=
A(z)
πR(z)2
(4.50)
the first term at the left side can be written as, using the product rule of
differentiation:
∂
∂t
Z
Z zk (t)
1
∂
2 φ(t)2
ρβ(z, t)vz (z, t)2 πR(z)2 dz =
ρ
dz
2
∂t z0
3 A(z)
z0
Z
Z zk (t)
4
∂φ(t) zk (t) 1
2
1
2 ∂
= ρφ(t)
dz + ρφ(t)
dz
3
∂t
A(z)
3
∂t
A(z)
z0
z0
zk (t)
(4.51)
2. The one-dimensional control volume has two areas with a flux, at z0 and at
zk (t). Therefore:
zk (t)
1
2
2
ρ α(z, t)vz (z, t) (vz,rel (z, t) · ez )πR(z)
=
(4.52)
2
z0
ρvz (z0 , t)2 (vz (z0 , t) − b(z0 , t)) · −1 A(z0 )
+ ρvz (zk , t)2 (vz (zk , t) − b(zk , t)) · 1 A(zk )
where b(z, t) represents the velocity of the boundary. The boundary at z0
is not moving, the boundary at zk is moving and equals the velocity of the
meniscus vz (zk , t) and the last term drops out of the equation. The equation
can be simplified to:
68
4.2
MODELING OF THE INK CHANNEL DYNAMICS
zk (t)
1
2
2
ρ α(z, t)vz (z, t) (vz,rel (z, t) · ez )πR(z)
2
z0
= ρvz (z0 , t)2 (vz (z0 , t) − 0) · −1 A(z0 )
= −ρvz (z0 , t)3 A(z0 ) = −ρ
(4.53)
φ(t)3
A(z)2
3. The first term on the right also only has to be evaluated at the two areas
at the boundaries.
−
Z
(p(ez · vz ))dA =
(4.54)
CS
− p(z0 , t)vz (z0 , t) · (−1)A(z0 ) − p(zk , t)vz (zk , t) · (1)A(zk )
If the pressure jump through the meniscus is neglected, the pressure at the
nozzle exit equals zero and the above equation reduces to:
−
Z
(p(n · vz ))dA = p(z0 , t)vz (z0 , t)A(z0 ) = p(z0 , t)φ(t)
(4.55)
CS
with p(z0 , t) the pressure at the nozzle entrance.
4. The fourth term of the energy equation is:
Z
v · (∇ · σ)dV
(4.56)
CV
It is assumed that the flow in the nozzle is Newtonian. Since the flow is
incompressible also, the viscous stress tensor σ in cylindrical coordinates
equals (see [Byr60]):

σrr
σ =  σθr
σzr
with:
σrθ
σθθ
σzθ

σrz
σθz 
σzz
(4.57)
4.2
THE TWO-PORT MODEL
∂vr
∂r
1 ∂vθ
vr +
= −2µ
r ∂θ
r
∂vz
= −2µ
∂z ∂ vθ
1 ∂vr
= −µ r
+
∂r r
r ∂θ
1 ∂vz
∂vθ
+
= −µ
∂z
r ∂θ
∂vz
∂vr
= −µ
+
∂r
∂z
69
σrr = −2µ
(4.58)
σθθ
(4.59)
σzz
σrθ = σθr
σθz = σzθ
σrz = σzr
(4.60)
(4.61)
(4.62)
(4.63)
To elaborate (4.56), we start by obtaining an expression for a Poiseuille
flow profile (4.46) in terms of flow φ(t) rather than pressure drop p. To that
purpose, the flow φ(t) of Poiseuille flow is computed by integrating the flow
profile over the surface:
φ(t) =
Z
0
R(z)
vz (r, z) 2πr dr = −
πR(z)4 ∂p
8µ ∂z
(4.64)
To write (4.46) in terms of the volume flow, the pressure gradient is expressed in terms of the volume flow:
∂p
8µφ(t)
=−
∂z
πR(z)4
(4.65)
(4.46) can then be written as:
R(z)2 8µφ(t)
r2
2φ(t)
r2
vz (r, z) =
1−
=
1−
4µ πR(z)4
R(z)2
πR(z)2
R(z)2
(4.66)
Recall that the velocity profile is assumed to be axis-symmetric. Hence, the
velocity in the θ direction as well as all derivatives with respect to θ are zero.
Furthermore, the velocity in the r-direction is assumed zero due to the one
dimensionality of the approach. Given these simplifications, the divergence
of the viscous stress tensor (4.57) can be given as:

0
∇·
0
z
−µ ∂v
∂r
0
0
0
 

∂ 2 vz
z
−µ ∂v
−µ ∂r∂z
dr
=

0
0
2
∂vz
∂v
∂
v
1
∂
−2µ ∂z
−µ r ∂r r ∂rz − 2µ ∂ 2 zz
(4.67)
70
4.2
MODELING OF THE INK CHANNEL DYNAMICS
Now, (4.56) can be written as:
Z
3
2
vz
0
−µ ∂∂r∂z
6
7
4
5
0
·4
0
5 dV
2
∂v
∂
v
∂
1
z
z
vz (r, z)
−µ r ∂r r ∂r − 2µ ∂ 2 z
v · (∇ · σ)dV =
CV
3 2
2
Z
CV
Z
vz (r, z) ∂
r
∂r
= −µ
r
∂vz
∂r
Z
dV − 2µ
CV
vz (r, z)
(4.68)
∂ 2 vz
dV
∂2z
CV
The first term in (4.68) can be elaborated using (4.66) as:
Z
−µ
vz (r, z) ∂
r
∂r
r
∂vz
∂r
Z
16φ(t)2
π 2 R(z)6
dV = µ
CV
CV
Z
zk (t)
=µ
z0
Z
zk (t)
=µ
z0
r2
R(z)2
1−
16φ(t)2
π 2 R(z)6
16φ(t)2
πR(z)4
1−
1−
dV
r2
R(z)2
r2
R(z)2
(4.69)
A(z)dz
dz
The second term in (4.68) cannot be elaborated further if the nozzle geometry R(z) is not known:
2µ
Z
∂ 2 vz
vz (r, z) 2 dV = 2µ
∂ z
2
2φ(t)
r2
∂ vz
1−
dV
πR(z)2
R(z)2 ∂ 2 z
Z
CV
CV
= 2µ
Z
zk (t)
z0
2φ(t) 1 −
r2
R(z)2
(4.70)
∂ 2 vz
dz
∂2z
Taking all four terms together, the energy equation equals in terms of flow φ(t):
4
∂φ(t)
ρφ(t)
3
∂t
Z
zk (t)
+µ
z0
Z
zk (t)
z0
16φ(t)2
πR(z)4
1
2
∂
dz + ρφ(t)2
A(z)
3
∂t
1−
r2
R(z)2
Dividing by φ(t) results in:
Z
zk (t)
z0
Z
zk (t)
dz − 2µ
1
φ(t)3
dz − ρ
= p(z0 , t)φ(t)
A(z)
A(z)2
(4.71)
2φ(t) 1 −
z0
r2
R(z)2
∂ 2 vz
dz
∂2z
4.2
THE TWO-PORT MODEL
4 ∂φ(t)
ρ
3 ∂t
Z
Z
zk (t)
z0
zk (t)
+µ
z0
2
∂
1
dz + ρφ(t)
A(z)
3
∂t
16φ(t)
πR(z)4
1−
r2
R(z)2
Z
zk (t)
z0
Z
φ(t)2
1
dz − ρ
= p(z0 , t)
A(z)
A(z)2
zk (t) dz − 4µ
1−
z0
r2
R(z)2
71
(4.72)
∂ 2 vz
dz
∂2z
Using a standard ODE solver of Matlab, equation (4.72) can be solved straightforwardly. The discussion of the resulting model is postponed until after the
discussion of the drop formation in Section 4.2.3.
2D modeling approaches: stream-function vorticity and CFD
Up to this point, two nozzle models have been introduced. Since the evaluation of
their accuracy is postponed until after the discussion regarding the drop-formation
in Section 4.2.3, the resulting accuracy cannot give rise to the exploration of somewhat more complex models yet. However, one specific property of both models
can be studied: their one-dimensional nature. Based on our interest in smaller
droplets, the meniscus shape in two dimensions becomes of importance. Therefore, two two-dimensional nozzle models are investigated.
For these 2D approaches, it is assumed that the nozzle can be regarded as axissymmetric. The implications and limitations of this assumption is already discussed in the introduction of this section. The first 2D approach is based on
solving the Navier-Stokes equation using stream-function vorticity method, see
[Poz97]. Rather than using the velocities and pressure as variables when solving
the governing equations, the stream-function and vorticity is used. Consequently,
the number of variables in the 2D problem can be reduced: from three (two velocity components and pressure term) to two (stream-function and vorticity). Furthermore, the surface tension is neglected. The application of the stream-function
vorticity was first proposed in [Fro84]. A detailed derivation of this approach to
the nozzle geometry at hand can be found in [Mar04] and [Mar06].
The second two-dimensional approach is based on the CFD package Flow3D.
Using the axis-symmetry of the nozzle, a 2D model is constructed. In contrast
to the previous three nozzle models, the surface tension is accounted for. Also,
drop formation is directly incorporated in the computations. Details with regard
to this model can be found in [Wij04].
4.2.3 The fluidic path: drop formation
Drop formation is a highly complex phenomenon. As a result, fulfilling the model
requirements as posed in Section 4.1 is far from trivial. With the exception of
[Dij84], the complexity of the models of Table 4.1 is too high. Therefore, the
72
MODELING OF THE INK CHANNEL DYNAMICS
4.2
approach to the modeling of the drop formation of [Dij84] is adopted in this thesis. In [Mar06], this approach is improved by incorporating the effect of friction.
The principles and derivation of the accompanying equations are presented in
this section, heavily based on [Mar06]. The suitability of an improved version of
[Dij84] for our PIJ printhead model is based on the following two key characteristics. First and foremost, by using an energy balance only to determine the course
of the drop formation process and the resulting drop properties such as velocity
and volume, computations are kept as simple as possible. Furthermore, rather
than maintaining a two-sided coupling with the fluid mechanics in the nozzle, a
one-sided coupling is adopted. Basically, the drop formation is computed as postprocessing step for all nozzle models except the Flow3D model. This latter model
namely already incorporates drop formation. Validation of this simplification will
be provided in the sequel of this section.
Figure 4.7: A typical sequence of drop formation as computed in Flow3D depicted
starting from t=18 µs in increments of 4 µs ([Mar06])
The drop formation model to be elaborated targets at the following. To start
with, it is to predict whether and if so, at what time instant a drop is formed.
Also, the resulting drop velocity and volume are to be predicted. Four stages can
be distinguished in the drop formation process:
1. Start up. During this first stage, a negative pressure wave hits the connectionnozzle interface, causing the free surface to be sucked into the nozzle. This
pressure course is required for the build-up of sufficient energy for the second
stage, see Fig. 1.7.
2. Drop initiation. At this stage, the pressure at the connection-nozzle interface becomes positive and the free surface is being pushed out of the nozzle,
see Fig. 4.7.
3. Thinning of the tail. The third stage starts when the pressure decreases
again. The free surface (called ligament from this point) has obtained
4.2
THE TWO-PORT MODEL
73
enough velocity and inertia to overcome the surface tension and does not
reverse its direction. This will in turn cause the ligament to become thinner.
4. Viscous loss in tail resorption. In the last stage, the ligament is broken and
a drop is created, traveling with a certain velocity and volume. The fluid in
the ink channel still oscillates slightly as discussed in Section 1.2.2. These
residual vibrations are damped out by viscous dissipation. In general, these
motions are too small to result in an additional drop.
h(r, t)
CV
z
r
Figure 4.8: Definition of the height of the free surface and the control volume
used in the drop formation model
These four steps are discussed in this section. The input for the drop formation
model forms the meniscus velocity vzk (r, t). Both the one- and two-dimensional
models can provide this input:
• 1D control volume model. The resulting average meniscus velocity vzk ,av (t)
is transformed back to a Poiseuille profile vzk (r, t). Note that vr (z, t) is zero
at all times due the one-dimensional character of the model.
• 2D stream-function vorticity. The two-dimensional stream-function vorticity model already outputs the required vzk (r, t). Furthermore, vr (z, t) at and
beyond the nozzle outlet is assumed null during the upcoming derivation.
As a consequence, since beyond the outlet the ink is supposed to form a cylinder,
mass conservation implies that ∂vz /∂z is zero beyond the nozzle outlet. From
this point, vzk (r, t) is written as vz . Based on these assumptions, the free surface
boundary can be written as:
74
4.2
MODELING OF THE INK CHANNEL DYNAMICS
∂h
= vz|h = vz
∂t
∂vz
1
1
p|h = µ
+σ
+
∂r
R1
R2
(4.73)
(4.74)
where σ is the surface tension, R1 and R2 the principal radii of the free surface,
and h the height of the free surface as depicted in Fig. 4.8. ph is the pressure
just under the free surface. The height of the free surface is a function of r and t.
(4.73) implies that the shape of the free surface can be found by time integration
of the velocity at the nozzle outlet. Now, a control volume (CV) is defined whose
boundaries are the surface of the nozzle outlet and the mentioned free surface,
see Fig. 4.8. The equation for the mechanical energy of this deformable control
volume reads as:
∂
∂t
Z
1 2
ρv dV +
2 z
CV
1 2
ρv (vz,rel · ez )dA
2 z
CS
Z
=−
Z
Z p(ez · vz )dA − µ
CS
(4.75)
vz ∂
∂vz
r
r ∂r
∂r
dV − σ
∂S
∂t
CV
where A is the closed surface around the CV. vz,rel is the velocity relative to the
CS. vz,rel is assumed to be equal to vz at the nozzle boundary and zero on the
meniscus that coincides with the free surface. n is the outward normal on A. The
physical interpretation of the various terms of (4.75) is similar to that of (4.32).
The additional term of (4.75) represents the enlargement of the free surface S.
Since the meniscus is a part of the CV, this term must be included in the energy
balance. The enlargement S equals:
Z
Rn
S = 2π
s
1+
0
∂h
∂r
2
rdr − πRn2
(4.76)
where Rn is the radius of the nozzle outlet. Hence, it is assumed that the reference
free surface at t = 0 equals the nozzle outlet surface.
To derive the condition for drop formation and the associated one for the drop
initiation, (4.75) is integrated with respect to time from zero to a certain time
instant τ :
Z
Rn
2π
0
Z
τ
1
ρhvz2 |t=τ rdr =
2
Z
2π
0
0
Rn
1 3
ρv rdr +
2 z
(4.77)
Z
Z
Rn
vz p|outlet rdr − µ
0
Rn
h
0
vz ∂
∂vz
r
r ∂r
∂r
rdr dt − σS(τ )
4.2
THE TWO-PORT MODEL
75
Here, the fact is used that the relative pressure just outside the free surface is
zero. At τ = 0, the CV is assumed to be zero since the free surface is equal
to the nozzle outlet. After τ = 0, kinetic and pressure energy is flowing in the
CV. Apart from the energy dissipation due to the viscous effects, the energy is
stored in the growing CV as kinetic and surface tension energy. The growth rate
of the height of the CV is equal to vz at the nozzle outlet. At a certain point in
time, the nozzle outlet velocity will again decrease. The velocity vz in the CV
will also decrease as long as the meniscus surface tension is capable to decelerate
the CV. If the speed reduction at the nozzle outlet is too strong the CV cannot
be slowed down to the outlet velocity by the surface tension in the meniscus. As
a result, the CV pushed out of the nozzle will thin out. This time instant t1 can
be determined as follows.
Suppose that at τ = t1 the kinetic energy accumulated in the CV (left hand side
of (4.77)) is exactly equal to the kinetic energy transported in the CV through
the nozzle minus the loss in viscous effects and in building the free surface energy.
In other words, time t1 is reached when:
Z
Rn
1
ρhvz2 |t=t1 rdr
(4.78)
2
0
(
)
Z t1 Z Rn
Z Rn 1 3
vz ∂
∂vz
= 2π
ρv rdr − µ
h
r
rdr dt − σS(t1 )
2 z
r ∂t
∂r
0
0
0
π
Comparing (4.77) and (4.78) it is observed that the instant t1 corresponds to the
moment when the energy given to the CV through the pressure forces at the nozzle
outlet is zero. After t1 the pressure energy flowing in the CV should become negative in order to satisfy the energy balance. In the actual drop formation process,
t1 for which (4.78) is satisfied corresponds to the instant when the outward velocity vz decreases but still is positive. Therefore, in order to have negative energy
from the pressure component, the pressure should become negative. Physically, it
means that at time t1 the nozzle outlet velocity, although still positive, has been
decreasing so much that the fluid outside cannot be decelerated fast enough by
internal dissipation and surface tension, and that a negative pressure would be
required to force the drop velocity to follow the reducing outlet velocity. This is
not possible and therefore t1 is the time instant when thinning takes place and the
drop formation starts, see Fig. 4.7. The instant t1 can be determined by monitoring the left and right hand side of (4.78) where vz is known from the flow model
in the nozzle, and S and h are computed from (4.73) and (4.76). In Fig. 4.9,
the condition is depicted graphically. Here, Ttransported represents the amount of
kinetic energy transported through the CS, Tsurface the surface tension energy,
Tviscous the viscous energy, Tnet the nett energy (Ttransported − Tsurface − Tviscous ),
76
4.2
MODELING OF THE INK CHANNEL DYNAMICS
and Tinstantaneous the instantaneous kinetic energy present in the CV.
−10
10
x 10
T
transported
T
+T
surface
viscous
T
net
Tinstantaneous
8
equilibrium
6
Energy [J]
4
2
0
−2
−4
−6
−8
0
2
4
6
8
10
Time [µs]
12
14
16
18
20
Figure 4.9: Graphical representation of (4.75) and (4.78): the various energy
terms involved in the drop formation
Computing precisely the thinning of the tail and the full drop formation is
very complex, not to mention computationally intensive. Again, a simple global
balance is used to estimate the resulting drop velocity and volume at the end of
the drop formation process. From an energy perspective, it can be assumed that
the kinetic energy of the CV is mostly converted into energy associated with the
drop (kinetic and surface tension). Part of the CV is returning to the nozzle.
Energy lost in viscous dissipation during the thinning of the droplet tail will be
considered as a correction later.
The instant t2 chosen for considering the drop as formed is taken as the moment
just before the droplet hits the paper. The drop creation is simply being modeled
as the creation of a new free surface which, considering the relatively long distance
the drop has to travel before hitting the paper, can safely be assumed to be
spherical. Simulations done with Flow3D validates this assumption, see [Wij04].
The energy balance between t1 and t2 can be expressed by:
4.2
THE TWO-PORT MODEL
2π
Z
0
Rn
1
1
ρ hvz2 |t=t1 rdr = E(t2 ) + ρVd vd2 + 4πσrd2
2
2
77
(4.79)
where Vd = 4πrd3 /3 is the final drop volume, rd is the drop radius, and E(t2 )
represents the energy of the part of the CV that was ejected out of the nozzle but
will return to the nozzle once the droplet breaks loose. This part usually is taken
as a certain percentage of the total volume pushed out of the nozzle. Typically,
around 70 % to 90 % of the ejected volume is assumed to be transformed in the
drop. Using (4.79), the drop velocity then can be computed as:
v
"
u
2/3 #
u 2π Z Rn
2
3
t
2
vd =
(4.80)
(hvz ) |t=t1 rdr −
E(t2 ) + 4πσ
Vd
Vd 0
ρVd
4π
The energy E(t2 ) remaining in the residual volume is composed of its kinetic energy and of the surface tension energy of its free surface generated after the drop
has separated from the main flow. The velocity of the residual volume is very
small for a realistic actuation. Indeed to form stable and repeatable drops the
actuation of the printhead has to be such that the nozzle flow returns quickly to
a rest once the drop is ejected so that the next actuation cycle can be started.
We can therefore neglect the kinetic energy of the residual volume. The surface
tension contribution to E(t2 ) can be estimated using (4.76) and assuming that
the free surface shape can be approximated by a quadratic function h(r), the free
surface height at the nozzle edges being zero and the total volume between the
free surface and the nozzle boundary being set to the residual volume.
When 100 % of the flushed volume is transformed in Vd , the free surface of the
residual volume is equal to the outlet surface and E(t2 ) = 0. If Vd is taken as 90 %
of the flushed volume, the surface tension energy of the residual volume was found
to be of the order of 1 % of the energy of the CV, see [Mar06]. Consequently, the
energy E(t2 ) can be neglected for the model.
One effect that was not accounted for in the energy balance (4.80) is the viscous dissipation related to the transformation of the fluid cylinder assumed at
τ = t1 to a spherical drop at τ = t2 . From Fig. 4.7 it is observed that shortly
after the drop formation process has begun most of the fluid volume is concentrated in the tip. The resorption of the trailing tail takes more time and generates
non-negligible deceleration of the drop. The velocity from (4.80) is obtained by
comparing the amount of energy at the point in time the drop creation just started
(t1 ) and at a point in time far from the point of drop break-up (t2 ). To include the
effect of elongation viscosity, a method similar to what was originally proposed
in [Dij84] is used. In this simplified drop model it is assumed that just after time
t = t1 the drop volume is a cylinder with the volume Vd . The radius R and length
l of the cylinder are changing between times t1 and t2 but satisfy πR(t)2 l(t) = Vd .
78
4.2
MODELING OF THE INK CHANNEL DYNAMICS
Initially at time t1 the radius is equal to the outlet radius of the nozzle Rn .
An increase of l(t) results in a decrease of R(t). The interface of the stretching
jet and the fluid inside the nozzle moves with a velocity equals to vz . Assuming
that the mass of the jet is concentrated in the tip of the jet and that the velocity
of the tip is equal to the drop velocity, the dynamic equilibrium between inertia
forces and viscous elongation forces yields:
ρVd
∂vd
vd − vz
= −3µ
πR(t)2
∂t
l(t)
t 1 < t < t2
(4.81)
The elongation rate has been assumed uniform over the cylinder and the elongations viscosity has been taken as the Trouton viscosity which is three times the
Newtonian viscosity µ. Since Vd = πR(t)2 l(t) is known and constant, (4.81) can
be written as:
ρ
∂vd
vd − vz
= −3µ
∂t
l(t)2
t 1 < t < t2
(4.82)
The change in length l(t) is related to the velocity difference between the cylinder
ends and equals:
dl(t)
= vd (t) − vz
dt
Substituting this relation in (4.82) yields:
ρ
∂vd
dl(t) 1
= −3µ
∂t
dt l(t)2
t 1 < t < t2
and integrating this relation between t1 and t2 results in:
3µ
1
1
−
vd (t2 ) − vd (t1 ) =
ρ l(t2 ) l(t1 )
(4.83)
(4.84)
(4.85)
Finally observing that l(t1 ) is significantly smaller than l(t2 ) we obtain:
vd (t2 ) − vd (t1 ) ≃ −
3µ 1
ρ l(t1 )
(4.86)
The velocity correction (4.86) must be added to the velocity obtained with (4.80)
to obtain the final velocity of the drop.
4.2.4 The fluidic path: a review
As discussed in Section 4.2.2, the nozzle block and the drop formation are linked
by a one-sided coupling only, except for the Flow3D model. To determine the
suitability of the various nozzle models in combination with the drop formation
model for use within the two-port model, they are to be evaluated with respect
4.2
THE TWO-PORT MODEL
79
to accuracy and model complexity. For the accuracy, the following procedure
is adopted. At this point, it is assumed that the Flow3D model is the most
accurate model of the complete fluidic path. This is confirmed by numerous experiments, see [Wij04]. The remaining three nozzle models in combination with
the drop formation model (hereafter referred to as fluidic path models) are therefore benchmarked against the Flow3D model.
To facilitate this, the response to a standard trapezoidal actuation pulse of the
various fluidic path models is computed. Given the nonlinear character of the
jetting process, comparison of any other property (such as a Bode diagram) is not
representative for a benchmark of these models. The simulation is programmed
as follows. The input to the various models is taken as the pressure history at
the nozzle entrance during the actuation with a standard trapezoidal pulse. This
history is obtained by tracing the response of the Flow3D model to this pulse at
the appropriate location. It provides a realistic pressure input for the benchmark
with the two-sided coupling being accounted for. Also, during the simulation, the
condition for the jetting of a drop (4.78) is monitored. If the criterion (4.78) is
fulfilled, the resulting drop speed and volume are determined. Also, the states
of the simulation are re-initialized and the simulation of the nozzle response is
continued. This way, the drop-formation is also accounted for, albeit on a very
rudimentary level. For example, the jetting of a droplet does not take place instantaneously, but more over a certain time span.
The described simulation procedure can only be applied if all fluidic models are
available in a time domain setting. Since the CV and stream-function vorticity (SV) models are formulated in the time domain, and the response of the 1D
impedance model can be easily obtained in the time domain, this is not a problem.
Prior to presenting the results of this simulation, the effect of the flow profile used
in the 1D CV approach is investigated. To that purpose, the response of (4.72) to
a standard trapezoidal actuation pulse is computed for a Poiseuille and a Womersley flow profile, see Fig 4.10. For the computation of the Womersley number
according to (4.45), the dominating frequency of the pressure input trajectory
has been used. Not surprisingly, this frequency corresponds to the channel’s first
eigenfrequency. As can be seen in Fig. 4.10, differences are small. Therefore, the
use of a Poiseuille rather than a Womersley flow profile is justified, simplifying
the computations considerably.
In Fig. 4.11, all four fluidic model responses are depicted. However, the response
of the Flow3D model is only shown for two time-intervals. Tracing the meniscus
in Flow3D equals tracing the tip of the drop that is being formed. Only after
the tail of the drop has been completely detached from the ink in the nozzle, the
’true’ meniscus position of the nozzle can be traced again. The accompanying
resulting drop speed and volumes are listed in Table 4.2, as well as an indication
80
4.2
MODELING OF THE INK CHANNEL DYNAMICS
30
20
Meniscus [µm]
10
0
−10
−20
−30
−40
0
10
20
30
40
50
Time [µs]
60
70
80
90
100
Figure 4.10: Meniscus response to an standard actuation pulse; using a Poiseuille
profile (gray) or a Womersley profile (Wo=18, black) within the 1D CV approach
1D
2D
impedance
CV
SV
Flow3D
vd (m/s)
Vd (pL)
5.13
10.29
4.07
4.23
16.09
13.67
15.38
15.30
model
accuracy
model
complexity
+
+++
−
−
++
−−
+++
−−−
Table 4.2: Evaluation of the fluidic path models
jetting %
100 %
75 %
95 %
-
4.2
THE TWO-PORT MODEL
81
of the relative model accuracy and complexity. Also, the used percentage of ink
that is jetted away is listed for all models except the Flow3D model. In case of
the Flow3D model, this percentage is not one of the parameters to be specified a
priori, since the drop formation process is completely determined by the Flow3D
computations. Note that these percentages have been tuned to match the Flow3D
drop speed and volume.
30
1D CV
1D impedance
2D SV
Flow3D
20
10
Meniscus [µm]
0
−10
−20
−30
−40
−50
0
10
20
30
40
50
Time [µs]
60
70
80
90
100
Figure 4.11: Meniscus response of the various fluidic path models to a standard trapezoidal actuation pulse; the 1D impedance (black dashed), 1D CV (gray
dashed), 2D stream-function vorticity (gray), and 2D Flow3D approach (black)
The 1D impedance nozzle model is quite accurate with respect to the Flow3D
response. Both the drop speed and volume are predicted reasonably accurately.
Also, the meniscus position is predicted quite satisfactorily, except for the slight
mismatch in the time instant of jetting. The 1D CV approach, however, is far
less accurate: only the predicted drop volume approximates the Flow3D outcome.
The meniscus trajectory, including the time of jetting, deviates considerably from
the Flow3D computations. Finally, the 2D SV approach is the most accurate of
the three fluidic path models benchmarked against the Flow3D model. Its nonzero value of the meniscus at t = 100 µs is the result of the re-initialization after
82
4.2
MODELING OF THE INK CHANNEL DYNAMICS
having fulfilled the requirement for the jetting of a drop. The volume in the CV
in front of the nozzle outlet is jetted away. However, due to the 2D character, this
only is an approximation. Therefore, too ’few’ is jetted away resulting in an offset
at t = 100 µs. Regarding the model complexity, the 1D impedance model is the
most simple, followed by the 1D CV and 2D SV model respectively. An additional
advantage of the 1D impedance model is its formulation in the frequency domain,
which is favorable for its incorporation in the two-port model to be constructed.
Note that the model complexity of the 2D SV approach should be weighted against
its ability to compute the meniscus profile. In Fig. 4.12, the meniscus profile of
the 2D SV model is compared to that of the Flow3D model. As can be seen,
these match quite accurately. The occurring inaccuracies can be explained as
follows. To start with, the surface tension is not accounted for in the 2D SV
approach. Additionally, the true location of the free surface is always assumed to
be on the geometrical outlet according to the transpiration approach, see [Mar06].
As a result, the computations are slightly in error. Overall, the 1D impedance
model will be incorporated in the two-port model due to its advantageous trade-off
between model accuracy and complexity.
a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
Figure 4.12: Comparison of the shape of the free surface computed with Flow3D
(solid) and the stream-function vorticity model (dashed) at certain times; t=6 µs
(a), t=8 µs (b), t=10 µs (c),t=12 µs (d),t=14 µs (e),t=16 µs (f),t=17 µs (g),t=18
µs (h),t=19 µs (i),t=20 µs (j) ([Mar06])
4.2.5 The actuation path
Since piezoelectric material acts as a two-port system quite naturally, major modeling difficulties with respect to the formulation of the actuation path within the
two-port framework are not to be expected. However, similar to the complexity
of modeling the nozzle dynamics, capturing the full three-dimensional (possibly
nonlinear) behavior of the piezo can be quite a challenge. In line with the deriva-
4.2
THE TWO-PORT MODEL
83
tions so far, a simple model of the actuation path is strived for in this section. In
this light, the following assumptions are done:
• It is assumed that the piezo actuator deforms according to its zeroth order
mode, A(x, s) = KA(s), with K the maximum displacement of the zeroth
order mode.
• Second, cross-talk is accounted for by means of the forcing function A(x, s).
This effect can be quantified using for example a FEM package.
• Furthermore, it is assumed that there are no significant structural dynamic
effects. This greatly simplifies the modeling of the piezo-unit. For the
application investigated in this thesis, this assumption does not introduce
great errors.
• Next, the electronic path is assumed to have no significant influence on the
behavior of the actuation path.
• Finally, the approach here is strictly one-dimensional. The so-called bimorf
effect, that occurs due to the fact that the piezo-unit deforms while glued
to the substrate, is neglected, see Fig. 4.13.
substrate
piezo unit
substrate
piezo unit
Figure 4.13: Illustration of the bimorf effect of the piezo-unit and substrate: not
actuated (left) and actuation (right)
For a detailed introduction into the behavior of piezoelectric material, one is referred to e.g. [Cra87] and [Waa91]. Here, supported by the assumptions listed
above, rather than accounting for the infinite dimensional character of a piezounit, a lumped parameter approach is adopted. Let us start with the accompanying full description of piezoelectric behavior:

