MICROSTRIP RESONATORS FORHIGH SPEED OPTICAL COMMUTATOR SWITCHESbyMINGCHE CHENM .A. Sc. Beijing Institute of TechnologyA THESIS SUBMITTED IN PARTIAL FULFILMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE FACULTY OF GRADUATE STUDIES(The Department of Electrical Engineering)We accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAMarch 1992© Mingche Chen, 1992In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission. Department of ^ELEC TR 1- CAL E/1/0147EER/A/The University of British ColumbiaVancouver, CanadaDate Apl. 2/ /Y/2DE-6 (2/88)AbstractUsing a planar microwave resonator as the electrode for modulating a high speedY-branch optical commutator switch is a novel idea. For this purpose a "cul-de-sac"shape microstrip resonator is suggested and studied both theoretically and experimentally.It has two substantially parallel legs unconnected on one end and connected via an openring on the other. It is the parallel legs that work as the electrode. Because of the storedenergy the potential difference between the two legs can be much higher than the sourcevoltage. Therefore, the resonator electrode has significant advantages over other kindsof electrodes. By using a resonator electrode the optical switch can operate at highfrequencies while consuming little power.The field distribution, discontinuity effects, and the radiation loss of the "cul-de-sac" resonator are discussed. The structure parameters of the "cul-de-sac" resonator, wereoptimized for maximum output voltage and minimum consumed power using a computerprogram. The source power can be coupled into the resonator by tap point couplingthrough a quarter-wave-length transformer. The input impedance of the resonator andthe coupling coefficient B were derived as functions of the coupling position 0 using anequivalent lumped circuit. Then an important relationship between the output voltage V 0 ,the source voltage V„ and the unloaded quality factor Q. is derived which tells us thatVo is proportional to the product of V, and the square root of Q. From this relationshipwe proved that the most efficient and practical coupling condition is B51.To verify the design theory experimental "cul-del-sac" resonators operating at 7GHz using Al203 substrates were designed, fabricated, and tested. Two new platingmethods for thick and fine metal strips, the double photoresist patterning and thealuminum mask methods, were tried to overcome the edge effects and thus reduce thelosses. Gold layers 6 A thick with smooth surfaces were plated successfully.Also a new method for measuring O. using a scalar network analyzer is described.From the plot of the return loss vs frequency Q u can be calculated directly. It wasproved that if the minimum return loss is 10.7 dB then Q u equals the ratio of the resonantfrequency to the 3dB bandwidth. The designed and measured Q, of the "cul-de-sac"resonator fabricated on Al203 were 123 and 179 respectively. The reasons for thisdifference are discussed and the error of the measurement was estimated to be about10%.Applying the design theory and the fabrication techniques developed here to a"cul-de-sac" resonator on a LiNbO 3 substrate, at an operating frequency of 15 GHz, asupply power of 100 mW should result in an output voltage of about 50 V between theparallel legs of the "cul-de-sac" resonator. It was, therefore, concluded that a "cul-de-sac" resonator used as the electrode on a LiNbO 3 substrate, could satisfy the requirementsof a high speed optical commutator switch.iiiTable of ContentsAbstract ^Table of Contents ^ ivList of Tables viList of Figures ^ viiAcknowledgements Chapter I Introduction ^ 11.1 Motivation of This Thesis ^ 11.2 The Electro-optic Switch 21.2.1 The Working Principle 21.2.2^Current Problems in High Speed OpticalSwitches ^ 41.2.3 A Resonator as the Electrode ^ 51.3 Organization of the Thesis ^ 8Chapter II Design Theory ^ 102.1 General Ideas ^102.1.1 Resonator Basics ^102.1.2 The "cul-de-sac" Resonator ^132.2 Analysis of the Microstrip Resonator ^172.2.1 Discontinuities ^182.2.2 Current and Potential Distributions ^212.3 Energy Stored and Power Lost in the "cul-de-sac"Resonator ^ 252.3.1 Stored Energy ^ 252.3.2 Attenuation Losses 262.3.3 Radiation Loss 272.4 Optimized Design of the Structure Parameters ^ 37Chapter III Coupling and Matching ^ 403.1 The Coupling Method Discussion ^ 403.2 Equivalent Circuit and Input Impedance 41iv3.3 Coupling Condition ^ 46Chapter W Fabrication of the Resonator ^ 524.1 Introduction 524.2 The Fabrication Method 534.2.1 Metallization ^534.2.2 Plating Problems and Solution ^ 544.3 Fabrication Procedures ^ 624.3.1 Mask Design 624.3.2 Preparation of Wafers ^ 644.3.3 Metallization and Patterning 644.3.4 Plating ^654.3.5 Cutting and Grinding the Sample ^ 67Chapter V Testing and Results Comparison ^ 695.1 Introduction ^695.2 The Measurement of Qu ^ 695.3 Measurement Procedure 825.3.1 The Measurement Set-Up ^ 835.3.2 The Measurement Principles 835.3.3 Measurement Procedure and Techniques ^ 855.3.4 Measurement of 8, fo and RL ^ 885.4 Measurement Results and Discussion 905.4.1 Calculating Qu ^905.4.2 Discussion ^915.5 Conclusion ^ 92Chapter VI Conclusion and Suggestions for Further Research ^ 936.1 Conclusions ^ 936.2 Suggestion for Further Research ^ 95References ^ 97List of TablesTable 4.1^The Designed Conductor Strip Thickness ^ 54Table 5.1^The measured return losses and bandwidths, and thecorresponding unloaded quality factor Qu ^viList of FiguresFigure 1.1^Y-branch optical switch ^ 3Figure 1.2^Y-branch optical commutator switch using a "cul-de-sac" resonatorelectrode ^ 6Figure 2.1^Equivalent circuit of the transmission line resonator with a tappoint coupling mechanism ^ 11Figure 2.2^Different kinds of microstrip line resonators ^15Figure 2.3^Top view of the "cul-de-sac" resonator^ 16Figure 2.4(a) An ideal microstrip half-wave linear resonator ^ 19Figure 2.4(b) The voltage distribution in the ideal microstrip half-wavelinear resonator ^19Figure 2.5(a) The equivalent microstrip linear resonator of the microstrip "cul-de-sac" resonator with impedance steps ^ 22Figure 2 . 5 (b) The voltage distribution on the "cul-de-sac" microstripresonator ^ 22Figure 2.6(a) The structure of stripline ^ 29Figure 2.6(b) The structure of microstrip line 29Figure 2.7^Coordinate system for calculating the radiation loss from the "cul-de-sac" resonator ^ 31Figure 2.8^Equivalent strip width and radius of the open ring ^ 33viiFigure 2.9 The voltage between the open ends of the parallel legs vs thedissipated power ^ 39Figure 3.1(a) The transformer tap point coupling to the "cul-de-sac" resonator ^ 42Figure 3.1(b) The transformer tap point coupling to the equivalent linear resonatorof the "cul-de-sac" resonator ^ 43Figure 3.2^Shunt transmission line equivalent circuit for "cul-de-sac" resonatorwith tap point coupling mechanism ^ 44Figure 3.3^Lumped element equivalent circuit for "cul-de-sac" resonator . • • 47Figure 4.1^The outline of the cross-section of the plated parallel microstriplines using photoresist mask ^ 58Figure 4.2 The cross-sections of the pattern during the plating procedure andelectrical charges distribution on the metal surface with photoresistmask 60Figure 4.3 The cross-sections of the pattern during the plating procedure andelectrical charges distribution on the metal surface with aluminummask 61Figure 4.4 The cross-section of the plated pattern recorded by TENCORprofilemeter ^ 63Figure 4.5^The plating system 66Figure 4.6 Fabricated samples ^ 68Figure 5.1(a) Transmitted power measurement system ^ 71viiiFigure 5.1(b) The equivalent circuit of the transmitted power measurementsystem ^ 71Figure 5.2(a) Reflected power measurement system ^ 73Figure 5.2(b) The equivalent circuit of the reflected power measurementsystem ^ 73Figure 5.3 A plot of normalized reflected power vs frequency recordedby network analyzer ^ 79Figure 5.4 The set-up for measuring the resonator parameters ^ 84Figure 5.5 Probing procedures ^ 87ixAcknowledgmentsI would like to express my gratitude to my supervisor, Dr. N. Jaeger, forsuggesting this project and providing support during the course of my work.I am also grateful to Dr. E.V. Jull, for his support in the first half year of mystudies.My thanks extend to our research engineer H. Kato for helping me to overcomemany of the fabrication difficulties that I encountered. My thanks also go to all myfriends in the solid state group and those individuals in the Electrical Engineeringdepartment who have helped me in various ways through out my studies.Last, but certainly not least, to my dear wife, I owe her infinite thanks for herhelp, understanding, and patience. Without her help I can not imagine how I wouldcoped.Chapter I. Introduction1.1 Motivation of This ThesisOptical switches and modulators are basic devices in optical communication.Various electro-optically modulated external modulators and switches fabricated inLiNbO3 have been reported, such as Ti diffused waveguide intensity modulators andpolarization modulators, directional coupler modulators, Mach-Zehnder interferometermodulators, and Y-branch modulators[1-6]. Among them, Y-branch optical modulatorshave been shown to have advantages over other types for high speed switches in opticalcommunications because of their non-interferometric nature[7]. They have been used asdigital optical switches and time division multiplexers/demultiplexers[8,9]. A Y-branchoptical switch consists of two parts, an optical waveguide in the form of a Y-branch anda modulating electrode. The design of the electrode plays an important role indetermining the operating speed and the power dissipation of the modulator. In thisthesis a new electrode configuration for use in integrated optical commutator switchesusing Y-branch optical modulators is studied. It is a "cul-de-sac" shaped microstripresonator, by which we mean an open ring with two parallel legs extending from theopening. The design theory, fabrication procedure, and measurement techniques aredescribed. Using such electrodes should allow one to realize a high speed optical switchwith a relatively low power consumption. Resonator electrodes designed here shouldwork up to 15 GHz with power dissipation of less than 100 mW. Operation at such high1frequencies will make it possible to take advantage of the wide bandwidth available usinglow loss and low dispersion single mode optical fibres.1.2 The Electro-Optic SwitchThe lay-out of an integrated optic Y-branch switch is shown in Fig. 1.1. An inputoptical waveguide branches into two arms at the Y-junction. Two electrodes, normallya section of coplanar waveguide or coplanar strip, are placed near or at the junction.When using such an optical switch, its switching speed, on/off ratio, percentage guidedpower, and power consumption are all factors that will determine its usefulness. Herethe on/off ratio is defined as the ratio between the power guided in the on and off statesand the percentage guided power is given by the total power that is transferred from theinput to the outputs. It should be pointed out that while in this thesis we assume the Y-branch to be dividing pulses between two waveguides, it can also be used inversely tocombine the pulses in the two waveguides to the main waveguide.1.2.1 The Working PrincipleThe integrated optic Y-branch switch operates by the linear electro-optic effect.Here we are concerned with devices formed by Ti indiffusion in LiNbO 3 . The Y-branchwaveguide is covered by a thin optical buffer layer. The electrodes are placed above thejunction to apply an electric field to the optical waveguide to change the refractive index2elect rode length leInput light output lightY - branchwavegu de^electrodeFig.1.1 Y-branch optical switch3distribution in the guiding region of the modulator. The change in the refractive indexobeys the following relation[10]:ti(—r = rEwhere n is the refractive index, r is the appropriate electro-optic coefficient, E is theapplied electric field. The distribution of the electric field is such that the refractiveindex of one branch, and its related part of the horn region, will be increased while thatof the other branch is decreased. The input optical wave, travelling down the mainwaveguide, will be steered alternatively into one or the other of the branches dependingon the modulating signal. When a resonator is used the light will be steered into eachof the branches on alternate half cycles of the resonant frequency. The modulating effectseen by the guided optical wave is proportional to the applied voltage.1.2.2 Current Problems in High Speed Optical SwitchesIn Y-branch modulators the on/off ratio increases with increasing branch anglewhile the percentage guided power decreases[11]. Both the on/off ratio and thepercentage guided power increase with increasing modulating voltage. Y-branch opticalswitches have been designed with a very small branch angle and long horn lengths to givea high percentage guided power[l 1]. Correspondingly, the required electrode is long.Having a long electrode results in two problems; first, it has a large capacitance whichreduces the switching speed[12], second, it introduces phase velocity mismatches between4the optical and the modulating signals at high frequencies. Both these problems can beovercome by increasing the branch angle so as to reduce the length of the horn and theelectrode. However, this results in a higher modulating voltage being necessary toachieve a large on/off ratio as well as a high percentage guided power. Normally,increasing the voltage results in increased power consumption, because the deliveredpower P is proportional to the square of the voltageD V2& =Zo(1.2)where V is the voltage and zo is the characteristic impedance of the transmission line.For Z0 =50 n, the power has to be increased by 5.5 W to boost the voltage from 25 to30 V. In fact in terms of power consumed per unit bandwidth these devices are amongstthe best[13]. Short lengths also have added advantages in terms of real estate andmaterial related costs and losses.1.2.3 A Resonator as the ElectrodeA large branch angle Y-branch optical modulator with a high on/off ratio and areasonably high percentage guided power has been realized[14]. It is clear that anelectrode, which can apply a high voltage but needs only a little power would be a keyelement for a high speed optical switch. We are proposing a planar resonator as onesolution. Fig. 1.2 shows a Y-branch optic switch with a "cul-de-sac" resonator electrode.The basic character of a resonator is that it can store electromagnetic energy. The quality5output I ightparallel legsopen ringcul-de-sac"resonator electrodeY-branch waveguideFig. 