M o d e lin g o f C o m p a ct A n te n n a s for W ir e le ss
C o m m u n ic a tio n in C o m p le x E n v ir o n m e n t
By
M d. M izanur R ahm an
B.Sc. Eng., Bangladesh University of Engineering and Technology, Dhaka, 1993 M.Sc. Eng., Bangladesh U niversity of Engineering and Technology, Dhaka, 1996
A D issertation Subm itted in P a rtial Fulfillment of the R equirem ents for th e Degree of
D O C T O R OF P H IL O S O P H Y
in th e D epartm ent of Electrical and C om puter Engineering
We accept this dissertation as conforming to th e required stan d ard
Dr. M. A /S tuchly, Supervisor (Dept, of Elec. and Comp. Eng.)
Dr. W . J,; R. Hoefer, D epartm ental M em ber (Dept, of Elec. and Comp. Eng.)
Dr. J. Bcpnemann, D epartm ental M em ber (D ept, of Elec. and Comp. Eng.)
Dr. N. D |ilali, O utside M em ber (Dept, of M echanical Engineering)
Dr. E. V. Jull,""E x tern al Exam iner (Dept, of Elec. and Comp. Eng., University of B ritish Columbia)
0 M d . M izanur R ahm an, 2001 University of V ictoria
All rights reserved. This dissertation m ay not be reproduced in whole or in part, by photocopying or other m eans, w ithout th e perm ission of th e author.
Supervisor: Dr. Maria A. Stuchly
A b stract
D em and for low-profile and com pact antennas has greatly increased due to m inia tu rizatio n of electronic devices. High perform ance for these antennas is also desired. T h e conflicting n atu re of th e requirem ents of high perform ance and com pact size m akes th e design of these antennas challenging. The prim ary focus of th is disserta tio n is to investigate and enhance th e perform ance of various com pact and low-profile antennas for wireless com m unications.
Two dual band antennas for handheld telephones have been designed for th e operation in AMPS and PCS bands and investigated in presence of th e u ser’s head. A ntenna perform ance is evaluated in term s of VSW R, far-fleld radiation p a tte rn s, and th e specific absorption ra te (SAR) of energy in th e u ser’s head. A finite difference tim e dom ain (FD TD ) code has been used for th e modeling of antennas and u ser’s head.
Two wide band circularly polarized patch antennas have also been analyzed using an FD TD code. A M om ent M ethod based code (Ensemble) has been used to verify th e perform ance of th e antennas. E xcitation of surface waves w ithin th e su b strate of patch antennas is one of th e m ain reasons for their low efficiency. R ecently developed 2D p lan ar photonic band gap (PB G ) stru ctu res can be used to prevent th e propagation of these unw anted surface waves w ithin a p articu lar frequency band. A n analytical m odel has been developed for two existing PB G structures th a t predicts th e band
and m ultiple stop bands has also been developed and m odeled using th e proposed analytical model. The analytical results have been com pared w ith FD TD com puted results and a good agreem ent has been found. Finally, a wide band circularly polarized patch has been fu rth er analyzed and integrated w ith a PB G stru ctu re. A significant im provem ent in th e an ten n a perform ance is obtained w ith th e use of PB G structure. The num erical results obtained are in excellent agreem ent with th e m easured data.
Examiners:
Dr. M. A. Stuchly, Supervisor (Dept, of Elec. and Comp. Eng.
Dr. W. J. R. Hoefer, D epartm ental M em ber (D ept, of Elec. and Comp. Eng.
Dr. J. B ornem ann, D epartm ental M em ber (D ept, of Elec. and Comp. Eng.)
