Spectral domain analysis of patch antennas and (M)MIC structures on lossy anisotropic substrates with possible applications to high-Tc superconductors

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ANTENNAS AND (M)MIC STRUCTURES ON LOSSY

ANISOTROPIC SUBSTRATES WITH POSSIBLE

APPLICATIONS TO HIGH-TC SUPERCONDUCTORS

DOCTOR OF PHILOSOPHY

in the Department of Electrical and Computer Engineering We accept this thesis as conforming

tc the required standard

Dr. J. Borndmann. Suoervlsor (Dent, of Elec. and Comp. Eng.)

Dr. R. Vahltfieck, Departmental Member (Dept, of Elec. and Comp. Eng )

Dr. P. Driessen, Departmental Member (Dept, of Elec. and Comp. Eng.)

Dr. Z.'Dqng Outside yiep/lbef (Dept, of Mech. Eng.)

Dr. V. K^FrrpStHV External Examiner (Dept, of Elec. and Comp. Eng.,

by Zhenglian Cai

B.S., Beijing Normal University, 1983 M.S., Beijing Normal University, 1986

u.j>*A Dissertation Submitted in Partial Fulfillment of the Requirement for the Degree of

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0

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ABSTRACT

In this work, an efficient algorithm to rigorously derive the spectral-domain impedance dyadic Green’s function for patch antennas and (M)MIC on general complex anisotropic substrates is developed. The main advantage of the applied technique is that it provides closed-form expressions for transverse propagation constants and related immittnnces in the spectral domain and, therefore, allows the following parameters and structures to be taken into account: dielectric and magnetic losses of anisotropic media, alternative directions for magnetic bias, the finite metallization thickness of conventional conductors and superconductors including their losses, microstrip, muhiconductor inter connects, coplanar-type structures, patch antenna and feed network. The theory is veri fied by comparison with previously published data. The flexibility is demonstrated for both superconductor and conventional conductor antennas and (M)MIC structures.

Examiners;

Dr. J. Bornemann, Superyispr (Dept, of Elec. and Comp. Eng.)

Dr. R. Vahldfeck, Departmental Member (Dept, of Elec. and Comp. Eng.)

Dr. P. Driessen, Departmental Member (Dept, of Elec. and Comp. Eng.)

DFTZTDongTOutSiCT

Dr. V. K. (Dep*. of Elec. and Comp. Eng.,

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List of Figures

Figure 2.1 Geometry of patch resonator... 5 Figure 2.2 Transmission line model of patch antenna... 8 Figure 3.1 Relationship between (x,y,z) and (v.y.u)

coordinate systems... 14 Figure 3.2 Equivalent transmission line circuits for TM-to-y and

TE-io-y fields... 17 Figure 3.3 Resonant frequency versus patch length (m m )... 29 Figure 3.4 Coordinate system for far-field calculations... 30 Figure 3.S Comparison of measured [70] and calculated radiation

patterns for a single tectangular patch.Parameters. £,=2.33, L=!.lcm , w=1.7cm, h=0.3175cm.

tl=t2=50|am, o=56S/pm, tan8=0.0012... 31 Figure 3.6 Configurations of microsirip line (a) and coplanar waveguide

(b)... 33 Figure 3.7 Normalized wavelength of microstrip line versus

frequency... 34 Figure 3.8 Characteristic impedance of microstrip line versus

frequency... 35 Figure 3.9 Normalized wavelength of coplanar waveguide versus fre

quency 36

Figure 3.10 Characteristic impedance of coplanar waveguide versus fre

quency 37

Figure 4.1 Geometry of patch resonator on lossy biaxial anisotropic sub strate... 40 Figure 4.2 Geometry of patch resonator with two layer lossy

biaxial anisotropic substrate... 49 Figure 4.3 Normalized propagation constant of microstrip line with

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biasing field H0 in z direction versus frequencies. Dimensions: hj=0.254mm, ii2=l.l5mm,w=0.9mni,

4nHmax=174CG, 47tHs=2300G, £rj=9.9, er2= l6 .f i...52 Figure 4.4 Slow wave factor and loss factor of multilayered CPW

with ferrite magnetized tn x direction. Dimensions: s=0.12mm, w=0.1mm, h|=lpm , h2=0.lmm,

£,•1=14.9, £,^=4.3, O j= 0 ,c2=0.1S/mm, 4nH,=725G,

Hx=500 Oers, AH=37 Oers... 53 Figure 4.5 Normalized propagation constant of CPW with

magnetized ferrite. Dimensions: h=!mm, w=lmm, s=0.5mm, e p ll.6 , Ms= 1800A/cm, Hx=300A/cm. Superconductor: t=0.2pm, on=200S/mm,

T/Tc=77/92.5, Xefp3000A ... 54 Figure 4.6 Loss factor versus frequency. Parameters as in Fig. 4 . 5 ... 55 Figure 4.7 Slow wave factor of microstrip line on

ferrite-dielectric substrates with magnetic bias in x direction. Dimensions: h]=lpm , h2=0.1mm, h3=0.5^m, w=0.1mm, 8,1=16.6, £,2=9.9, <jg=40S/pm, Ci =0.1 S/m, o 2=0.LS/m, 4itHs=870G, Hx=22000ers. AH=50Oers.

Superconductor: t^0.2pm, cr„=200S/mm,

T/Tc=77/92.5, A ^O O O A ...56 Figure 4.8 Loss factor of microstrip line on ferrite-dielectric

substrate. Parameters as in Fig. 4.7... 57 Figure 4.9 Slow wave factor of CPW on ferrite-dielectric substrates

with magnetic bias in x direction. Dimensions: s=0.12mm, h]=0.1pm, h2=0.1mm, h3=0.5pm, w=0.1mm, e,i=23, 8,2=12, o g=40S/pm, Oj=0.4S/m, o 2=0.1S/m, 4nH9=14300G

Hx--770G, AH=10Oers. Copper conductor: o g=40S/pin, t=0.1pm. Superconductor: t=0.2^m, o r=200S/mm,

T/Tc=77/92.5, Xeff=3000A...58 Figure 4.10 Loss factor of CPW on ferrite-dielectric substrate

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Figure 4.1 i Figure 4.12 Figure 4.13 Figure 4.14 Figure 4.15 Figure 4.16 Figure 4.17 in Fig. 4.9...59 Normalized propagation constant of microstrip

line with biaxial anisotropic substrate. Dimensions: hl=1mm, w= 1.27mm. Superconductor:

t=0.2pm, on=2tiOS/mm, T/Tc=0.5, Xefp30G0A; a: ex=3.5, e -3.5, ez=5.5, m=3.5, lty=5.5, |iz=4.0. b: ex=4.5, Ey=5.5, Ez=6.5, ^ x=4.5, py=5.5, ^= 6.0.

c: ex=5.5, ey=6.5, ez=7.5, p x=5.5, Py=6.5, jiz=7.0... 60 Resonant frequency of patch resonator veisus anisotropic ratio ey=2.35, £z=£x, L= 1.0cm, h=0.158cm...61 Q'factor of patch resonator versus substrate thickness

with dielectric loss tangent as parameter; £y-Ex=Ez=25, L=6.0mni,w=i.5mm. Superconductor: :=0.5pm,

o n=200S/mm, T/Te=77/92.5, Xeff=300CA... 62 Q-factor of patch resonator versus substrate thickness

with conductor materials as parameter. Parameters: £y=£x=Ez=25, L=6.0mm, w= 1.5mm.

