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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Problem formulation 1 1 8 9 9 11 16 21 2.1 The research objective . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2 A decomposition in research questions . . . . . . . . . . . . . . . . 22 2.3 The structure of this thesis . . . . . . . . . . . . . . . . . . . . . . 24 3 Experimental exploration 25 3.1 Description of the experimental setup 3.1.1 Piezo sensor signal . . . . . . . 3.1.2 CCD camera . . . . . . . . . . 3.1.3 Laser-Doppler interferometry . . . . . 25 27 31 31 3.2 Description of the experimental printheads . . . . . . . . . . . . . 33 3.3 Identification method . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.4 Piezo-based experimental identification . . . . . . . . . . . . . . . . 3.4.1 With bridge-structure . . . . . . . . . . . . . . . . . . . . . 3.4.2 Without bridge-structure . . . . . . . . . . . . . . . . . . . 40 40 43 3.5 Laser-vibrometer based experimental identification . . . . . . . . . 45 3.6 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 47 iii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 49 53 55 60 71 78 82 86 88 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 91 92 94 96 99 . . . . . . . 101 101 102 107 112 112 116 118 . . . . . . . . 119 119 123 123 128 131 133 136 141 5 Model validation 5.1 5.2 5.3 5.4 5.5 Introduction . . . . . . . . . . . . . Piezo-based validation . . . . . . . Laser-vibrometer based validation Discussion . . . . . . . . . . . . . . Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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. 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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 163 164 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 167 168 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. 169 170 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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