Dr. N. Djilali, Qjftside M em ber (D ept, of M echanical Engineering)
Dr. E. V. dull, E xternal Exam iner (D ept, of Elec. and Comp. Eng., U niversity of B ritish Columbia)
C o n te n ts
A b s tr a c t ii
T a b le o f C o n te n ts v
L ist o f F ig u res x
L ist o f T ab les x v ii
L ist o f A c ro n y m s and A b b r e v ia tio n s x v iii
A c k n o w le d g e m e n ts x ix
D e d ic a tio n x x
1 I n tr o d u c tio n 1
1.1 M o tiv a tio n ... 1
1.2 O bjective and C o n trib u tio n s ... 3
2 L iteratu re R ev iew 7
2.1 M odeling of H andset A ntennas ... 7
2.2 B road B and P atch A n te n n a s ... 12
2.3 PB G Structures for A ntenna A pplications ... 15
2.4 Concluding R e m a r k s ... 20
3 M o d e ls an d M e th o d s 22 3.1 Finite-Difference Tome-Domain Method ... 23
3.2 C om putational M ethod . ... 28
3.2.1 A ntennas on H a n d s e t s ... 28
3.2.2 Broad B and P atch A n te n n a s ... 31
3.2.3 Photonic B and C ap Structures ... 33
3.3 M o d e ls ... 36
3.3.1 H a n d s e t ... 36
3.3.2 H um an H e a d ... 37
3.4 Verification of SAR Evaluation ... 38
4 D u al B an d A n ten n as for P C S H an d sets 43 4.1 I n tr o d u c tio n ... 43
4.2 Description of th e A ntennas ... 44
4.4 R e s u lts ... 48
4.4.1 Antenaa D e s ig n ... 48
4.4.2 A ntenna C h a r a c te r is tic s ... 51
4.4.3 S A R ... 53
4.5 C o n c lu s io n s ... 57
5 B road B and P atch A n ten n as 59 5.1 I n tr o d u c tio n ... 59
5.2 A ntenna Design and A n a l y s i s ... 60
5.3 Conclusion ... 74
6 P la n a r P h o to n ic B a n d G ap S tr u c tu r e s 76 6.1 In tr o d u c tio n ... 76
6.2 Modeling PB G S t r u c t u r e s ... 77
6.2.1 A nalytical M o d e l ... 77
6.2.2 Com parison w ith N um erical Modeling ... 85
6.3 C om pact and M ultiple Stop B and S tr u c tu r e s ... 90
6.3.1 Design and A nalytical Model ... 91
6.3.2 Com parison w ith N um erical M odeling ... 93
6.4 Conclusions ... 94
7.1 D escription of th e A ntenna and PB G S t r u c t u r e ... 97
7.2 N um erical R e s u lts ... 98
7.3 E xperim ental R e s u lts ... 100
7.4 Conclusions . ... 102
8 C on clusions and Future W ork 103 8.1 C o n c lu s io n s ... 103 8.2 Future W o r k ... 105 8.2.1 A ntennas ... 105 8.2.2 P B G Structures ... 105 B ib lio g r a p h y 107 A A n te n n a P a r a m e te r s 121 A .l R adiation P a tte rn s and Half-Power B e a m -W id th ... 121
A .2 In p u t Im pedance and V SW R ... 123
A .3 D irectivity, Efficiency and G a i n ... 124
A .4 B a n d w id th ... 125
A .5 Polarization ... 125
B A n te n n a M e a s u r e m e n ts 127 B .l Input Impedance, Return Loss and V S W R ... 127
B.2 R adiation P attern s, Axial R atio, and G a i n ... 129
L ist o f F ig u res
2.1 PB G stru ctu re consists of square array of square m etal plates w ith shorting pins [65]... 18
2.2 PB G stru ctu re consists of array of square m etal plates w ith connecting branches [50]... 18
2.3 G rounded dielectric m aterial w ith square lattice and finite height [55]. 19
3.1 Yee cell in th e FD T D m e th o d ... 25
3.2 FD TD m odeling of coplanar lines using (a) a coarse grid, (b) a fine grid, (c) a sub cell grid, and (d) a nonuniform grid... 27