Superconductor: t=0.5|im, o n=200S/mm,

TA>77/92.5, X^fpSOOOA...63 Resonant frequency of patch resonator versus

substrate thickness with conductor materials as parameter. Parameters as in Fig. 4.14 (all three

curves fall within the plotting accuracy)... 64 Resonant frequency of a patch on biaxial anisotropic

substrate; Parameters: ey=9.6, Ez=0, Ho=0.14T, HS=0.15T, h= 1.0mm, w=2.0mm, tan8=10‘s. Copper

conductor; t=0.2pm, o=40S/pm... 65 Normalized E- and H-plane pattern of a patch resonator. Parameters: w= 1.0cm, L=f .2cm, h=3.15mm, ex=2.3, ey= 2.3, £z=0, |ix= ^,=1, pz=0. ideal conductor and

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tan8B0.93,frs:8.12G H z... 67

Figure 4.19 Same as in F g . 4.17 but including conductor und ground plane losses only; for the patch: t|-0.2min, ground plane: ti*0.3mm, cMQS/pm, fr= 8.14GHz...68 Figure 4.20 Same as in Fig. 4.17 but with superconducting patch;

t,=0.3pm, o n=200S/mm, T/Tc=77/92.5, \ cf,=3000A.

fr= 8.15GHz... .69 Figure 4.21 Seme as in Fig. 4.17 but including all losses, patch

thickness: tp0.2m m , t2=0,3mm, o=40S/pm, taii8~0.03,

ff=8.12GHz...70 Figure 4.22 Normalized radiation pattern of a patch resonator on a

substrate with different anisotropy ratio. Parameters:

w=0.2cm, L= 1.0cm, h=1.58nim, tan8=0.002; for the patch: ti=0.2pm , ground plane: tj-O Jm m , o=40S/pm, Ez=(i/ =0,

px=py.“ l, ex=2.35; (i) V=9.6GHz; (ii) fr=8.!5GHz.

(iii) fr=6.98GHz... 71 Figure 4.23 Normalized E-plane radiation pattern with different

magnetic anisotropic ratios. Parameters: w=0.2cm, h=0.158mm, L=lcm, tan8-0.001, py=4, pz=0, O=40S/|J,m; (i) fr=10.47GHz; (ii) fr= 10.25GHz;

tiii) fr=l 0.12GHz...72 Figure 4.24 H-plane pattern corresponding to E-plane pattern of

Fig. 4.23...73 Figure 4.25 E-plane radiation pattern of a patch resonator on

biaxial anisotropic substrate of different height. Parameters: w=0.5cm, L= 1.0cm, tan8=0.001, ex=6.4, ey= 2.3, ez=6.0, p x= 1.5, Jty=2.5, p z=2.0; for the patch:

tf-O.Jmm; ground plane: t2=0.3mm, o=40S/pni; (i) h=0.07mm, f^ .filG H z ; (ii) h=1.5mm, fr=7.27GHz;

(iii) hs-2.0mm, fr=6.62GHz...74 Figure 4.26 H-plane patterns corresponding to E-plane patterns

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Figure 4.27 Resonant frequency of a patch on double-layered anisotropic substrate versus substrate height. h2=0.253 mm, L=8.00mm, w=1.00mm, e,i=16.6,

£,2-9.6, HS=0.16T, Hmax= 0.1 OT... 76 Figure 4.28 Influent J direction and amplitude of magnetic bias

on resonant frequency of a rectangular patch resonator. Parameters: £^=16.6, tanfclO*3, w=2.5mm, L=8.0mm,

h= 1.0mm, t -0.2mm, o=40S/pm, HS=0.16T... 77 Figure 4.29 Influence of direction and amplitude of magnetic bias

on Q-factor of a rectangular patch resonator.

Parameters as in Fig. 4 .2 8 ...78 Figure 4.30 Influence of direction and amplitude of magnetic bias

on res .nant frequency of a square patch resonator. Parameters: w=L =8.0mm, other parameters as in

Fig. 4.29...79 Figure 4.31 Influence of direction and amplitude of magnetic bias

on Q- factor of a square patch resonator.

Parameters as in Fig. 4 .3 0 ...80 Figure 4.32 Influence of different magnetic b r s on radiation pattern

of patch resonator. Parameters: h=lmm, L=8mm, w=2.5mm, £^=16.6, HS=0.16T, o=0.0005S/m; superconducting patch: t=0.5|im, on=200S/mm, T/Tc-77/92.5. Xeff=1500A; ground plane: O=40S/pm, t=0.5mm. Resonant frequencies for different bias:

f,=6.48GHz, f2=7.81GHz, f3=9.08GHz...81 Figure 5.1 Illustration of coupled patch antenna... 84 Figure 5.2 Resonant frequencies of even and odd mode versus

ratio of 2s/h for coupled patches. Parameters: h=0.5mm, 2L=6mm, 2w= 1.5mm, ex/ t y=9.4/11.6, ez=0, c=0.05S/m, superconductor: t=0.5pm, o r=200S/mm, T/Tc=77/92.5,

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Figure 5.3 Q-factor of coupled patches of Fig. 5.2... . .89 Figure 5 . 1 Radiation patterns of coupled patches for even mode

excitation. Parameters: h=lmm. 2L=8mm, 2w=2.5mm, 2s=0.5mm, Ef=23, o=0.01S/m; superconductor: t=0.5pm, o n=200S/mm, T/Tc=77/92.5, Xeff=1500l< , und plane: 0=40 S/pm, t=0.5mm. f,=4.14GHz...90 Figure 5.5 Radiation patterns of coupled patches for odd mode

excitation. Parameters as in Fig. 5.4...91 Figure 5.6 Effective dielectric constant of a coupled

microstrip line versus frequency with different

permeability ratios. Parameters: w=0.6mm, s=0.4mm,

h=0.635mm, ex=Ez=9.4, ey= 11.6, px=pz, p y= 1 .0 ...93 Figure 5.7 Effective dielectric constant of asymmetric

coupled lines versus frequency with different widths dimensions, h=hj+h2=0.6mm, h] =0.4 mm, W|=0.3mm,

W2=0.6mm, exi=7 A eyi= 7.6, ezj=7.4, e2=5.3... 94 Figure 5.8 Effective dielectric constant of a coupled microstrip

line versus frequency. eX2=eZ2=9.4, eX|=EZ|=eyl=2.4,

w=0.6mm, h]=0.3mm, h2=0.6mm, O|=0.3S/pm, o 2=0.5S/pm; ground plane: t2=0.5mm, c g=40S/pm; superconductor:

t=0.5pm, o n=200S/mm, T/Tc=77/92.5, Xeff=1500A... 95 Figure 5.9 Attenuation constant of coupled microstrip line versus

frequency. Parameters as in Fig. 5 . 8 ... 96 Figure 5.10 Characteristic impedance of coupled microstrip line versus

frequency. Parameters as in Fig. 5 . 8 ... 97 Figure 5.11 Effective dielectric constant of an asymmetric three-line

structure versus substrate height ratio.