1.2 Y-branch optical commutator switch using a "cul -de-sac" resonator electrodefactor of a planar resonator is defined as the ratio of the stored energy to the averagepower lost per cycleStored EnergyQuality factor Q = 2ir^power lost per cycleFor our purposes a quality factor of 100 is needed, while this may not be seen as highfor its type, i.e., a planar microstrip resonator at the frequencies proposed, it is. If noother loss mechanism exists the power lost inside the resonator should equal to the sourcepower, that is to say that the stored energy is more than 100 times the energy suppliedper cycle by the source. This energy is stored in the resonator in the form of bothvoltage and current. Therefore, the resonator may be used as a voltage transformer,having a much higher potential difference between its ends than that on the line feedingit. The following relationship is derived in chapter HI of this thesisV;^4p^2Q17,2. (1.3)Z, 0+02 n Z,orv...^2V, 2Q„ Z, (1.4)(1 +3)\7E^Zswhere Vr is the voltage in the resonator ( amplitude of the standing wave ), Z r is thecharacteristic impedance of the transmission line constituting the resonator, V, is the7voltage output by the source, Z. is the source impedance, B is the coupling coefficientbetween the source and the resonator(it can be designed so that B =1). Using such aresonator, with the source power of 0.1 W, we can obtain a potential difference of about50 V between the ends of the parallel legs of the "cul-de-sac" resonator, for a frequencyof 15 GHz and perhaps higher frequencies. Voltages of about 50 volts are sufficient foruse with certain Y-branch optical modulators[15]. Whereas, in a normal 50 II coplanarstrip electrode with the same source power only 3.2 volts can be obtained between thetwo conductor strips. Compared with the coplanar trips, our resonator increases thevoltage by a factor of 16.1.3 Organization of ThesisThis thesis consists of six chapters. Chapter I is the Introduction.Chapters II and III are devoted to the design theory of the resonator for use as theelectrode for a Y-branch optical commutator switch. In chapter II we concentrate on thestructure of the resonator. In chapter III we deal with the coupling and matchingproblems between the resonator and the external power system. After studying in detailthe voltage, current, and power distributions inside the resonator, the calculation of thestored energy, the ohmic and dielectric attenuations, the radiation loss, the unloadedquality factor, the input impedance, and the coupling coefficient, a computer programwas written to optimize the design and structure of a "cul-de-sac" shaped resonator.Chapter IV is a discussion of the fabrication techniques used to make our samples.8In Chapter V we describe the testing procedures used and discuss the results andmeasurement accuracy. Also, a new method for measuring the unloaded quality factorQu using a scalar network analyzer is presented.Finally, in Chapter VI the conclusions and suggestions for further work are given.9Chapter II Design Theory2.1 General Ideas2.1.1 Resonator BasicsGenerally, a resonator is described by its quality factor Q. Q is defined by thefollowing ratio= wws^(2.1)1Where W, is the stored energy and P 1 the average power lost per cycle in the resonator.It is usual to define three types of quality factor, each of which depends on the lossmechanism[15]. If P1 is the loss inside the resonator the Q is called the unloaded qualityfactor Qu. If the resonator is coupled to an external load and P 1 denotes the powerabsorbed thereby the Q is called the external qiinlity factor Q, u. When P1 is the totalloss, both internal and external, the Q is the loaded quality factor QL. Using thesedefinitions we have the following relationship[16]1^1^1QL QN Qat(2.2)A transmission line resonator can be described by its equivalent circuit as shownin Fig.2.1, in which a parallel resonant circuit with lumped elements resistor R o ,capacitor Co, and inductor Lo is coupled to the generator on the left. The coupling100Fig. 2.1 Equivalent circuit of the transmission line resonator with a tap point couplingmechanism11mechanisms will be discussed in next chapter. In Fig. 2.1 an ideal N:1 transformer isused. The coupling coefficient B is another parameter of a resonant system, and it isdefined as the ratio of the input impedance (equal to R 0/N2) of the resonator to theimpedance of the input transmission line feeding the resonator Z o[16]P = RoN2Z0(2.3)where at the resonant frequency the input impedance of a resonator is a real value andthe internal impedance of the generator is assumed to be matched to the inputtransmission line. The coupling coefficient can also be written as(2.4)and QL, as^ z, = Q.(1 4- P)^ (2.5)The peak values of the stored electric energy and magnetic energy are[17]^= Co Yr^ (2.6)and1WIN = 2 4//` (2.7)respectively, where V*, and I* are the complex conjugates of V and I.122.1.2 The "cul-de-sac" ResonatorAs described in Chapter I, in order to modulate our optical switch, the requiredpeak potential difference between the two electrodes is 50 volts and the space betweenthem is 4 gm. Therefore, the resonator used as the electrode must have parallel legs,separated by about the width of a diffused wave guide, which in our case is about 4 ism,and the potential difference between them should be 50 volts. It was our goal to limitthe power supplied to the resonator to about 100 mW. This, of course, means that thetotal loss of the resonator at the resonant frequency should be less than this value,supposing that the loss on the input transmission line might be neglected. Also, theresonator must be designed so that the power from the generator will be efficientlycoupled into it.In designing such a resonator we should aim to reduce the power loss to thegreatest extent possible while keeping the voltage between the legs high. The ohmic skinloss[18] in the resonator is due to the resistance of the conductor and is proportional tothe length of the transmission line from which it is made. Therefore, to minimize thetotal length, the resonator should be a half-wave resonator[19]. The radiation loss iscaused by the discontinuities in the resonant system. It is well known that an infinitelylong ideal transmission line will not radiate any power[20]. Therefore, the resonatorshould have as few discontinuities as possible. It turned out, after careful computation,that coplanar strip transmission line resonators[21] would not satisfy our goal of 100 mWlost power due to their high losses. Actually, since they do not have a ground plane,13these structures act like antennas. In the case of the microstrip, there is a ground plane,thus the electromagnetic field is confined between the strip line and the ground plane andless power is radiated. Hence, microstrip was chosen to form the planar resonator.Several designs were numerically modelled for the resonator for the electrode, themain ones being the hairpin[22], and the open ring track-like microstrip resonators[23-26], as shown in Fig. 2.2. However, their attenuation losses were too high. In the twolegs of the hairpin resonator the conductor attenuation is much higher than in a simple,straight section of microstrip, because the ohmic loss is increased as a consequence ofcurrent crowding at the inner edges of the strips, as in coupled microstrip lines. In theopen ring structure, if the radial length of the gap is long enough to serve as theelectrode, the stripe will be too wide and the capacitance will be too large, so that theenergy is stored as a large current rather than a large voltage. For somewhat the samereason the losses in a track-like resonator were high too.The best design that we studied was a mixture of the hairpin and open ringresonators, we named it a "cul-del-sac" resonator. It is shown in Fig.2.3. In it the twoends of an open ring are joined to two straight microstrip lines. It is these two parallellines, as mentioned in Chapter I, that will work as the electrode of the opticalcommutator switch. In this figure, the two open ends of the half wavelength resonatorare bent together so that these two radiation sources form a dipole with a separation ofonly 4 Am so that the dipole moment is very small. It is well known that the radiationfrom a dipole is much smaller than that from a single element. Because of the imageeffect of the ground plane, each dipole forms a quadrupole with its image, that will14V zez ■ erir Arzommeze:e v/-4 •^r^Af.f^414# 4# :02 ,•(a) a track like resonator (b) a hair-pin resonator(c) an open-ring resonatorFig. 2.2 Different kinds of microstrip line resonators15Fig. 2.3 Top view of the "cul-de-sac" resonatorreduce the radiation even further. For the same reason, there would be almost noradiation loss from the two 90° ends either. The parallel microstrip line portion of the"cul-de-sac" is short, minimizing the additional attenuation due to current crowding inthe "coupled microstrip lines" [27] so as not to influence the total loss seriously. The"cul-de-sac" shape helps to suppress the losses, although there are more discontinuitiesin it than in other types of resonators. Hence, we feel that the "cul-de-sac" design hasthe best shape.2.2 Analysis of the Microstrip ResonatorIn the following analysis the quasi-static method is used. It is well known that atrue TEM wave does not propagate on a microstrip transmission line because there areseveral modes present, therefore, the normally used quasi-TEM method[27] is only anapproximation. For a high dielectric constant substrate material the boundary conditionsare more stringent than for low dielectric constant ones, further limiting thisapproximation[16]. However, designing and fabricating a microstrip resonator is a verycomplicated procedure with numerous uncertainties. In order to obtain an optimal designone expects to go through several iterations of modelling, fabrication and testing. Herewe are interested primarily in "the proof of principle" and have limited our workaccordingly. Needless to say the model used allowed the design of resonators withsufficient performance to be applicable to the intended purpose, see Chapter VI.172.2.1 DiscontinuitiesIn a resonator the electromagnetic field is a standing wave field. In an ideal,linear, half wavelength transmission line resonator, ignoring the open end effects, thecurrent distribution is a 1/2 period of a cosinusoidal wave while the voltage issinusoidal[28], where the origin is assumed to be the middle of the half wavelength lineas shown in Fig.2.4 (a) and (b). The distribution of both current and voltage in an openring resonator, along the longitudinal direction of the arc, are also cosine and sinefunctions[29] respectively. In the case of the "cul-de-sac" resonator, there arediscontinuities such as open ends, curvature, junctions, and bends. These discontinuitieswould disturb the distribution of the electromagnetic field, causing radiation, as well aschanging the current and voltage distribution from pure sinusoidal and cosinusoidalfunctions[30]. The effects of the discontinuities can be expressed by excessimpedances[31], of which the real parts correspond to the radiation resistance and theimaginary parts will influence the propagation constant and introduce mismatches andreflections[19]. The radiation will be discussed later; at this point we will concentrateon the influence of the discontinuities on the structure of the resonator.The effect of an open end can be treated by adding an excess capacitance C.between the strip and the ground plane. The value of this capacitance can be calculatedusing Silvester's method[32]. The corresponding excess impedance is18A/4(a)Vvo --X/4^ X/4(b)Fig. 2.4^(a) An ideal microstrip half-wave linear resonator;(b) The voltage distribution in the ideal microstrip half-wave linear resonatorA/419weit(0)-w-W+1 +(fifs)2(2.11)Z - ^1 ^(2.8)j2itfC.According to transmission line theory this impedance is equivalent to an excess lengthA1[30] added to the open end which is determined byZ. = Zocot(k.61)^ (2.9)and Al is solved as1^Zoe/ = k— arctan(---=). (2.10)where Zo is the characteristic impedance of the microstrip as a resonator.The curvature of the ring constitutes another discontinuity. The curvatureincreases the distributed capacitance of the microstrip line so that the characteristicimpedance and the propagation constant of the curved part is different from that of astraight portion. This curvature effect is taken into account by using an effective