 
S1
 S2  
  
 S3  
  
 S4  
  
 S5  = 
  
 S6  
  
 D1  
  
 D2  
D3
sE
11
sE
21
sE
31
sE
41
sE
51
sE
61
d11
d21
d31
sE
12
sE
22
sE
32
sE
42
sE
52
sE
62
d12
d22
d32
sE
13
sE
23
sE
33
sE
43
sE
53
sE
63
d13
d23
d33
sE
14
sE
24
sE
34
sE
44
sE
54
sE
64
d14
d24
d34
sE
15
sE
25
sE
35
sE
45
sE
55
sE
65
d15
d25
d35
sE
16
sE
26
sE
36
sE
46
sE
56
sE
66
d16
d26
d36
d11
d12
d13
d14
d15
d16
ǫT11
ǫT21
ǫT31
d21
d22
d23
d24
d25
d26
ǫT12
ǫT22
ǫT32
d31
d32
d33
d34
d35
d36
ǫT13
ǫT23
ǫT33


T1
  T2 
 
  T3 
 
  T4 
 
  T5 
 
  T6 
 
 E1 
 
 E2 
E3
(4.87)
84
4.2
MODELING OF THE INK CHANNEL DYNAMICS
~ and T~ represent the applied electrical field and the stress, respectively. D
~
Here, E
~ stand for the electric displacement and strain, respectively. Furthermore,
and S
~ to the applied
d is the piezoelectric charge constant, relating either the strain S
~ in the absence of mechanical stress, or the electric displacement
electrical field E
~ to the applied stress T~ in a zero electric field. sE is the compliance for a conD
~ Finally, ǫT is the permittivity under zero stress T~ . In
stant electrical field E.
Fig. 4.14, the designation of the axes and directions of deformation are depicted.
poling axis
z
3
6
y
2
5
x
4
1
Figure 4.14: Designation of the axes and directions of deformation
For the printhead under investigation, the piezoelectric material is used in the socalled d33 -mode (plane stress). This implies that for both the actuator and sensor,
the electrodes are perpendicular to the poling axis, see Fig. 4.14. Actuation then
takes place by expansion of the piezo-unit along the poling axis. Sensing in this
case occurs by measuring the stress along that same poling axis. In case of d33 mode and given the fact that a one-dimensional model is aimed at, (4.87) can be
simplified to:
S3
s33 d33
T3
=
(4.88)
D3
d33 ǫ33
E3
Up to this point, piezoelectric material in general was considered. However, when
modeling a piezoelectric unit with thickness hp and wet surface Sp , the same
behavior can be written in terms of:
• the tensional force applied to the unit along the poling axis F = Sp T3 ;
• the expansion of the unit along the poling axis u = hp S3 assuming constant
strain over the piezo;
• the voltage applied to the electrodes V = hp E3 ;
• and the charge measured at the electrodes q = Sp D3 .
(4.88) then becomes:
u
q
=
1/k
d
d
C
F
V
(4.89)
4.2
THE TWO-PORT MODEL
85
with d, k, and C are the piezoelectric charge coefficient, the stiffness of the piezo,
and the piezo capacity, respectively. Here:
k=
Sp
hp s33
(4.90)
d = d33
Sp ǫ33
C=
hp
The values of the parameters (4.90) depend on the specific piezo material used.
Also, their values are highly influenced by the specific structure that surrounds
the piezo actuator, such as for example the substrate to which the actuator is attached. Therefore, the so-called ’effective’ value of these parameters can best be
determined using a FEM package Ansys or Femlab. Irrespective of the complexity
of the actuator, as long as the effective parameters can be computed (4.89) can
be used as two-port model of the actuator path. The resulting block diagram of
the actuation path is depicted in Fig. 4.15. As can be seen in Fig. 4.15, the piezo
capacity is omitted in the two-port model of the actuation path. As discussed in
detail in Section 3.1.1, the piezo’s capacity is compensated for in the measurement
setup.
q
V
6
d
d
1/k
F
+
+
u
Figure 4.15: Block diagram of the actuation path
One important issue concerning piezo behavior is the following: the tendency of
piezoelectric material to nonlinear behavior. To that purpose, the behavior of
the piezo is more closely inspected. The relation between electrical charge (polarization) and applied electrical field for a piezoelectric material is depicted in
Fig. 4.16. Apparently, a hysteresis effect is present. If the electrical field is increased above a certain value, this will not result in an increase of polarization,
since the saturation polarization Ps is reached. Suppose that from that point the
86
4.3
MODELING OF THE INK CHANNEL DYNAMICS
electrical field is decreased, then the polarization decreases too. However, the
polarization will not become zero but assumes a certain remanent polarization
Pr . If the electric field is increased in the opposite direction, the polarization
first drops to zero and later to −Ps . If the electric field is then increased again,
via −Pr the curve goes back to Ps . For the piezo-unit used in this thesis, the
nonlinear behavior is avoided. Due to the use of multilayer piezoelectric material,
the used voltages can be kept low. Consequently, one remains within the linear
operating range of the piezo-unit, see Fig. 4.16.
D (C/m2)
Ps
Pr=Dr
E (V/m)
-Pr = -Dr
-Ps
Figure 4.16: The dielectric hysteresis curve and the linearization around its operating point
Finally, the following remarks are noteworthy. First, the fluid-structure interaction is taken into account via the stiffness of the piezo. A displacement of the piezo
results via the ink in a force sensed by the piezo. This force on its turn causes a
displacement of the piezo via the piezo’s stiffness. This way, the fluid-structure is
accounted for, see Fig. 4.15. Secondly, multiple ink channel models can be coupled
to form a complete printhead model by adjusting the forcing function A(x, s).
4.3 The bilateral coupling
In the previous subsections, the subsystems that make up the inkjet channel model
have been discussed. To couple the various subsystems, normally one uses a staggered scheme of some kind. For an overview of the use of staggered schemes, see
e.g. [Fel01]. For example, a sequential staggered scheme is depicted in Fig. 4.17
4.3
87
THE BILATERAL COUPLING
and comprises the following steps. First, the response of system 1 to a certain input is computed. Second, this response is used as input for system 2. Next, after
having computed the response of system 2 to this input, system 1 can be provided
with a new input a timestep ∆t later. The timestep used forms a crucial factor
in staggered schemes. To start with, for accuracy the Courant number should be
chosen with care (see Section 4.1). Also, the timestep should be sufficiently small
to avoid staggering errors. As a result, the computational load is usually very
high, especially if more than two systems are to be coupled.
system 1
system 2
step length
time
Figure 4.17: Sequential staggered solution procedure
For most PIJ printhead models, a CFD package is used to model the behavior of
the acoustic and fluidic path and a FEM package for the actuation path. In this
paper, only first principle modeling has been used such that analytical expressions
are available for the formulation of the two-port systems. One major advantage
of the presented approach is that the use of staggered scheme can be avoided.
Instead, the Redheffer star product can be used, see [Red60; Red62]. Given
two two-port systems as depicted in Fig. 4.18, the coupled system can then be
computed according to:
a1
v5
+
v3
a2
v3
+
+
+
c1
b1
c2
b2
+
v6
v1
+
d1
+
v4
v4
+
v2
d2
Figure 4.18: The coupling of two subsystems using Redheffer’s star product
v1
v6
=
a2 (1 − c1 b2 )−1 a1
b1 + d1 b2 (1 − c1 b2 )−1 a1
c2 + a2 c1 (1 − b2 c1 )−1 d2
d1 (1 − b2 c1 )−1 d2
v5
v2
(4.91)
88
MODELING OF THE INK CHANNEL DYNAMICS
4.4
Also, blocks with more than two input and output signals can be coupled, only
slightly more complex Redheffer relations are needed. The two-port model of an
ink channel is constructed by coupling all subsystems and applying the various
boundary conditions. Validation of the resulting model and a discussion of its
properties is treated in the next chapter. In anticipation thereof, one observation
with respect to the behavior of an ink channel is discussed in this section already.
As a result of the coupling of the various two-port subsystems, the infinite dimensional character is converted to a finite dimensional one. Even more specifically,
the behavior of the resulting system turns out to be representable by an extremely
low dimensional resonating system, at most 4th order system. If the system is
interpreted as an equivalent mass-spring-damper system, this can be explained
as follows. Apparently, after coupling maximally two masses play a role in the
dynamics of an ink channel. Possibly, one originates from the ink in the channel,
coupled with that of the connection and the piezo-unit. The other may represent
the mass of ink in the nozzle. The coupling provides the necessary elasticity to
the dynamics of both masses. This interpretation of the working of an ink channel will be further discussed and illustrated in the upcoming chapter. Note that
this behavior as low dimensional resonating system can already be observed in
the results presented in Section 4.2.4. As can be seen in Fig. 4.11, the response
is governed by the first channel resonance frequency only. The second resonance
frequency already seems absent.
4.4 Concluding remarks
In this chapter, the necessity of a new PIJ printhead model has been demonstrated. Given the requirements concerning accuracy and model complexity, a
modeling approach based on the notion of bilaterally coupled systems has been
proposed. To that purpose, the ink channel has been divided in several functional
subsystems. Each of these subsystems have been modeled as two-port systems
that have been derived using first principle modeling only. As a result, the model
complexity could be kept low. Simultaneously, it has been shown that this does
not necessarily imply poor accuracy, e.g. in case of the nozzle dynamics. Finally,
as another major advantage of the chosen modeling strategy, it has been demonstrated that the computational demanding coupling via staggered schemes can
be avoided by the application of Redheffer’s star product. In the next chapter,
the resulting two-port model will be validated using the measured frequency responses as presented and discussed in Chapter 3.
Anticipating on the validation and accompanying discussion presented in the next
chapter, the low dimensional resonating character of an ink channel has been introduced. It has been argued that due to the coupling of the subsystems and
incorporation of the boundaries, the resulting system’s behavior equals that of an
4.4
CONCLUDING REMARKS
89
extremely low dimensional resonating system. This observation will be further
elaborated in the next chapter.
To improve the resulting two-port model, the following research directions can
be explored. First, an upgrade of the current one-sided coupling between the
nozzle dynamics and the drop formation to a two-sided one should be investigated.
Though the one-sided coupling does not form an obstacle in the use of the twoport model for the (re-)design and control purposes in this thesis, the presence of
a two-sided coupling may be desirable for future investigations. Second, the use of
more accurate models for the nozzle dynamics requires further research. Now, the
exact meniscus profile is not modeled, whereas this is essential for the research into
e.g. the jetting smaller drops. Other effects, such as the presence of air-bubbles,
also require the incorporation of more advanced nozzle models. Finally, the piezounit should be modeled more accurately. To start with, the bimorf effect should
be accounted for. Also, cross-talk effects and hence the coupling with other ink
channel models should be further investigated. Note that the governing equations
of the ink channel model are capable of modeling these effects: it only requires the
use of more complex forcing functions. However, choosing these forcing functions
properly is an unresolved issue.
Chapter 5
Model validation
This chapter starts with the validation of the theoretically derived two-port model
of Chapter 4 using the measured frequency responses as presented in Chapter 3.
Next, several properties of the system are critically reviewed, including the suitability of the resulting model in light of the requirements posed in Chapter 4, the
low dimensional approximation of the behavior of an ink channel, and several of
its fundamental limitations.
5.1 Introduction
In this chapter, the theoretically derived two-port model of Chapter 4 is put to the
test by validating it against measured frequency responses (FRs). As presented
and discussed in Chapter 3, two sets of measured FRs are at our disposal: one
obtained via the piezo-unit and the other via the laser-vibrometer. After having
coupled the acoustic, fluidic, and actuation path using the proposed Redheffer
product, an analytical expression of the transfer function of the two-port model
between certain predefined inputs and outputs becomes available. These inputs
and outputs can be selected as desired. Given the measured FRs available, the
voltage sent to the piezo-unit is chosen as input whereas the measured electric
charge and meniscus speed are chosen as outputs. By substitution of an appropriate frequency vector in the theoretical transfer function, both two-port FRs
can be obtained.
To enable a sensible comparison between the measured and theoretical FRs, they
must be adjusted for various measurement devices present in the setup. The
subsequent two sections will elaborate on this in detail. In addition, the twoport FRs are provided with additional modal damping. To accomplish that, a
weighted least square approximation algorithm was applied to the two-port FRs,
see [Sch94]. The resulting state space descriptions are in the controllable canoni91
92
5.2
MODEL VALIDATION
cal form. These state space descriptions are then each transformed into the real
Jordan canonical form, see [Hor85]. Next, without altering the natural frequencies, the poles can be shifted away from the imaginary axis to change the amount
of damping. This way, each of the resonances can be tuned individually. The
necessity of adding damping to the two-port model is further addressed in the
sequel of this chapter.
This chapter is organized as follows. In Section 5.2, the two-port piezo-based FR
is validated against the measured FR. The same is done for the laser-vibrometer
based approach in Section 5.3. In Section 5.4, the resulting two-port model is
critically reviewed. Finally, Section 5.5 ends this chapter with some concluding
remarks with respect to the modeling of an inkjet channel.
5.2 Piezo-based validation
Magnitude [dB]
40
30
20
10
0
4
10
5
10
frequency [Hz]
6
10
0
Phase [Deg.]
−20
−40
−60
−80
−100
4
10
5
10
6
10
Figure 5.1: Measured FR of the Krohn-Hite 7602 amplifier
Prior to the validation of the piezo-based two-port FR, the presence of the following measurement equipment is compensated for:
• Piezo amplifier. The measured FR includes the Krohn-Hite 7602 amplifier
used for the amplification of the generated pulses by the waveform generator,
see Fig. 3.1. This amplifier is not accounted for in the two-port model.
Therefore, the theoretical FR is extended with this amplifier, whose FR is
shown in Fig. 5.1.
• Low-pass filter. As discussed in Section 3.3, the Krohn-Hite 7206 low-pass
filter is used during the various upcoming ILC experiments. Since from this
5.2
93
PIEZO-BASED VALIDATION
point it will be present in the FRs, it is added here to both the theoretical
and measured FRs. The FR of this filter with a cut-off frequency of 500
kHz is depicted in Fig. 3.12.
• Differential action. As discussed in detail in Section 3.1.1, since only changes
in the electric charge can be measured, basically a differential action is
included in the measurement loop. Since this is not accounted for in the
two-port model, a differentiator is to be added to the theoretical FR. The
differential action is approximated by an appropriate lead-lag filter.
• Piezo-sensing device. Finally, the measured FR contains the piezo-sensing
device dynamics, see Fig. 3.6. For comparison, this effect is also accounted
for in the theoretically obtained model.
After having incorporated the adjustments listed above, the theoretical and measured FRs are compared, see Fig. 5.2. In Fig. 5.3, the measured and the simulated
response to a standard trapezoidal actuation pulse are shown.
−15
Magnitude [dB]
−20
−25
−30
−35
−40
−45
−50
5
6
10
10
Frequency [Hz]
200
Phase [Deg.]
0
−200
−400
−600
−800
5
10
6
10
Figure 5.2: FR from the piezo actuator to the piezo sensor; measured (293e02,
black) and model (gray)
Based on Fig. 5.2, it is concluded that the two-port model matches the measured
FR from the piezo-unit used as actuator to the piezo-unit used as sensor quite
94
5.3
MODEL VALIDATION
1
0.8
0.6
Sensor signal [V]
0.4
0.2
0
−0.2
−0.4
−0.6
0
50
100
150
Time [µs]
Figure 5.3: Sensor signal resulting from a standard trapezoidal pulse (black dotted, scaled); measured (293e02, black) and model response (gray)
accurately, especially with respect to the location and magnitude of the resonance
frequencies. In Fig. 5.3, it is shown that the measured and simulated response to
a standard trapezoidal actuation pulse match accurately as well.
These results are not trivial. During the measurement of the piezo-based FR, the
amplitude of the sinusoids has been chosen such that the ink channel was not
jetting. In contrast, the measured response results from a jetting ink channel.
During the derivation of the two-port model, the nonlinear effect of the jetting
of a drop was neglected. Now, whereas a match of the two-port model with
the measured FR might be expected, it certainly is not trivial for the measured
and simulated responses. Since the match is still accurate in the latter case, the
supposedly nonlinear effect of jetting a drop as seen from the piezo is indeed
negligible. When Fig. 5.3 is inspected closely, a small increase of the resonance
frequency of the measured response can be observed though. This is due to the
decrease in ink in the nozzle, causing a slight increase of this frequency. Despite
this small mismatch, the behavior of the ink channel can be regarded linear for
the piezo-based case from a control perspective.
5.3 Laser-vibrometer based validation
Similar to the piezo-based approach, the laser-vibrometer based FRs are adjusted
with respect to the following devices:
• Laser-vibrometer. The laser-vibrometer introduces a considerable phase lag
5.3
LASER-VIBROMETER BASED VALIDATION
95
according to (3.4). At 1 MHz, this already amounts to 467 kHz. Prior to
the comparison to the theoretical FR, the measured FR is compensated for
this phase lag.
• Piezo amplifier. Similar to the piezo-based case, the theoretical FR is adjusted for the presence of the Krohn-Hite 7602 amplifier in the measurement
loop.
As mentioned in Section 3.5, the measured FR at 2.5 V is used throughout this
thesis for validation purposes. Recall that the amplitude of the first resonance
frequency is dependent on the used excitation voltage. After having adjusted
both FRs for the various measurement devices, the measured and two-port FRs
are compared, see Fig. 5.4.
0
Magnitude [dB]
−10
−20
−30
−40
−50
−60
4
10
5
10
Frequency [Hz]
0
Phase [Deg.]
−200
−400
−600
−800
−1000
4
10
5
10
Figure 5.4: FR from the piezo actuator to the meniscus velocity; measured at 2.5
V (233e01, black) and model (gray)
As can be seen in Fig. 5.4, the two-port FR matches the measured FR quite
satisfactorily, except for the first and most important resonance frequency. This
mismatch can be explained as follows. In the laser-vibrometer based case, from
a two-port perspective, the nozzle dynamics are measured coupled with a certain output impedance. This output impedance lumps the connection, channel,