Dimensions hj=0.5mm, W ]=0.8m m, w2=l .Omm, W3= 1.2mm, S]=0.4mm, s2=0.6mm, exj=2.4, ey)= 2.6,

ezl=2.8, ey2=9.4, ey2= 11.6, ez2=9.4, o^0.3S/m , o 2=0.5S/m, ground plane: t2=0.5mm, o g=40S/pm;

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Figure S. 12 Figure 5.13 Figure 5.14 Figure 6.1 Figure 6.2 Figure 6.3 Figure 6.4 Figure 6.5 Figure 8.1 Figure 8.2 Superconductor: t=0.2pm, c n=200S/mm, T/Tc=77/92.5, ^jplSOOA, M O .O G H z... 98 Losses of an asymmetric three-line, structure versus different substrate height ratio. Parameters as in Fig. 5.U ... 99 Effective dielectric constant of four lines versus

frequencies. Parameters: exi-2.6, eyi= 1.6, eZ]=2.5, ®x2=®z2ss^’^* £y2= l ^ M’y 1=5.6, |lz 1 “ 1 * wj=w3= 1.0mm, W2=W4=0.9mm, si=s3=1.0mm, S2=0.9mm, hpl.Omm, h2=1.0mm, Oj =0.3 S/m,

O2=0.2S/m; copper strips and ground plane: t2=0.5mm,

o s=40S/pm, t]=0.3mm, Og=40S/pm...100 Attenuation constant of four lines versus frequency.

Parameters as in Fig. 5 .1 3 ... 101 Patch antenna fed by a microstiip line (a) and a coaxial

cable (b )... 104 Equivalent TE and TM wave circuit models for a microstrip structure fed by a planar source embedded in the biaxial

anisotropic layer at y = -h ... 106 Equivalent TM circuit model for microstrip structure fed by slice of three-dimensional source embedded in biaxial a lisotropic layer at y = -h ... 113 Radiation patterns of an ideal patch resonator with z-oriented source at different distances: a) h=0.0mm, b) h=0.2mm, c) h=0.4mm. Parameters: d= 1.0mm, L=8.0mm, w=2.5mm, ey=9.6, ez=£x=l 1.4, fr=6.22GHz... 116 Radiation patterns of a y-oriented source at different

distances, a: h=0.0mm, b: h=0.2mm, c: h=n.4mm. Other

parameters as in Fig. 6 . 4 ... 117 Configuration of a microstrip patch antenna fed by a

coplanar-slot line configuration...122 Aperture-coupled patch antenna...123

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Acknowledgments

*

I would like to take this opportunity to thank my supervisor, Dr. J. Bornemann, Professor of the Department of Electrical and Computer Engineering, University of Victoria, for his advice during my whole research and study at the University of Victo ria. Dr. J. Bornemann suggested this research topic. I am grateful to him for his strong support, encouragement and invaluable assistance as well as technical advice through out the development of this dissertation. I am also grateful to him for his financial sup port to make it possible to finish my Ph.D program and this dissertation.

I would also like to thank Dr. R. Vahldieck. Dr. P. Driessen and Dr. Z. Dong for serving on my supervisory committee, providing valuable suggestions, kindly assist ing and patiently reviewing this research throughout my Ph.D program.

In addition, I would like to thank the members of the LLiMic group in the Department of Electrical and Computer Engineering, University of Victoria with members of which I have performed, discussed and published research.

Finally, I would like to thank the faculty members and staff at the Department of Electrical and Computer Engineering, University of Victoria, from whom I got much help.

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Dedication

This dissertation is dedicated to my wife Dong and

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Chapter 1 Introduction

The concept of a microstrip antenna was discussed by Deschamps in 195311], und has been developed and used quite extensively since 1974 for many applications where low profile, light weight and high frequency devices are requiredf2j. Previous work in microstrip antennas has been largely confined to designs on isotropic substrates. There has been a growing interest, however, in analyzing microstrip structures on anisotropic substrates[3]-[4]. Many of the materials used in microstrip structures which have been treated as isotropic are actually slightly anisotropic. For example, sapphire, alumina and a variety of alumina substrates are anisotropic, as are a number of commonly used glass- filled and ceramic-filled polymeric materials such as Duroid and Epsilam[5]-[8]. Particu larly, the wide variety of possible low-loss and low-dispersion applications of high-Tc superconductor-film MMIC’s (Monolithic Microwave Integrated Circuits) with anisotro pic substrates offer attractive solutions in practice, such as in microwave resonators, fil ters, delay lines and antenna systems[9]-[l 1].

Therefore, there is definitely a need to rigorously investigate the characteristics of structures with anisotropic behavior. Moreover, an increasing level of integration in mod em microwave structures, such as (M)MIC, does not allow any changes, modifications or tuning once the circuit is manufactured. A powerful CAD tool is becoming more and more urgent to accurately predict the actual performance of circuits for industry.

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It is not only the purpose of this thesis to rigorously investigate the characteristics of patch antennas and planar waveguide structures based on anisotropic substrates, but also, and more importantly, to provide design engineers with a highly flexible analysis tool. Emphasis is placed on practical relevance, i.e. providing fast and reliable results within several minutes on modem workstations but, at the same time, maintaining the accuracy known from highly computer intensive models and field solvers.

The key component of this work is the development of an efficient algorithm to rigorously derive the spectral-domain impedance dyadic Green’s function for patch antennas and related MMIC structures including uniaxial and biaxial anisotropic sub strates and all circuit losses relevant in practice. Of course, the material parameters will not always simultaneously influence all practical design parameters, but it will be shown that at least some of them are strongly affected. For example, the radiation pattern of a single patch resonator m relatively insensitive to the anisotropy of the substrate whereas

the resonant frequency is highly influenced, and substrate, metallization and surface- wave losses contribute to a reduction in Q factor and radiation efficiency. Note that although the material characteristics (e.g. tensor elements) are not always known, new developments in material technology will require procedures which are capable of includ ing these parameters.

Chapter 2 contains a brief review of both the history of microstrip antennas, and the analytical techniques commonly used for these antennas. Also included in Chapter 2 is a review of electric and magnetic anisotropy as well as superconductors, as it pertains to this work.