widthweff when designing the microstrip[33-35]where w is the actual width of the strip, f is the working frequency, w eff(0) is determinedby the following equationhno w.tt(0)=iCiTirZo(2.12)20and fs corresponds approximately to the cutoff frequency of the first transverse resonancein the microstripfr 2wdo)F4rec,^(2.13)where Eeff is the frequency dependent effective relative dielectric permittivity and h is thethickness of the substrate of the microstrip, and n o is the intrinsic impedance of the air.The most serious discontinuity is located at the junction where the microstrip ringjoins the parallel microstrip lines. It is more complicated than the open end. Theimpedance of the parallel lines is smaller than that of the ring, so that in addition to thegeometrical bend there is a step in the impedance. The discontinuity at the junctiondisturbs the distribution of the electromagnetic field in the resonator and the reflectedpower, reducing the energy stored in the parallel lines. The effect of the 90° bend canbe reduced by chamfering with an arc or a 45° straight line[361. Chamfering can preventserious reflection and junction impedance, but the change• of characteristic impedancefrom the ring to the parallel lines can not be cancelled. Its influence on the current andvoltage distribution will be discussed in the next section.2.2 Current and Potential DistributionsIn order to discuss the current and voltage distributions it is convenient to firststretch the "cul-de-sac" to a straight line. Fig.2.5(a) is the straight microstrip lineequivalent circuit for the "cul-de-sac" resonator, where section AB corresponds to the21F C^A^01,^i(^1 e ^141(a)0E1Vv, --A / a^ X 4(b)Fig. 2.5 (a) The equivalent microstrip linear resonator of the microstrip 'cul-de-sac'resonator with impedance steps(b) The voltage distribution on the 'cul-de-sae microstrip resonator22ring microstrip, CA and BD correspond to the parallel lines, BE and DF are the excesslength caused by the open ends, and 0 is the mid point of the resonator. The parallellines are made of so called "coupled microstrips", of which the characteristic impedanceZ,, is smaller than that of the ring part Zr, therefore the corresponding equivalent straightmicrostrip is wider.Although there are steps in the "cul-de-sac" resonator, the electromagnetic fielddistributed in the resonator at resonance is still a standing wave and for the dominantmode the current wave is still cosinusoidal while the voltage wave is sinusoidal. Becauseof the different characteristic impedances the propagation constants in the two sectionsof the resonator in Fig. 2.5(a) are different. Let k r be the one for the ring microstrip andice be that for the parallel lines. Now the voltage can be written as a piecewise smoothfunctionV(l) ={Vrsin(k,x) , 0 < Ix1 <xliVcsin(kx) , xs<lxi<xp(2.14)(2.15)where V, is the voltage amplitude in the ring section and V, is that in the parallel legs,xA and xit are the coordinates of the junction and xc and xr, are the coordinates of theopen ends. At the open end the voltage should be equal to the designed maximum outputvoltage V0 , that gives a boundary conditionVcsin(kxD)=^ (2.16)from which we get23VV,- ^°sin(k,xD)(2.17)At the junction the voltage should be continuous, so we have a joining conditionV esin(k 3). V csin(kx)^ (2.18)That gives V,Vcsin(kerB)sin(k,x8)Vosin(k,xi)sin(krxit)sin(k,xD)(2.19)(2.20)To design such a resonator one has to determine the structure parameters such asthe arc length of the ring lT and the length of the parallel lines lc , corresponding to ABand BD. Adding the excess length Al caused by the open end, the phase length of theresonator should be n, therefore, we have the additional conditionsin(k c(x 8+ AID = 1^ (2.21)therefore,reXi= "'""""'•2k, -At . (2.22)The length of the parallel lines should be at least as long as the of the electrode le shownin Fig. 1.1, so(2.23)Or24XA SXB -1e •^ (2.24)In practice is and 1, are determined by minimizing the total loss of the resonator.Using the same method one can get the distribution for the current simply byreplacing sin by cos.2.3 The Energy Stored and the Power Lost in the "cul-de-sac" ResonatorKnowing the current and voltage distribution in the resonator, we can calculatethe stored energy, the conductor and radiation losses, and then the unloaded qualityfactor.2.3.1 Stored EnergyThe energy can be stored in the resonator in two forms, electric and magnetic, thetime average of the two being equal. The total stored energy is the sum of the electricand magnetic energy, or twice either. Here we are interested in the output voltage andonly need to calculate the electrical energy. The electrical energy stored in a section oftransmission line is given by[17]we= f 4cv2d1^(2.25)25where 1 is length of the line and C is the distributed capacitance per unit length of theline. The capacitance can, in turn, be expressed in terms of the characteristic impedanceand the propagation constantC- k . (2.26)6)0ZoThe total stored energy W, is 2W e . Since there are two kinds of microstrip each havinga different voltage distribution, the energy density in each is different. Therefore, theintegration should be written as^ws=-If xAcy,414-3-f^V.d14 -x4^-^4 xs^k, f x^k^x _ .2=^A K2. (*IX +^D v;(x)dx^2 cooZr -x4^woZc x1(2.27)(2.28)^• u_ ^4k,^26)o;I ,^sin(k,1,)}+ kcK2 • sinik,(1,+2,1A-sin(k1,)) .^(2.29)' caoZ, 4 '4/0 ^2.3.2 Attenuation LossesThere are two kinds of attenuation losses, namely dielectric loss in the substrateand ohmic skin losses in the conductor strip and ground plane. These losses are a smallproportion of the transmitted power P t . The attenuation factor a is the loss per unitlength [18]26a- dP/dx2P(x)(2.30)where a =ad +a. is the sum of the dielectric attenuation factor a d and the ohmicattenuation factor a„ and both a d and a„, can be calculated following Gupta's formula[271.The dielectric attenuation factor a d for LiNbO3 and Al203 are less than one percent of theohmic attenuation factor, so that the latter dominates. The attenuation loss in theresonator isP. =2 a f P(x)dx1 f x4 -2^2 fx, ..2(x)dx= a — v-(x)dx+ a — v-Z, -xAa ,sin(k1r) 1+ a ,V! u sin[kc,(1,+201-sin(kcl,)Z, 2^4k,^Z^2k(2.31)(2.32)(2.33)where a, and a, are the total attenuation factor for the ring and the parallel legs,respectively.2.3.3 Radiation LossRadiation loss is not usually the dominant loss mechanism for microstrip circuits,but it tends to be relatively large in planar resonators for use at millimetre-wavefrequencies. This is because the amount of radiation increases rapidly with the ratio ofthe substrate thickness to the wavelength. Furthermore, the resonator discussed here has27serious discontinuities which cause additional radiation loss. Therefore, the radiation lossshould be calculated carefully when designing the resonator.The method used to estimate the radiation loss was based on Lewin's[20] orPauw's[37] theory. The one only considers the radiation from discontinuities[20], andthe other involves the assumption of a thin substrates[37]. What constitutes a thinsubstrate was hard to quantify. The accuracies of both these methods are difficult toestimate, the difference between the results from the two being about 30% [38]. It was,therefore, necessary to develop a method for calculating the radiation loss from ourmicrostrip resonator. S.E. Schwarz et.al.[39] calculated the radiation from a linearresonator using a Green's function method and suggested a parallel microstrip linesresonator with very low radiation loss. The total Q of this dual resonator, in fact, maybe lower than that of a conventional single resonator, for the same reason discussed inSection 2.1.2 of this thesis[39). Hence, this resonator cannot be used as an electrode andthe Green's function for current in a linear resonator does not apply to our "cul-de-sac"resonator either.The approach adopted here is based on fundamental electromagnetic theory, theGreen's image method[40]. The radiation vector N[41] and then the radiated power P rare calculated directly from the current distribution in the resonator. Using the quasi-static assumption, the field in the resonator is assumed to be a TEM wave, and themicrostrip resonator is considered as a strip-line surrounded by a dielectric having aneffective dielectric constant ceff. Fig. 2.6(a) is a sketch of a microstrip line and Fig.2.6(b) is that of a strip-line. This assumption has been widely used[20,28,37,38] and it28groundplanedielectric^conductor strip( 0)conductor stripor if 11/11*WWWWWWWWWWWWWWWWWWWZAggAgggdielectric^ground plane(b)Fig. 2.6 (a) The structure of microstrip line(b) The structure of stripline29has been shown that the capacitance, characteristic impedance, and phase velocityobtained using this equivalent strip-line model are in fairly good agreement with themeasured values. We believe it is reasonable to analyze our resonator using the quasi-static model for we are concerned here only with the far-field radiation, where the fielddistribution is, in fact, a true TEM wave. The error introduced by this method is thatthe field below the strip might be under-valued, while that above might be over-valued.This method will, if anything, overestimate the total radiation loss since the energydensity is proportion to the dielectric constant, which above the strip line is smaller thanthe effective value and below the strip line is larger.We will give the general radiation formula for a planar resonator and then use itfor the "cul-de-sac" resonator. A spherical coordinate system is used, as shown in Fig.2.7, where h denotes the thickness of the substrate. In the "cul-de-sac" resonator, theparallel lines form a quadrupole with their image in the ground plane. The radiationfrom this quadrupole, compared with that from the open ring, is small enough to beneglected. What remains is an open ring. We will denote the internal and external radiiof the ring by ri and ro respectively. While there is a displacement current in thegap[24], its contribution to the radiation vector N is also very small and can be neglecteddue to the gap angle being small, i.e., much smaller than it. Therefore, the conductioncurrent is continuous and forms a current ring. The thickness of the metal strip has beentaken into account when introducing the effective dielectric constant E ar, so we supposethe stripe is infinitely thin when considering the current distribution. In the radialdirection the rigorous solution of the current distribution is a Bessel function[24], but in30Fig. 2.7 Coordinate system for calculating the radiation loss from the "cul-de-sac" resonator31the radial direction the microstrip is very narrow, therefore, from the far-field point ofview, the current on the strip is considered as a line current[37] in a ring having anequivalent radius re , as shown in Fig.2.8,andw wrrrr -r ^2 ra^11V —IV^1'r =ra^2+ eff (2 1 ) ," aw(w.„-w)r =41+ ^ 4`'^]r(2r+w)r=-1 (r,+ra) .2 •(2.34)(2.35)(2.36)(2.37)Referring to the formula derived in the last section, the current distribution alongthe ring can be written aswhere1i=0/osinffi (1)^+ c1.8(r-re) (3.38)g= It /2 - k A^(3.39)is the phase at the junction of the ring and the parallel line.Using Green's method of images, the influence of the ground plane can be takeninto account by using the image current i'32Fig. 2.8 Equivalent strip width and radius of the open ring33(z +2h)^ (3.40)where z=r cost). The total current it isit.../0[1 -8 (z +2h)] sinR2^+ i gj (r-rs) .^(2.41)Knowing the current distribution we can calculate the radiation vector N•^N =12 teib °cc.* r eth p 0^(3.42)where (r0 , 00, Od is the coordinate of the current while (r,0,13) is the coordinate of theradiation field andcos* =cost) toseo + sin() idnOo•..os(q) - To)^(2.43)is the cosine of the azimuth angle of the considered point in the radiation field. Theabove integration cannot be solved directly. In the ring segment of the "cul-de-sac"resonator we have rek < 1/2, andrehine^< (2.44)so the exponential term can be expanded in a Taylor series and we keep the first twoterms; the error caused by this approximation is only a few percent. Now we haveNatF12 1E^^.[1 + j kre sme cos(* - * 0) +...14,(ycos*0-isimpo)red4ro^(2.45)0 where34F.euukvasi2sin[khcose j .^(2.46)This integration was solved and the resulting x and y components of the radiation vector,Nx and Ny areand-327rjrk wog) ssings sine^115 it -26, j^o(2.47)Ny'Ro4(20 -1t)cosp _g jkr ,{287c -164p dr.omp g 'cos" sin() )^(2.48){ 3^11 -40 1^15-260grespectively. The 0 and 0 components of the radiation vector, Ns and No , can beexpressed asNo =Nxcosecos, +Nycos0sin0^(2.49)andNt^+Nyco:up .^(2.50)The radiation power isnoe Pr.^-: Joo 2[INe12]+ IN, Ildedv0,22fe.121:h2k44_1ri2k2o 2+ C + 3.2 B2 — A C 7.2k2} 'COO (i) g7 ^35(2.51)(2.52)where35-327: (2.53)15n-26v g4(2" --7r) (2.54)37t.-4tp t28n -16,^ (2.56)157c--26(p sWhen obtaining Eq.(2.52), following approximation was used:sin(khsinecosp)-khsinecovp .^(2.57)The relative error caused by this approximation is less than 3%.If os is much smaller than n, i.e. 01 c IT, so that it can be neglected, Eq.(2.51)can be solved asP= 101,2 {1 sin(2kh) }^(2.58)2khThis equation gives us a reasonable approximation of the radiation loss. Contrary to theattenuation loss, the radiation power is proportional to the square of the thickness of thesubstrate h2 . To design a high Q resonator the thickness of the substrate should beoptimized, giving a trade-off between the radiation loss and the attenuation loss.362.4 Optimized Design of the Structure ParametersThe parameters to be designed are the height of the substrate h, the width of thestrip wr for the ring and w, for the parallel lines, the length of the parallel legs lc , andthe radii of the open ring I-, and r0 . They are optimised for the lowest loss not the highestQ. The reason for this follows.From Eq.