reservoir, and piezo-unit dynamics into one single output impedance condition.
96
MODEL VALIDATION
5.4
Apparently, the two-sided coupling between the nozzle dynamics and this output impedance is incorrectly accounted for causing the modeling errors as can
be seen in Fig. 5.4. So, either the nozzle dynamics or the output impedance are
not accurately modeled. Validation of the fluidic path in Section 4.2.4, however,
has shown that the nozzle dynamics are modeled rather accurate, see Fig. 4.11.
The validation has been carried out using an output impedance that has been
determined by a Flow3D model. By tracing the pressure history at the nozzle
entrance, an input signal was obtained that certainly accounts for the two-sided
coupling properly. In contrast to this procedure, here the output impedance is
determined by the two-port model itself. Therefore, it seems a valid conclusion
that there are some modeling inaccuracies present in the output impedance causing the encountered mismatch. In the piezo-based case, the piezo-unit is coupled
with an output impedance representing the channel, connection, reservoir, and
the nozzle dynamics. Based on the fact that the results obtained in this case are
accurate, this impedance apparently is more correct.
The impact of the incorrect two-sided coupling seems to be limited to the first
resonance frequency only, see Fig. 5.4. As discussed in Section 3.2, the nozzle
acts more as open rather than a closed end for low frequencies. Therefore, the
two-sided coupling between the nozzle and the remainder of the ink channel plays
a more prominent role for these frequencies. For higher frequencies, the nozzle
dynamic behavior becomes rather autonomous.
As a result of the discussed model inaccuracies, the usability of the two-port model
in the laser-vibrometer based case is rather limited. As discussed previously, the
response to an actuation pulse is largely determined by the ink channel’s first
resonance frequency. Since this is not modeled correctly, a considerable mismatch
results between the measured and simulated responses to a certain pulse. Therefore, a comparison between the measured and simulated response is not useful
and is omitted.
Based on the encountered dependency of the dynamics on the applied input voltage, laser-vibrometer based operation of a PIJ printhead cannot be regarded as
linear. The two-port model does not give rise to an adjustment of this statement.
Still, since it only involves a soft nonlinearity, the linearity assumption will again
be reviewed when control is applied to an ink channel, see Chapter 7.
5.4 Discussion
In this section, the resulting two-port model and the accompanying system is critically reviewed. First, it is discussed to what extent the resulting model actually
fulfills the requirements as posed in 4. Second, several important model properties
are addressed, such as the low dimensional approximation of system’s behavior
5.4
DISCUSSION
97
and the encountered differences in accuracy in the piezo- and laser-vibrometer
based case. Third, the necessity of adding damping is discussed. Finally, several
shortcomings of the system itself are addressed.
In Section 4.1, the objectives for the modeling of an ink channel have been formulated as achieving high accuracy while having low model complexity. This
way, the suitability of the resulting model for control and (re-)design purposes
is enforced. Based on the validation presented in the previous two sections, it is
concluded that the resulting overall accuracy is satisfactory, except for the inaccuracy of the first resonance frequency in the laser-vibrometer case. To guarantee
suitability for the application of control in this case, the two-port model should
be improved with respect to this transfer function. Due to the sole use of first
principles for the modeling of various blocks and their coupling via the Redheffer
star product, the two-port model has a low model complexity. In addition, the
employment of bilaterally coupled systems offers valuable insight in the working
of an ink channel that partly already have been and partly will be discussed in
(the sequel of) this thesis. In conclusion, the two-port model meets the predefined goals set in the beginning to a large extent. With the work presented in
the previous and current chapter, a satisfying answer can be provided to the first
research question as formulated in Chapter 2.
Concerning the resulting two-port model, the following remarks are in order:
• Sensor locations. In the previous section, two sensor locations and accompanying transfer functions have been investigated. Based on the obtained
results, it is concluded that the system properties depend on the specific
sensor location. In terms of our two-port approach, the output impedance
encountered by the specific subsystem that incorporates the sensor functionality clearly is different. This explains the fact that different results are
obtained.
A related issue concerning the sensor location is the following. The adopted
internal structure of the two-port model, based on the physical structure
of an ink channel, cannot be validated using only the two measured inputoutput relations. For that, additional measurements are required that could
not be performed due to the limitations concerning the printhead structure.
For example, measuring the flow or pressure at the channel-connection transition is not possible for the printhead under consideration without destruction of the printhead.
• Low dimensional character of the ink channel system. In Section 4.3, the low
dimensional character of the ink channel’s dynamics has been introduced.
As a result of coupling various subsystems and the application of boundary
conditions, the infinite dimensional character is replaces by a finite dimen-
98
MODEL VALIDATION
5.4
sional one. Based on Fig. 5.3, two dominating resonance frequencies can
be distinguished. One equals the channel’s first eigenfrequency. The other
equals a higher order resonance mode of the system. Given these observations, the system’s behavior can be described by an equivalent 4th order
mass-spring-damper system.
• Damping. Another issue concerning the two-port model is the necessity
of the adding of damping as described in Section 5.1. Clearly, there are
some modeling inaccuracies regarding damping. As discussed earlier, the
majority of the damping occurs in the nozzle. However, given the accuracy
of the 1D impedance model as shown in Section 4.2.4, it is expected that the
damping in the nozzle block is taken into account correctly. In our view, the
additional damping originates from the other boundary: the reservoir. In
Chapter 4, the reservoir was assumed to act as an open end. In practice, the
reservoir is not a genuine open end and, more importantly, it contributes to
the damping. Further research is necessary to verify whether improving the
reservoir block with respect to the damping renders the adding of damping
as described in this chapter superfluous.
Based on the derivation of and the investigations into the two-port model, the
following properties of the ink channel system have come up:
• Linearity of the jetting process. As discussed above, the jetting process can
be assumed to be linear for the piezo-based case. For the laser-vibrometer
based case, the jetting process cannot be considered linear. This is due to
the earlier established nonlinear behavior, see Section 3.5. However, this
does not imply that the linear control techniques are useless, as will be
demonstrated and argued in Chapter 7.
• Limits on the jetting frequency. The currently used standard trapezoidal
actuation pulse is completely geared to the channel’s first resonance frequency. This is due to several reasons. For one, it is the most energy
effective, see [Wij06], making use of interfering traveling waves. As a result,
the actuation voltage can be kept within admissible limits. From our twoport perspective, the use of one of the two dominating resonance frequencies
of the system is a logical choice. Drawback of this approach is the fact that
the minimum time required for jetting a droplet is fixed by a channel’s first
eigenfrequency, limiting the attainable jetting frequency. Also, the residual
vibrations are dominated by the same frequency. Alternatively, if multiple
piezo-units were present, a pulse train could be used that does not require
the use the channel’s first eigenfrequency. Then, the limits with respect to
the jetting frequency could be lifted. Upon using such a pulse train, the
energy can be added to the pressure wave gradually without the need to
exceed the admissible actuation voltage.
5.5
CONCLUDING REMARKS
99
• Controllability and observability. An ink channel is a distributed parameter
system. However, after coupling of the various subsystems, the two-port
model becomes a lumped parameter model. Though in general this simplification is very useful, in some cases it does not suffice. One important case
for example concerns the controllability and observability of a distributed
system, since it concerns a definite spatial property. Therefore, the notions
of spatial controllability and observability would be more appropriate to
consider here, see e.g. [Jaı̈88] and [Tzo94]. Given the limitations of the twoport model with respect to this issue, a discussion concerning controllability
and observability is restrained to the following remarks:
1. Sensor and actuator position. Drop formation takes place in the nozzle.
For control purposes, therefore, having a sensor and actuator located
in the nozzle would be ideal. Both are not present in the current design
at that specific location. Note that the laser-vibrometer forms only a
temporary solution, since it can only measure in non-jetting situations.
This will be discussed in detail in Chapter 7. Therefore, measuring
and control of the meniscus is in the hands of the piezo-unit, which is
only an indirect way. Furthermore, as can be seen in Fig. 5.4, several
anti-resonances are present. Apparently, some meniscus trajectories
cannot be generated. This effect can be explained by the occurrence
of destructive interference, see Section 3.2.
2. Length of the piezo-unit. The piezo-unit senses the force that results
from the pressure distribution in the channel acting on the piezo’s
surface, see Section 3.1.1. This force thus represents an average value.
Consequently, it is not easily possible to track traveling waves. Also,
certain standing waves will be difficult or impossible to measure. The
piezo-unit acting as actuator also has similar disadvantages due to
its length. Most importantly, it is impossible to generate all wave
patterns. As a consequence, one is restricted in the actuation. For
example, creating a wave front capable of jetting a drop needs to be
generated using interference.
The mentioned issues with respect to controllability and observability can
be resolved by using multiple piezo-units rather than one. The two-port
model can serve as starting point for further investigations into these issues,
for example by coupling several channel blocks, see Fig. 4.3
5.5 Concluding remarks
The following conclusions are drawn concerning the modeling of an ink channel:
• The two-port model fulfills the requirement with respect to the accuracy, except for the first eigenfrequency of the system in the laser-vibrometer based
100
MODEL VALIDATION
5.5
case. The cause of this inaccuracies has been attributed to modeling errors
of the ink channel itself. The requirements regarding the model complexity
and the related computational load are met as well. Altogether, the twoport model forms a suitable starting point for the control and (re-)design
purposes in mind.
• The behavior of an ink channel can be approximated by a low dimensional
system. In our case, a 4th order linear model is capable of describing the
system dynamics accurately.
• Several (fundamental) limitations of a PIJ system have been identified.
First, the dominance of the channel first eigenfrequency limits the attainable
jetting frequency. Second, the geometry and location of the sensor hampers
the control of an ink channel.
In the upcoming two chapters, our attention shifts to feedforward control of the
a PIJ printhead. The insight in the working of an ink channel obtained in the
preceding chapters forms a valuable tool for the application of control. Several
issues that have been discussed, such as linearity and the fundamental limitations
of current PIJ printhead designs, will be revisited at the end of Chapter 7.
Chapter 6
The control framework
Motivated by the repetitive character of the inkjet printing process, Iterative Learning Control (ILC) is chosen as feedforward control strategy to enable the switch
to a controlled environment for PIJ printheads. In preparation for the implementation of ILC presented the next chapter, the ILC framework is introduced in this
chapter. After having discussed the adopted lifted ILC control structure, the control goals for the PIJ printhead under investigation are formulated. Finally, ILC
controller design is discussed. With the theoretical background on ILC presented
in this chapter, and the insight obtained the previous two chapters in the inkjet
system, an excellent starting point is provided for the successful implementation
of ILC.
6.1 Introduction
From a systems and control perspective, virtually all PIJ printheads are uncontrolled systems. As discussed extensively in Chapter 1 and 2, a switch to a controlled environment is investigated in this thesis. The aim is twofold: to push the
printhead performance to its limits in face of the current operational issues, and
to simultaneously establish the corresponding fundamental limitations of a certain printhead design. The switch is performed by the application of feedforward
control to the PIJ printhead under investigation. The choice for feedforward control is motivated by the following. Given the fact that a PIJ printhead performs
the same task over and over again, application of feedforward control generally
yields considerably more performance improvement than feedback control. Additionally, feedback control is not required to stabilize a passive system such as
a PIJ printhead. Furthermore, given the small timescales involved in the jetting
process, feedback control is considered computationally too demanding.
The repetitive character of the jetting process gives also rise to another choice,
101
102
6.2
THE CONTROL FRAMEWORK
namely that for Iterative Learning Control (ILC) as feedforward control strategy.
ILC is par excellence suited for systems that have to perform the same task time
and again. It is a control strategy used to iteratively improve the performance of
these systems by updating the command signal from one experiment to the next.
This update is based on measured data from previous trials, hence the term learning. Two remarks are in order. First, only the trial invariant part of the error can
be reduced by ILC. Second, application of ILC requires that the system returns
to the same initial condition in between the consecutive command applications.
If this condition is not met, repetitive control should be applied. It is assumed
that the PIJ printhead fulfills this requirements at all times. For an overview
on ILC, one is referred to [Moo93], [Moo98], [Bie98], or [Lon00]. ILC has been
successfully applied to a wide variety of applications in many different engineering areas, ranging from its original ([Ari84]) application of robotics (e.g.[Tay04])
to servo-mechanical applications (e.g. [Dij04] and [Roo97]) and chemical batch
processing (e.g. [Lee96]).
Similar to Chapter 3, the application of ILC is performed for both the piezo- and
laser-vibrometer based cases. Though the general ILC approach is the same for
both cases, there are several small differences. This is clearly indicated whenever
appropriate in this chapter. Starting with the lifted control structure, the control
goals are formulated next. Synthesis of the controller is then discussed. Special
attention is given to the robustness of resulting controller against model inaccuracies as well as the constraints imposed by the actuator of PIJ printheads. This
chapter ends with some concluding remarks.
6.2 The lifted ILC control structure
yref − d
+
uk+1
z −1 I
uk
integrator
-
+
∆uk
+
yk
H
ek
γ
L
Figure 6.1: ILC control structure in the trial domain in the piezo-based case
Of several ILC structures available, the lifted ILC structure ([Pha88]) is adopted
in this thesis, see Fig. 6.1 and 6.2. For a derivation, one is referred to e.g. [Dij04].
The choice for the lifted is based on the following two arguments. First, the lifted
system description accounts for the finite character of the intervals in contrast to
6.2
THE LIFTED ILC CONTROL STRUCTURE
103
descriptions based on infinite time considerations, like for example the standard
setting. In the latter case, the nonzero value of the error used by the ILC at the
start and end of the trajectory causes problems that need to be handled separately often resulting in rather heuristic approaches. These nonzero values may
be caused by e.g. system noise. Design in the lifted setting has as main advantage
that the solution explicitly takes into account states of the plant at the beginning
and end of the trajectory. Second, the lifted setting allows for the use of standard
classical (optimal) control methods for the analysis and design of learning update
schemes, see [Tou01]. A final, more application related, advantage of the lifted
setting are its numerically favorable properties. This will be further discussed in
Section 6.4.
yref − d
+
uk+1
uk
z −1 I
H
γ
L
+
∆uk
yk
-
+
ek
Figure 6.2: ILC control structure in the trial domain in the laser-vibrometer case
The lifted ILC structure for the piezo-based and laser-vibrometer based case are
depicted in Fig. 6.1 and 6.2, respectively. The mapping H is the impulse response
matrix of the plant having a state space representation (A, B, C), for an LTI system a lower triangular Toeplitz matrix. The learning matrix, that still has to be
designed, is represented by L and may be non-causal and time-varying. z −1 is one
trial delay operator and can be seen as memory block. The trial length N equals
1000 given a sample rate of 10 MHz and the DOD frequency of 10 kHz. Signal
uk is a vector containing the system’s inputs or states of the ILC system. Signal yk is the system output, ŷref the reference trajectory, and d the disturbance.
Throughout this thesis, the effect of the noise d is assumed to be negligible. The
effect of noise is discussed in e.g. [Nor01]. ek is the error output. The update
of the system’s input is ∆uk and uk+1 is the input for the next trial k + 1. At
the k-th trial, signal uk is provided to the system, resulting in the (integrated)
output yk . The output yk is then subtracted from the reference yref to obtain the
error ek . Based on this error, the learning controller computes the adjustments
to the input ∆uk that, added to the previous input, forms the input for the next
trail uk+1 . Apparently, the ILC controller functions as a feedback controller in
the trial domain.
104
6.2
THE CONTROL FRAMEWORK
In case of a MIMO system, the above control structures are the same. In case
of two channels, the signals in (6.1) have dimension 2N × 1. H has dimension
2N × 2N . The various signals and impulse response matrix are then structured
as follows:

and:
ykA (0)
ykB (0)
ykA (1)
ykB (1)
..
.





yk = 



 y A (N − 1)
k
ykB (N − 1)

hA (0)
hAB (0)


uA
k (0)
uB
k (0)
uA
k (1)
uB
k (1)
..
.










 uk = 







 uA (N − 1)
k
uB
k (N − 1)
hBA (0)
hB (0)
0
0




hA (1)
hBA (1)
hA (0)


H =
hAB (1)
hB (1)
hBA (0)


.
.
..
..
..

.

 hA (N − 1) hBA (N − 1)
...
hAB (N − 1) hB (N − 1)
...


eA
k (0)
eB
k (0)
eA
k (1)
eB
k (1)
..
.










 ek = 







 eA (N − 1)
k
eB
k (N − 1)
0
0
...
...
hBA (0)
...
hB (0)
..
.
...
..
.
...
...
...
...
For a larger array of channels, (6.1) and (6.2) are adjusted
played structure.
0
0
..
.
..
.
..
.











(6.1)
0
0
..
.
..
.
..
.












hA (0) hBA (0) 
hAB (0) hB (0)
(6.2)
according to the dis-
In the piezo-based case, the measured sensor signal represents the derivative of
the pressure in the ink channel, see e.g. Section 5.2. Bringing the derivative
of the channel pressure to zero, however, does not imply that the channel is at
rest. Therefore, the measured output is numerically integrated as can be seen in
Fig. 6.1. Control then is focussed on the channel pressure itself. In Section 6.4,
it is shown that adding an integrator also is numerically advantageous. In the
laser-vibrometer case, the integrator is omitted, see Fig. 6.2. As a result, the
meniscus speed is controlled rather than its position. Since drop formation is
highly dependent on the meniscus velocity rather than on its position, adding
an integrator is not necessary. For a study into refill as well as stability, the
availability of the meniscus position becomes important. In that case, adding an
integrator can be considered. However, integration requires the availability of a
correct initial state, which in case of the meniscus is not trivial as apposed to the
channel’s initial state. After all, the channel pressure can be measured at any
time instant whereas the meniscus position cannot be determined at all. Finally,
6.2
THE LIFTED ILC CONTROL STRUCTURE
105
a discussion on the effect of the scaling γ is postponed until Section 6.4.
1
observation
0
N
1
actuation
N1
0
Figure 6.3: Illustration of the actuation and observation time windows
For some applications, the actuation and observation time intervals are not equal
to the complete trial length N . Though in case of a PIJ printhead the observation
window does cover the the complete trial length, the actuation is to be restricted
to a limited time window. This is depicted in Fig. 6.3. Restriction of the actuation is necessary to enable the increase of the jetting frequency. After all, the
higher this frequency, the shorter the available actuation time interval. To enable
the restriction of the actuation and observation windows, the lifted ILC control
approach can be adjusted according to the following two methods.
The first approach adjusts to the impulse response matrix H. To that purpose,
(6.2) is structured as follows:

yk (0)
..
.



 yk (N1 )

 yk (N1 + 1)


..

.
yk (N − 1)



 
 = H11

H21




0
H22
uk (0)
..
.



 uk (N1 )

 uk (N1 + 1)


..

.
uk (N − 1)










(6.3)
where N1 is the time instant for the actuation to stop. In our case, the tracking
behavior of the complete trial is important, yk , but the actuation, uk , is restricted
to a certain time period. Therefore, (6.3) can be reduced to:
106
6.2
THE CONTROL FRAMEWORK

yk (0)
..
.



 yk (N1 )

 yk (N1 + 1)


..

.
yk (N − 1)





uk (0)
 

..
 = H11 


.

 | H{z21 }
uk (N1 )