Chapter 3 lays the foundation for the work by presenting a unified treatment of a patch antenna on a single layer of isotropic material including a detailed analysis using the new scheme based on the spectral-domain immittance approach (SDIA). A procedure

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known as Galerkin’s method is then reviewed. In this method, basis functions are chosen which approximate the current distribution on the patch or microstrip conductors. There fore, this technique allows one to determine the resonant frequency of the patch from the immittance function and, finally, the radiation pattern.

In Chapter 4, the extended SDIA is used to derive the impedance Green's function of a patch on anisotropic and biaxial anisotropic substrate with different magnetic biasing field. Also included is the fact that a modified impedance dyadic Green’s function can be obtained by considering different metallizations. Finally, two layered impedance Green’s functions in the spectral-domain based on anisotropic substrates are derived. Results of planar circuits and patch resonators are given.

In Chapter S, the characteristics of a coupled patch resonator based on anisotropic substrate ore investigated. The analysis of the interacting microstrip resonant structures is performed with the snectral-domain technique. The resonant frequencies and radiution patterns of patch antennas in the even and odd resonance modes are evaluated from the numerical solution. The presented results also include effective dielectric constants and losses for multiconductor structures.

In Chapter 6, the dyadic Green's function in the spectral domain is evaluated in the general ewe of a planar integrated structure fed by a distribution of three-dimensional electrical sources arbitrarily located within the biaxial anisotropic substrate. The elements of the spectral dyadic Green’s function are evaluated via SDIA to a general three-dimen sional source excitation.

Chapter 7 gives the conclusions based on the above analysis.

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Chapter 2 Patch antenna review

2.1 Introduction

A microstrip antenna is, in its simplest configuration, a conducting patch separated from a ground plane by a dielectric or magnetic substrate, as shown in Fig. 2.1. Radiation from the antenna occurs from the fringing fields between the edge of the conducting patch and the ground p!ane[12]-[13]. Although the conducting patch can be of any shape in gen eral, only rectangular or circular shapes are common. The structure is generally excited by one of two different methods. A microstrip line on the same substrate can be run into the patch at some point along the patch perimeter. An alternative method consists of running a coaxial tine up from underneath the ground plane. The outer shield of the line connects to the ground plane while the inner conductor passes through a hole in the ground piane and substrate and electrically connects to the patch. Depending on the dielectric and magnetic constants and thickness of the substrates, the shape of the patch, and how and at which locations the patch is fed, various radiating characteristics, input impedances and resonant frequencies of the antenna can be achieved, including circularly polarized radiation.

Microstrip antennas were first suggested by Deschamps in 1953[1], However, due to the limitations in the technology of integrated circuits d of photoetching, the first practical antennas were iater developed in the early 1970’s by Howell et.al. [14]-[15] and others. Some of the main advantages of microstrip antennas are their light weight, their

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low volume, thesr low profile and their planar configurations which can be made confor- mal to the surface of other structures. This w:>s demonstrated by Munson[2] in his work on low profile, flush mounted microstrip antennas for rockets and missiles. This practical use of microstrip antennas gave birth to a new antenna industry.

Microstrip antennas are used in the broad frequency range of !(X)MHz-50GHz[13] and have several advantages over conventional microwave antennas, in addition to those already mentioned. These advantages includefli]:

1. The fabrication cost is low, and they are readily amenable to mass production. 2. They can be made thin and therefore do not perturb the aerodynamics of the

host aerospace vehicles.

II

Figure 2.1 Geometry cf patch resonator.

3. Linear or circular (left or right hand) polarizations are possible. 4. Dual frequency antennas can be made eisily,

5. No cavity backing is required.

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7. Feed lines and matching networks can be fabricated simultaneously with antenna structure.

Using the advantages listed above, microstrip antennas have been developed for the following app!ications[12]-[13]:

1. Biomedical radiators. 2. Satellite communications.

3. Feed elements in complex antennas. 4. Satellite navigation receivers.

5. Environmental instrumentation and remote sensing. 6. Doppler and other radars.

7. Radio altimeters. 8. Command and control.

However, patch antennas are not perfect. They are rather inefficient radiators and only work over narrow bandwidth for single layered structures. Also rigorous proce dures to predict the radiation properties of these antennas have yet to be developed.

2.2 Analysis techniques

During the past several years many techniques have been developed to analyze the patch antenna. Some of the more prominent methods will be described briefly. They can be classified in two categories, the first one is that of approximate methods, the second one can be called full wave analyses. Two excellent reviews of the subject are the article by Carver and Mink[17] and the book by Bahl and Bhartia[13].

2.2.1 Transmission line model

Probably the simplest model of a patch antenna is the transmission line model. First proposed by Munson[2], this model works well for rectangular patch antennas

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excited in a quasi TEM mode. Work on this technique has also been published by Dern- eryd[18],[19] and improved by Sengupta[20],[21]. The antenna is modeled as a microstrip transmission line which is fed at some place along the line, as shown in Fig. 2.2. The transmission line is terminated at both ends by its “radiating admittance”. The exact value of this radiating admittance is not known. However, it is small compared to the character istic admittance of the line since the microstrip line essentially ends in an open circuit. The advantage of this technique is its simplicity. Disadvantages include ignoring the field vari ation along the radiating edge and the current distribution on the patch. Moreover, it is only applicable to patches of rectangular shape.

2.2.2 Modal cavity model

A technique more general than the transmission line model is the cavity model for patch antennas. This technique can be applied to any patch antenna whose patch is con formable to some orthogonal co-ordinate system, such as a rectangular or circular patch. This technique was suggested by Deschamps[l] and first worked on by Lo et.al.[22]-[27]. Work in this area has also been published by Van de Capelle et.al. [28]-[29]. The patch antenna is treated as a cavity, the patch and ground plane forming perfectly conducting walls. The aperture in the dielectric around the perimeter of the patch is assumed to be a magnetic wall. To solve for the fields inside this cavity, as well as the current on the patch, a modal or eigenfunction expansion of the fields is used.

2.2.3 Modal leaky cavity model

One of the disadvantages of the cavity model is that it does not predict the true behavior of the antenna near its resonance. Since the patch antenna does not have perf ect magnetic aperture walls, some energy radiates away from it. Unfortunately, since the patch antenna is in general a poor radiator, these antennas are operated at resonance for all

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Yr

- L / 2

L / 2

Figure 2.2 Transmission line model of patch antenna.

practical applications. Therefore, the cavity model does not work well in exactly that region in which the antenna is to be operated. In order to improve the results computed with the cavity model, Carver and Coffey[30-32] have proposed using an impedance boundary condition for the aperture instead of magnetic walls. This technique is called “Modal Leaky Cavity Model” or “Modal Expansion Model”. It not only keeps the fields finite at resonance but also slightly shifts the resonances. The difference between the two approaches is that in the “Cavity Model" the effect of radiation and oth ,'r losses is repre sented as an artificially increased loss tangent of the substrate while in the “Modal Expan sion Model” these effects are taken into account by appropriately using the impedance boundary condition at the walls.