3.3 Transmission through P B G stru c tu re ... 34
3.4 C om putation of th e phase of the reflected signal from th e PB G stru c
tu re . . . . 35
3.5 Model of th e hum an head (resolution 1.1 mm up to th e chin, 3.6 m m below )... 39
3.6 Different dimensions of th e spherical head m odel and dipole antenna. 40
3.8 Com parison of m easured and FD TD com puted SARs a t 1900 MHz.
SARs are normalized to a radiated power of 0.5 W ... 42
4.1 Sleeve-monopole an ten n a and th e handset: (a) external view, and (b) different dimensions of th e a n ten n a ... 45
4.2 D ual-m eander an ten n a w ith sleeves and th e handset: (b) different di mensions of th e antenna, and (a) external view for two different posi tions of th e an ten n a on th e h a n d se t... 46
4.3 O rientation of th e handset w ith respect to th e u ser’s h e a d ... 47
4.4 P aram etric studies of th e sleeve-monopole an tenn a param eters on its input im pedance, (solid line: resistance, dashed line: reactance), (a) Effect of sleeve spacing. I — 30mm constant, separation s varied from 14mm to 22mm. (b) Effect of sleeve length. Separation S = 18mm constant, length varied from 26mm to 34m m ... 50
4.5 In p ut im pedance of th e sleeve-monopole w ith and w ithout the presence of u ser’s h ead ... 52
4.6 V SW R characteristics of sleeve-monopole w ith and w ithout th e pres-ence of user’s head... 53
4.7 E-plane radiation p a tte rn s of th e sleeve-monopole (solid line: in free space, dashed line: in presence of th e user), (a) at th e lower resonant frequency, and (b) at th e higher resonant frequency... 54
4.8 H -plane radiation p attern s of th e sleeve-monopole (solid line: in free space, dashed line: in presence of th e user), (a) at th e lower resonant frequency, and (b) at th e higher resonant frequency... 55
4.9 V SW R characteristics of th e dual-m eander-sleeve an ten n a in two posi- tions on the handset (solid line: in free space, dashed line: in presence of th e user). G ray areas show th e allocated bandw idth, (a) Antenna at th e center of th e box. (b) A ntenna in th e edge of th e box... 56
5.1 Dimensions of th e modified square patch an ten n as... 60
5.2 Effect of different param eters of th e modified patch (a n te n n a # 1) on its in p u t resistance: (a) Position of th e feed point, %, withæo= 6 m m , w =14 m m , 1—28 m m (b) Length of th e sh o rter side, 1, w ith xq= 6 m m , 1/0=18 m m , w =14 m m , and (c) W id th of th e shorter side, w, w ith Xo= 6
m m , ^0 = 1 8 m m , 1=28 m m ... 62
5.3 V SW R characteristics of th e modified patch (a n te n n a # 1 ) (solid line) com pared w ith stan d ard square patch (d o tted lin e)... 63
5.4 R adiation p attern s Eq (solid line) and (do tted line) of the modified patch (a n te n n a # l) : (a) & (b) at th e lower resonant frequency in xz and yz plane, respectively, and (c) & (d) at the higher resonant frequency in xz and yz plane, respectively... 64
5.5 Ez field p a tte rn under th e modified p atch (a n te n n a # l) : (a) Lower resonant frequency, and (b) Higher resonant frequency... 65
5.6 R adiation p attern s E$ (solid line) and (do tted line) of th e slot loaded modified patch ( a n te n n a ^ l): (a) & (b) at th e lower resonant frequency in xz and yz plane, respectively, and (c) & (d) at the higher resonant frequency in xz and yz plane respectively... 67
5.7 jBz held pattern under the slot loaded modihed patch (antenna#!): (a) Lower resonant frequency, and (b) Higher resonant frequency... 68
5.8 Polarization loss factor (P LF) of th e slot loaded m odified patch (an- t e n n a ^ l) for an incident R C P wave: (a) Lower resonant frequency, and (b) Higher resonant fr e q u e n c y ... 69
5.9 V SW R characteristics for the modified patch (a n te n n a # 2 )... 70
5.10 (a) Axial R atio (A R), (b) Polarization Loss Factor (P L F) for th e m od ified patch (an tenna#2)... 71
5.11 R adiation p attern s Ee (solid line) and E^ (dotted line) of th e m odified patch ( a n te n n a ^2): (a) & (b) a t th e lower resonant frequency in xz and yz plane, respectively, and (c) & (d) at th e higher resonant frequency in xz and yz plane respectively... 72
5.12 Polarization loss factor (PLF) of th e modified patch (a n te n n a # 2 ) for an incident R C P wave: (a) Lower resonant frequency, and (b) Higher resonant frequency... 73