H∗
(6.4)
This adjusted H is now used during the design of the learning filter L, see Section 6.4. The incorporation of actuation and observation intervals in the design
of the learning filter L as demonstrated can only be facilitated by the lifted ILC
setting.
yref − d
+
uk+1
uk
z −1 I
H
γ
L
integrator
-
+
∆uk
+
yk
Wi
ek
Wo
Figure 6.4: ILC control structure with weightings in the trial domain in the piezobased case
A second approach expands the ILC control structure with weighting filters, see
Fig. 6.4. Wi and Wo serve as weighting on the inputs and outputs of the system,
respectively. If these weightings are taken diagonal, they act as time-weights on
the signals. A very small weight on certain parts of the input signal ensures that
the ILC controller does not generate control signals in that range. Similarly, a
very small weight on the output ensures that the ILC algorithm does not try to
reduce the errors in that range. To illustrate the choice of the weighting filters,
consider the following choice:
IN1
0
IN1
0
Wi =
Wo =
(6.5)
0
0N −N1
0 IN −N1
Using these filters, the same objectives are strived for as in the first time windows
approach:
∗
H =
IN1
0
0
IN −N1
H11
H21
0
H22
IN1
0
0
0N −N1
=
H11
H21
0
0
(6.6)
6.3
THE CONTROL GOALS
107
The range for the choice of the weighting filter is seemingly endless. Consequently,
compared to the approach of time windows, the use of weighting filters offers more
flexibility. For example, allowing the ILC algorithm to generate slightly more error in a time interval where it is of less importance, generally yields better overall
performance.
A final remark concerning the restriction of the actuation window is the following.
The restriction is limited, e.g. due to avoiding too high actuation voltages. As a
result, the increase of the jetting frequency is bounded also. Measures to overcome
this include the following. For example, linearity of the jetting process can be
assumed and the ILC actuation pulses can be superimposed. Alternatively, ILC
actuation pulses can be learned for a sequence of drops. The former solution is
adopted in this thesis.
6.3 The control goals
In Section 1.2.2, the performance requirements of a PIJ printhead as well as the
corresponding limitations have been discussed in detail. In this thesis, the focus
lies on improving the performance with respect to the following two requirements:
• Enhancing the productivity. The productivity of a PIJ printhead is mainly
determined by the jetting frequency and the amount of nozzles per inch, see
e.g. [Bru05]. As discussed in Chapter 1.2.2, the attainable jetting frequency
is limited by the residual vibrations. The amount of nozzles per inch, also
referred to as npi-ratio, is limited by the measure to minimize the effect of
cross-talk, see Section 1.2.2 also. Therefore, to improve the productivity of
a PIJ printhead, the residual vibrations and cross-talk effects are to be minimized. Changing/varying dynamics and robustness against disturbances
do not affect the productivity directly.
• Improving the drop-consistency. Apart from the specific requirements with
respect to drop properties such as speed, volume, shape, and straightness,
consistency of these properties is the most important property of all. Meeting current consistency requirements limits the operation of PIJ printheads.
For example, jetting at 10 or 20 kHz yields inadmissible variations in e.g.
drop-speed and -volume. Actuation of a random combination of neighboring channels generally yields too large variations in drop properties as well.
Again, the residual vibrations and cross-talk are the major performance limiting phenomena when considering consistency, see Section 1.2.2. The other
performance limiting phenomena affect the drop-consistency much less.
Other requirements as formulated in Section 1.2.2, such as stability and dropspeed and -size, are not considered in this thesis. However, this does not imply
108
6.3
THE CONTROL FRAMEWORK
that ILC cannot be used to improve the performance concerning these requirements. At the end of this section, it is shortly indicated how ILC can be employed
to improve the performance concerning these issues as well.
Apparently, both improving the productivity and consistency demand for the
minimization of the residual vibrations and cross-talk. In the sequel of this section,
therefore, the attention is shifted from the control goals as formulated above to
the minimization of the residual vibrations and cross-talk. To measure the effect
of ILC with respect to these two issues, the following performance indicators are
used:
• IAE of the resulting error signal. The error signal indicates to what extent
the reference trajectory is attained. The error can be expressed in terms of
the Integrated Absolute Error (IAE):
IAE =
N
X
i=0
|ek (i)|
(6.7)
Given an appropriate choice for the reference trajectory, attaining this trajectory implies that the residual vibrations and cross-talk are effectively
minimized. Therefore, the IAE serves as indicator for the performance.
• DOD-speed and -volume curves. A Drop-On-Demand (DOD) curve shows
the relation between drop-speed or -volume and the used jetting frequency.
For example, a DOD-speed curve is depicted in Fig. 1.9. Elimination of the
residual vibrations leads to an improvement of the DOD curve, as discussed
in Section 1.2.2. Therefore, the DOD-speed and volume curves can be used
as performance indicators for the minimization of residual vibrations. Ideally, a DOD curve is a horizontal line.
• Cross-talk curve. The influence of cross-talk on the performance of a PIJ
printhead is assessed in a cross-talk curve, see e.g. Fig. 1.10. It depicts
the resulting drop-speed of one particular channel when in turn neighboring channels are actuated simultaneously. If the cross-talk is eliminated
completely, the cross-talk curve is a horizontal line. For an array of two
channels, the cross-talk curve reduces to a table.
The link between the formulated objectives and the adopted control framework
is formed by the reference trajectories. More specifically, these are to be constructed such that minimization of the residual vibrations and cross-talk is enforced. The observation that drop properties are completely determined by the
meniscus trajectory forms the starting point in the formulation of suitable reference trajectories. Note that this observation implicitly has served as basis for
the derivation of the governing equations for the drop formation in Section 4.2.3,
6.3
THE CONTROL GOALS
109
where the meniscus speed serves as input for the computations. This observation
greatly facilitates the implementation of ILC. Instead of formulating the control
objectives in terms of drop properties such as drop-speed or -volume, information
that is available only at certain discrete time instances, a continuous objective can
now be adopted. Still, the relationship between the resulting drop properties and
the meniscus trajectory is far from trivial and cannot be characterized straightforwardly. For example, various meniscus trajectories may result in similar drop
properties whereas some drop properties may not be realizable for any meniscus trajectory. Consequently, choosing a suitable meniscus reference trajectory
remains a non-trivial matter. Another complicating matter concerns the usage
of the laser-vibrometer, the most sensible sensor when aiming at realization of a
certain meniscus trajectory. As discussed in Section 3.1.3, there are a number
of practical disadvantages associated with the use of the laser-vibrometer as sensor. Alternatively, the piezo-unit can be chosen as sensor functionality. Then, a
channel pressure trajectory can be used for the control purposes in mind. However, given the fact that the pressure trajectory is only an indirect measure of
the realized meniscus trajectory, specification of a proper reference trajectory for
the channel pressure might be even more difficult. Nevertheless, both options are
considered and are used in the sequel of this thesis.
Theoretically, the following procedure is to be utilized to construct a suitable reference trajectory. Based on the required drop properties, a corresponding meniscus
trajectory can be computed using the relations derived in Section 4.2.3 in general
and the inverse of (4.78) in particular. Basically, this amounts to computing the
inverse of the drop formation model. Once this trajectory has been computed,
measures to counteract the residual vibrations and cross-talk can be incorporated.
If desired, the corresponding pressure trajectory can be obtained using both the
piezo-based and the laser-vibrometer based transfer functions, see [Bos05]. Based
on these TFs, the TF between the channel pressure and the meniscus velocity
can be computed, see Fig. 6.5. Using the inverse of this computed TF, the corresponding channel pressure trajectory can be computed given a certain meniscus
trajectory.
From a practical point of view, the above procedure is rather complex (computing
the inverse of the drop formation model) and sensitive to modeling errors (computing the corresponding channel pressure trajectory). An alternative simple yet
effective approach is the following. The starting point is a measured meniscus
velocity or channel pressure response to a standard trapezoidal actuation pulse,
see Fig. 6.6. Suppose that the corresponding drop properties are according to the
specifications. The trajectories are then adjusted as follows:
• Eliminating residual vibrations. The measured trajectories are supposed to
consist of two parts. During the first part, up to the point where condition
(4.78) is fulfilled, is left unchanged. In this way, the drop formation is
110
6.3
THE CONTROL FRAMEWORK
replacemen
V
TFV 2p
channel
pressure
vmeniscus
TFp2v
vmeniscus
V
TFV 2v
Figure 6.5: Schematic overview of the TFs playing a role during computation of
a channel pressure reference trajectory
left undistorted. During the second part, the response is governed by the
residual vibrations. By forcing the meniscus velocity or the channel pressure
to a rest, this operational issue can be eliminated.
• Eliminating cross-talk. If the responses are measured while only one ink
channel is actuated, the measured trajectories are cross-talk free. Subsequently, when these references are used during the ILC computations,
cross-talk is effectively eliminated.
There are two important constraints for the construction of the reference trajectories. First, to ensure the refill of the nozzle the fluid-dynamics are not brought
to a rest immediately after the ejection of the drop. Details with respect to refill
can be found in e.g. [Yan04]. Second, the fluid-dynamics are brought to a rest
somewhat gradually to avoid too high actuation voltages. The sketched approach
is illustrated in Fig. 6.6.
In the next chapter, it is shown that this somewhat pragmatic approach to the
construction of reference trajectories is in fact a very successful one. Still, it is
emphasized that this is just one possible choice for the reference trajectory. A
complete analysis based on the theoretical approach would provide valuable insight in the limitations of current printhead designs, e.g. with respect to the drop
properties that are feasible.
The question arises whether the piezo- or laser-vibrometer based approach is more
suitable for the realization of the control goals. Intuitively, one could argue that
the adoption of the laser-vibrometer based approach leads to better results. Eventually, the meniscus determines the performance for a large part. The ink channel
pressure remains an indirect indicator. Additionally, the limited (spatial) controllability might prevent the realization of some meniscus movements using solely
the piezo-unit, see Section 5.4. A comparison of both approaches provides insight
6.4
THE CONTROL GOALS
111
−6
4
x 10
1
0.5
2
Meniscus velocity [m/s]
Integrated sensor signal [Vs]
3
1
0
0
−1
−0.5
−2
0
20
40
60
Time [µs]
80
100
0
20
40
60
Time [µs]
80
100
Figure 6.6: Measured sensor signals (black) and reference signals (gray dotted)
for the piezo- (left) and laser-vibrometer case (right)
concerning the limitations of certain PIJ printhead designs.
In addition to the control goals formulated above, the following two operational
issues can be handled by ILC also by adopting the right reference trajectories:
1. Drop-speed and -volume (modulation). Drop properties such as speed and
volume can be adjusted by changing the trajectory. This can possibly be
done during operation, enabling drop speed or size modulation. The effectiveness of these measures can be established by measuring the resulting
drop-speed and -volume using the CCD camera.
2. Stability. Stability of the jetting process is among other things dependent
on the retraction of the meniscus. By adjusting the reference trajectories in
this respect, this could be realized easily. The larger the retraction, the more
the possibility that instabilities occur. Mainly connected with robustness
also, especially dirt particles and air-bubbles. Therefore, ILC can even be
invoked to improve stability. Stability can be checked by bitmap tests.
Note that drop-shape and straightness have not been discussed here. Though
both might very well be controllable with the ILC approach discussed in this
thesis, the 1D approach adopted throughout this work restricts our scope to the
drop properties considered so far. If the approach is extended to 2D, these issues
could be resolved also.
112
THE CONTROL FRAMEWORK
6.4
6.4 ILC design
In this section, the design of ILC controllers in the lifted setting is discussed. For
the synthesis of ILC controllers various approaches can be adopted. Here, LQoptimal ILC design is treated. It is shown that this method can be used for the
design of both SISO and MIMO ILC controllers. Also, special attention is paid to
issues such as robustness of the resulting controller and limiting the observation
and/or actuation interval. Implementing the resulting ILC controllers usually
results in rather complex ILC actuation pulses. Since the Application Specific
Integrated Circuits (ASIC) can only handle actuation pulses that are limited in
complexity, an adjusted ILC algorithm is proposed. This so called constrained
ILC constructs actuation pulses that are composed of a predefined number of
piece-wise affine functions.
Note that throughout this section it is assumed that the structure of the various
signals as in (6.1) and (6.2) is adopted consequently.
6.4.1 LQ-optimal control
In this section, LQ-optimal ILC design is discussed. The derivations presented in
this section are based on the work of [Tou01] and [Dij04]. Starting point forms
the following ILC system description:
uk+1 = uk + ∆uk
ek = −yk + yref = −Huk + yref
∆uk = Lek
(6.8)
with u0 = 0. Recall that the noise d is neglected, as assumed in Section 6.2. Let
us first verify whether the conditions for the existence of a solution of the optimal
LQ-problem are fulfilled. For that, the system must be both stabilizable and detectable. If a system is not stabilizable, then obviously it cannot be stabilized. If
a system is not detectable, there exist state feedback controllers that do not stabilize the system but hide the instabilities from the output. Stability then cannot
be guaranteed. A sufficient condition for stabilizability is that the system is controllable. A sufficient condition for detectability is that the system is observable.
Given the presence of a bank of integrators in the control structure, controllability
is automatically fulfilled. In contrast, observability is not a trivial matter. In case
the output matrix H is singular or nearly singular this criterion is not fulfilled.
This might occur if the underlying plant contains time-delays or non-minimum
phase zeros. To resolve this, define the singular value decomposition of H as:
T
Σ1 0
V1
T
H = U ΣV = U1 U2
(6.9)
0 Σ2
V2T
6.4
ILC DESIGN
113
where U and V are unitary orthogonal matrices and Σ is a diagonal matrix with
the singular values on the main diagonal ordered from large to small. Furthermore,
V V T = V T V = U U T = U T U = I. If Σ2 contains singular values from Σ that are
(nearly) zero, H can be approximated as:
H ≈ U1 Σ1 V1T
(6.10)
Note that U1 and V1T will in general not be square. Incorporating V1 and V1T into
the control structure, see Fig. 6.7, renders (6.8):
uk+1 = uk + ∆uk
ek = −yk + yref = −HV1 uk + yref
∆uk =
V1T Lek
(6.11)
∗
= L ek
yref − d
+
uk+1
z −1 I
uk
V1
integrator
-
+
∆uk
+
yk
H
ek
γ
L∗
Figure 6.7: Adjusted ILC control structure in the piezo-based
The conditions for the existence of an LQ-optimal control solution now have been
fulfilled for the system (6.11). The design of the ILC controller is formulated in
terms of the following optimal control problem:
J=
N
X
ykT Qyk + ∆uTk R∆uk
k=1
=
N
X
uTk V1T H T QHV1 uk + ∆uTk R∆uk
(6.12)
k=1
For an array of n channels, the summation is extended to nN . Furthermore,
weighting matrices R and Q must be positive-definite. R has to be positivedefinite to prevent infinite input amplitudes. If Q is not positive-definite then
there may be unstable closed-loop modes that have no effect on the performance
index. Choosing Q = I and R = βI, results in:
114
6.4
THE CONTROL FRAMEWORK
V1T H T IHV1 = Σ1 U1T U1 Σ1 = Σ21
(6.13)
and (6.12) reduces to:
J=
N
X
uTk Σ21 uk + β∆uTk ∆uk
(6.14)
k=1
The solution to the discrete LQ-optimal control problem (6.14) is:
∆uk = −(βI + X)−1 Xuk
(6.15)
with X the stabilizing solution of the Discrete Algebraic Riccati Equation (DARE):
−X(βI + X)−1 X + Σ21 = 0
(6.16)
Since Σ1 is diagonal, the solution X to the Riccati equation (6.16) will be diagonal
as well. With σi and xi denoting the i-th elements of Σ1 and X, respectively, the
solution is:
s
!
1 2
4β
xi = σi 1 + 1 + 2
(6.17)
2
σi
The feedback interconnection matrix L∗ becomes:
T
L∗ = (βI + X)−1 XΣ−1
1 U1
(6.18)
The resulting closed loop system can be analyzed by its closed loop system matrix
I − L∗ HV1 and equals:
T
−1
I − (βI + X)−1 XΣ−1
X = βI(βI + X)−1
1 U1 HV1 = I − (βI + X)
(6.19)
having closed loop poles:
λi =
Thus:
λi ≈
β
≈0
σi2
β
β
q
=
β + xi
β + 12 σi2 1 + 1 +
√
β
β
√
√
≈
≈
≈1
β + σi β
β + σi
4β
σi2
(6.20)
for σi2 ≫ β
for σi2 ≪ β
(6.21)
For large singular values, LQ-optimal control approximately provides dead-beat
performance with poles in the origin. For small singular values, the gain in the
6.4
ILC DESIGN
115
feedback loop is almost zero. Apparently, β can be viewed as tuning parameter
determining which dynamics are taken into account in the ILC algorithm. To
clarify this, let us zoom in on the relation between the dynamics in terms of the
FR and the singular values. There exists a fundamental difference between the
FR and the singular value characterization of a system. The FR is the result of
Fourier transforms of -in principle- an infinite time signals. The SVD description
of a system is based on a finite time impulse response. The question remains
to what extent a FR describes the system’s behavior for a finite time, as is the
case in the lifted ILC framework. However, if the resulting singular values are
ordered according to their frequency content, the FR is approximately obtained,
see Fig. 6.8. For a more detailed treatment, one is referred to [Dij04]. In conclusion, with β the relevant system dynamics can be selected, even though being
based on the singular values.
−20
Magnitude [dB]
20*log10(σi) [dB]
−20
−30
−40
−50
−40
−50
200
400
600
800
Element number i
−60
0
10
1000
−130
−130
−140
−140
Magnitude [dB]
20*log10(σi) [dB]
−60
−30
−150
−160
−170
−180
1
2
10
10
Frequency [kHz]
3
10
−150
−160
−170
200
400
600
800
1000
−180
0
10
1
10
2
10
3
10
Figure 6.8: Piezo-based FR and the corresponding SVD; without (above) and
with (below) integrator
By choosing a large β, only the most dominant system dynamics are used in
the ILC algorithm. Focussing on these dynamics only renders the ILC controller
robust against model inaccuracies. Therefore, the β parameter also is a tuning
parameter to enhance the robustness of the resulting ILC controller.
116
THE CONTROL FRAMEWORK
6.4
Based on Fig. 6.8, another advantage of the added integrator in the piezo-based
case becomes apparent. Due to the +1 slope of the FR in combination with a
certain β, the wrong dynamics would be taken into account. Rather than selecting the channel first eigenfrequency around 40 kHz, higher frequent dynamics are
selected. By adding an integrator, this problem can be resolved. The first eigenfrequency then correspond to the larger singular values. Apart from a physical
necessity of the added integrator, it is favorable from a numerical point of view as
well. Note that in the laser-vibrometer based case, the integrator is not required
from a numerical perspective.
Note that to facilitate the computations, the solution (6.16) can be approximated
by:
s
!
1 2
4β
xi = σi 1 + 1 + 2 ≈ σi2 + β
(6.22)
2
σi
This approximation holds as long as:
4β
≪1
σi2
(6.23)
In Fig. 6.1 and 6.2 and various other control structures, a scalar learning gain
γ is visible. A gain of γ < 1 can be used to increase the robustness of the ILC
controller against model uncertainties by shifting the closed-loop closer to 1. Note
that this does not affect the final attainable error, see [Dij04].
The LQ-optimal ILC design presented here has several drawbacks. Most importantly, the associated computations become increasingly difficult if not impossible
for long reference trajectories, mainly due to numerical issues. Therefore, in Appendix A, an alternative ILC design procedure is discussed that can handle these
long trajectories: the Hamiltonian based ILC design.
6.4.2 Constrained ILC
For the implementation of an actuation pulse on a PIJ printhead, use is made
of an Application Specific Integrated Circuit (ASIC). In contrast to a Field Programmable Gate Array (FPGA), an ASIC is capable of handling the high voltage
actuation pulses required for PIJ printheads. Unfortunately, an ASIC can handle signals that consist of a limited number of piece-wise affine functions. Since
ILC pulses usually contain high frequency components, they fail to meet the requirements for implementation on an ASIC. Though choosing a suitable β solves
this issue to a certain extent, the complexity of the resulting ILC pulse simply
cannot be reduced sufficiently. In this section, therefore, another simple yet effective modification of the ILC algorithm is discussed that allows for the design
6.4
ILC DESIGN
117
of ILC actuation pulses that fulfill the requirements for ASIC implementation:
constrained ILC.
For the design of simplified actuation pulses within the ILC framework several
strategies varying in complexity can be followed. To start with, given the number
of switching instances a non-linear optimization problem can be formulated that
determines the switching instances in time and amplitude, e.g. see [Hat04]. Interpolation between those points then gives the actuation pulse. However, formulation within an ILC framework is not trivial and the computational complexity
makes it unsuitable for implementation on a PIJ printhead. Second, by utilizing a certain set of basis functions the non-linear optimization problem can be
transformed into a linear optimization problem within the ILC framework, see
e.g. [Pha96; Gor97]. However, since a high number of basis functions is usually
needed to obtain reasonable performance, quite complex actuation pulses result
that still are infeasible for ASIC implementation.
yref − d
+
uk+1
uk
z
−1
I
+
yk
H
integrator
+
∆uk
least squares
approximation
ek
L
Figure 6.9: Constrained ILC control structure in the trial domain in the piezobased case
Alternatively, rather than using a high number of basis functions for the construction of a simplified actuation pulse, an optimized basis is adopted that is based
on known limitations concerning the implementation on an ASIC and physical
insight in the working of a PIJ printhead. This is accomplished by the following adjustment of the ILC algorithm, see Fig. 6.9. The resulting ILC controller
computes, based on the resulting error signal ek , an update ∆uk of the actuation
signal uk . The actuation signal (and the update accordingly), is to be transformed
into a simplified signal. Given a certain number of switching instances tsw that
are fixed in time and determined a priori, a nonlinear least-squares algorithm
([Mar63; Lev44]) is used to approximate the update ∆uk with function F (tsw , p):
min |∆uk − F (tsw , p)|
p
(6.24)
where p is the amplitude of the approximation function at tsw . The switching
118
THE CONTROL FRAMEWORK
6.5
instances tsw are chosen such that the first eigenmode of the ink channel can be
effectively damped by the ILC algorithm. As discussed in Section 3.2 and Chapter 5, this eigenmode dominates the response and hence forms a suitable choice.
If the actuation is changed such that other modes become dominant, the switching instances should be adjusted accordingly. Typically, around twelve switching
instances are chosen. Note that omitting this projection step, the unconstrained
lifted ILC framework is obtained.
6.5 Concluding remarks
In this chapter, the theoretical background for the implementation of ILC on a
PIJ printhead has been presented. In the next, both piezo- and laser-vibrometer
based ILC is implemented on the experimental setup. This setup is not equipped
with an ASIC such that there are no limitations with respect to the ILC actuation