2.2.4 Integral equation approach

In 1979, rigorous mathematical techniques were published. Mosig and Gardiol[33] determined the field produced by a horizontal electric dipole located on top of a grounded substrate. The characteristics of the antenna are evaluated using numerical techniques,

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e.g., the Method of Moments[34].

2.2.5 Dyadic G reen's function method

Another technique is the dyadic Green’s function method. Using a Hertzian dipole printed on the grounded substrate, Alexopoulos et.al.[35] developed u dyadic Green’s function which can be used, via superposition, to determine the field from an arbitrary source distribution.

2.2.6 Spectral domain methods

Another group of techniques can be classified as “Spectral Domain Methods”. As the name implies, these techniques are simply those which formulate the problem in the spectral (or Fourier transform) domain. Many of the technique j discussed previously yield a set of integral equations in the time domain which must be solved. However, for many types of problems, a formulation in the spectral domain is more adequate since the equations to be solved are algebraic. In the literature, there are many variations of utiliz ing the Fourier transform technique and of types of problems solved with these tech niques. The types of problems solved include microstrip transmission lines by Itoh and Mittra, 1973, 1974[36]-[38]; Knorr and Tufekcioglu, 1975(39]; Davies and Mirs^ekar- Syahkal, 1977(40]; Itoh, 1980(41]; Itoh and Menzel, 1981 [42]; Hornsby, 1982[43]; Sharma and Bhat, 1983[44]; Zhang, et. al., 1985[45]; Khanna, et at., 1986(40]; uniaxial patch antennas by Pezar 1989[47], and Nelson 1990(48]. One of these techniques [42] will be discussed in depth in th? following chapters, applied and extended to micro strip and antenna structures.

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All of the techniques described above for analyzing microstrip antennas have been for the cases of isotropic or uniaxial dielectric substrates. However, a growing interest has been observed in the analysis of microstrip patch antennas on uniaxial or biaxial anisotropic substrates[3], [4], [47], [48]. This interest is due in part to some of the intrin sic properties of these materials, such as high reproducibility, low loss exhibited in some dielectric or magnetic materials, and because many of the materials used as substrates for modern microwave integrated circuits exhibit dielectric as well as magnetic anisotropy. For example, sapphire, alumina and a variety of alumina substrates are anisotropic, as are a number of commonly used glass-filled and ceramic-filled polymeric materials such as Duroid and Epsilam[5]-[8], [47], [48]. Especially the newly proposed superconductor p,-»ch antennas show attractive properties compared to the conventional conductor patch antenna[9]-[l 1], [49], [SO]. Therefore, from a practical point of view, there is a definite need to analyze microstrip patch configurations on lossy anisotropic or electric and mag netic biaxial anisotropic materials either for superconductors or for conventional conduc tors. Moreover, losses caused by substrates and conductors need to be considered for the accurate characterization of such structures.

The reason for materials behaving electrically and magnetically anisotropic is either because of their natural crystalline structure, or because of the way they are manu factured. In senerai, an anisotropic dielectric and magnetic material is characterized by permittivity or permeability tensors of the forms:

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< H > = (2.2)

Complex quantities are used to account for losses in the magnetic and dielectric materials. If nondiagonal elements are zero, then the crystal is called a biaxial anisotropic crystal[48]. If, in addition, tensors have two equal diagonal elements, then the crystal is called a uniaxial anisotropic crystal[48]. In this work, the materials considered will be either uniaxial or biaxial anisotropic materials.

2.4 Conclusion

In this chapter, a brief introduction to microstrip antennas has been presented including a review of the major analytical methods. Also, a short explanation of the con cept of anisotropy has been given. In the next chapter, the spectral-domain techniques will be introduced to determine the resonant frequency, Q-factors and the radiation pat terns for patch antenna and effective propagation constant for MMIC structure. The con cepts illustrated in Chapter 3 will then be extended in later chapters to cases involving anisotropic substrates and conventional conductors or superconductors.

i L

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Chapter 3 Spectral-domain method

3.1 Introduction

In Chapter 2, various methods, which can be used to analyze rectangular microstrip patch antennas are briefly described. One of these methods is the spectral domain approach. This method is commonly used for the full-wave analysis of micros trip patch antennas as well as microstrip transmission lines [42]. For patch antennas, the technique is used to solve for the resonant frequency and Q-factors first, and radiation pattern subsequently. When analyzing microstrip transmission lines, the method yields the propagation constant and the loss factor.

In order to distinguish between different procedures, one technique will be called the (standard) spectral domain approach (SDA). The second technique, called the spec tral domain immittance approach(SDIA), uses an equivalent transmission line to simplify the analysis. The SDIA usually is more efficient than SDA as it enables an easy solution for structures by decoupling the TE and the TM components. This is an important factor for the analysis of multilayered structures.

The purpose of this chapter is to illustrate how the technique of SDIA is used for solving our problems. For simplicity, it will be assumed that a rectangular patch of finite thickness rests on a single layer of isotropic substrate. The structure and the coordi

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nate system are shown in Fig. 2.1 The anisotropic versions including the finite thickness of the patch will be discussed in subsequent chapters.

3.2 Spectral domain immittance approach (SDIA)

The spectral domain immittance approach has been widely used[42). In this method, the structure is transformed into an equivalent modal transmission line problem. The SDIA is summarized below.

*

The y component of the electric held can be written as:

<3' ,) " 0 0 « * 0 O

Ey is the electric field component in the spectral domain, where a and (5 are propagation constants in x and z directions, respectively. Note that

Eye~j{ttx**z) (3.2)

is a plane wave traveling in the direction given by the vector [51]

oea, + phz (3.3)

Therefore, Ey can be thought of as a superposition of waves propagating in the direction given in (3.3).

It turns out to be advantageous to use a new coordinate system which has one of its axes in the direction given by (3.3). As can be seen in Fig. 3.1, the v axis forms an angle 5 with the z axis, where

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X

Figure 3.1 Relationship between (x,y,z) and (v,y,u) coordinate systems

n

8 = cos ( £ ) (3.4)

and u = zsin 8 -* c o s8

v = zcosS + xsinS (3.5)

T = ( a 2 + |32) 1/2 (3.6)

The y axis is identical in both coordinate systems. Any electromagnetic field can be represented by the combination of fields with either Ey = 0 or Hy - 0. These two fields could be called TM-to-y and TE-to-y modes, respectively.

Since an arbitrary field can be decomposed into TM and TE modes, it is conve nient in this case to decompose the field into TM-to-y modes and TE-to-y modes. The TM-to-y modes have components (Ev, Ey, H u) , while the TE-to-y modes have compo nents (Hv, H yfEu) as shown in Appendix B.

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Note that on a .nicrostrip line, the current can be represented as J - J vd v + Judu. Using the boundary condition at y=0 and assuming zero patch thickness.