6.1 High im pedance surfaces (a) array of square m etal plates w ith shorting pins [65], (b) array of square metal plates with connecting branches [66], and (c) equivalent circuit of each resonator section... 78
6.2 Equivalent circuit of the periodic stru c tu re ... 79
6.3 (a) Dispersion diagram (dashed line represents th e wave in a m icrostrip line), and (b) Transmission coefficients (S2 1) and reflection coefficients
(S'il) for the stru ctu re in Fig. 6.1a w ith the dimensions a = 6 .5m m , w = 6.6m m , g = 6.5m m , t = 3.0m m , d = 1.0mm, and Sr = 4.2... 86
6.4 (a) Dispersion diagram (dashed line represents the wave in a m icrostrip line), and (b) Transm ission coefficients (S'21) and reflection coefficients (S'ii)for th e stru ctu re in Fig 6.1b w ith th e dimensions a = 7.0m m , w = 6 .5 m m ,g = s = 6 .5 m m ,g l = 0.75mm, 6 = 1.25m m, 1 = 3.0m m , d — 3.5m m , and Sr = 4.2... 87
6.5 Phase of th e reflection coefficient for th e stru ctu re in Fig. 6.1a w ith th e dimensions a = 6.5m m , u; = 6 .6 m m ,g = 0.5m m , t = 3.0m m , d = 1.0mm 6r = 4.2. (shaded region is the analytically obtained band g a p ) ... 88
6.6 Phase of th e reflection coefficient for th e stru ctu re in Fig. 6.1a w ith th e dimensions a = 7.0m m , w = 6 .5m m , g = s = 0.5m m ,p l = 0.75mm, 6 = 1.25mm, t = 3.0m m , d ~ 3.5m m , and — 4.2. (shaded region is the analytically obtained band g a p ) ... 88
6.7 Two layered PB G stru c tu re ... 91
6.8 (a) Single layered com pact and m ultiple stop band P B G stru ctu re, (b) single cell of th e stru ctu re, and (c) equivalent circuit of each resonator section... 92
6.9 (a) Dispersion diagram (dashed line represents the wave in a microstrip line), and (b) Transmission coeScients (^'gi) and rejection coe&cients (5 ii) for th e stru ctu re in Fig. 6.1a w ith th e dimensions a ~ 6.5m m , w = 6.0mm , p = = 0.5mm, luZ = 4.0m m ,t = 3.0mm , and d = 1.0m m ... 94
7.1 Modified square patch antenna surrounded by PB G c e l l s ... 97
7.2 N um erically com puted (a) VSW R, (b) Axial ratio (A R), and (c) po larization loss factor (P L F) of the reference and th e PB G patch. . . . 99
7.3 N um erically com puted Eg (solid line) and (dotted line) component of the radiation p a tte rn in th e x z plane for (a) reference patch, and (b) PBG patch... 100
7.4 M easured characteristics of th e reference and PB G patch antenna, (a) V SW R, and (b) Axial R atio ... 101
7.5 M easured spin-linear p attern s of (a) reference patch (8.6GHz), and (b) PBG patch (9.0GHz)... 102
A .l Spherical coordinate sy stem ... 122
B .l Experimental setup for measuring the antenna input impedance, VSWR, and re tu rn loss... . 129
B.2 E xperim ental setup for m easuring th e an ten n a radiation patterns, and axial ra tio ... 130
C.2 Measured return loss of (a) reference patch, and (b) PBG patch. . . . 133
C.3 Measured radiation patterns of (a) reference patch (8.6GHz), and (b) PB G patch (9.0GHz) in E-Plane . ( E g is th e solid line and E^ is th e
d o tted l i n e . ) ... 134
C.4 M easured radiation patterns of (a) reference patch (8.6GHz), and (b) PB G patch (9.0GHz) in H-Plane . { E g is th e solid line and E ^ is th e
L ist o f T ab les
3.1 Dielectric properties of th e tissues in th e head m odel at 835 MHz . . 38
3.2 Dielectric properties of the tissues in th e head m odel a t 1900 MHz . . 39
3.3 Dielectric properties of th e spherical p h a n to m ... 41
4.1 A ntenna dimensions for Sleeve-Monopole an ten na (See also Fig. 4.1). 49
4.2 A ntenna dim ensions for a Dual-M eander-Sleeve an ten n a for b o th po sitions on th e handset (See also Fig. 4.2)... 51
4.3 Specific A bsorption R ates (SAR) of th e designed antennas norm alized to 1 W ... 57
4.4 Specific A bsorption R ates (SAR) of th e designed antennas norm alized to m axim um typical o u tp u t pow er... 57
5.1 Dimensions of the various antennas (See also Fig. 5.1)... 74
5.2 Perform ance of th e various antennas ... 75
6.1 Com parison of analytic m odel predictions w ith num erical com putations. 89
L ist o f A cr o n y m s and A b b r e v ia tio n s
ABC Absorbing Boundary Condition AM PS Advanced Mobile Phone Service A R Axial R atio
CAD C om puter Aided Design C A TR C om pact A ntenna Test Range DCS D igital Cellular System
FC C Federal C om m unication Commission (U.S. FD T D F inite Difference Tim e Dom ain
FE M F inite Elem ent M ethod FV T D F in ite Volume T im e Dom ain
GSM Global System for Mobile C om m unication G TD G eom etrical Theory of Diffraction