pulses. Nevertheless, in preparation of the implementation of ILC to a commercial
PIJ printhead, constrained ILC is implemented as well for the piezo-based case.
The performance is benchmarked against the performance of the unconstrained
ILC algorithms. At the end of the next chapter, the in Chapter 5 discussed
fundamental limitations are revisited.
Chapter 7
Application of feedforward control
This chapter demonstrates the use of lifted ILC to improve the printhead’s performance. To that purpose, both piezo- and laser-vibrometer based ILC are applied
to various PIJ printheads to reduce the residual vibrations and cross-talk. Next
to the realization of a performance improvement, more fundamental limitations
of current printhead designs become apparent. After having presented the experimental results, these results and their implications are discussed in detail. As it
turns out, several findings will confirm the suppositions stated in earlier chapters
concerning the printhead’s limitations.
7.1 Introduction
For the implementation of ILC on the various PIJ printheads, use is made of the
measured FRs rather than the theoretically obtained FRs. Though the theoretically obtained piezo-based FR is sufficiently accurate for the frequency range of
interest, the laser-vibrometer based FR is not (as argued in Section 5.3). To adopt
a similar approach to the implementation of ILC throughout this chapter, only
measured FRs are used as starting point. In addition, to enhance the general
applicability of our proposed ILC approach, the employment of measured FRs
not only guarantees the usage of the most accurate system descriptions available,
but also lifts the necessity to model a printhead theoretically. Still, an example of
the successful utilization of the theoretically obtained FR in the piezo-based ILC
approach can be found in [Gro05b]. All the same, based on the measured FRs as
presented in Chapter 3, transfer functions are fitted using weighted Output-Error
(OE) least-squares approximations, see [Sch94]. For the piezo-based approach,
the 293e02 and DG074 PIJ printheads are used for the SISO and MIMO case, respectively. The measured FR from the piezo actuator to the piezo sensor and the
accompanying fitted transfer function is depicted in Fig. 7.1 and Fig. 7.2, respectively. To assess the quality of both models, it has been validated using measured
119
120
7.1
APPLICATION OF FEEDFORWARD CONTROL
sensor signals, see Fig. 7.3 and Fig. 7.4. These sensor signals are the result of
actuating a channel with a standard trapezoidal pulse at a jetting frequency of
10 kHz. Based on Fig. 7.3 and Fig. 7.4, we conclude that the piezo-based dynamics are modeled satisfactorily. Note that the sensor signal of the non-actuated
ink channel in Fig. 7.4 (cross) oscillates in anti-phase to the sensor signal of the
actuated ink channel in Fig. 7.4 (direct). This corresponds to the fact that a decrease of one channel induces an increase of its neighboring channels and provides
a physical explanation of the obtained sensor signals. For the laser-vibrometer
based approach, the 233e01 printhead is used. The corresponding FR and TF
from the piezo actuator to the meniscus velocity is displayed in Fig. 7.5. The accompanying measured and simulated response to a standard trapezoidal actuation
pulse are depicted in Fig. 7.6. Note that the differences between the measured
and simulated responses shown in this section can be handled by ILC.
−10
Magnitude [dB]
−20
−30
−40
−50
−60
0
10
1
2
10
10
3
10
Frequency [kHz]
200
Phase [Deg.]
0
−200
−400
−600
−800
−1000
0
10
1
10
2
10
3
10
Figure 7.1: Frequency response of the 293e02 from the piezo actuator to the piezo
sensor; measured (black dotted) and model (gray)
The piezo-based MIMO case will be elaborated for an array of two channels. As
discussed in Section 3.2, it is assumed that all ink channels are identical. Consequently, the MIMO case is simplified. Rather than having to take four transfer
functions into account (Ha , Hb , Hab , and Hba ), now two suffice (Ha = Hb and
Hab = Hba ). The validity of this assumption and the possible consequences for
the attainable performance are subject of discussion in subsequent sections of this
chapter. The laser-vibrometer based MIMO case is not investigated here due to
INTRODUCTION
−15
−20
−20
−25
−25
−30
Magnitude [dB]
Magnitude [dB]
7.1
−30
−35
−40
−45
−35
−40
−45
−50
−50
−55
5
5
10
Frequency [Hz]
10
Frequency [Hz]
0
200
−100
100
Phase [Deg.]
Phase [Deg.]
121
−200
−300
−400
−500
0
−100
−200
−300
−600
−400
5
5
10
10
Figure 7.2: Frequency response of the DG074 from the piezo actuator to the piezo
sensor, direct (left HA and HB ) and cross (right HAB and HBA ); measured (black
dotted) and model (gray)
the availability of only one laser-vibrometer.
1
0.8
0.6
Sensor signal [V]
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
0
10
20
30
40
50
Time [µs]
60
70
80
90
100
Figure 7.3: Response of the 293e02 to a standard trapezoidal actuation pulse;
measured (black) and simulated (gray)
This chapter is organized as follows. To start with, the piezo-based ILC approach
is elaborated in Section 7.2. The SISO case serves as demonstration of the use of
ILC for the reduction of residual vibrations. Next, the MIMO case is employed
to show the minimization of cross-talk effects. Then, the same MIMO setting is
adopted for the implementation of the constrained ILC framework. During each
of the treated cases, the performance measures discussed in Section 6.3 are used.
In Section 7.3, the laser-vibrometer based ILC approach is discussed. In corre-
122
7.2
APPLICATION OF FEEDFORWARD CONTROL
1
0.15
0.8
0.1
0.6
0.05
0.4
0
Sensor signal [V]
Sensor signal [V]
spondence with earlier discussions, the experiments are conducted using 2.5 V as
actuation voltage. For the implementation, the resulting learned ILC pulses are
scaled to the appropriate jetting voltage. After having presented the experimental results, the obtained results and their implications are discussed in detail in
Section 7.4. This chapter ends with conclusions regarding the implementation of
ILC to PIJ printheads.
0.2
−0.05
0
−0.1
−0.2
−0.15
−0.4
−0.2
−0.6
0
0.1
0.2
0.3
0.4
0.5
Time [s]
0.6
0.7
0.8
0.9
1
−0.25
0
−4
x 10
0.1
0.2
0.3
0.4
0.5
Time [s]
0.6
0.7
0.8
0.9
1
−4
x 10
Figure 7.4: Response of the DG074 to a standard trapezoidal actuation pulse;
direct (left) and cross (right), measured (black) and simulated (gray)
magnitude [dB]
0
−20
−40
−60
−80
4
10
5
10
6
10
0
phase [Deg.]
−500
−1000
−1500
−2000
4
10
5
10
frequency (Hz)
6
10
Figure 7.5: Frequency response of the 233e01 from the piezo actuator to the
meniscus velocity at 2.5 V; measured (black) and model (gray)
7.2
PIEZO-BASED ILC
123
1
0.8
Meniscus velocity [m/s]
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
0
0.2
0.4
0.6
0.8
1
Time [µs]
1.2
1.4
1.6
1.8
2
−4
x 10
Figure 7.6: Response of the 233e01 to a standard trapezoidal actuation pulse at
2.5 V; measured (black) and simulated (gray)
7.2 Piezo-based ILC
7.2.1 SISO ILC: reducing residual vibrations
In this section, ILC is applied to one ink channel of the 293e02 PIJ printhead to
reduce the residual vibrations. The used control structure is depicted in Fig. 6.1.
The reference trajectory is constructed according to the procedure discussed in
Section 6.3. Starting with an integrated sensor signal of a PIJ printhead jetting
at 10 kHz resulting from a standard trapezoidal actuation pulse, the first part
up to the firing of a drop at 30 µs is copied. After that, the residual vibrations
are eliminated by forcing the reference trajectory to zero, see Fig. 7.7. Note that
the damping is not enforced too quickly after 30 µs to avoid too high actuation
voltages and to ensure the refill of the nozzle.
The controller synthesis is performed based on the identified transfer function as
depicted in Fig. 7.1 plus an added integrator. The presence of an integrator has
been motivated extensively throughout this thesis, e.g. see Section 6.2 and 6.4.
β has been chosen such that the printhead dynamics up to approximately 250
kHz are taken into account. Beyond 250 kHz, there are no relevant printhead
dynamics. γ is chosen as 0.25. Recall that this only affects the convergence speed
only. Furthermore, the length of the reference trajectory allows for the use of the
LQ-optimal approach for the design of the ILC controller, see Section 6.4.
For a PIJ printhead, attaining the reference trajectory is of importance during
124
7.2
APPLICATION OF FEEDFORWARD CONTROL
−6
4
x 10
Integrated sensor signal [Vs]
3
2
1
0
−1
−2
0
10
20
30
40
50
Time [µs]
60
70
80
90
100
Figure 7.7: Integrated sensor signal; without ILC (black), with ILC (gray), and
chosen reference trajectory (black dotted)
35
30
25
Input [V]
20
15
10
5
0
−5
−10
0
10
20
30
40
50
Time [µs]
60
70
80
90
100
Figure 7.8: Actuation pulse; standard trapezoidal (black dotted) and resulting
ILC pulse (gray)
7.2
PIEZO-BASED ILC
125
the complete duration of one jetting cycle. For a jetting frequency of 10 kHz,
this amounts to 100 µs. In contrast, the actuation is restricted to a certain time
interval to be able to increase the jetting frequency without the immediate necessity of overlapping actuation pulses. In this case, the actuation is limited to
the first 60 µs. Consequently, the jetting frequency can be increased to 16.6 kHz
without overlapping actuation signals. Note that the adopted restriction of 60
µs is not the absolute minimum length of the actuation window. Nevertheless, a
further decrease of this window deteriorates the attainable performance considerably. For jetting frequencies beyond the 16.6 kHz, the superposition principle for
linear systems is used. Having assumed linearity of the jetting process at least in
the piezo-based case, this is a valid approach.
The sensor signal resulting from a standard trapezoidal and the learned ILC pulse
are shown in Fig. 7.7. The accompanying actuation pulses are shown in Fig. 7.8.
Based on Fig. 7.7, the conclusion is drawn that the reference trajectory is attained satisfactorily. Since the first part of reference trajectory up to the firing of
a drop is the same as realized by the standard trapezoidal pulse, it is not surprising that the learned ILC pulse resembles the standard trapezoidal pulse for the
first part. After that, the ILC controller adjusts the actuation pulse such that
the fluid-mechanics follow the desired trajectory in presence of the restriction of
the actuation interval. In Fig. 7.8, it can be seen that the ILC actuation pulse
counteracts the pressure oscillation. The peaks just before 60 µs originate from
the fact that the ILC controller cannot actuate beyond 60 µs while it is required
that the channel is in rest after 60 µs nonetheless. If desired, these peaks can be
suppressed by additional weightings.
−4
2.5
x 10
2
IAE
1.5
1
0.5
0
2
4
6
8
10
12
Iteration number
14
16
18
20
Figure 7.9: Integrated absolute error of the error signal against the trial number
126
APPLICATION OF FEEDFORWARD CONTROL
7.2
−6
1.2
x 10
1
CPSD [V]
0.8
0.6
0.4
0.2
0
1
10
2
10
Frequency [kHz]
Figure 7.10: Cumulative power spectrum of the error signal; standard trapezoidal
(black) and ILC pulse (gray)
The IAE criterion for the discussed ILC experiment is depicted in Fig. 7.9. Though
convergence occurs monotonously here, in general this is not the case. Especially
during the early stages of learning the IAE might temporarily deteriorate compared to a previous trial. Though this might affect the drop properties during
operation negatively, usually only a few iterations or equivalently a couple of microseconds are involved. In Fig. 7.10, the cumulative power spectrum (CPS) of
the error of the first and last trial is depicted. Based on Fig. 7.10, it is concluded
that the largest error reduction takes place around the channel’s first resonance
frequency at 45 kHz. This is in correspondence with our observation that the
channel’s response and thus the residual vibrations are governed by the this first
resonance frequency.
Finally, the DOD-speed curve is obtained to assess the effect of minimization of
the residual vibrations on the attainable jetting frequency and hence productivity.
In Fig. 7.11, the DOD-speed curve is depicted for the standard trapezoidal and
the ILC learned actuation pulse. Note that for frequencies beyond 16.6 kHz, the
ILC actuation pulses are superposed as discussed above.
The location of the local minima and maxima of the DOD-speed curve in Fig. 7.11
for the standard trapezoidal actuation pulse can be linked to the occurring residual
vibrations as follows. In Fig. 7.7, these residual vibrations are depicted. Typically, it takes approximately 150 µs for these residual vibrations to be completely
damp out. If the jetting frequency increases, the time between two successive
pulses decreases. Therefore, at a certain jetting frequency, the channel is not
rest anymore if the consecutive actuation pulse is given. Assuming linearity, the
7.2
PIEZO-BASED ILC
127
5
4.5
Droplet speed [m/s]
4
3.5
3
2.5
2
5
10
15
DOD frequency [kHz]
20
25
Figure 7.11: DOD (drop-on-demand) curve; standard trapezoidal (black) and ILC
pulse (gray)
response of an ink channel can be obtained by superposing two responses as depicted in Fig. 7.7 at the appropriate time instant. For example, at 11.8 kHz and
12.8 kHz, or a time between two actuation pulses of 85 µs and 78 µs equivalently,
the overlapping responses amplify and attenuate each other, respectively. Consequently, a local maximum and minimum results. A similar reasoning holds for
the local maximum at 15.8 kHz (63 µs) and minimum at 18.2 kHz (55 µs) and
the subsequent minima and maxima. Additionally, structural modes of the PIJ
printhead itself influence the course of the DOD-curves also.
Since residual vibrations are minimized with the ILC actuation pulse, the phenomenon of attenuating or amplification is eliminated, theoretically at least up
to a jetting frequency of 16.6 kHz. Based on Fig. 7.7, it is concluded that this
is the case. For jetting frequencies beyond 16.6 kHz, the ILC actuation pulse
still outperforms the standard actuation pulse. Typically, 15 % deviations from a
nominal drop-speed are allowed given the desired print quality. Given a nominal
drop-speed of 3.5 m/s, the lower and upper bound on the drop-speed are 3.0 and
4.0 m/s, respectively. These boundaries are indicated in Fig. 7.11. As can be
seen, the ILC learned actuation pulse reduces the speed variations such that the
jetting frequency can be increased up to 25 kHz.
Finally, both DOD curves show a positive linear trend for frequencies up to approximately 15 kHz. This trend is caused by the wetting of the nozzleplate, see
128
7.2
APPLICATION OF FEEDFORWARD CONTROL
[Nag06]. Wetting is the phenomenon that the nozzleplate is covered with a thin
layer of ink. Among other things, it slows down the resulting drop and deteriorates the jet straightness. Since the wetting decreases with an increase of the
jetting frequency, the positive trend can be explained. The only effective measure
to counteract this phenomenon aims at developing a non-wetting nozzleplate.
3
3
2.5
2.5
2
2
1.5
1.5
sensor signal B [V]
sensor signal A [V]
7.2.2 MIMO ILC: minimizing cross-talk
1
0.5
1
0.5
0
0
−0.5
−0.5
−1
−1
−1.5
−1.5
0
10
20
30
40
50
Time [µs]
60
70
80
90
100
0
10
20
30
40
50
Time [µs]
60
70
80
90
100
Figure 7.12: Integrated sensor signal of channel A (left) and channel B (right);
without ILC (black), with ILC (gray), and chosen reference trajectory (black
dotted)
In this section, MIMO ILC is applied to an array of two ink channels, A and B,
to simultaneously minimize the effect of cross-talk and residual vibrations. The
same control structure as in the SISO case can be adopted here, albeit with an
adjustment to the structure of the various signals and matrices. The reference
trajectories for both channels are constructed as follows. Starting point forms the
response of each channel to a standard trapezoidal pulse without the neighboring
channel jetting. This guarantees the absence of cross-talk. From this point, the
construction is equal to that in the SISO case. In Fig. 7.12, the resulting reference
signals are depicted in case both channels are to be jetting.
Despite the fact that the impulse response matrix is doubled in size compared to
the SISO case, the LQ-optimal ILC design approach can still be used. The β and
γ values are the same as those in the SISO piezo-based case. Given our focus on
the minimization of cross-talk, the limitations concerning the actuation interval
is omitted. Starting point for the ILC synthesis form the transfer functions as
depicted in Fig. 7.2.
The resulting sensor signals from the standard trapezoidal and learned ILC actuation pulses are shown in Fig. 7.12. The accompanying actuation pulses are
7.2
PIEZO-BASED ILC
129
40
35
30
25
Input A [V]
20
15
10
5
0
−5
−10
0
10
20
30
40
50
Time [µs]
60
70
80
90
100
Figure 7.13: Actuation pulse; standard trapezoidal (black dotted), the resulting
ILC pulse for channel A (black) and channel B (gray)
depicted in Fig. 7.13. In Fig. 7.12, small differences between both reference trajectories are visible. Apparently, ink channel A and B are not completely identical as assumed. Consequently, the learned ILC pulses for channel A and B differ
slightly also. This is not bothersome, since both reference trajectories are attained
satisfactorily and the required performance level is met. As discussed previously,
the first part of reference trajectories up to the firing of a drop is the same as
realized by the standard trapezoidal pulse. As a result, the learned ILC pulses
resemble the standard trapezoidal pulse for the first part, though there are some
considerable deviations. They can be accounted for by the fact that the ILC controller is counter-acting the cross-talk effects. After the jetting of the drops, the
ILC controller adjusts the actuation pulses such that the fluid-mechanics follow
the desired trajectory for the damping of the residual vibrations.
The convergence of channel A and B in terms of the IAE is depicted in Fig. 7.14.
The CPS of the resulting error signals of the standard and learned ILC actuation pulses are depicted in Fig. 7.15. As can be seen in Fig. 7.14, convergence is
achieved in approximately 20 iterations. An error reduction of a factor of 3.4 and
2.6 is achieved for channel A and B, respectively. Based on Fig. 7.15, it concluded
that the largest error reduction takes place around the first resonance frequency at
45 kHz. This is similar to our findings in the SISO case. The differences between
channel A and B in both Fig. 7.14 and 7.15 can be attributed to the differences
between the channels.
In Table 7.1, the cross-talk curve for channel A and B is listed. The resulting
130
7.2
APPLICATION OF FEEDFORWARD CONTROL
−4
1.8
x 10
1.6
1.4
IAE
1.2
1
0.8
0.6
0.4
5
10
15
Iteration number
20
25
Figure 7.14: Integrated absolute error of the error signal against the trial number;
channel A (black) and channel B (gray)
−6
1
x 10
0.9
CPS standard A
CPS ILC A
CPS standard B
CPS ILC B
0.8
0.7
CPS [V]
0.6
0.5
0.4
0.3
0.2
0.1
0
1
10
2
10
Frequency [kHz]
Figure 7.15: Cumulative power spectrum of the error signal of channel A and B;
standard trapezoidal (black) and ILC pulse (gray)
7.2
PIEZO-BASED ILC
131
drop-speed if only one ink channel is actuated is listed first. The drop-speed of
this ink channel if a neighboring channel is actuated simultaneously is listed in the
following columns using the standard trapezoidal and ILC learned actuation pulse.
The deviations in drop-speed in both cases are a measure of the drop-consistency.
In case of the standard trapezoidal actuation pulse, the drop-speed consistency
is considerably less than in case an ILC approach is adopted. A similar result is
obtained for the drop-volume consistency. In conclusion, MIMO ILC can be used
to minimize the effect of cross-talk and consequently improve the drop-consistency.
channel A
channel B
standard
single
3.96 m/s
3.37 m/s
standard
double
3.28 m/s
2.58 m/s
ILC
double
4.20 m/s
3.56 m/s
standard
variation
17.2 %
23.4 %
ILC
variation
6.1 %
5.7 %
variation
reduction
64.7 %
75.9 %
Table 7.1: Comparison of drop-speed with and without ILC
Up to this point, it has been assumed that the ink channels of a PIJ printhead are
identical. Based on the results presented in this section, this assumption has been
proven to be not valid. To investigate the influence of these differences on the
attainable performance, the same ILC experiments have been conducted using all
four transfer functions Ha , Hb , Hab , and Hba . The results show that although the
convergence rate is slightly higher, the performance is the same. However, the
experiments conducted in this section have been performed on two neighboring
channels. Channels that are located further apart from each other may show more
differences in channel dynamics and hence performance. Altogether, for the array
of channels considered in this section, the assumption still holds.
7.2.3 Constrained MIMO ILC
The final piezo-based ILC experiment discussed in this section is the implementation of constrained ILC. To establish the effect of the imposed constraints, the
same experiments are conducted in the unconstrained setting. As argued in Section 6.4, ASIC limitations require the use of piece-wise affine actuation signals
only. To that purpose, the ILC algorithm has been adjusted such that only ILC
pulses are learned that fulfill this requirement. In Fig. 6.9, the adopted control
structure is depicted. For the implementation of constrained MIMO ILC, the reference trajectories are chosen such that channel A is jetting and channel B is at
rest. The residual vibrations present in channel A are damped. The corresponding reference trajectories are constructed similarly to the previous two cases and
are depicted in Fig. 7.16.
The same ILC controller as used for the unconstrained MIMO ILC case has been
132
3.5
0.25
3
0.2
2.5
0.15
0.1
2
0.05
sensor signal B [V]
sensor signal A [V]
7.2
APPLICATION OF FEEDFORWARD CONTROL
1.5
1
0.5
0
−0.05
−0.1
0
−0.15
−0.5
−0.2
−1
−1.5
−0.25
0
10
20
30
40
50
Time [µs]
60
70
80
90
100
0
10
20
30
40
50
Time [µs]
60
70
80
90
100
Figure 7.16: Integrated sensor signal of channel A (left) and B (right); without
ILC (black), with constrained ILC (gray), with ILC (gray dotted), and chosen
reference trajectory (black dotted)
used for the experiments shown in this section. For the constrained case, the
algorithm is adjusted as discussed in Section 6.4. Furthermore, the number of
switching instances is chosen as 11. The location of these instances is determined
based on physical insight: the switching instances are tuned to the channel’s first
resonance frequency, that dominates the residual vibrations.
35
3
2.5
30
2
25
1.5
20
Input B [V]
Input A [V]
1
15
0.5
0
10
−0.5
5
−1
0
−1.5
−5
0
10
20
30
40
50
Time [µs]
60
70
80
90
100
−2
0
10
20
30
40
50
Time [µs]
60
70
80
90
100
Figure 7.17: Actuation pulse for channel A (left) and B (right); standard trapezoidal (black), the resulting constrained ILC pulse (gray), and the ILC pulse (gray
dotted)
The resulting sensor signals from the unconstrained and constrained ILC actuation pulses are shown in Fig. 7.16. The accompanying actuation pulses are
depicted in Fig. 7.17 for both channel A and B. The small nonzero values of
7.3
LASER-VIBROMETER BASED ILC
133
the actuation pulses at 100 µs do not influence the performance negatively. These
nonzero values can be avoided by the use of additional constraints. It is concluded
that the reference trajectories are attained satisfactorily. Given the fact that the
resulting sensor signals are quite similar, it is concluded that constrained ILC is
capable of attaining similar performance as its unconstrained version. Apparently,
the actuation signal can be simplified considerably to meet the requirements for
the implementation on an ASIC without sacrificing too much performance. The
corresponding CPS for channel A and B are shown in Fig. 7.18.
−6
1
−7
x 10
5
4.5
0.8
4
0.7
3.5
0.6
3
CPS B [V]
CPS A [V]
0.9
0.5
2.5
0.4
2
0.3
1.5
0.2
1
0.1
0.5
0
1
10
x 10
2
10
Frequency [kHz]
0
1
10
2
10
Frequency [kHz]
Figure 7.18: Cumulative power spectrum of the error signal of channel A (left)
and B (right); standard trapezoidal (black), constrained ILC pulse (gray), and
ILC pulse (gray dotted)
7.3 Laser-vibrometer based ILC
In this section, laser-vibrometer based SISO ILC is implemented on the 233e01
printhead. As discussed previously, the major limitation of the accompanying
experiments is the restriction to non-jetting regimes only. The measurement configuration does not allow the jetting of a drop. This would cause the measurement
to stop. Therefore, all experiments are carried out using 2.5 V. To obtain the
DOD-speed and -volume curves, the resulting ILC actuation pulse is scaled to a
jetting voltage. Note that various consequences of this restriction is discussed in
detail in the next section.
The adopted control structure is shown in Fig. 6.2. In Fig. 7.19, the used reference
trajectory at 2.5 V is depicted. The construction of this trajectory is performed
the same way as in the piezo-based ILC case. The ILC controller has been designed using the LQ-design approach based on with the fitted transfer function
134
7.3
APPLICATION OF FEEDFORWARD CONTROL
1
0.8
Meniscus velocity [m/s]
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