&y x ( H z - Hj) = J _(3.7)

implies that J v = Hu l - H uX at y=0

Ju = HvX- H vl at y=0 (3.8)

where 1,2 refer to regions I, II, respectively, in Fig. 2.1.

Therefore, since Hu and Hv are present only in TM-to-y and TE-to-y modes, the

Note that this convenient distinction of each current density giving rise to only one set of modes would not occur in the (x,y,z) coordinate system.

Using this current density distinction, equivalent transmission line circuits for the TM-to-y and TE-to-y fields are shown in Fig. 3.2 [42]. In Fig. 3.2, the wave admittances in region j = 1,2 are defined as:

where the “t ” corresponds to the part of the field propagating in the positive y direction. Combined with Maxwell’s equations in the spectral domain, (3.9) yields

current components Jv and Ju will give rise to TM-to-y and TE-to-y modes, respectively.

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T M j y j

_ yj _(3.10)

where

f .

= a 2 + P2 - co

\xot oz r}

(3.11)

is the propagation constant in the y direction in the j-th region. Mote that in this case of an isotropic substrate, the same expression is obtained for yjt regardless whether it corre sponds to TE-to-y or TM-to-y modes. According to Fig. 2.1, er2 = 1, so that y2 = yo. The boundary conations for the TM-to-y and TE-to-y modes are such that the tangential electric fields are zero at y=-h, as denoted by the short circuit in Fig. 3.2. At y=0, the tan gential electric fields are continuous. The tangential magnetic fields are discontinuous at y=0 due to the current density, which is modeled by the current source in Fig. 3.2. For y>0, the open microstrip patch is modeled as a transmission line of infinite length in the y direction.

At y=0, the electric fields Ev and Eu can be written as

(3.12)

where Ze and z!" are the input impedances at y=0 of the; TM-to-y (electric) and TE-to-y (magnetic) modes, respectively. Therefore,

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y = 0

y= "h

y= -h

T M - to - y

T E - t o - y

Figure 3.2 Equivalent transmission line circuits for TM-to-y and TE-to-y fields

where Y$ , F? are the input admittances looking up and down, respectively, at y=0 in the TM-to-y circuit, and Y%, T” are the respective admittances in the TE-to-y circuit(c.f. Fig. 3.2).

From transmission line theory, the input admittance of a line of length L of char acteristic admittance Y0 and load admittance YL is given as

(3.14)

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yf = yrAflcoth(Y, ( - h ) )

r = yr£1co th(Y ,(-A )) (3.15)

When looking up (towards y -> «*»), the input admittance is that of an infinitely long line, which equals the characteristic admittance. Therefore,

K ~ Y TM2 K = y TE2 (3-16) Substituting (3.15), (3.16) in (3.13) yields T v '

J

m 0 (T, + Y0ercoth (Yi ( - * ) ) ) y - i m i ° o 17, T, + T| coth(T1( - * ) ) ,-’ 1"

The task now is to derive the immittance functions using the above relations. To do this, Ev, Eu and J v, J u need to be mapped from the (v,y,u) coordinate system to the (x,y,z) coordinate system. Fig. 3.1 yields

x = - m c o s8 + vsinS

z » -« sin 8 + vcosS (3.18)

or in terms of unit vectors,

&x = -fiBcos8 + fivsin8

= fiusin8+ fivcos8

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&u

= SgSinS-dxCosS

&v = dzcos8 + flvsin8 (3.19)

Therefore,

Ex ~ E • &x = (£ va v + Eyay f Euau) »&x = sin8£v - cos8Eu

Ez = E»& z - ( £ va„ + Eyhy + Euau) »&z = cos8£„ + sin8£(< (3.20)

Similarly

J v « (Jx&g + J ^ ) »&v = sin87^ + cos8}2

}„ = ( J ^ x + Ji&g) •&„ - -c o s 8)* + sin8)j, (3.21)

Substituting (3.12), (3.13) and (3.21) in (3.20) yields

£ , = [sin28Ze + cos /* + cosSsinS [Z* - z ” ] J z

Ez = cosSsinS [Z* - z"1] >, + [ cos 28Z* + sin28z"'] Jz (3.22)

In order to obtain a matrix formulation of the form

£ = Z J (3.23)

(3.22) can be rewritten as

Ex - ^XX^X + Zxz^z

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where Zxx = —r-J—= [ a 22f + P2^ ]

o r + PZ

7 = 7 - f Z e - Z , ” l * f , r ft* + p2 l J Z« = - j i - 5 tP 2^ + “ 2Z” l (3 25)

o r + PZ

The immittance matrix Z has now been derived. The next step is to use Galer- kin’s method to solve the matrix equation given in (3.24).

Once the immittance functions Zj,j, Z y and are determined for any given problem, the algebraic equations given in (3.24) can be solved. Note that the following procedure is quite general and may be used to solve any problem which yields equations of the form given in (3.24).

The first step is to note that the cunent densities Jx ( a , (3), ~JZ ( a , (3) are the Fou rier transforms of the actual distributions Jx (x, z) , Jz (x, z) on the conducting patch or microstrip line at y=0 (see Fig. 2.1). The current densities Jx and ~JZ are expanded in terms of linear combinations of known basis functions, Jxm and Jzm, as indicated in (3.26). M ) , s T c 7 m o 1 N <3.26) n = 1

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Jxm (*, z ) , Jzn (•*, z) which are nonzero only on the conducting patch. Substituting (3.26) in (3.24) yields M N &x ~ Z xx 2 ) c m^xm + Z xz d~Jzn m = 1 n = 1 (3.27) M N &z = Zzx c m^xm + Z zz d~J;n m = 1 n = I (3.28)

There are M basis functions for the current density in the x direction, and N basis functions for the current density in the z-direction. The choice of basis function is dis cussed in the next section.

The next step is to multiply (3.27) by J xp, p=l,2,... M and (3.28) by J zq,q=l,2,... N and then integrate both expressions with respect to a and (3 from - « to + « . This yields oo oo oo oo r M “1 oo oo

J J

ExJxpd& dfi =

J J

Z xx ^ c m^xm J Xp d a d $ +

J J

Zxz ■m = 1 N 5 ) d„Jt n Ln = 1 J xpd a d $ for p=l,2,3,... Mand (3.29) 00 00 r M

J

J E ^ d a d fi

=

\ j i „

X

' j m

j j K

Lm = 1 CO °° r N

X

dn ~JZn ■n = I JzqdadV for q=l,2,3,... N. Next recall one form of Parseval’s theorem [SI]:

(3.30)

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J J EzJzqd a d ^ = (2 n )2 J J EzJzqdxdz (3.31)

Note that both Jxp (x, z) and Jzq (x, z) are zero everywhere except on the patch

( |x| £ Izl <> ~ ) (c.f. Fig. 2.1) for all values of p and q. Also due to the boundary

condi-Jm «

tion of the tangential electric field, Ex (x,0, z) and E (x,Q,z) are both zero for (|x| <, Izl <> ~ ) . Therefore, the products E (x, 0, z) J (x, 0, z) _xp' and Ez (x, 0, z) Jzq (x, 0, z) vanish everywhere so that