MoM M ethod of M oment
PB C Periodic B oundary Conditions P B G Photonic B and Gap
PCS Personal Com m unications Services PE C Perfect Electric C onductor
PIEA P lan ar Inverted F A ntenna P L F Polarization Loss Factor PM C Perfect M agnetic C onductor PM L Perfectly M atched Layer PTD Physical Theory of Diffraction TLM Transmission Line Matrix SAR Specific Absorption Rate VNA Vector Network Analyzer VSW R Voltage Standing Wave Ratio
A c k n o w le d g e m e n ts
I would like to express m y h eartiest appreciation to m y supervisor, Dr. M. A. Stuchly for her continuous guidance and encouragem ent shown throughout this re search work and th e process of w riting this dissertation. I can n o t th an k you enough for your patience and forbearance w ith me. Special thanks to Dr. M. Okoniewski for his valuable suggestions as related to this work, especially on th e num erical modeling. I would like to th an k m em bers of my dissertation com m ittee Dr. W. J. R. Hoefer, Dr. J. B ornem ann, and Dr. N. Djilali for taking tim e from th eir busy schedule, and for th e valuable suggestions th ey provided.
I wish to th a n k Mr. K. C aput a for his day to day help w ith th e com puters in th e lab and also w ith th e m easurem ents. My other colleagues in th e lab. Elise, Mike, Trev, Asad, and K im m o deserve special thanks for th e ir valuable suggestions and help.
I would also like to express my appreciation to Dr. Lot Shafai, U niversity of M anitoba for allowing m e to use th e anechoic cham ber for the an tenn a m easurem ents. Special thanks go to Brad for his help with the measurements. The financial support of the Canadian Commonwealth Scholarship and NSERC is greatly acknowledged.
Finally, I wish to express my deepest gratitu d e to m y wife, Sharm in for her contin uous support and encouragem ent. Sharm in, w ithout your understanding, sacrifices, and prayers th e dream would never come to reality, thanks.
C h a p ter 1
In tr o d u c tio n
1.1
M o tiv a tio n
W ith th e recent advances in th e wireless com m unications, th e need for com pact, low- profile, and high perform ance antennas has greatly increased. The greatest dem and for these antennas is from personal com m unication system s (e.g. cellular telephones, pagers, m obile d a ta system s and global positioning system s) and other m obile appli cations (e.g. autom obiles, trains). Depending on applications, th ere are differences in an ten n a perform ance requirem ents (e.g. gain, bandw idth, polarization). How ever, com pact and low-profile antennas are essential for such applications, for either m echanical reasons or due to th e m iniaturization of electronic equipm ent in general.
It is well known th a t, as th e size of th e an ten n a is reduced, th e efficiency tends to degrade and th e bandw idth becomes narrow er [1]. So, th e conflicting n a tu re of th e requirem ents of high perform ance and com pact size makes th e design of these antennas very challenging. In addition, th e interaction of th e an ten n a w ith its com plex environm ent also effects its perform ance. These environm ents m ay include the presence of the user's body or other complex structures.
Since two frequency bands have been allocated for personal wireless communica- tion (e.g. AMPS & PCS in North America and GSM & DCS in Europe) an antenna for handsets is needed th a t can operate in b o th th e allocated frequency bands. These anten n as are typically a few m illim eters to 2-3 centim eters away from th e u ser’s body. This has consequences for b o th th e antenna and th e user. Since a t microwave frequen cies th e u ser’s body behaves like a lossy dielectric, its presence modifies th e antenna characteristics. Also, a significant fraction of the radiated energy is absorbed by the u ser’s body, resulting in lower antenna efficiency of th e antenna and possible health risks for th e user. So, from b o th an ten n a design and u ser’s h ealth risks points of view, th e interaction of th e hum an body w ith th e antenna m ust be investigated and under stood. Linear wire antennas have been m ost often used in th e current generation of wireless telephones. For dual band operation, th e use of two separate antennas has been reported in th e lite ra tu re [2], [3]. A single an ten n a th a t can operate in b o th the allocated bands would offer an a ttractiv e alternative.