0
10
20
30
40
50
Time [µs]
60
70
80
90
100
Figure 7.19: Meniscus velocity at 2.5 V without ILC (black), with ILC (gray),
and chosen reference trajectory (black dotted)
at 2.5 V, see Fig. 7.5. Finally, note that the actuation window is limited to 50 µs.
3
2.5
2
Input [V]
1.5
1
0.5
0
−0.5
−1
0
10
20
30
40
50
Time [µs]
60
70
80
90
100
Figure 7.20: Actuation pulse without ILC (black dotted), with ILC (gray)
The resulting sensor signal from the standard trapezoidal and learned ILC actuation pulse at 2.5 V are shown in Fig. 7.19. The accompanying actuation pulses
are depicted in Fig. 7.20. Similar to the previous cases, the first part of reference
7.3
LASER-VIBROMETER BASED ILC
135
trajectory up to the firing of a drop is roughly the same as realized by the standard trapezoidal pulse. Therefore, it is not surprisingly that the learned ILC pulse
resembles the standard trapezoidal pulse for the first part. The higher voltage
during the first part result mainly from the differences as discussed at Fig. 3.18.
The convergence is depicted in Fig. 7.21. The CPS of the resulting error signal of
the standard and learned ILC actuation pulse is depicted in Fig. 7.22. In Fig. 7.21,
it can be seen that convergence is achieved in approximately 10 iterations. A reduction of a factor of 5.2 is achieved. In Fig. 7.22, it can be seen that the largest
error reduction takes place at the first resonance frequency around 45 kHz as in
the previous cases.
70
60
IAE
50
40
30
20
10
0
5
10
15
Iteration number [−]
20
25
Figure 7.21: Integrated absolute error of the error signal against the trial number
For the measurement of the DOD-speed and -volume curves, the laser-vibrometer
configuration has been taken away. The learned actuation pulse at 2.5 V has been
scaled up to 30 V and implemented on the experimental setup. Using this scaled
pulse, the DOD-speed and DOD-volume curves have been measured, see Fig. 7.23.
Based on Fig. 7.23, it is concluded that the variations are reduced considerably.
A number of remarks are noteworthy. First, the drop-speed is considerably higher
compared to the DOD-speed curve measured with the 293e02 printhead. This is
caused by the differences in both printheads. Second, the DOD curves measured
with the learned ILC pulse show a certain offset compared to the DOD curves
obtained with the standard trapezoidal actuation pulse. The main reason for this
offset lies in the used reference trajectory. In contrast to the reference trajectories
used previously, the damping of the residual vibrations is imposed approximately
10 µs earlier, see Fig. 7.19. Since the drop-formation process is still ongoing at 25
µs, see Fig. 4.7, the drop is so to speak hold back. More specifically, the tail of the
drop that is still connected to the drop is decelerated and slows down the drop
136
APPLICATION OF FEEDFORWARD CONTROL
7.4
itself as well as reduces its volume. Third, the linear trend visible in Fig. 7.11
is almost not present here. The direct control of the meniscus itself rather than
the related channel pressure provides a better mean to control the wetting of the
nozzleplate. This can be explained as follows. The meniscus trajectory is now
confined such there hardly is any overfill. In the piezo-based case, the meniscus
position simply cannot be controlled so directly. Apparently, in the latter case
there is (more) overfill and thus wetting.
0.4
0.35
0.3
CPSD [V]
0.25
0.2
0.15
0.1
0.05
0
1
10
2
10
Frequency [kHz]
Figure 7.22: Cumulative power spectrum of the error signal; standard trapezoidal
(black) and ILC pulse (gray)
Based on the results presented in this section, it is concluded that the nonlinearities can be handled by the proposed ILC approach. Despite the undeniable
presence of these nonlinearities and the limitations of the measurement setup,
the performance can still be improved considerably by the application of laservibrometer based ILC. Although for the application of ILC considered here the
system was assumed to behave linear, the validity of the linearity assumption
needs additional research. To further enhance the performance of this particular
ILC approach, current research strive for the integration of a sensor in the nozzle
to replace the laser-vibrometer sensor. Details can be found in [Gro06a].
7.4 Discussion
In this section, the ILC approaches presented in the preceding sections are reviewed. To start with, the experimental results are evaluated in light of the
formulated control objectives. Next, several issues concerning the implementation of ILC to PIJ printheads are considered. Then, various subjects for further
research regarding the current application of ILC are brought up. Finally, some
7.4
137
DISCUSSION
42
9
8.5
40
8
38
7.5
Drop volume [pl]
Drop speed [m/s]
36
7
6.5
6
34
32
5.5
30
5
28
4.5
4
8
10
12
14
16
18
20
Jet frequency [kHz]
22
24
26
28
30
26
8
10
12
14
16
18
20
Jet frequency [kHz]
22
24
26
28
30
Figure 7.23: DOD (drop-on-demand) speed- (left) and volume (right) curve of
the 233e01; standard trapezoidal (black) and scaled ILC pulse (gray)
fundamental limitations of the current PIJ printhead design are discussed.
To assess the performance of ILC, the experimental results are evaluated based
on the following ILC objectives as formulated in Section 6.3:
• Enhancing the productivity. In the previous two sections, it has been demonstrated that the productivity of a PIJ printhead can be improved in two
ways. First, due to the active damping of the residual vibrations, the jetting
frequency can be increased up to approximately 25 kHz. Second, minimization of cross-talk by means of ILC renders the use of the so called bridge
structure obsolete. Consequently, the npi-ratio can at least be doubled.
Both the use of higher jetting frequencies and the increase of npi enhances
the productivity considerably.
• Improving the drop-consistency. It has been shown that application of
MIMO ILC can improve drop-consistency. More specifically, drop-speed
variations have been reduced from 20.3 % to 5.9 % on average.
Apart from the actual realized performance improvements, it is demonstrated
that ILC is a suitable control strategy to overcome the current boundaries with
respect to at least productivity and drop-consistency. Furthermore, it is expected
that performance with respect to these two formulated control objectives can be
increased even further. In the sequel of this section, various research directions
are pointed out to further enhance the performance. First, however, the following
ILC-printhead related issues are considered:
• Piezo- versus laser-vibrometer based ILC. From a performance point of view,
the question arises whether piezo- or laser-vibrometer based ILC is more favorable. As argued in Section 6.3, eventually, the drop-formation is the
138
APPLICATION OF FEEDFORWARD CONTROL
7.4
most important performance determining process of a PIJ printhead. If the
meniscus would be completely observable and controllable using the piezounit, using the piezo- or laser-vibrometer based ILC approach would not
make any difference. However, as discussed in previous chapters, this is
not the case. Therefore, one might be inclined to attribute one’s preference to the laser-vibrometer based approach. On the other hand, based on
the experimental results presented in this chapter, the conclusion must be
drawn that both approaches achieve almost similar performance, neglecting the small differences in the various printheads for convenience. However, whereas the piezo-based approach is quite straightforward, the laservibrometer based approach certainly is not. This can be attributed mainly
to the limitations imposed by the laser-vibrometer as sensor. It is therefore expected that the attainable performance in the laser-vibrometer based
approach can be further increased if the sensor functionality is improved.
Altogether, further research in the controllability and observability of the
meniscus and the development of more suitable sensor functionality in the
nozzle is needed to decide in this issue.
• Linearity of the jetting process. In Section 7.2 and 7.3, linearity of the
jetting process has been assumed. The obtained experimental results did not
provide reasons to question the validity of this assumption. Whereas this is
not surprising for the piezo-based case, validity is certainly not trivial for the
laser-vibrometer based case. After all, in Chapter 5, nonlinear behavior has
been demonstrated in the laser-vibrometer based case. Irrespective of which
ILC approach is adopted, it remains to be seen to what extent linearity
can be assumed. For example, the use of superposition of ILC actuation
pulses to obtain the DOD curve at high frequencies or to decouple a (large)
array of ink channels may lead to new insights with respect to the validity
of the linearity assumption. More specifically, if superposition gives rise
to actuation signals of high actuation voltages, linearity may be lost. In
conclusion, for the experiments conducted thus far, linearity of the jetting
process has been a valid assumption. However, further research is required
to reach a final conclusion.
• Uniformness of ink channels. In this thesis, it is assumed that all ink channels are identical. The validity of this assumption can be questioned based
on the observation that two neighboring channels already show differences
in channel dynamics, albeit small. Measurement of the frequency response
of ink channels further apart show that the variations in channel dynamics
can indeed become considerable. Nevertheless, the main channel resonance
frequency usually only differs a few kHz. At this point, insufficient experiments have been conducted to support a finite conclusion regarding the
uniformness of ink channels. Further research is required. It is expected
that the assumption regarding channel uniformness can be used at least in
7.4
DISCUSSION
139
one particular case. If the β parameter is chosen such that the ILC algorithm only takes the channel’s first eigenfrequency into account, the use of
this assumption is justified.
• Robustness of the ILC approach. In case of a PIJ printhead, there are two
cases of robustness to be considered:
1. Robustness against model uncertainty. Model uncertainty can originate
from for example model mismatches (e.g. due to wrongfully assuming
channel uniformness) or aging. As discussed in Section 6.4, the robustness against model uncertainty of the ILC approach can be increased
by the parameter β. Given the experiments presented in this chapter,
it is concluded that the robustness against model uncertainty of the
current ILC approaches is sufficient.
2. Robustness against disturbances. Disturbances that can occur include
dirt-particles entering the nozzle and air-bubbles sucked in the nozzle.
To improve the robustness against disturbances the reference trajectories can be used. For example, by limiting the retraction of the
meniscus in the nozzle, the chance of sucking in air-bubbles is reduced.
Still, it is not trivial how to improve the robustness in these cases and
requires further research.
Various subjects for future research have been discussed above. In addition, the
following more general topics are of interest:
• Further optimization of the ILC approaches. The ILC approaches as employed in this thesis can be improved with respect to several issues. First
and foremost, the design of the reference trajectories can be improved. The
trajectories used to obtain the results in this chapter are based on the response to a standard actuation pulse. As indicated Section 6.3, there are
many alternatives to be investigated for various purposes. Next, only the
use of actuation windows has been investigated. As discussed in Section 6.2,
weight filters can be employed to the same purpose offering more freedom
in the design of the ILC pulses. Additional research is required to establish
the possible advantages of this approach. Third, the actuator and sensor
functionality is to be improved, especially the sensor used for the meniscus.
Finally, the constrained MIMO ILC approach can be developed further.
Though the fixation a priori of the number and location of the switching
instances renders the corresponding optimization problem linear, it also possibly limits the performance of the constrained ILC algorithm. Extending
the algorithm to allow the algorithm itself to determine the number and
location of the switching points can improve the constrained ILC approach
considerably.
140
APPLICATION OF FEEDFORWARD CONTROL
7.4
• Application of ILC to improve stability and enable DSM. As discussed in
Section 6.3, it is expected that ILC can be used to improve the jet stability
and enable the use of DSM. Research in both subjects is required.
• Development of a decoupling strategy for large arrays of ink channels. Application of ILC to a large array of ink channels calls for the development
of a decoupling strategy. Given the fact that an inkjet typically consists of
an array of 100 to 300 ink channels and that depending on the data to be
printed many different actuation schemes are used, it is simply not realistic
to learn for every possible occurring situation. One option is to assume linearity and superpose the various actuation signals. However, as mentioned
above, validity of the linearity assumption remains to be seen in case too
high actuation voltages are present. There are many alternatives, e.g. the
use of a (static) decoupling matrix. This requires additional research.
Finally, some of the fundamental limitations discussed in Chapter 5 are revisited
given the experimental results obtained in this chapter:
• The channel’s first eigenfrequency. As discussed previously, the dynamic
behavior of an ink channel is dominated by its first eigenfrequency. Under
certain conditions, the most energy-efficient actuation pulses are tuned to
that particular frequency. The PIJ printheads considered in this thesis form
no exception to both observations. The minimum required time for one jetting cycle is thus also determined. For example, if the first eigenfrequency
is 45 kHz, a minimum jetting cycle of a multiple of 22 µs (typically two to
three) is required to jet a drop and damp the residual vibrations without too
high actuation voltages. This roughly corresponds to the results obtained
in this section. As a result, the attainable jetting frequency is limited, even
when ILC is applied. This also has been demonstrated in this chapter. The
DOD curves can be improved up to a certain jetting frequency. Beyond that
frequency, the DOD curve based on an ILC curve deteriorates also.
There are a few solutions possible. First, rather than designing an actuation pulse for one drop only, pulses can be designed for multiple drops.
This requires research in reference trajectories, see the discussion above.
Second, the design can be adjusted to facilitate higher jetting frequencies.
For example, the channel’s length can be decreased such that the its first
eigenfrequency is decreased. Another adjustment to the printhead’s design
concerns the piezo-unit as actuator. A division of the piezo-unit in multiple
piezo-units allows for the use of a completely different actuation strategy of
an ink channel. For example, a drop could be extruded out of a channel. As
a result, one avoids the use of the channel’s eigenfrequency thereby lifting
the corresponding constraint in attainable jetting frequency.
7.5
CONCLUDING REMARKS
141
• Spatial controllability and observability of the jetting process. As discussed
in Chapter 5, the spatial controllability and observability of the PIJ printheads investigated in this thesis is limited. Without proper actuation and
sensing functionality, the attainable performance is limited. For example, if
the piezo-sensor indicates that the ink channel is at rest, there still may be
traveling pressure waves present. Various experiments with shorter piezounits confirm this observation. To enhance both the controllability and
observability, the following adjustments to the PIJ printhead design is suggested. First, the piezo-unit is to be divided in multiple piezo-units. Second,
a sensor in the nozzle is to be incorporated.
7.5 Concluding remarks
Based on the experimental results presented in this chapter, the following main
conclusions are drawn:
• The suitability of ILC as control strategy to enhance a PIJ printhead’s
performance in face of commonly encountered operational issues has been
demonstrated. Minimization of the residual vibrations and cross-talk has
been proven to be very profitable in terms of the productivity and dropconsistency of a PIJ printhead. For the further exploration of ILC many
research directions have been pointed out.
• The operation of a PIJ printhead can be regarded as linear for the experiments conducted in this thesis. Further research is necessary to establish
the validity of this assumption in face of various other experiments to be
conducted in the near future.
• The two major limitations of current PIJ printheads relate to the first eigenfrequency of a particular design and the limited spatial controllability and
observability of the jetting process. To overcome both issues, a re-design
of PIJ printheads with respect to the actuator and sensor functionality is
of crucial importance. For the design at hand, a division of the current
piezo-unit in multiple piezo-units as well as the development of a sensing
device in the nozzle is advised.
Chapter 8
Conclusions and recommendations
In the beginning of this thesis, the importance of inkjet technology was sketched.
The commonly encountered limitations of PIJ printheads were discussed and a
solution strategy was pointed out (Chapter 1) that led to the research objective
specified into three research questions (Chapter 2). In this chapter, the research
is concluded and the recommendations are presented.
8.1 Conclusions
The conclusions presented in this section are categorized according to the three
research questions as formulated in Chapter 2.
Question 1: How should a PIJ printhead be modeled given its intended use for
the proposed systems and control approach?
Based on the research objective of this thesis, several requirements for the modeling of a PIJ printhead have been formulated. First and foremost, a suitable
model is to provide insight in the working of a PIJ printhead. Second, the model
complexity is to be kept as low as possible while maintaining the model accurate
enough for the use for control and (re-)design. Current available models fail to
satisfy these requirements simultaneously and are therefore not completely suited
for the purposes in mind.
The key to the successful modeling of an ink channel forms the use of bilaterally
coupled systems (BCS). For one, this concept not only fixes an appropriate internal model structure, but also provides an explicit role for the surroundings acting
on a system. To apply this concept to an ink channel, the following - generally
applicable - approach to the modeling of PIJ printheads has been followed. An ink
channel is divided in several functional blocks. Main guidelines for the division
143
144
CONCLUSIONS AND RECOMMENDATIONS
8.1
chosen is the geometry of an ink channel. Subsequently, the dynamic behavior of
these blocks is modeled using first principles only. During the various derivations,
it is assumed that the jetting process behaves linearly. For each block, the resulting dynamical equations are transformed into the two-port formulation as part
of BCS. Finally, the coupling of all the blocks is performed by the application
of Redheffer’s star product. A model of an array of channels can be obtained
by the coupling of an arbitrary number of ink channel models. In that case, the
coupling as well as the accompanying cross-talk effects are facilitated by means
of the actuating function of the actuator.
The resulting so called two-port model fulfills the requirements for the modeling
of a PIJ printhead as formulated a priori to a large extent. To start with, due to
the chosen modeling strategy, the resulting model has relatively low complexity.
Also, experimental validation shows that the resulting model is accurate for the
piezo-based case. In the laser-vibrometer based approach, however, the two-port
model is to be improved with respect to the first and most important resonance
frequency. Third, the two-port model provides insight in the working of an ink
channel as well from a systems and control perspective. The obtained results presented in this thesis show the two-port suitability for the control and (re-)design
purposes in mind.
Based on the resulting two-port model, one important observation concerning the
ink channel dynamics is the following. Apparently, the dynamic behavior of an
ink channel can be represented by an extremely low dimensional system, in our
case a 4th order. Further research is required to further explore this possibility.
To further improve the two-port model, several research directions have been indicated. These include the use of more complex nozzle models, the upgrading of the
one-sided coupling between the nozzle and the drop formation to a two-sided one,
the further development of the piezo-unit modeling, and the adding of damping
to the reservoir block.
Question 2: Can we design actuation wave forms which will be implemented
as feedforward control such that the performance of current PIJ printheads is
improved?
The three most prominent performance criteria for a PIJ printhead are its productivity, drop-consistency, and stability. The focus of the research presented in this
thesis lies on the former two. The attainable performance with respect to these
two issues is limited by two commonly encountered operational issues: residual vibrations and cross-talk. In this thesis, it has been demonstrated that feedforward
control, more specifically ILC, is a suitable control strategy to overcome these two
issues and hence increase the performance of PIJ printheads considerably beyond
current limits.
8.1
CONCLUSIONS
145
Given the available sensor functionalities, two ILC approaches have been investigated: piezo- and laser-vibrometer based ILC. Also, the limitations for the implementation of ILC as posed by the ASIC have been resolved. Based on our
exploration of the possibilities of feedforward control of a PIJ printhead, the following conclusions are drawn:
• Productivity. The productivity of an individual ink channel as well as a complete array can be increased in two ways by the implementation of ILC. To
start with, due to the active damping of the residual vibrations, the attainable jetting frequency can be increased with a factor 2.5 for the printheads
under consideration. Given an admissible deviation in drop-speed of ± 0.5
m/s from a nominal value, the jetting frequency can be increased from 10
up to 25 kHz. A second effect results from the minimization of cross-talk.
Since the bridge-structure becomes redundant, the npi-ratio can at least be
doubled. As a result, one has more nozzles per inch available for jetting.
Both effects contribute to an enhancement of the productivity.
• Drop-consistency. The drop-consistency can be considerably improved by
the application of ILC. It has been demonstrated that the variations in dropspeed can be reduced from 20.3 % to 5.9 % on average for the printheads
under investigation.
Although it has not been experimentally demonstrated in this thesis, it is expected
that feedforward control can also improve a PIJ printhead’s performance with
respect to the following issues:
• Stability of the jetting process. Feedforward control can be applied to design
actuation pulses such that the stability of the jetting process is improved.
Stability is closely related to the meniscus retraction. For example, it has
been argued that limiting this retraction reduces the risk of entrapping an
air-bubble leading to nozzle failure.
• Drop-speed and -volume (modulation). The ILC control structure uses a
certain reference trajectory to learn an actuation pulse that results in a
drop of some predefined properties. By switching between various reference
trajectories, drop properties such as speed and volume can be varied during
operation.
Given the exploratory character of the research presented in this thesis, it is
expected that the performance can be further increased with respect to the productivity and drop-consistency of a PIJ printhead. Several recommendations are
provided in the next section. Stability and the on-demand realization of certain
drop properties by means of feedforward control is to be investigated.
146
CONCLUSIONS AND RECOMMENDATIONS
8.1
Two important assumptions have been used throughout this thesis and in particular during the application of control, namely linearity of the jetting process and
the uniformness of the ink channels. From a control perspective, the former has
proven its validity during the implementation of ILC, at least in the considered
cases. Further research is required to establish to what extent this assumption
remains valid. It is noted that the implications of the validity of the linearity
assumption are eminent: it facilitates the application of a systems and control
approach and the accompanying range of (optimization) tools considerably. Validity of the latter assumption has not been conclusively determined in this thesis.