J j E j xrdaA$ = / J = 0 (3.32)

Equations (3.29) and (3.30) can therefore be rewritten as M

s

m = I J J %XXJxmJxpd a d $ N oo oo 'm

_{+ i}

n = 1 J J ZxzJznJxpdctd$ dn = 0 (3.33) for ps=l,2,...M M

1

m = I J J %zxjXmJzqd f t r OO OO cm+ £ n - 1 J J ZzzJznjzqdQ-d$ dn = 0 (3.34) for q=l,2,...N.

where Zxx, Zxv Zxz, Zzz, J xm, J xp, J zn and Jzq are all functions of a and {3. Equations (3.33) and (3.34) can be written in a more compact form:

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for p= 1,2,.. for q= 1,2,.. where M and N, M N - 0 0 35) m = I n ~ I

C = J J

= J { Z„.)„,7!/ a d P (3.36)

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r s X X K \\ t f X X 21 K]\ . . K\xm K\\ K\\ • • I f X X m J f X X V X l v - x z K 22 K 2 M 21 K 22 r s X X KM1 K« I S Z X 21 « I f Z X KNl j f X X - jw rt.it jw rtZ _ . KM2 * * KMM KM1 *M2 * * *12 • • * u , * n *12 • • * 1 v x z A2A 22 • • *2M A1 zz K N2 * * * A W * A / I * V 2 * * A]Jr ’ZZ I C 1 o 1 1 J c 2 0 • • • c \l CM 0 d x 0 d 2 0 • • • • Jj 0 (3.37)

which represents the characteristic equation of the patch antenna or planar structures. Here it is necessary to point out that the equation (3.37) is either suitable /or two- dimensional structures, such as planar waveguide structures, or three-dimensional struc tures like patch resonators. The difference for these two cases is the way to solve the ele ments of (3.36). For the two dimensional problem, the unknown element in the matrix (3.37) is p, the propagation constant in z direction. Therefore, the two-dimensional inte gral in the form of (3.36) can be transferred into a one-dimensional integral, which can be solved by a summation instead of an integration for a discretized a in the form of nn/ 2L, where L is the dimension in x direction. For the three-dimensional problem, the unknown element of the matrix is the resonant frequency which is obtained after solving the two- dimensional integral in (3.36).

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3.3 Patch resonator

Equations (3.27) and (3.28) present expressions for the x and z components of the Fourier transform of the electric field. However, these expressions are exact only if Jx and Jz are completely represented by the summations given in (3.36). This would in gen eral require both M and N to be infinite. However, if Jxm and Jzn are chosen such that the inverse Fourier transforms are close approximations to the unknown current distribu tions, then it is possible to use only a few basis functions to obtain good results with reduced computation time[42].

Several different basis functions have been used by various authors for the patch antenna or the regular microstrip line[37], [42], [43]. The basis functions used in this work are given as [42] (c.f Fig. 2.1)

where f 0N(x) is an odd function and nonsingular at |jc| = W /2 , and f o s (z) is odd and singular at Izl = L / 2 . These are given as

sin \2rnx/W] J {Xt z) m { f 0N <*> f o s <*> w s W /2 Izl <5 y 2 Jzn(x,z) = { }Es(x) /e n(z) 0 \x\ <, W / 2 Izl £ L / 2 |x| > W / 2 |z! > L / 2 (3.38) [ (W /2 ) 2 - x2] / o s ( z ) - s in [ 2 ( s - 0 .5 ) n z /L ] (3.39) [ ( L / 2 ) 2 - z 2] 1/2

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Also f ES (*) is an even function and singular at |x| = W / 2 , and f EN(z) is even and nonsingular at Izl = L /2 .

, , x c o s [ 2 ( r - \ ) n x / W \ f E s W * ---:— T~UT

[ (W /2 ) 2 - x 2] 2 , / x cos [2 ( s - 0.5) n z/L ]

/ " W " u i s * - * ™ ( )

Note that this choice of basis functions satisfies the requirement that Jx (x, z ) and Jz ( x, z) are nonzero only on the patch (|*| < W /2 , Izl £ L / 2 ) . Given the basis func

tions in (3.38) to (3.40), the next step is to determine their Fourier transforms Jxm and ~Jzn. These derivations are given in Appendix A, with the results shown below:

/* m = /it* , m ^ x z , in

Jzn = Jzx, nJzz, n (3.41)

where

j a m - \ \ J A p t / 2 + ( s - 0 .5 ) * ) - J J & L / 2 - ( s - 0 .5 )* )]

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J:z,n = ? [ / tf( P I / 2 + ( s - 0 . S ) n ) + J o 0 L L / 2 - ( 5 - 0 . 5 ) * ) ] (3.42)

J0 is the zero-order Bessel function of the first kind.

Any combination of r and s provides a specific basis function. For example, if N=2, M =l, equation (3.26) gives: Jx - CjJjfi Jz = d xJzX+ d 2Jz2 (3.43) or equation (3.41) Jx = C | Jxx% fyJxz, 1 Jz = d xJzx< xJZZt j + djjzjc, iJzz, 2 (3.44)

Choosing r=s=l for m=l and r=s=l for n=l, and r=s=2 for n=2 yields

Jx = c x { n / 2 ) 2 [J0 ( a W / 2 - n ) - J0 ( a W / 2 + n )]

[J0 ( $ L / 2 + it/2) - J 0 ( $ L / 2 - n / 2 ) ]

Jz = d x( n / 2 ) 22J0 ( a W / 2 ) [J0 ( $ L / 2 + n / 2 ) +J0 ( $ L / 2 - n / 2 ) ]

+ d x ( n / 2 ) 2 [J0 ( a W / 2 + n ) + J a ( a W / 2 - n) J

[J0 ( p /- /2 + 3 * /2 ) +Ja (|3L/2 - 3 n /2 ) ] (3.45)

The results for patch resonators are obtained with M=0 and N=l, using r=s=l for n=l. This simple choice of basis functions gives satisfactory results as compared with

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measurements[42]. We find that the deviation for resonant frequencies between one and two or three basis functions is less than one percent. The computation time, however, is much longer than using one basis function. Therefore, from a practical point of view, only one basis function is chosen throughout the calculations.