Among th e low-profile antennas, m icrostrip patch an ten n a is a strong candidate for different applications due to some a ttra ctiv e features (e.g. lightweight, com pact size, th in profile, ease of fabrication, and especially reduced SAR in th e u ser’s head). Along w ith these advantages patch antennas have some inherent disadvantages, nam ely narrow im pedance bandw idth and low efficiency [4]. A num ber of techniques for increasing th e bandw idth have been reported in th e literatu re, bu t m ost of them are achieved by increasing th e volume of the antenna. Techniques for increasing the bandw idth w ithout increasing th e size of th e an ten n a deserve fu rth er research.
One of the main reasons for low efi&ciency of patch antennas is the excitation of surface waves. Recently, a high-im pedance electrom agnetic surface has been inves
tig ated and shown to prevent th e propagation of surface waves w ithin a p articu lar frequency band. These high-im pedance surfaces are basically frequency selective sur faces th a t are referred to as 2D planar photonic band gap (PB G ) structures. These stru ctu res offer prom ising perform ances when used as th e ground plane for antennas. Investigations of antennas on PB G surfaces are relatively recent. B etter u n d erstan d ing of th eir properties and m ore efficient design m ethods for 2D PB G stru ctu res are needed.
1.2
O b je c tiv e an d C o n tr ib u tio n s
T he general objective of this thesis is to investigate perform ance of various an ten nas for wireless com m unications. T he antennas considered for this research require num erical m odeling for th e evaluation of th eir perform ances because of th e environ m ent in which they operate. In m ost of th e cases due to th e presence of th e u ser’s body, which is a heterogeneous lossy dielectric, num erical sim ulation is necessary. Therefore, th e finite difference tim e dom ain (FD T D ) m ethod is used for num erical modeling. The contributions of th e thesis in antennas for wireless com m unication include:
1. D u a l-B a n d A n ten n a : Design and characteristics evaluation of antennas for handsets th a t can operate in two allocated frequency bands (824-894 MHz and 1850-1900 MHz) of the personal communication service have been perform ed. Two different dual-band antennas have been designed, namely the sleeve-monopole and the dual-meander-sleeve. Qualitative per- formances of th e antennas have been evaluated using th e FD T D m eth o d
taking th e proxim ity of th e u ser’s head into account. T he specific absorp tion rates (SAR) of th e energy w ithin th e u ser’s head from th e designed antenna are also evaluated.
2. B ro a d B a n d P a tch A n ten n a : Two broad band m icrostrip p atch an ten nas w ith circular polarization have been developed. To design th e antennas num erical m odeling is perform ed using th e FD T D m ethod.
3. M o d e lin g o f P B G str u c tu r e s: An analytic model for the existing 2D planar PB G structures has been developed. The transm ission properties of th e stru ctu res are com puted num erically and com pared w ith th e results obtained by th e analytical model. A new 2D planar PB G stru ctures w ith lower operating frequency and m ultiple stop bands has also been developed and m odeled using th e proposed analytical model.
4. A p p lic a tio n o f P B G str u c tu r es: One of th e existing p lan ar PB G structures has been used to enhance th e perform ance of th e circularly polarized broad band patch antenna.
1.3
O u tlin e
C h ap ter 2 reviews th e previous research related to th e antennas for handsets and th eir interactions w ith th e user. Some atten tio n is devoted to th e techniques for im proving the bandwidths of microstrip patch antennas. It also presents some literature review related to the development of planar PBG structures and their application in antenna systems and explains how this work builds upon the previous knowledge.
work has been carried out using FD TD technique, a brief description of th e m eth o d is given first. The different aspects of th e m odeling criteria are described. The m odel of th e hum an head and th e handset is described th a t has been used for handset an ten n a analysis. N ext, the method for numerical evaluation of the specific absorption rate (SAR) has been validated by com paring th e num erical results w ith th e experim ental results. Finally, th e experim ental m ethods for m easuring th e an ten n a properties have been described.
C h apter 4 describes th e results regarding th e dual-band handset antennas. F irst, a description of the antennas are given. Next, antenna performances in free space and in th e presence of th e user’s head are presented and explained. It also presents th e data regarding th e SAR of rad iated energy w ithin th e u ser’s head from th e designed an ten n a and com pares those w ith recom m ended safety standards.