(2.6) we know1;12=2 w e^ (2.58)=2 QPI .^ (2.59)caCHere the maximum output voltage is determined by the capacitance and the stored electricenergy, or the unloaded Q, and the lost power P1 . The greater the stored energy and thesmaller the lost power, the higher the quality factor Q, but the output voltage may notbe so high if the capacitance is too high. Therefore, the unloaded quality factor Q. is notthe only parameter, needed to evaluate the resonator for use as an electrode in our case.In fact it is more appropriate to optimize the structure of such a resonator by minimizingthe lost power and maximizing the output voltage.A computer program was written to optimize the design. The optimized structureand the operating parameters are listed below:37h = 0.3 mm,^is the thickness of the substrate;w = 0.02 mm,^is the width of the ring potion microstrip;r = 0.79 mm,^is the radius of the open ring;4 = 0.35 mm,^is the length of the parallel legs;ws = 0.30 mm,^is the width of the each of the legs;ss = 0.004 mm,^is the width of the gap between the legs;fo = 15 GHz,^is the resonant frequency;e t. = 34.7 ,^is the effective dielectric constant of the LiNbO 3 substrateas defined below*;eeff = 20.5 ,^is the effective dielectric constant for microstrip on ourLiNbO3 substrate at the resonant frequency f0[42];V. = 59 V,^^is the calculated potential difference between the open endsof the parallel legs;Q. = 71 ,^is the calculated unloaded quality factor;P1 = 0.1 mW,^is the calculated dissipated power;* LiNbO3 is an anisotropic crystal, of which the two dielectric constants in the x(or y)and z directions are e x =43 and ez =28 respectively. The equivalent dielectric constantused here is calculated by following formula[27]:Z =iii-l^X Z38We have plotted the curve of the output voltage vs total the loss in Fig. 2.9.E. Hammerstad's computation method[42] is used to calculate the characteristicimpedances, propagation constant and effective dielectric constant.Vo80.0070.00 _w^-_-*-> _=-5 _60.00 =.."^=a)em^7.ocs 50.00..) 5^=_=40.00 --a.^_-_.4.) _o 30.00 -?...."20.00 ^1^i^IIIIIIII^i^1^rill^1^1^i^1^i^I^1^P i0.00^0.05^0.10^0.15^0.20Lost Power in WattsFig. 2.9 The voltage between the ends of the parallel lines and the dissipated power39^,Chapter III Coupling and Matching3.1 The Coupling Method DiscussionCoupling and matching are important topics in microwave transmission linecircuits. In the previous chapter we discussed the structure and parameters of theresonator without addressing the connection between the resonator and the power source.Obviously, the resonator must be a part of a microwave circuit and be excited bycoupling electromagnetic energy into it from outside. That is the topic of this chapter.There are three types of coupling mechanisms to a half-wave microstripresonator: capacitive gap, coupled line, and direct-coupled tap point. Because the outputof the system, namely the modulating voltage, is just the potential difference between thetwo parallel legs, which are a part of the resonator, no load is coupled to the resonator.The whole system is terminated at the resonator. The coupling mechanism should allowthe resonator to obtain the maximum energy from the generator, therefore, the resonatorshould be matched to the input microstrip line coming from the generator. Another pointthat should be mentioned is that the output voltage at the open ends of the resonator ishigher than the voltage input to it from the source, therefore the energy should becoupled into the resonator at a point at which the voltage is low and the current is high.According to the voltage and current distributions derived in the last chapter, such a pointis located near the middle of the resonator. To both realize this condition and reduce theradiation loss the direct-coupled tap point coupling method is chosen. The input40microstrip is joined to the open ring section of the resonator directly. A quarter-wavelength transformer is used to minimize the discontinuity effect of the T-junction aswell as to improve the match between the input microstrip and the resonator. Fig. 3.1(a)shows the quarter-wavelength transformer tap point coupling mechnism to the "cul-de-sac" resonator and Fig. 3.1(b) shows same coupling mechanism to its linear resonatorequivalent circuit.3.2 Equivalent Circuit and Input ImpedanceThe resonator can be modeled as two shunt open-circuited transmission lines, ofwhich the equivalent circuit is shown in Fig. 3.2. Here 0 is the electrical length fromthe coupling point to the centre of the resonator; Z r is the characteristic impedance of thetransmission line which is the same as that of the section of microstrip constituting thering; Zr is the characteristic impedance of the section of the microstrip constituting thetransformer; and Z, is the impedance of the input microstrip line which is the same as theinternal impedance of the power source; y =a +jk is the effective complex propagationconstant of the wave on this microstrip, of which the real part a includes both theconductor attenuation and the radiation loss from the "cul-de-sac" resonator, and theimaginary part k is the same as that of the microstrip ring of the "cul-de-sac" resonatorkr. Since the total electrical length of the two shunt transmission lines in Fig.3.2 equalsall the imaginary part of the discontinuities were also taken into account. Therefore41Figure 3.1(a) The transformer tap point coupling to the "cul-de-sac" resonator42 NFigure 3.1(b) The transformer tap point coupling to the equivalent linear resonatorof the "cul-de-sac" resonator43I.^ e Fig. 3.2 Shunt transmission line equivalent circuit for "cul-de-sac" resonator with tap pointcoupling mechanism.44this circuit is equivalent to the "cul-de-sac" resonator, if we are concerned only with theinput impedance looking into a point on the ring.Let the left hand portion of the transmission line in Fig. 3.2 be shorter than thaton the right, that is 0 < 7r, then the input impedance of the left part Z il is:Zu = Zreoth(a/+je)^ (3.1)where 1 is the geometric length of the left part. In our resonator the loss is very small,that is a<1, so we havetanh(al) • alWith this approximation the input impedance can be written aszi-zral+jtan0RI-cos20whereRI=Zoal^ (3.5)B=Zrtan0 (3.6)since (a1)2z0. Using the same procedure the input impedance of the right portion of thetransmission line in Fig. 3.2 zir can be written as1+jaltan0 (3.2)(3.3)(3.4)45R,Zir - cos2e +jB (3.7)whereR, - Zo a(2-1).^ (3.8)If we treat Z11 and Zir as the impedances of two lumped devices we can obtain theequivalent circuit shown in Fig. 3.3. The total input impedance Zi is the parallelimpedance of Zir and zil :Z =—e,sinksI R2Z,Q.. sin2eit(3.9)(3.10)where R=12,.=R I corresponds to the total loss in the resonator, which is independent ofthe coupling mechanism, and Q, is the unloaded quality factor of the resonator.3.3 Coupling ConditionThe transformation factor of the 1/4 transformer isN = Zr 1^ ( 3.11); (ifAt the coupling point the source impedance Z, is transformed to Z',46Fig. 3.3 Lumped equivalent circuit for the input impedance of the "cul-de-sac" resonator47(3.12)ZsBy definition, the coupling coefficient B is the ratio of the input impedance of theresonator to the transformed source impedance ( the impedance looking towards thesource from the coupling point ) Z'„ so we haveZsZr 2Q* shoeZ2T(3.13)The condition for the critical coupling is B=1, which gives the matching condition22 Qs^zrz—sibeeZs(3.14)Two variables in this equation affect the coupling condition, 0 and Z T. Thecoupling position 0 determines the input impedance Z i( Eq. (3.10 )) and the potential atthe coupling point V(0)V(0)= Vesin0 (3.15)Zr determines the transformed source impedance (Eq. (3.12)) and the transformed sourcevoltage V',. When the system is matched the transformed source impedance and voltageare48ZT—Zsand(3.16)(3.17)Z =v,' v -1*Zsrespectively. The potential at the centre of the junction is continuous, that is to say,V(0)=V%. This condition holds for both matched and mismatched coupling. Formatched coupling ( critical coupling ) we haveVssin = V^.'ZsEq. (3.14) and Eq. (3.18) hold simultaneously only whenV, 2Q.(3.18)(3.19)orVs =ZS 2Q.i Zs •(3.20)Eq. (3.20) gives the relationship between the output voltage of the resonator and thesource voltage. This voltage transformation is very important for designing a resonatorto be used as a modulating electrode. For a given source power this equation determineshow high a Q, and what characteristic impedance Z the resonator needs to obtain therequired modulating voltage.49If the system is not matched there will be some power reflected back into thesource. Let r be the voltage reflection coefficient at the boundary between the inputmicrostrip and the transformer and let Vb be the potential at that boundary, then Vb canbe written asVb = Vs (1 +r) .^ (3.21)The transformed source voltage at the coupling point V' s becomes(3.22)Zs aand Eq. (3.18) becomes=v. -2-Z .2\1 1-3 -'Zs^1+p (3.23)Vrsin0 = V. 2 \I1^.- Z,^+13(3.24)Combining this equation with the equation for the critical coupling condition Eq. (3.13),we get11 = ^4p 2 Q. F,2:^(3.25)Z,^(1+13)2 itWe can see, from Eq. (3.25), that the mismatch results in the additional factorf(B)=48/(1 +13)2 on the right hand side of Eq. (3.19). So the output voltage of theresonator depends on the coupling mechanism too. It is easy to show504p s 1^ (3.26)(14-p)2where the equality holds only for B =1. That means that both over-coupling and under-coupling would reduce the output voltage of the resonator.Because of the tolerance in the design and fabrication of the resonator, the criticalcoupling condition is not easily satisfied. In order to find out whether over-coupling orunder-coupling is preferred for obtaining a high output voltage we take the derivative ofthe factor f(B) and getdf(13) _ 4(1-0 .d3^(1.403 (3.27)It was found from this equation that f(B) changes more slowly for B <1 than that forB > 1. Therefore, over-coupling is preferred to under-coupling and is actually used in ouroptimization program, i.e., B1 is used as the coupling condition.There is another point that should be taken into account in the program. The T-junction introduces an imaginary impedance in series with the ring and the transformer.To overcome the influence of this impedance on the resonant frequency and on thematching, the electrical length of the resonator, which is not exactly ir, and the lengthof the transformer, which is not A/4, should be slightly modified.51^.Chapter IV Fabrication of the Resonator4.1 IntroductionThe purpose of using a resonator as the electrode in the integrated opticscommutator switch is to reduce the required power. As mentioned in Chapter II, atmicrowave frequencies both conductor attenuation and electromagnetic radiation losseshave to be taken into account when designing and fabricating the resonator.Correspondingly, the patterning and the metallization have to be dealt with very carefullyduring the processing. This is because both the density of, and types of impurities in,the deposited metal influence its conductivity which determines the attenuation loss, whilestagger of the strip edge and the roughness of metal surface increase the radiation loss.Therefore, the fabrication is an important part of this work.To verify the design theory presented in Chapter II, as well as to find the best wayto fabricate the device, experiments were performed on Al 203 at a lower frequency of 7GHz with a larger electrode spacing (12 Am), because the large-scale mask is muchcheaper and easier to make than the one for LiNbO 3 at 15 GHz, and since Al203 wafersare also cheaper and easier to handle than LiNbO 3 wafers.There are many methods in microelectronic device processing for defining thinmetal structures, each of which has advantages over other techniques for particularapplications[45]. In this chapter, the problems and their solutions, as regards thepatterning and electrolytic plating of fine patterns are discussed, the procedure for52fabricating the resonators is outlined which includes: designing and making the mask,preparing the wafer, metallization by thermal evaporation, patterning usingphotolithographic techniques, electrolytic deposition, etching, cutting, and polishing.4.2 The Fabrication Method4.2.1 MetallizationPhotolithographic photoresist mask and thin film technology[46] is used tofabricate the resonator. Gold and copper were tried as the conductor, because of copper'shigh conductivity and low cost and because of gold's resistance to environmental effectslike oxidation and corrosion as well as the ease with which one may make contact to it.Both gold and copper electroplating are very easy to control. Because they adhere withdifficulty to inorganic substrates, an adhesive layer of chromium followed by a thinconductive layer of gold or copper is evaporated to the substrate[46]. A photoresist maskis made photolithographically on it, and the exposed pattern is then galvanically thickenedto the desired value.The metal layer thickness of the microstrip is a crucial factor for the ohmic loss.To minimize this loss the thickness of the metal layer d should be at least ten times theskin depth 6[47]. If the metal layer is not thick enough it will not obtain a surfaceresistivity approaching that of the bulk metal. From Ramy et. al. [47] one