To solve the matrix equation of (3.37), the determinant of the matrix must equal zero. Note that each element of the determinant is the double integral of an immittance function and two basis functions as given in (3.36), and each element of the determinant is a function only of co for given dimensions and material constants. Solving for the first zero of the determinant of (3.37), one obtains, in general, a complex number

where a)r is the resonant frequency of the patch, and on. or the Q-factor represent the loss in the system. Since to. is complex even for a lossless structure, co. not only represents heat dissipation (see section 4.4) but also losses due to surface waves [42], Note also that the matrix of (3.37) will be of size (M + N) x (M + N ) , where M is the number of basis function needed to approximate Jx, and N is the number of basis functions needed to approximate Jz. Therefore, to solve for the resonant frequency of a patch resonator, one must derive the immittance matrix appropriate for the particular problem and then solve for the determinant of the (M + N) x (M + N) matrix given in (3.37). The real part of this complex frequency coc = cof +j(d{ is the first resonant frequency. Higher-order

reso-3.4 Resonant frequency of patch resonator

CD, = (0r +j(0. (3.46)

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£

g .

S '

§

* [71]

er = 2.65, h=1.27mm

w=4.0mm

m i

o-&

1 e

o

s

oe

Patch length (mm)

Figure 3.3 Resonant frequency versus patch length (mm) nances can also be calculated if required.

To check the theory given above, the resonant frequency of a single patch resonator on lossless isotropic substrate is investigated. Fig. 3.3 gives the results of resonant fre quency versus patch length for a patch located 1.27mm above the ground plane. The patch width is 4.0mm. The basis functions are given by (3.45). As can be seen from the figure, numerical results obtained with this method for the single layer patch on isotropic material agree quite well with those given in [71]. For further dependencies of patch reso nator performance on dimensions or material characteristics, the reader is referred to sec

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tions 4.7,4.9, 5.3 and 6.3.

3.5 Radiation pattern of patch resonator

The evaluation of the electric held in the far zcne of the sources requires the Fou rier transform of the tangential distribution of the electromagnetic field on the interface plane(y=0). The standard spherical coordinates (r, 0, $) are superimposed on the (x,y,z) system so that the radiation pattern is expressed in terms of 0 and <J> shown in Fig. 3.4. <|> is measured from the z axis, and 0 is the angle measured in die xy plane. The top surface of the substrate is taken to be y=0.

Once the complex resonant frequency is obtained according to section 3.4, far field radiation patterns of the resonator are obtained from Ex and Ez as they are the Fou rier transforms of the electric field. Such an approach avoids the evaluation of Sommer- field-type integrals when calculating the far fields. Similar to [42], the far fields can be expressed as:

L

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R

el

at

iv

e

am

pl

it

ud

e

/

d

B

Ee (0 ,0 ) oe sin0£x (a , P) + cos0£z (a , P) (3.48) (0 ,0 ) oe cos0cos0E*(a, P) - cos0sin0£j ( a , P) (3.49)

wheie a and p are given in spherical coordinates

-10

-15

This method (E - plane)

Measured [70] (E - plane)

-20

This method (H - plane)

•25

Measured [70] (H - plane)

•30

-80

-60

-40

-20

Theta (degrees)

Figure 3.5 Comparison of measured [70] and calculated radiation patterns for a single rectangular patch. Parameters: 6^=233, L=l.lcm , w= 1.7cm, h=0.3175cm. tj=t2=50)j,m, o=56S/nm, tan8=0.0012

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a = Arosin<t>sin0 |3 = &ocos0sin0

and kQ is the free space wavenumber. Ex and Ez are given in (3.22) and (3.41). Note that Ex, Ez on the patch are nonzero in the spectral domain. In (3.48) and (3.49), the quantities Ex ( a , 0, (3) and Ez ( a , 0, (3) are immediately obtained through the elements of the spectral impedance Green's function matrix evaluated in the interface plane.

The classification of far-field components in the two major planes of radiation has long been a controversially discussed issue. In principle, four different patterns can be cal culated.

a. E-plane: £ e (0) with 0 = 0, b. H-plane: £ e (0) with <|> = n / 2 . c. E-plane: £ ^ (0 ) with <|> = 0, d. H-plane: £<j,(0) with <j> = n / 2 .

Definitions (a) and (d) have been widely accepted in earlier papers. However, modem antenna technology incorporating applications for dual (linear horizontal - verti cal or circular left hand - right hand) or quadruple polarization (all of the above) distin guishes between copolar (definitions a, b) and cross-polar (definitions c, d) patterns as introduced by Ludwig's third definition [74].

While the E-plane pattern according to definition (a) is indisputable, the earlier definition of the H-plane pattern (d) is used in this work as a compromise between the cal culation of this pattern as such and reasonable CPU time for user-oriented applications. Indeed, both patterns (a) and (d) can be calculated with a minimum of basis functions con sidered on the patch. Since for this minimum number, £ e ( 0 ,0 = Jt/2 ) vanishes, definition (b) as H-plane pattern can only be calculated at the expense of increasing the number of basis functions. However, this would significantly increase the CPU time (approximate by a factor of ten or higher).

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data [70] is shown in Fig. 3.S. Close agreement is observed. The agreement is good up to angles of more than 80 degrees and can be considered sufficient for the following investi gations.

3.6 Planar waveguide structures

A similar procedure for the current distribution can be used for planar waveguide structures. We present a microstrip line and a coplanar structure shown in Fig. 3.6 as examples. The current distributions on strips are chosen as the basis functions of a microstrip line, which can be expressed as:

cos ( ( n - 1) n - ( x / ( w / 2 ) + 1)) ( — ) ' w / 2 (3.50) = sin (nn ( x / ( w / 2 ) + 1))

f -

<572>' (3-SI)

where n=l,2,3..., and w as shown in Fig. 3.6.

W

r

1--- ►

Spectral domain analysis of patch antennas and (M)MIC structures on lossy anisotropic substrates with possible applications to high-Tc superconductors

ANTENNAS AND (M)MIC STRUCTURES ON LOSSY

ANISOTROPIC SUBSTRATES WITH POSSIBLE

APPLICATIONS TO HIGH-TC SUPERCONDUCTORS

331 UMI

0

ABSTRACT

Table of Contents

List of Figures

Acknowledgments

Dedication

Chapter 1

Introduction

Chapter 2

Patch antenna review

2.1 Introduction

2.2 Analysis techniques

2.2.1 Transmission line model

2.2.2 Modal cavity model

2.2.3 Modal leaky cavity model

Yr

- L / 2

L / 2

2.2.4 Integral equation approach

2.2.5 Dyadic G reen's function method

2.2.6 Spectral domain methods

2.4 Conclusion

Chapter 3

Spectral-domain method

3.1 Introduction

3.2 Spectral domain immittance approach (SDIA)

X

n

f .

= a 2 + P2 - co

\xot oz r}

y = 0

y = 0

y= "h

y= -h

T M - to - y

T E - t o - y

J

&u

o r + PZ

o r + PZ

J J

J J

J J

J

J E ^ d a d fi

=

\ j i „

X

' j m

j j K

X

s

+ i

1

C = J J

3.3 Patch resonator

reso-3.4 Resonant frequency of patch resonator

£

g .

S '

§

* [71]

er = 2.65, h=1.27mm

w=4.0mm

o-&

1

e

o

s

Patch length (mm)

3.5 Radiation pattern of patch resonator

L

R

el

_{+ i}