In view of the intended extension of the ILC framework to a complete printhead,
it certainly deserves further research. For an array of two channels considered
here, the assumption was valid.
Since the proposed ILC feedforward control strategy is generally applicable, the
results and conclusions presented in this thesis are not limited to the specific
printhead design that has been investigated.
Question 3: Can we improve current PIJ printheads such that some basic limitations with respect to the attainable performance are lifted?
During the derivation of the two-port model and the implementation of ILC and
the accompanying discussions, several new limitations of PIJ printheads have
become apparent. The following more fundamental limitations have emerged:
• Observability of the jetting process. The observability of the jetting process
is limited by the current sensor functionality (piezo-unit) in two ways. To
start with, the placing of the sensor functionality is not optimal. As argued
in this thesis, the preferred sensor location is in the nozzle where the actual
drop-formation takes place. Adding or relocating sensor functionality would
enhance the observability of the process. Second, the resolution of the piezo
used as sensor is limited. Several smaller wave-forms are not sensed due to
the fact that the nett contribution in pressure distribution over the piezo’s
surface is zero. Incorporating multiple smaller piezo-sensors would improve
the observability of the jetting process.
• Controllability of the jetting process. The controllability of the jetting process
is limited by the current actuator (piezo-unit) in three ways. First, the
length of the piezo-unit is too long to be able to generate several pressure
wave patterns. Ultimo, this limits the performance of a PIJ printhead. The
length now equals the length of the ink channel. Incorporating multiple
smaller piezo-actuators can lift this limitation regarding the controllability.
Second, the controllability of the meniscus movements using the piezo-unit
as actuator is limited. Some movements simply cannot be generated in the
nozzle. Consequently, some drop properties cannot be formed. Third, with
8.2
RECOMMENDATIONS
147
the current printhead design it is very difficult to simultaneously damp both
the ink channel and nozzle. Again, this affects the performance negatively,
in particular the drop-consistency. Incorporating an additional actuator in
the nozzle can lift this limitation.
• Dominancy of the first eigenfrequency. Considering the piezo-unit’s constraints with respect to the admissible actuation voltage and utilizing the
energetically most favorable actuation mode, the actuation is tuned on the
ink channel’s first eigenfrequency. Consequently, the residual vibrations are
then dominated by the same frequency. It is demonstrated that the damping is limited by this frequency also. Altogether, the attainable jetting
frequency of a printhead therefore is limited depending on the used eigenfrequency during actuation. To overcome this boundary, the design is to be
adjusted such that the eigenfrequency is increased (shorter ink channels) or
multiple piezo-units are incorporated rather than just one.
8.2 Recommendations
The recommendations for further research are formulated as follows:
• Further development of the two-port model. The accuracy of the resulting two-port can be improved with respect to the following issues. First,
the extension of one ink channel to an array of multiple channels (a PIJ
printhead) is to be performed. This requires the proper incorporation of
cross-talk effects. Since the derived equations are in principle capable of
handling these effects, only the exact determination of cross-talk is to be
further investigated. Second, to enable future investigations into formation
of smaller drops, the incorporation of more complex nozzle models is to be
considered. A related issue concerns the coupling of the nozzle dynamics
with the drop formation. At present, these two are linked by a one-sided
rather than a two-sided coupling is adopted. The quality of the two-port
model would be enhanced if the correct two-sided coupling were incorporated. Third, the boundary condition that represents the reservoir is to be
improved with respect to the damping it introduces. The current necessity
to add damping then would become superfluous. Further development of
the two-port model with respect to these points will improve the insight
that is obtained by the systems and control approach to the modeling of a
PIJ printhead.
• Further exploration of control (ILC). The application of feedforward control to a PIJ printhead is to be further explored. First, ILC can be employed to improve the PIJ printhead performance with respect to several
uninvestigated issues such as stability. Second, the robustness against varying or changing dynamics and disturbances of the ILC approach requires
148
CONCLUSIONS AND RECOMMENDATIONS
8.2
research. Third, the application of ILC to enhance the productivity and
drop-consistency is in need of further optimization. In our view, the attained performance with respect to these two issues can be even further
increased. The key to all these issues, irrespective whether it deals with
further extension or improvement of the ILC approach, lies in the design of
the reference trajectories. It is expected that tuning of these trajectories
based on physical insight is highly profitable. Finally, the application of ILC
to an array consisting of more than two ink channels is to be investigated.
• Employment of the insight obtained from the derivation of the two-port model
and the application of control for PIJ printhead (re-)design. The insight that
results from the derivation of the two-port model and the application of control is to be used for the (re-)design of PIJ printheads, e.g. the established
fundamental limitations. Also, a design can be evaluated from a systems
and control perspective using the two-port model approach. This way, the
performance of a design can be optimized a priori.
Appendix A
Hamiltonian ILC design
In this appendix, an alternative for the LQ-optimal ILC design approach as presented in Section 6.4 is discussed: the Hamiltonian based ILC design. An important motivation for considering an alternative is found in the length of the
reference trajectory. The longer the trajectory, the numerically more difficult the
LQ-optimal computations become. The Hamiltonian approach offers a numerically attractive alternative.
Two major differences between the Hamiltonian and the LQ-optimal ILC design
approach can be distinguished. To start with, an alternative method is used to
select the observable part of the impulse response matrix H, thereby avoiding
the singular value decomposition. Second, the computations for the update law,
requiring the inverse of a matrix of size H, are handled differently.
Rather than using the singular value decomposition, it is possible to select the
observable part of H using a non-square identity matrix I˜ = [I 0]T of size N ×
(N − m), where m is the number of (nearly) zero singular values. Note that this
matrix replaces V1 in the sense that it removes the last columns of H. Define the
new output and feedback matrix as:
H̃ = H I˜
L̃ = I˜T L
(A.1)
respectively. Using H̃, the measurement horizon is m samples longer than the
control signal. The optimal control problem (6.12) with Q = I and R = βI
becomes:
149
150
APPENDIX A
J=
N
X
ykT Qyk + ∆uTk R∆uk
k=1
=
N
X
uTk H̃ T H̃uk + β∆uTk ∆uk
(A.2)
k=1
The corresponding Riccati equation then equals:
−X̃(βI + X̃)−1 X̃ + H̃ T H̃ = 0
(A.3)
The solution of (A.3) can be approximated by:
X̃ = H̃ T H̃ + βI
(A.4)
Substitution of this approximate solution in the Riccati equation shows that it
is a solution of an optimal control problem with a slightly different weighting
Q̃. Since it still is a solution of an optimal control problem (Q̃ > 0), it gives a
stable solution and hence a convergent ILC. The feedback interconnection matrix
L̃ equals:
L̃ = X̃ −1 H̃ T = (H̃ T H̃ + βI)−1 H̃ T
(A.5)
and the corresponding update law:
∆uk = (H̃ T H̃ + βI)−1 H̃ T ek
(A.6)
For long trajectories, matrices H̃, L̃, and X̃ can become very large. To avoid
numerically intensive computations, e.g. to compute the inverse of (A.4), it is
possible to obtain the responses for any length of the trajectory with a simulation
of a Hamiltonian system, hence the naming Hamiltonian based ILC design. This
system comprises two linked difference equations, one with a forward (causal) recursion and one with a backward (anti-causal) recursion, based on a state space
model of the process and the weighting parameter β. It is assumed that the system has not a relative degree of zero and thus has a throughput matrix D = 0.
Our aim is to obtain a realization of (A.5). A block diagram of the update law
(A.6) is depicted in Fig. A.1. Suppose that H̃ has a state-space representation
(A, B, C) with zero state initial condition at k = 0, simulated with a forward
recursion:
xk+1 = Axk + B∆uk
ỹk = Cxk
(A.7)
HAMILTONIAN ILC DESIGN
151
ek
H
ỹk
+
∆uk
+
ṽk
H̃
− β1
Figure A.1: Graphical representation of the update law
x and k denote the system’s state and time, respectively. H̃ T has a state space
representation (AT , B T , C T ) with zero state initial condition k = N −1, simulated
with a backward recursion:
qk−1 = AT qk + C T (ỹk + ek )
1
∆uk = − B T qk
β
(A.8)
Combining the linked state space descriptions (A.7) and (A.8) leads to the following system:
1
BB T qk
β
qk−1 = AT qk + C T (Cxk + ek )
1
∆uk = − B T qk
β
xk+1 = Axk −
(A.9)
Suppose that A is invertible, then:
1
BB T (A−T qk−1 − A−T C T Cxk − A−T C T ek )
β
qk = A−T qk−1 − A−T C T (Cxk + ek )
1
∆uk = − B T (A−T qk−1 − A−T C T Cxk − A−T C T ek )
β
xk+1 = Axk −
or equivalently in state space:
(A.10)
152
APPENDIX A
qk
xk+1
=
∆uk =
A−T
−A−T C T C
qk−1
− β1 BB T A−T A + β1 BB T A−T C T C
xk
−T T
−A C
+
ek
(A.11)
1
T −T T
BB
A C
β
qk−1
1
− β1 B T A−T β1 B T A−T C T C
+ B T A−T C T ek
xk
β
(A.11) contains a Hamiltonian matrix as system matrix. This matrix has n stable and n anti-stable eigenvalues. There exists a similarity transformation that
separates the stable and anti-stable part:
I
−S
0
−I
A−T
−A−T C T C
A + β1 BB T A−T C T C
!
A−T C T C
A
− β1 BB T A−T
A
=
−T
0
I
−S
0
−I
(A.12)
with S = S T the stabilizing solution of the DARE:
S = ASAT − ASC T (I + CSC T )−1 CSAT +
1
BB T
β
(A.13)
T
and A the stable closed-loop matrix:
T
A = AT − C T (I + CSC T )−1 CSAT
The corresponding transformed variables are:
qk−1
I
0
qk−1
qk−1
=
=
wk
−S −I
xk
−xk − Sqk−1
(A.14)
(A.15)
such that (A.11) becomes:
qk
wk+1
−T
=
∆uk =
A
0
−T
A
T
C C
A
!
− β1 B T A−T (C T CS + I)
+
−A−T C T
+
(S − β1 BB T )A−T C T
qk−1
− β1 B T A−T C T C
wk
qk−1
wk
1 T −T T
B A C ek
β
ek
(A.16)
with initial and terminal conditions w0 = −Sq−1 and qN −1 = 0, respectively. The
result is the sequence ∆uk which is the new input to the memory block.
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Glossary of symbols
Symbols
Ach
Aco
b
c
ceff
cw
C
d
ek
ez
f
h
hp
H
k
K
l
L
Lch
Lco
Ln
n
N
p
Pr
Ps
q
Q
r
R
Cross-sectional area of the ink-channel
Cross-sectional area of the connection
Boundary velocity
Speed of sound
Effective speed of sound
Wave propagation velocity
Piezo capacity
Piezo-electrical charge constant
Error signal at trial k
Outward normal in the positive z-direction
Sample frequency
Height of the free surface
Piezo thickness
Discrete time Hankel matrix
Piezo stiffness
Maximum displacement of the piezo’s zeroth order mode
Length of the ink cylinder
Learning filter
Length of the ink-channel
Length of the connection
Length of the nozzle
Outward normal
Trial length
Pressure
Remanent polarization
Saturation polarization
Electric charge
Weighting on ILC error/input
Nozzle radius
Weighting on ILC control effort
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164
GLOSSARY OF SYMBOLS
sE
S
t
tsw
u
uk
U
v
vav
vd
vr
vz
V
Vd
Wi , Wo
xi
X
yk
yref
z −1
z1 , z2
Z
α
β
γ
∆uk
∆φ
ǫT
λ
λi
µ
ν
ρ
σi
σ
Σ
φ
ω
Compliance for a constant electrical field E
Surface area
Time
Switching instant
Displacement
Input signal at trial k
Matrix with singular output vectors
Velocity
Average velocity
Drop velocity
Relative velocity
Meniscus velocity in the z-direction
Matrix with singular input vectors
Drop volume
Input and output weighting filters
Diagonal entries of matrix X
Stabilizing solution of a DARE
Output signal at trial k
Reference trajectory
Discrete time delay operator
Flow
Impedance
Kinetic energy correction factor
Momentum-flux correction factor, or ILC tuning parameter
Scalar learning gain
Update of the input signal at trial k
Phase lag of the laser-vibrometer
Permittivity under constant stress T
Wave length
Closed-loop poles in the trial domain
Dynamic viscosity
Surface tension
Density
i-th singular value
Viscous stress tensor
Diagonal matrix with singular values on the main diagonal
Flow
Angular frequency
Abbreviations
ASIC
Application Specific Integrated Circuit
GLOSSARY OF SYMBOLS
BCS
CCD
CFD
CIJ
CPS
CS
CV
DARE
DOD
DSM
ETFE
FEM
FPD
FPGA
FR
FRD
FRF
FSI
HP
IAE
IBM
ILC
LCD
LTI
LQ
MAC
MIMO
NPI
OE
PCB
PID
PIJ
PLED
RCA
REI
RFID
SISO
SRI
SV
SVD
TF
TIJ
VOF
165
Bilaterally Coupled System
Charge Coupled Device
Computational Fluid Dynamics
Continuous Inkjet
Cumulative Power Spectrum
Control Surface
Control Volume
Discrete time Algebraic Riccati Equation
Drop-on-Demand
Drop Size Modulation
Empirical Transfer Function Estimate
Finite Element Method
Flat Panel Display
Field Programmable Gate Array
Frequency Response
Frequency Response Data
Frequency Response Function
Fluid Structure Interaction
Hewlett Packard
Integrated Absolute Error
International Business Machines corporation
Iterative Learning Control
Liquid Cristal Display
Linear Time Invariant
Linear Quadratic
Marker And Cell
Multiple-Input Multiple-Output
Nozzles Per Inch
Output Error
Printed Circuit Board
Proportional, Integrating, and Differentiation feedback control
Piezo-electrical Inkjet
Polymer Light Emitting Diode
Radio Corporation of America
Recognition Equipment Institute
Radio Frequency Identification
Single-Input Single-Output
Stanford Research Institute
Stream-function Vorticity
Singular Value Decomposition
Transfer Function
Thermal Inkjet
Volume Of Fluid
Summary
Inkjet printhead performance enhancement by feedforward input design based on two-port modeling
Inkjet technology is an important key-technology from an industrial point of view.
Its ability to deposit various types of material on a substrate in certain patterns
makes it a very versatile technology. Not surprisingly, the variety of applications is very wide, ranging from standard document printing to the fabrication of
flat panel displays. Applications of inkjet technology are often accompanied with
tight performance criteria. Usually, these include specifications concerning several
drop-properties, such as speed and volume, and the consistency of those properties. Also, requirements for the jetting process itself are frequently imposed, e.g.
with respect to the productivity and stability. Whereas current performance criteria are quite stringent already, they are expected to become even tighter in the
near future.
A typical design of a piezo-electrical inkjet (PIJ) printhead comprises a large array
of piezo-actuated channels. The shape of the corresponding actuation pulses is
determined by manually tuning based on physical insight such that the requested
drop-on-demand results. However, this approach in combination with printhead
designs has become mature and its possibilities have been exhausted, especially in
face of some operational issues that are generally encountered: residual vibrations
and cross-talk. The former issue relates to the fact that the ink in a channel is
usually not at rest immediately after drop ejection. On average, it takes approximately 100 µs for the pressure waves to be damped such that a next drop can be
fired. Cross-talk refers to the fact that if one channel is actuated, the fluid mechanics in neighboring channels are also actuated. This results in different drop
properties if neighboring channels are actuated simultaneously or shortly after
one another. Altogether, both phenomena limit the productivity as well as the
drop-consistency, and hence the performance, of PIJ printheads considerably.
In this thesis, a systems and control approach to the functioning of PIJ printheads is proposed to break current boundaries. The aim is threefold. First, such
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SUMMARY
an approach to the modeling of an inkjet printhead provides good insight in its
working that can be used for the control and redesign purposes in mind. Second,
application of feedforward control is a very cost-effective method to improve a PIJ
printhead’s performance, e.g. concerning its productivity and drop-consistency.
Finally, the proposed approach helps identifying several more fundamental limitations of PIJ printheads that can be taken into account in future designs.
The key in the modeling of an ink channel from a systems and control perspective
is to view the system as a series of bilaterally coupled subsystems. Additionally,
to keep the model complexity low, the dynamics of each of these subsystems are
modeled using first principles only. This is also achieved by coupling the various blocks by using the Redheffer star product rather than staggered schemes.
Despite the low model complexity of the resulting so called two-port model, it
is still accurate enough to serve as starting point for the intended control and
redesign purposes. For one, the two-port model provides sufficient physical insight in the jetting process to facilitate the implementation of feedforward control.
Given the repetitive character of the jetting process, the Iterative Learning Control (ILC) framework is used as feedforward control strategy. In this framework,
reference trajectory design plays a crucial role in achieving the control objectives, i.e. the minimization of residual vibrations and cross-talk. The chosen ILC
framework also enables a reduction of the time required for actuation while still
attaining the formulated control objectives. Furthermore, a modified ILC algorithm is presented that allows for the design of piece-wise affine actuation pulses.
This is necessary to overcome the limitations posed by the electronics of a PIJ
printhead, that can only handle extremely simplified actuation pulses. Finally,
ILC is implemented on various PIJ printheads using either the pressure in an
ink channel or the meniscus velocity as sensor signal. The experimental results
demonstrate that by meeting the control objectives a considerable improvement
of the performance with respect to the drop-consistency and the productivity can
be achieved.
Upon using the systems and control approach for PIJ printheads, several more
fundamental limitations of the design emerge, e.g. concerning the maximally attainable jetting frequency and the spatial observability and controllability. At the
same time, based on the insight obtained several adjustments to the design are
proposed to overcome even those.
M.B. Groot Wassink
Samenvatting
Prestatieverbetering van inkjet printkoppen door het via feedfoward
technieken ontwerpen van aanstuursignalen op basis van tweepoort
modellering
Inktjet technologie vormt een belangrijke sleuteltechnologie voor de industrie.
De mogelijkheid om verschillende soorten materiaal op een substraat te kunnen
printen in zekere patronen maakt de technologie tot een zeer breed inzetbare. Het
mag daarom geen verassing heten dat het spectrum aan toepassingen zeer breed
is, variërend van het printen van documenten tot de fabricage van zogenaamde
platte beeldschermen. Doorgaans gelden er voor de toepassingen van inktjet technologie strikte prestatie-eisen. Zo is het gebruikelijk dat er eisen worden gesteld
aan diverse druppeleigenschappen, zoals snelheid en volume, evenals de consistentheid daarin. Daarnaast worden vaak eisen gesteld aan het jet-proces zelf,
zoals bijvoorbeeld betreffende de productiviteit en de stabiliteit. Ondanks het
feit dat de huidige prestatie-eisen al vrij hoog liggen, wordt verwacht dat deze
steeds strenger worden in de nabije toekomst.
Een typisch ontwerp van een piezo-electrische inktjet (PIJ) printkop omvat een
aanzienlijk aantal piezo-geactueerde kanalen naast elkaar. De bijbehorende actuatie pulsen worden vastgesteld door handmatig tunen op basis van fysisch inzicht
zodat de gewenste druppel resulteert. Echter, deze aanpak in combinatie met
verschillende printkop ontwerpen is uitontwikkeld en de mogelijkheden die het
biedt zijn uitgeput, zeker gezien enkele veel voorkomende operationele problemen: residuale trillingen en overspraak. Het eerstgenoemde probleem betreft het
verschijnsel dat de inkt in een kanaal niet direct in rust is nadat er een druppel is
gejet. Het duurt gemiddeld gezien ongeveer 100 µs voordat de drukgolven zodanig
zijn uitgedempt dat een volgende druppel kan worden gejet. Overspraak is de benaming voor het verschijnsel dat een bepaald kanaal niet geactueerd kan worden
zonder dat de buurkanalen dit ook worden. Een gevolg is dat druppeleigenschappen variëren wanneer buurkanalen gelijktijdig of kort na elkaar worden geactueerd.
Al met al, beide verschijnselen beperken de productiviteit en de druppelconsistentie, en daarmee dus ook de prestatie van PIJ printkoppen aanzienlijk.
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170
SAMENVATTING
In dit proefschrift wordt een systeem en regelaanpak voor het functioneren van
PIJ printkoppen voorgesteld om de huidige grenzen te doorbreken. Het doel daarvan is driedelig. Ten eerste biedt een dergelijke aanpak voor het modelleren van
een inkjet printkop goed inzicht in de werking dat weer gebruikt kan worden voor
regel en herontwerp doeleinden. Ten tweede vormt feedforward regelen een prima
kosten-effectieve manier om de prestatie van PIJ printkoppen te verbeteren, bijvoorbeeld als het gaat om de productiviteit en de druppelconsistentie. Tot slot
helpt deze aanpak om de meer fundamentele beperkingen van PIJ printkoppen
aan het licht te brengen. Kennis op dat vlak kan weer gebruikt kunnen worden
bij toekomstige ontwerpen.
De sleutel tot succes bij het modelleren van een inkt kanaal vanuit een systeem en
regelperspectief is door het systeem als een serieschakeling van tweezijdig gekoppelde systemen te beschouwen. Om de complexiteit van het model laag te houden,
wordt vervolgens de dynamica van elk van deze subsystemen gemodelleerd door
uitsluitend gebruik te maken van first principles. Dit wordt ook bereikt door de
diverse blokken te koppelen met behulp van het Redheffer star product in plaats
van staggered schemes. Ondanks de lage complexiteit van het resulterende zogenaamde tweepoort model, is het nog steeds nauwkeurig genoeg om als startpunt
te dienen voor de beoogde regel en herontwerp doeleinden. Zo biedt het tweepoort model voldoende fysisch inzicht in het jet-proces om de implementatie van
de feedforward regeling te vergemakkelen. Gegeven het repeterende karakter van
het jet-proces wordt het Iterative Learning Control (ILC) raamwerk gebruikt als
feedforward regelstrategie. In dit raamwerk speelt referentie trajectorie ontwerp
een cruciale rol in het bereiken van de regeldoelen, namelijk het minimaliseren
van residuale trillingen en overspraak. Verder maakt het gekozen raamwerk het
mogelijk om de benodigde tijd voor actueren te reduceren zonder dat dit het behalen van de geformuleerde regeldoelen aantast. Vervolgens wordt een aanpast
ILC algoritme geı̈ntroduceerd waarmee stuksgewijs-affine actuatie signalen kunnen worden ontworpen. Dit is noodzakelijk om goed om te kunnen gaan met de
beperkingen van de electronica van een PIJ printkop, die slechts extreem vereenvoudigde actuatie pulsen aankan. Tot slot wordt ILC toegepast op verschillende
PIJ printkoppen waarbij als sensor signaal ofwel gebruik wordt gemaakt van de
druk in een kanaal of de meniscus snelheid. De experimentele resultaten laten
zien dat door de regeldoelen te behalen de productiviteit en de druppelconsistentie aanzienlijk verhoogd kunnen worden. Door gebruik te maken van een systeem en regelaanpak voor PIJ printkoppen komen een aantal meer fundamentele
beperkingen van de ontwerpen aan het licht, zoals bijvoorbeeld de maximaal haalbare jet-frequentie en de ruimtelijke regel- en waarneembaarheid. Tegelijkertijd
worden op grond van het verkregen inzicht verschillende aanpassingen aan het
ontwerp voorgesteld waarmee deze overwonnen kunnen worden.
M.B. Groot Wassink
Curriculum Vitae
January 15, 1978
Born in Leiden, The Netherlands
1990 - 1996
VWO (pre-university education), Stedelijk Gymnasium, Leiden, The Netherlands
1996 - 2002
MSc student Mechanical Engineering at Delft University of
Technology, Delft, The Netherlands, with a specialization in
Systems and Control. Graduated cum laude with a MSc thesis on Linear Parameter Varying control for a wafer stage, for
which research was conducted at Philips Center for Industrial
Technology, Eindhoven, The Netherlands
2002 - 2006
PhD student Mechanical Engineering, Systems and Control group, at Delft University of Technology, Delft, The
Netherlands. This PhD research was sponsored by OcéTechnologies, Venlo, The Netherlands
2006 - present
Project manager at Boer & Croon Young Executives, Amsterdam, The Netherlands
171
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