can find thatthe difference between the quality factor of half wave length resonators with conductor53layer thicknesses of 78 and 108 is about 30%[47]. The formulae that this thesis followedto calculate the conductor ohmic loss in Chapter II holds only for conductor layers ofthickness more than five times the skin depth of the conductor[27].Table 4-1 lists the resistivities of gold and copper, the skin depth, and the requiredstrip thickness t at 7 GHz and 15 GHz for LiNbO3 and Al203 substrates.Table 4-1. The Designed Conductor Strip ThicknessMaterialResistivityp (fl cm)Frequencyf (GHz)Skin Depth8 (hm)Designed StripThickness t (pm)Gold 1.367 0.9 715 0.63 6Copper 17 0.8 5.515 0.55 4.44.2.2 Plating Problems and SolutionPlating by the electrolytic technique is well-known. The control factors likecurrent density, contents and concentration of the solution, and agitation methods areavailable in many texts. But when used in depositing metal onto very fine patterns,say on the order of few square millimetres, problems occur; preventing one fromobtaining the desired results.54Two important problems due to stress and burned deposit are discussed below:Stress. The deposited metal grains will have stresses in the plated layer aswell as between the plated layer and the base metal[48]. This may arise from themismatch of lattice parameters or from incorporation of foreign material such as,oxides, hydroxides, water, sulphur, carbon, hydrogen, or other metals. It may alsobe due to ohmic heating, if the plating current is too high. The stress can causecracking of the deposit or loss of adhesion to the substrate. When plating very finelines, these problems are much more likely to occur.Burned Deposit. When the surface of the deposited layer becomes dark it iscalled a burned or powdery deposition[48]. In this circumstance the deposited grainsare too big and the content is not pure gold(or copper) but contains impurities. Theloose structure and the impurities significantly reduce the conductivity resulting inincreased losses. Burning is also caused by high current densities and incorrectsolution concentrations.In practice the above two problems often occur simultaneously. Ourexperiments showed that if the deposited grains are too large, fine lines would crack.In this case the surface of the plated pattern looks dark red and not smooth. Whenobserved through a 400x microscope, the grain size was found to be about 0.2-0.5Am. The large grains may be caused by several reasons such as: impurities, lowsolution concentration, and high current density. The high current density may alsocause too high a deposition speed and ohmic heat. The resulted stress would act to55peal off the plated strip at the longitudinal edges, so that before the structure was thickenough the pattern had balled up. To prevent these effects one has to be very carefulabout controlling the concentration and contents of the solution, above all, one mustcontrol the cathode current density.The contents of fresh, commercially-available solutions are very pure makingthe concentration easy to control, therefore, only the cathode current density needs tobe controlled. This requires accurate knowledge of the total exposed area on thewafer. For very small areas the plating current should be significantly smaller than isusually required. This is difficult to control accurately without special equipment. Anold approach in this circumstance is to simultaneously plate a much larger area, thecurrent being determined by the large area. In our experiments we found that, withthe current suitable for the large area, the fine patterned structure was still burnedsince the current tended to concentrate on it. We were able to obtain good results forour areas using the following criteria: the area of the accompanied piece was aboutone hundred times that of the structure to be patterned and the cathode current waschosen assuming an area ten times that of the structure. This will be discussed indetails in following section.A third important problem that needs to be considered is the edge effects:Edge Effects. Edge effects are the most serious problem in plating a finepattern. The deposition rate is proportional to the current density which is related tothe distribution of the static electrical field. In the electrolytic plating bath the electric56field is concentrated at sharp edges and points, where, therefore, the current density ismuch higher than in the central areas of the pattern. Hence the edges become thickerthan the central areas. Fig. 4.1 shows the outline of the intersection of a fine linewith sharp edges measured and plotted using a TENCOR Alphastep 200 profilometer.The sharp edges look like wings. In our application these will act as antennaeincreasing the radiation loss.Another edge effect that presents a problem in our case is lateral growth of thestructure. All parts of the structure, exposed to the plating solution, will growgalvanically. That is to say that while the strips grow thicker they also grow wider ifthere is no mask suppressing lateral growth. As mentioned before for the resonator onLiNbO3 , the gold layer should be about 7 m thick but the gap should be about 4 itwide. To plate such a deep and narrow gap successfully, the photoresist mask shouldbe at least 6 i.t thick. We were not able to obtain such a photoresist pattern using theequipment available.To overcome these problems two methods were developed; double patterningand aluminum masking. In double patterning an initial 3 At thick photoresist layer isapplied to the metallized wafer. Then the exposed pattern is grown galvanically to thesame height as that of the photoresist mask, which can be monitored using theprofilometer. The wafer is then rinsed in a diluted gold etchant to remove the sharpedge and smooth the surface. Another layer of photoresist is then applied and thepatterning, plating, and etching are repeated. During the etching procedure, thechemical reaction at the sharp edges is much faster than in the central areas, so that57,11 •-,4 ^.11•:tI31 00^ :430-^umI ENCON I NS I IttrIEtt ISLEVEL.Fig. 4.1 The outline of the crosssection of the plated microstrip ring using photoresistmask58the thickness of the gold layer is not significantly reduced after the sharp edges areremoved, however, this step still has to be controlled very carefully.In aluminum masking an aluminum mask is used instead of a photoresist mask.The deposition and etching of aluminum is straightforward; allowing the aluminummask to be made by standard photolithographic procedures. Gold does not plate ontothe aluminum because of the mismatch of the lattices of the two metals. Yet,aluminum is a conductor, so that the mask alters the electric charge distribution overthe wafer. Fig. 4.2 and Fig. 4.3 show the charge distribution over, and the growthprocedure of, the cross-section of the resonator. It is clear that in the case with thephotoresist mask, shown in Fig. 4.2(a), the electric field concentrates at the edges ofthe pattern, while in the aluminum masking method, shown in Fig. 4.3(a), the edgesof the pattern become inner corners, where the electric field is weaker than in otherareas, and the resulting current density is less. During the plating procedure thedeposition rate at the edges of the pattern is slower than in other areas, so theresulting cross-section is like a bun instead of a sharp wing. These procedures areillustrated in latter figures in Fig. 4.2 and Fig. 4.3. The aluminum masking methodis more efficient than the double patterning method and became the preferred method.In both the cases discussed above, if the plated layer is too thick, the gold stripwill eventually be wider than the mask and the resulting cross-section begins to looklike a mushroom. Hence, the upper part of the gap is narrower than the lower part sothat the electric field concentrates in the air and not in the substrate, as we would like.59(a)^a•l•tt•• charges^•••oorstosa MO surface wig low(b)pislia, sof/item',^ O1•110►•111111 MO**11111KMI.4°07441:"4AIM:0 sOstrot^ 0o••iorstit•sold low(C)Fig. 4.2 The cross-sections of the pattern and electrical charges distribution during theplating procedure using photoresist mask1■11•11,shstw••t , 0••60plating solutionAl masknegative charges^evaporated gold layera.b. e we g a em .■ 1 z^6‘1.,‘IlMil•■• 411111115111111■0C.iFig. 4.3^The cross-sections of the pattern and electrical charges distribution duringthe plating procedure using aluminum mask61To improve this, the extra part hanging out of the "mushroom" can be etched off,with the help of a negative mask which protects the desired pattern. The cross-sectionof the resulting structure in our experiments is shown in Fig. 4.4, which is alsorecorded using the TENCOR profilometer.4.3 Fabrication Procedures4.3.1 Mask designBoth patterning methods were tried, the resulting sample discussed in thisthesis was obtained using the double patterning method.The masks used in this experiment were made in-house. A positive rubylithmask, 20 times larger than the intended dimensions was cut. The mask was thenphotographically reduced using high contrast, high resolution black and white film. Aseries of exposure and development times was tried and the resolution limit of thismethod was measured. It turned out to be about 12 Am, introducing an error of about3-4 i (it is difficult to define a sharp edge using a silver based emulsion film). Forthe tests on Al203 this accuracy was acceptable. A positive mask made in this waywas used for patterning the "cul-de-sac" resonator.The mask used for cleaning the extra part hanging out of the "mushroom" wasmade by transferring the positive mask to a negative chromium mask using wetetching techniques.6206/17 04,56ID111.70R 5.940vmAlITYPEITI 6.165vmRe 2.485vm"TR!'R 1,2.0'.mArelPWRISCAM MENU 4um s/um10000 .22000 .2 1lair 4SCAM ta4OsseDIR.-->STYLUS 11mg0 400v* LEO EL^06'17 04.52 um^JimIDR 5.605.mA..141 WEITIR 5.855umRa 845.mmTUIPPalR 118.0vmArtirffigiSCAN MENU 4um s/um10000 .22000 .2 13SCAM ts40ste^2DIR.-->STYLUS 11mg4 Mum LE1 .1EL411211of.....^300yEticripFig. 4.4 The cross-section of the plated pattern recorded by TENCOR profilemeter.634.3.2 Preparation of wafersWe used one inch Al203 wafers, optically polished on one side, that arecommercially available. These were cleaned with a series of cleaners and solvents inthe sequence of boiling Alconox, boiling DI water ( several times, then blow driedwith N2 gas ), boiling tricbloroethylene, boiling acetone, boiling isopropanol. Inevery step the samples were ultrasonically agitated for 1-2 minutes. The wafers werethen blow dried using a N2 gun.4.3.3 Metallization and PatterningBoth sides of the cleaned wafers were first metallized with a thin layer ofchromium, about 350 A thick, and a layer gold, about 450 A thick, by thermalevaporation. The evaporation was done in a Carl Herman & Associates thermalvacuum evaporator. Then Shipley positive photoresist PR 1400-32 was spun onto oneside (the optically polished side) at 3000 rpm for 40 sec giving a 3.2 Am thick layer.After pre-baking the photoresist at 70°C for 25 min the patterning was done using apositive mask and Karl-Siiss MJB3 contact mask aligner. The exposure time was oneminute and the UV intensity was 25 mW/cm 2 . Then the pattern was developed inMicroposit developer MR-316 for 5 min and rinsed in a DI water cascade bath forabout 1 min. In order that the photoresist withstand high temperatures, as well as along immersion time in the plating solution, it was hard baked at 110°C for 25 min.644.3.4 PlatingActually, plating is also a metallization step. This step is so important that wewill describe it in detail in its own section.Fig. 4.5 is a photograph of the set-up for the plating system. It included atunable current source, a ALA meter, a petri dish for the bath of plating solution, aXYZ positioner, a platinum net anode, and a simple agitator made from a rubberblower and a plastic dropper. A 30x microscope was used to observe the contact andthe plating rate. The plating solution was the "OROPTMP 24" bufferred cyanideneutral 24Kt gold solution from Technic Inc. The plating procedure and conditionswere as follows:1. Pickled in warm hydrochloric acid, 30 percent by volume, at 50°C for 1.5min.2. Rinse in running DI water.3. Blow dry roughly.Once the sample was ready to be plated it was placed flatly on the bottom ofthe petrex dish. The tip of the probe was placed onto the sample 5 mm away from thepattern and the plating solution was slowly poured into the dish until it was 3/4 full.While agitating the solution we turned on the current source and increased it slowly toabout 150 AA. The deposition rate was observed by measuring the change inthickness of the gold layer using the profilometer. The correct deposition rate forsuch an agitating condition is about 1500-2000 A per hour.65Figure 4.5^The plating system664.3.5 Cutting and Grinding the SampleWhen the resonator pattern and the ground plane are plated we applied aphotoresist layer, about 2µm thick, on both sides and hard baked them at 120°C for30 min to form a hard protecting layer. The sample was then out with a diamondwire saw in such a way that the space between the edge of the ring and the edge ofthe substrate was 1 mm. This edge was then ground until that space became 25 Atm,to allow contact using our microstrip probe. The sample is ground by hand becausethe substrate is very fragile and breaks easily. A plexiglass jig was made so that thesample could easily be held against the grinding plate.Fig. 4.6 shows the top view of a finished sample. This sample was made byusing the double patterning method. Actually, both aluminum mask and doublepatterning methods were tried during my experiments. Because of burned deposit thesample using aluminum mask failed half way, and the final successfully tested onecame from the second method. But we do believe that aluminum mask would workbetter and it is the best way to fabricate the resonators on LiNbO 3 substrates, as it willbe suggested in Chapter VI.67Figure 4.6^Fabricated sample68(5.1)Chapter V. Testing and Results Comparison5.1 IntroductionIn this chapter a new method for measuring a planar resonator using a scalarnetwork analyzer is introduced, a formula for calculating the quality factor from thereflected power is derived, the set-up for testing is described, and the measurementaccuracy is discussed. The unloaded quality factors obtained from the tests are comparedto the design values and the reasons for their differences are discussed.5.2 The Measurement of Q t,As discussed in Chapter I it is usual to evaluate the properties of a resonator bymeasuring its quality factor, or Q, which is defined as the ratio of the energy stored inthe resonator to the average power lost by the resonator per cycle[15]:where W, is the energy stored in the resonator and P, is the power lost by the resonator.To describe a resonant system completely, one has to know two other parameters; theinput impedance and the coupling coefficient. The input impedance tells us theimpedance characteristics of the internal construction of the resonator while the couplingcoefficient relates to its connection to the power source and any potential load.69A typical circuit for measuring the Q is shown in Fig. 5.1 where a ring resonator,as an example, is tested. Fig. 5.1(a) depicts the measurement set-up where S is thelevelled frequency sweep source and G is a power meter used to detect the powertransmitted through the resonator. Fig. 5.1(b) is the equivalent lumped element circuit.The input impedance or admittance of the resonator is a function of the frequency, as isthe transmitted power detected. By measuring the transmitted power at differentfrequencies one can obtain the impedance of the resonator. Ginzton's book[15] describesthis transmittance measurement method in detail. Khanna[49] and Kajfez[50] have alsodescribed some methods for measuring the quality factors. The unloaded quality factorQ, is the ratio of the resonant frequency to the 3dB bandwidth, i.e., the differencebetween the frequencies at which the transmitted power is half of the value at the resonantfrequency[15]o^. `8- A co^ (5.2)The loaded Q factor Q1 is[16]^(21,^ (5.3)Qi= l+pwhere B is the coupling coefficient, which is defined as the ratio of the resistance of theresonator to the characteristic impedance of the line delivering power to the resonator[161^a— zoN2^ (5.4)If no transformer is used, as in our measurement set-up, Eq.(5.4) becomes70Fig. 5.1 (a) set-up for measuring transmitted power(b) the equivalent circuit71(5.5)Yo^ (5.6)Gwhere Yo is the characteristic admittance of the input transmission line and G is theconductance of the resonator.When the coupling between the main line and the resonant circuit is weak themeasurement can be quite accurate[15]. With a vector network analyzer one even canobtain values of both the real and imaginary parts of the input impedance, giving the lostpower and the stored energy, and subsequently the quality factors.The structure of the resonator electrode system shown in Fig. 5.2(a), is differentfrom that shown above. The resonator is directly connected to the source through amicrostrip line so that the resonator itself is the termination of the circuit. This is asingle port network, i.e., there is only an input port. Hence, we cannot use thetransmission measurement method. Therefore, a new method had to be developed.The parameters of interest are the power loss P I and the output voltage V0 .However, it is very difficult to measure them directly at microwave frequencies. Whatcan be tested, with the scalar network analyzer available in our lab, is only the relative72SS•■•••••41.a.DRb.Fig. 5.2 (a) Reflected power measurement system(b) the equivalent circuit73reflected power. Therefore, to test the properties of the resonator, it was necessary forus to find the relationship between the relative reflected power and the voltage distributedalong the parallel legs.The resonant structure in Fig. 5.2(a) can be described by a lumped elementparallel resonant circuit as shown in Fig. 5.2(b)'. The input admittance of the resonatorin Fig. 5.2(b) is1^1^.yo„)).-.-+-- +.1(.0%.R jcaLwhere co is the angular frequency, L is the inductance, C the capacitance and R theresistance of the lumped circuit. Comparing Eq.(5.7) with Eq.(3.4)-Eq.(3.9) in ChapterDI, we haveR= 2Z---Lsin20a Ao(5.8)C=Zrcot(0)^ (5.9)L=Zrtan(0)^ (5.10)At the resonant frequency co o , woL = 1/6)0C, therefore the input admittance is purely real* Both Fig. 5.2(b) and Fig. 3.3 are equivalent circuits of the "cul-de-sac" resonator. Forsmall 6 the input impedances calculated from them are the same. While Fig. 3.3 isconvenient for calculating the input impedance, Fig. 5.2(b) is convenient for calculatingthe quality factor Q.(5.7)74(5.11)V2 (5.15)P'' 2Rat the frequencies co very close to the resonant frequency w o , that is to say, when A co =co - wo is very small, the admittance of the circuit can be written as:Y(6))=-1-+jwC(--f--1 -) - 6-22) (5.12)R^coo^(..)=—+0C (5.13)whereRis the bandwidth being considered.8.2(4)-(00) (5.14)Following the method laid out in Coffin's book[17], when a voltage V is appliedto the resonator, the average power lost iswhile the maximum electrical energy stored in the resonator's capacitor isW,= V2C- s= 2 '(5.16)and therefore, at the resonant frequency co o the unloaded quality factor ()„ is,(4=w0CR .^ (5.17)The input admittance can be written as a function of Q, as75Or176))=-L+j a Qs,R coo R(5.18)Yi(w)=G(l+j-8 Q.)^ (5.19)(Jowhere G =1/R.If the network analyzer is matched to the measuring probe and the total systemis calibrated properly, the relative reflected power in dB can be interpreted as the returnloss RL, which is defined asRL=-10log(-1)Pi=-101ogra(5.20)(5.21)where r is the reflection coefficient, P r is the reflected power, and P i is the input power.r= yo Yi^ (5.22)Yo+YiThe return loss is the directly measured value in the measurement method used in thisthesis.It is seen that the reflection coefficient r is a function of the frequency co, and itis a complex number. The square of its amplitude can be written as76^8^2Yo-G(1+1 Q„).)Ir^8(6012— I^o ^iYo+G( 1 ti—QH)wo8^2(Yo-G)2+(- (2.G)(ao(Y0+02+(-8 Q, G)200( 3-1)2441-Q1Y(i)o(5.23)(5.24)(5.25)(134-1)2 +(---Q,a)2coowhere B is the coupling coefficient.At the resonant frequency co o, 6=0, so the reflection coefficient is real and thesquare of its amplitude reaches the minium value r.:2_ YO-G 2r. —[+_]Yo u_ (R-1)2(p+1)2where the coupling coefficient B can be solved as14-rP =^m .1—T„(5.26)(5.27)(5.28)Eq.(5.27) shows that at the resonant frequency co o the resonator and the generator arematched better than at any other frequency and the power reflected by the resonator is77a minimum. Using Eq.(5.28) we can rewrite Eq.(5.27) as(13 4- D2r„,+(-15-Qd2(00IR(0)1 2-^(P+1)2+(-6 Q"(0 0(5.29)Eq.(5.29) is the basic formula in the reflected power measurement. From this formulaQu can be solved for as(22=0 + 02 ( 6)0; r2-r .8^i-r2or substituting Eq.(5.28) into Eq.(5.30) and taking the square root(5.30)_ 6)0 2 \ P2-r2m8 i-rm i-re (5.31)Both Eq.(5.30) and Eq.(5.31) can be used to calculate Qu. All of the variables on theright hand sides of these two equations are measurable directly or indirectly, where 8 isgiven by Eq.(5.14), B by Eq. (5.28) and r can be solved from Eq.(5.21)ALr=10 20 (5.32)A plot of the return loss vs frequency, using the scalar network analyzer, is shown inFig. 5.3. To determine Qu from this curve, four parameters have to be measured. Thefirst is the maximum return loss which will give the minimum reflection coefficient r m .The second is the resonant frequency coo. The third is the return loss at another pointproperly selected to calculate r. The fourth is the frequency difference 6 between this78co^0.0 -- c-• •NIOW^ 2.0 --LON.o -4.0 -a.-00^-6.0 --r)a)^-4 : :a)^-8.0-CC ..-C)^..4.) -10.0 -N-6^-E -12.a -L._^„o "z^,-14.0 ^5.514- 3dB BW=55.4 MHz'\ fo=7.1225 GHz6.0 1^ '^' ' 'O 7^.5' ' '1 ^I'^is.b^8^' ^'^'^I6 T . 5Frequency in GHzFigure 5.3 A plot of normalized reflected power vs frequency recordedby network analyzer79selected point and the point having the same return loss, i.e., the bandwidth at theselected point.The measurement of the two parameters, RL and 0 co, mentioned above, can berepeated at several locations on the curve. The resulting Qu should be the same eachtime. In this way one can check whether the measurement is correct or not, which wedid and found to be the case. The choice of the location of these points is notsignificantly important and will be discussed later on in the section where the accuracyof this method is discussed. The method used in this thesis is described below.Using the half power concept, the return loss is measured at a point at which thereflected power is twice the minimum reflected power, that is where1'4=2.14u,^ (5.33)Rewriting both side of the above equation in dB we find-10100'2= -10log2+logrt,^(5.34)or in return lossRL=RL.+3dB^ (5.35)Eq.(5.34) shows us that the difference between the measurement point and theminimum return loss point ( the bottom of the return loss vs f curve ) is 3dB. Knowingthat, it is very easy to find the correct position on the RL vs f curve. This method issomewhat similar to the transmitted power method in which the frequency bandwidth isalso measured at the position 3dB from the top of the curve, but, the physical conceptsin the two method are different, resulting in a formula to calculate Q, that is different as804 ^It —r n)2 1_2r2,(5.37)241.6)well. Substituting Eq.(5.34) into Eq. (5.31) we getwo)2 4 ^r2m 8 0 —r„)2 1-2rt,(5.36)The fractional 3dB bandwidth 8 is the commonly used quantity for evaluating theQ of a resonator. As mentioned when discussing the transmission measurement method,the loaded quality factor Q is just the ratio of the resonant frequency to the 3dBbandwidth. It is meaningful to calculate the Q at this point, where it is pretty close tothe bottom of the curve, so that the linear relationship between the input admittance ( orimpedance ) in Eq. (5.18) holds well. This would prevent some measurement error.Making use of this concept, we can obtain a rough idea of Q, using the 3dB bandwidthfrom the RL vs frequency curve. Furthermore, from Eq.(5.35) we can see that if wecouple the resonator in such a way that the minimum reflection coefficient rm satisfiesthe following conditionEq.(5.35) becomesQ.= (L)o^(5.38)(5.39)where O co =00 - (03413*81Eq.(5.38) gives a simple relation between the Qu , the centre resonant frequency,and the 3dB bandwidth. With the help of a computer Eq.(5.37) is solved and the twosolutions for rm are -0.517 and 0.319. The corresponding minimum return losses RL mare 5.7 and 10.7. That means that when the minimum return loss is 10.7(when reflectioncoefficient r rn < 0) or 5.7 (when rm >0), the ratio of the resonant frequency to the 3dBbandwidth is the Qu . This makes the reflection method more straight forward and easierto use than transmission method, because in the latter the 3dB bandwidth only gives theloaded quality factor, Q. In order to obtain Q u using the transmission method, one hasto determine the coupling coefficient B first, and then calculate Q u using Q1 and a SmithChart. It is known that direct measurement of Q u , using the transmission method, cancause considerable errors[49], but no problems were found with the reflectionmeasurement method using the scalar network analyzer. In this sense, the methodintroduced here has its advantages over other methods.5.3 Measurement ProcedureIn the previous section some general ideas of the measurement method werediscussed. In practice attention must be paid to some of the detailed procedures in orderto obtain good measurement accuracy. This will be discussed in this section.825.3.1. The Measurement Set-UpFig. 5.4 illustrates the set-up used for the measurement of the resonatorsparameters and Fig. 5.2(a) is its sketch. Here S is a levelled synthesized sweeposcillator(HP8341B), of which the output sweep-frequency signal can be selected in anyband between 10 MHz and 20 GHz. D is a directional bridge(HP85027E) which teststhe difference of the forward and backward power by testing the voltage signals.Actually, the out-of-balance signal of the bridge is proportional to the reflectioncoefficient of the device under test(DUT)[51]. G is a scalar network analyzer(HP8757C).P is a microstrip probe from Design Technique(MSP10-01). Basically, the microstripprobe is a coaxial-to-microstrip launch which is attached to a XYZ positioner. It containsa 50 n microstrip transmission line on a length of 0.010 alumina substrate and an integralspring loaded ground pin. The ground pin has 0.015 inches of vertical travel whichensures that the probe will make contact on substrates of different thicknesses.All the measurement procedures are performed under a 30x microscope since thedimensions of both the structures and allowed movements are on the order of only a fewthousandths of and inch.5.3.2 The Measurement PrinciplesDuring measurement the swept signal from the generator arrive at to the DUT by83Figure 5.4^The set-up for measuring the resonator parameters84passing through the directional bridge. A portion of the swept signals reflected by theDUT, passing through the directional bridge, will be coupled to the network analyzer sothat the reflected power may be measured.5.3.3^Measurement Procedures and Techniques1. Warming up the Network analyzerBefore taking any measurements the system should be turned on and allowed towarm up for about half an hour. Distinct shifts in the reflected power vs frequencycurves were observed after some ten minutes without warming up.2. CalibrationThe calibration is done with the measurement circuit both open and shorted.Following the instructions, the probe was connected to a short and then opened. Sinceno ideal short is available in our lab, a piece of copper was used. The open circuit wassimply realized by leaving the probe in the air without contacting anything. Thiscalibration with short and open circuits ignores the noise from the connections andenvironment. It is used to set a reference for the reflection measurement. When thecircuit is either open or shorted, the reflection coefficient is unity and the reflected poweris equal to the power delivered to the sample. The average power, in dB, of the tworeflected signals is stored in the network analyzer as a reference of the input power to thesample. During measurement the power reflected by the sample is compared to thisreference and the return loss RL is just the difference between the two in dB.853. Probing the sampleThe probing technique is very important as regards obtaining accuratemeasurements. Improper probing leads to a contact impedance which increases thereflection and radiation loss as well as shifts the resonant frequency.In normal operation the ground pin extends well beyond the bottom of themicrostrip substrate and retracts as the probe is lowered into position to make contact tothe DUT. The probing technique basically uses a plastic spacer as a feeler gageindicating when the probe tip is in close proximity to the DUT. The most visibleevidence of successful contact can be found on the screen display of the networkanalyzer. The sequence in Fig. 5.5 illustrates the probing technique as discussed below.(1). First lower the probe using the z-axis vernier of the X,Y,Z-positioner untilthe probe tip is in the focal plane of the microscope but not in contact with the resonator.Looking through the microscope, align the probe so that the edge of the probe tipsubstrate is directly above and in-line with the edge of the test substrate, see Fig. 5.5(a).(2). Using the scale of the x-axis vernier of the X, Y, Z-positioner as a guide movethe probe so the edge of the probe tip substrate is 0.008" inside the edge of the testsubstrate, Fig. 5.5(b). At this stage the probe is still open and the return loss displayedon the network analyzer is a zero dB straight line.(3). Insert the spacer between the probe tip and the microstrip ring as shown inFig. 5.5(c). While moving the plastic shim in an in/out motion across the top of themicrostrip ring of the resonator slowly start lowering the probe. When the probe tip startsto grab the shim stop lowering the probe and remove the shim from under the probe tip.86'FrostcowT&cr•^ Pamirsuss,*.Tc TMCARRIERRETRACTABLEGRGLRIB RIR\ \\: .\\\ \.\\:`\\N--- CARRIER CARRIER CLAMP(a)^(b)PQM/SUBSTRATEC CN TACT(c) (d)Fig. 5.5 probing procedures(as supplied by DESIGN TECHNIQUE INTERNATIONAL, INC)87At this point, the probe tip is separated from the top of the resonator by the thickness ofthe plastic shim, about 0.002". This air gap forms a capacitor between the microstripring and the probe tip through which the electric field can be coupled. A peak can beseen on the display of return loss at a frequency of about 15 GHz, corresponding to thesecond harmonic resonance.(4). Contact could now be made to the resonator by lowering the probe 0.004"(thethickness of the shim plus 0.002") using the scale on the z-axis vernier. While loweringit, a peak at about 7 GHz was found on the return loss curve. This peak became sharperand sharper. When successful contact was made, the curve stabilized, so that furtherlowering of the probe had no obvious effect on the return loss curve. Now themeasurement could be made. The correct probing was relatively easy to repeat, and itwas found that there was a tunable range of positions within which the contact impedancemight be neglected, i.e., no obvious difference was found in the measured reflectedpower. Therefore, if the radiation loss, caused by the contact impedance was smallenough to be neglected, the probing condition was not terribly critical.5.3.4 Measurement of 8, fo and RLThe measurement is very simple using the facilities in our network analyzer.Pushing the "minimum" button both the return loss and the resonant frequency at thelowest point of the curve are displayed on screen, and the "bandwidth" button will givethe bandwidth at any selected point on the curve.88Although the measurement itself is simple, the selection of the position on the"cul-de-sac" resonator at which the probe should be put and points of the plotted curveat which the data are collected determines the accuracy of the measurements.Theoretically, a plot of the return loss vs frequency recorded at any coupling point shouldgive the same Qu , but different Qu 's were found for different coupling position. Thereason is that Eq.(5.31) for calculating Q is only accurate for a certain range of couplingcoefficients. If the minimum return loss r m and r are either too small or too big, theresulting Q may not be accurate. On the other hand, Q is a constant for a fixedresonator and independent of the positions on the RL vs frequency curve where onemeasures the bandwidth S and reflection coefficient r. In the other words, given a plotof RL vs frequency, different r's and d's will result in the same Q by using Eq.(5.31).This basic principle can be used as a criterion to check if the coupling position wasselected properly and if the probing was correct. It turned out in our experiments thatthis criteria holds very well for the minimum return loss of about 10.7 dB.In practice the probing was done at several points along the microstrip ring, soas to find the point where the minimum return loss was about 10.7 dB, and the 3dBbandwidth 63dB was then measured. The other merit of this coupling point, as provedearlier in Section 5.2, is that the ratio of the resonant frequency co o to the 3dB bandwidth63dB will directly give the Q ( see Eq.(5.38) ). This measurement method introduces lesserror.895.4 Measurement Results and Discussion5.4.1 Calculating Q uFollowing the method described above, a graph of the return loss RL vs sweptfrequency was displayed on the screen of the network analyzer. In order to get a clearplot, the data collected by the network analyzer is transferred to an IBM compatible 486computer and printed using a HP LaserJet IIP laser printer. Fig. 5.3 shows thenormalized reflected power vs swept frequency. It is the same as a plot of the return lossvs frequency but upside down. At the peak of the curve the return loss is 10.72 dB andthe resonant frequency is 7.1225 GHz. The bandwidth at 3dB from the peak is 0.055457GHz. The ratio of the resonant frequency to 3dB bandwidth is 128.4. Q u calculatedfrom Eq.(5.31) is 118.0. This value should be corrected using values measured at othernear-by points, as shown in Table 5.1.Table 5.1 The measured return losses and bandwidths, and the correspondingunloaded quality factor Q.Return Loss RL (dB) Bandwidth A f (MHz) Unloaded quality factor Q.6 0.081 1177.72 0.056 1189 0.049 12310 0.020 13090The final value of Qu was determined as the mean of the four measured values:Q=123. The calculation error is less than 5%, that is about the same order ofaccuracy as those of the network analyzer and the directional bridge. Therefore, theaccuracy of this measurement can be estimated to be about 10%.5.4.2 DiscussionThe quality factor calculated from the structure dimensions actually measured,gave a Qu of 179. The difference between the measured and the calculated value isabout 30% i.e., the measured Q u is about 70% of the calculated value for the structurefabricated. Because the voltage is proportional to the square root of the power, theoutput voltage of the resonator can reach 84% of the design value.Besides. the errors introduced in the modelling, other reasons for this differencemay be attributed to:1. Fabrication. Since we used an emulsion mask, the definition of themicrostrip edges was rather poor. The plating solution was not filtered and theagitation condition was not ideal. Also, the power source was not stable resulting invariable platting current densities. This resulted in a plated surface that was notsmooth. During plating the surface changed colour several times indicating that theconductor was not pure gold. Such defects increase both radiation and attenuationlosses and, of course, result in a lower Q.2. Measurement. As discussed in last section the accuracy of the91measurement of Q i, was about 10%. This may lower the measured value of Q, aswell.Under improved conditions we would expect a significant improvement in Q.That will be discussed in next chapter.5.5 ConclusionDespite the difference in the Q u 's this resonator increases the voltagesufficiently to be used in the intended application without unacceptable increases indrive power. Applying the design theory and the fabrication techniques developedhere to a "cul-de-sac" resonator on a LiNbO 3 substrate at an operating frequency of15GHz, a supply power of 100 mW should theoretically result in an output voltage ofabout 60 V. Taking the various factors, described above, that would act to lower Q„,into account, there would still be an output voltage of 49 V between the parallel legsof the "cul-de-sac" resonator. Therefore, a "cul-de-sac" resonator fabricated onLiNbO3 substrate, for use as the electrode, can satisfy the requirements of an opticalcommutator switch.92Chapter VI. Conclusions and Suggestions for Further Research6.1 ConclusionsIn this thesis a novel resonator to be used as the electrode for a high speed opticalcommutator switch is studied both theoretically and experimentally. A practicalresonator, a "cul-de-sac" shape resonator, is suggested to be the best resonator structurefor such a purpose. It has two substantially parallel legs unconnected on one end andconnected via an open ring on the other. It is the parallel microstrip lines that work asthe electrodes. Because of the stored energy the potential difference between the two legscan be much higher than the source voltage. Therefore, the resonator electrode can havesignificant advantages over other kinds of electrodes. By using a resonator electrode theoptical switch can operate at higher speed while consuming little power.The "cul-de-sac" resonator was designed using electromagnetic field theory andtransmission line theory. The field distribution in the "cul-de-sac" resonator is apiecewise smooth sinusoidal function. The effects of discontinuities in the resonator weretaken into account, by including additional impedances, and were reduced by chamferingthe sharp corner and by adjusting the total microstrip length. In order to calculate theradiation loss from such a planar resonator, which is as important as the conductorattenuation loss, new formulae were derived using Green's image method. The storedenergy, the radiation loss and the ohmic skin depth loss, and the unloaded quality factorQu were calculated. The structural parameters of the "cul-de-sac" resonator, such as the93length of the parallel lines, the radii of the open ring, the widths of the microstripsconstituting the parallel lines and the open ring, and the thickness of the substrate wereoptimized for maximum output voltage and minimum consumed power.The coupling between the "cul-de-sac" resonator and the power source was alsostudied. The coupling point should be at a position where the current is high and voltageis low. Tap point coupling was chosen and a quarter-wave-length microstrip transformerwas used to reduce the reflection and disturbance to the field distribution inside theresonator due to the discontinuity effects of the T-junction. The whole resonant systemcan be described by its equivalent lumped circuit as shown in Fig. 3.3, from which theinput impedance of the resonator and the coupling coefficient B were derived as functionsof the coupling position 0. Then an important relationship between the output voltageVo, the source voltage V„ and the unloaded quality factor Q, was derived telling us thatVo is proportional to the product of V, and the square root of Q. Also, from thisrelationship we proved that the most efficient and practical coupling condition is B51.To verify the design theory, experimental "cul-del-sac" resonators, operating at7 GHz using Al203 substrates, were designed, fabricated and tested. Two new platingmethods for plating thick and fine metal strips, double photoresist patterning andaluminum mask methods, were tried, to overcome the edge effect so as to reduce thelosses. Gold layers 6 thick with smooth surfaces were plated successfully.New methods for interpreting the unloaded quality factor Q u , probing the sample,and sampling the data were developed, so that Q, could be measured using a scalarnetwork analyzer. It was proved that if the minimum return loss is 10.7 dB, then Q, is94equal to the ratio of the resonant frequency to the 3dB bandwidth. The designed andmeasured Qu of the "cul-de-sac" resonator fabricated on Al 203 were 179 and 123respectively. The reasons of this difference were discussed and the accuracy of thismeasurement was estimated to be about 10%.While the measured Qu is about 70% of the designed value, the actual outputvoltage should be about 84% of the designed value. Despite the difference in the Q u 'sthis resonator increases the voltage sufficiently to be used in the intended applicationwithout unacceptable increases in drive power. Applying the design theory and thefabrication techniques developed here to a "cul-de-sac" resonator on a LiNbO 3 substrateat an operating frequency of 15GHz, a supply power of 100 mW should theoreticallyresult in an output voltage of about 60 V. Taking the various factors, described above,that would act to lower Q., into account, there would still be an output voltage of 49 Vbetween the parallel legs of the "cul-de-sac" resonator. Therefore, a "cul-de-sac"resonator fabricated on LiNbO3 substrate, for use as the electrode, can satisfy therequirements of an optical commutator switch.6.2 Suggestions for the Further ResearchThe "cul-de-sac" resonator should next be fabricated on LiNbO 3 . In order toimprove the design, fabrication, and measurement techniques, further investigation in thefollowing areas is suggested:1. Mask design. Because the modelling is based on ideal conditions, the design95method itself introduces errors so that the resulting parameters of the resonator structuremay not be the best although they were optimized. Therefore, a series of masks shouldbe tried using various structural parameters close to the optimized values.2.Plating conditions. The agitation conditions should be improved and a betterpower source should be used to control the current density.3. jvleasurement. As regards of the different Q u's found at different samplingpoints on the plot of return loss vs frequency, the data processing should be done usinga computer program which optimizes the calculation of Q, by sampling the data fromthe network analyzer during the measurements. In this way the incorrect probing can beidentified and corrected during the measurement.4.Shielding To reduce the radiation loss further more one could shield the entireresonant system.5. The "cul-de-sac" resonator could be used as electrode for optical switches athigher frequencies up to 50 GHz. At higher operating frequencies, lower conductorattenuation and higher radiation loss are expected, and, as the result, the total losses willnot increase seriously. If the whole system can be shielded the radiation loss will notincrease the total loss at all, so that a supply power of 100 mW should result in an outputvoltage of about 50 volts. Therefore, we suggest that an Y-branch optical commutatorswitch using the "cul-de-sac" resonator fabricated on LiNbO3 substrate should work ata frequency as high as 50 GHz, with about the same on/off ratio and percentage ofguided power.96References1. S.K. Korotky, G. Eisenstein, et al, "Optical Intensity Modulation to 40 GHzUsing a Waveguide Electro-optic Switch," Appi. Phys. Lett. vol. 50, no. 23,pp.1631, 1987.2. R.C. Alferness and L.L. Buhl, "High-speed Waveguide Electro-optic PolarizationModulator," Optics Left. vol. 7, no. 10, pp. 500-502, 1982.3. J.J. Veselka and G.A. Bogert, "Low-loss TM-pass Polarizer Fabricated by ProtonExchange for Z-Cut Ti:LiNbO3 Waveguides," Electron. Lett. vol. 23, pp. 29-30,1987.4. K. Kubota, J. Noda and 0. Mikami, "Travelling Wave Optical Modulator Usinga Directional Coupler LiNbO 3 Waveguide," 'MFR Journal of QuantumElectronics, vol QE-16, no. 7, pp. 754-760, 1980.5. R.A. Becker, "Broad-Band Guided-Wave Electrooptic Modulators," IEEE Journalof Quantum Electronics, vol. QE-20, no. 7, pp. 723-727 1984.6. N. A. F. Jaeger and L. Young, "Voltage-Induced Optical Waveguide modulatorin Lithium Niobate," IEEE J.Quantum Electron., vol. 25, no. 4, pp. 720-728,1989.7. Y. Silberberg, P. Perlmutter, and J. E. Baran, "Digital Optical Switch," Appl.Phys. Left., vol. 51, no. 16, pp. 230-1232, October 1987.8. P. Granestrand, et al., "Integrated Optics 4x4 Switch Matrix with Digital OpticalSwitches," Electron. Lett., vol. 26, no. 1, pp. 4-5, January 1990.979. A. DjupsjObacka, "Time Division Multiplexing Using Optical Switches," IEEEJ. Select. Areas Commun., vol. 6, no. 7, pp. 1227-1231, August 1988.10. A. Yariv and P. Yeh, "Optical Waves in Crystals," John Wiley & Sons, NewYork, pp. 220-274, 1984.11. Winnie C. Lai, "The Effects of Branch angle on a Y-branch Optical Modulator,"The First Graduate Student Conference on Opto-Electronics Material, Devices,and Systems, pp. 30, McMaster University, Hamilton, Ontario, 1991.12. T. Tamir(editor), "Guided-Wave Optoelectronics," Springer Series in Electronicsand Photonics 26, Sprinter-Verlag: Berlin Heidelberg. 1988.13. I.P. Kaminow, L.W. Stulz, and E.H. Turner, "Efficient Strip-waveguideModulator," Appl. Phys. Lett., vol. 27, no.10, pp. 555-557, November 1957.14. Nicolas A.F. Jaeger and Winnie C. Lai, "Y-branch Optical Modulator," SHEvol. 1583, Integrated Optical Circuits, pp.202-209, 1991.15. E.L. Ginzton, "Microwave Measurements," New York: McGraw-Hill, pp. 391-461, 1957.16. K. Chang(editor), "Handbook of Microwave and Optical Components," JohnWiley & Sons, New York, pp. 191-219, 1989.17. Robert E. Collin, "Foundations for Microwave Engineering," McGraw-Hill, NewYork, pp. 65-147, 1966.18. R.A. Pucel, D.J. Masse, et al," Losses in Microstrip," WEE vol. MTT-16,no.16, pp. 342-350 June 1968.19.^E. Belohoubek and E. Denlinger, "Loss Consideration for Microstrip Resonators,"98IEEE Transactions on Microwave Theory and Techniques, pp. 522-526 June1975.20. L. Lewin, "Radiation from discontinuities in stripline," Proc. Inst. Elect. Eng.,pp. 163-170, 1960.21. Y.X. Wang, "Coplanar Waveguide Shout-Gap Resonator for MedicalApplications," IEEE Transaction on Microwave Theory and Techniques, vol.MTT-33 no. 12 pp. 1310-1312 1985.22. R.J. Roberts and B Easter, "Microstrip Resonators having Reduced RadiationLoss," Electronics Letters vol. 7, no. 8, pp. 365-368, April 1971.23. P. Troughton, "Measurement Techniques in Microstrip," Electronics Letters vol.5,pp. 25-26, 1969.24. I. Wolff and V. Tripathi, "The Microstrip Open-Ring Resonator," IFF.F.Transaction on Microwave Theory and Techniques, vol. MTT-32, pp. 102-107,1984.25. K. Chang and S. Martin, et al, "On the Study of Microstrip Ring and Varactor-Tuned Ring Circuits," JEFF. Transaction on Microwave theory and Techniques,vol. MTT-35, no. 12, pp. 1288-1295, 1987.26. I. Wolff and N.ICnoppik, " Microstrip Ring Resonators and DispersionMeasurement on Microstrip Lines," Electron. Lett. vol. 7, pp. 779-781, 1971.27. K.C. Gupta, R. Garg,.and LI. Bahl, "Microstrip lines and Striplines," ArtechHouse, Norwood, 1979.28.^B. Easter and R.I. Roberts, "Radiation from Half-Wavelength Open-Circuit99Microstrip Resonators," Electon. lett. vol. 6, pp. 573-574, 1970.29. V. Tripathi, and I. Wolff, "Perturbation Analysis and Design Equations for Open-and Closed-Ring Microstrip Resonators," IEEE Transaction on Microwave Theoryand Techniques, vol. 32, no. 4, pp. 405-409, 1984.30. R. Garg and I.J. Bahl," Microstrip Discontinuities," Mt. J. Electronics, vol. 45,no. 1, pp. 81-87, 1978.31. W.J.R. Hoefer, and A. Chattopadhyay, "Evaluation of the Equivalent CircuitParameters of Microstrip Discontinuities Through Perturbation of a ResonatorRing," IEEE Transaction on Microwave Theory and Techniques, vol. 23, pp.1067-1071, 1975.32. P. Silvester and P. Benedek, "Equivalent Capacitances of Microstrip OpenCircuits," IEEE Transaction on Microwave theory and Techniques, vol. MTT-20,no. 8,pp. 511-516,1972.33. A.-M. Khilla, "Computer Aided Design for Microstrip Ring Resonators," inProc.11th Eur.microwave Conf.(Amsterdam), 1981.34. R.P. Owens, "Curvature Effect in Microstrip Ring Resonators," Electron. Lett.,vol. 12, no. 14, pp. 356-357, 1976.35. G. Kompa and R. Mehran, "Planar Waveguide Model for Calculating MicrostripComponents," Electron. Left., vol. 11, pp. 459-460, 1975.36. H. Groll, and W. Weidmann, "Measurement of Equivalent Circuit Elements ofMicrostrip Discontinuities by a Resonant Method," NTZ, vol. 28, pp. 74-77,1975.10037. L.J. Van der Pauw, "The Radiation of Electromagnetic Power by MicrostripConfigurations," IEEE Transaction on Microwave Theory and Techniques, vol.MTT-25, pp. 719-725, 1977.38. M.D. Abouzahra and L. Lewin, "Radiation from Microstrip Discontinuities,"IEEE Transactions on Microwave Theory and Techniques, vol. MTT-27, no.8,pp. 722-723, 1979.39. S.E. Schwarz, M.D. Prouty, and K.K. Mei,"Radiation from Planar Resonators,"IEEE Transaction on Microwave Theory and Techniques, vol. MIT-39, no. 3,pp. 521-525, 1991.40. R.F. Harrington,"Time-Harmonic Electromagnetic Fields," McGraw-Hill, NewYork, pp. 95-132, 1961.41. S.Ramo, J.R. Whinnery and T.Van Duzer, "Fields and Waves in CommunicationElectronics," (2nd) John Wiley & Sons, New York, pp. 579-606, 1984.42. E. Hammerstad and 0. Jensen, " Accurate Models for Microstrip Computer-Aided Design," International Microwave Symposium Digest, Washington, pp.407-409, 1980.43. M. Dydyk, "Efficient Power Combining," IEEE Transaction on MicrowaveTheory and Techniques, vol. MIT-28, no. 7, pp. 755-762, 1980.44. M. Dydyk, "EHF Planar Module for Spatial Combining," Microwave Journal,pp. 157-176, May 1983.45. R. Williams, "Modern GaAs Processing Methods," Artech House, Norwood,1990.10146. R.K. Hoffmann, "Handbook of Microwave Integrated Circuits," AfTech House,Inc., Norwood, pp. 23-47, 1987.47. J.P. Ramy and M.t. Cotte, "Optimization of the Thick- and Thin-FilmTechnologies for Microwave Circuits on Alumina and Fused Silica Substrates,"MEE Transaction on Microwave Theory and Techniques, vol. M1T-26, no. 10,pp. 814-820, 1978.48. F.A. Lowenheim, ed. "Modern Electroplating," Third Edition, John Wiley andSons: New York, 1974.49. A. Khanna, and Y. Garault, "Determination of Loaded, Unloaded, and ExternalQuality Factor of a Dielectric Resonator Coupled to a Microstrip Line, "IERETransaction on Microwave theory and Techniques, vol. MTT-31, no. 3, pp. 261-264, 1983.50. D. Kajfez and E.J. Hwan, "Q-Factor Measurement with Network Analyzer,"TEFF. Transaction on Microwave Theory and Techniques, vol. MTT-32, no. 7,pp. 666-670, 1984.51. "Microwave Measurement," Chapter 10, edited by A.E. Bailey, Peter PergrinusLtd., pp. 209-247, England, 1989.102BIOGRAPHICAL INFORMATION NAME:^/71171/00-/E GHE/1/MAILING ADDRESS:^2 7/ C OyAivi A COW y^vER 8PLACE AND DATE OF BIRTH:^Oec^/Y -57^BEIIIAA , Loz/vAEDUCATION (Colleges and Universities attended, dates, and degrees):CE^T TU E of TECO vez-& 0Y^Sep^935^/01,4y,de^/?/I A . S4W1g4;1HUAZHONGU/ lVER51 Ty^ScIen/CE and T E0-010 L 0 El' y friARCe--1 ,^,TA/v,^ Y 28 A, Se,POSITIONS HELD:RESEARCH 45.5 I S FAA/TPUBLICATIONS (if necessary, use a second sheet):AWARDS:Complete one biographical form for each copy of a thesis presentedto the Special Collections Division, University Library.06-5
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Microstrip resonators for high speed opitical commutator switches Chen, Mingche 1992
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Title | Microstrip resonators for high speed opitical commutator switches |
Creator |
Chen, Mingche |
Date Issued | 1992 |
Description | Using a planar microwave resonator as the electrode for modulating a high speed Y-branch optical commutator switch is a novel idea. For this purpose a "cul-de-sac" shape microstrip resonator is suggested and studied both theoretically and experimentally. It has two substantially parallel legs unconnected on one end and connected via an open ring on the other. It is the parallel legs that work as the electrode. Because of the stored energy the potential difference between the two legs can be much higher than the source voltage. Therefore, the resonator electrode has significant advantages over other kinds of electrodes. By using a resonator electrode the optical switch can operate at high frequencies while consuming little power. The field distribution, discontinuity effects, and the radiation loss of the "cul-de-sac" resonator are discussed. The structure parameters of the "cul-de-sac" resonator, were optimized for maximum output voltage and minimum consumed power using a computer program. The source power can be coupled into the resonator by tap point coupling through a quarter-wave-length transformer. The input impedance of the resonator and the coupling coefficient β were derived as functions of the coupling position θ using an equivalent lumped circuit. Then an important relationship between the output voltage V0,the source voltage Vs„ and the unloaded quality factor Qu. is derived which tells us that Vo is proportional to the product of Va, and the square root of Qu. From this relationship we proved that the most efficient and practical coupling condition is β<1. [More abstract follows] |
Extent | 5687315 bytes |
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Thesis/Dissertation |
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Text |
FileFormat | application/pdf |
Language | eng |
Date Available | 2008-12-23 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0064797 |
URI | http://hdl.handle.net/2429/3308 |
Degree |
Master of Applied Science - MASc |
Program |
Electrical and Computer Engineering |
Affiliation |
Applied Science, Faculty of Electrical and Computer Engineering, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 1992-05 |
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UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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