C h ap ter 5 analyzes two novel m icrostrip patch antennas w ith broad impedance bandw idth and circular polarization. A fter a brief description of th e antennas, differ ent aspects of th e an ten n a design and analysis are described. N ext, a discussion of th e analysis and results is presented.
C h ap ter 6 describes various high-im pedance surfaces for th e low-profile antenna, which are basically 2D planar PB G structures. A fter a brief description of th e ana lyzed structures, a simple analytical model of th e 2D planar PB G stru ctu re is pre sented. The analytic design is com pared w ith th e results obtained by num erical modeling. Finally, it presents a new structure with a lower operating frequency and multiple stop bands. This new structure has also been modeled using the proposed analytical model and compared with the numerical results.
C h ap ter 7 exam ines th e use of a planar PB G stru ctu re for th e perform ance en hancem ent of circularly polarized patch an ten n a described in C hapter 5. F irst th e design and analysis of bo th th e reference and P B G p atch an ten n a using FD T D m ethod are described. N ext, experim ental results for these antennas are presented, followed by a discussion.
C hapter 8 closes the thesis w ith a few concluding rem arks and description of th e possible future extensions of this work.
C h a p ter 2
L itera tu re R e v ie w
A brief review of lite ra tu re related to this work is given. F irst, th e research related to th e m odeling of handset antennas and th eir interaction w ith th e user is reviewed. Design and characterization of bo th single and dual band antennas are discussed. N ext, th e advantages and lim itations of m icrostrip patch antennas are discussed. As th ey have narrow im pedance bandw idths, different techniques for im proving their bandw idth are reviewed. Finally, th e background m aterial relevant to th e develop m ent of planar PB G stru ctures and th eir applications in different an ten n a systems are discussed.
2.1
M o d e lin g o f H a n d se t A n te n n a s
Wireless com m unication system s, especially those for th e cellular com m unications have experienced enorm ous grow th over th e last decade. A ntennas for handsets are receiving increasing interest as th ey co nstitu te an im p o rtan t p a rt of these system s [1]. Since the size of the handset units dramatically decreased in the last few years, the main design efforts have been devoted to maintaining approximately the same antenna
performance (e.g. a gain of 0 dBi, bandwidth of ^ 10%, and nniform coverage over th e horizontal plane), while ensuring th a t th e an ten n a is sm all enough. Different antennas are currently used for handsets th a t usually operate in a single frequency ban d (e.g. AM PS or GSM). The popular antennas for handsets include a monopole, a sleeve dipole, a norm al m ode helix, and a planar inverted F antenna (FIFA ) [1]. P atch antennas have recently come into use. Perform ance analysis of m ost of th e antennas has been done on an infinite ground plane. For these antennas, simple, closed form analytical formulas are available for evaluating th e antenna perform ance (e.g. an operating frequency, radiation p attern s, gain) [5]. For exam ple, a m onopole a n ten n a on an infinite ground plane m ay be m odeled by im age theory as a dipole with one-half of the input impedance and double the peak directivity of the dipole [5]. B ut when th e antennas are placed on a handset, th e m etallic p art of th e casing of th e handset acts as a radiating elem ent. A lternatively, one m ay consider th e m etal casing as th e ground plane. Due to th e small size of th e ground plane, its edges act as scatterers which diffract th e incident field. The diffraction from th e edges modifies th e current on th e ground plane and alters th e rad iatio n p a tte rn , causing scalloping [9] or nulls [10] in th e forward radiation, th e presence of back rad iatio n (radiation behind th e ground plane) [11], and higher cross polarization levels [10]. O th er an tenn a param eters m ay be affected as well. The sm aller the ground plane, th e stronger th e currents on its back side, resulting in stronger back radiation. The m etal casing of th e handset becomes an integral p a rt of th e an ten n a system itself.
M any analytical tools can be used to model th e diffraction of fields from ground plane edges. These include such techniques as th e m ethod of m om ents (MoM) [14], the geometrical theory of dif&action (GTD) [7] and the physical theory of diffraction
(PTD ) [7]. The MoM and the GTD are sometimes used together.
A monopole an ten n a on a conducting box has been studied in [10] and [12]. It has been shown th a t as th e monopole an ten n a is moved from the center of the to p surface towards the edges or a corner, the magnitude of the conductance decreases by 50%, the resonance frequency increases by a few percent, and deep nulls are produced in the radiation p attern . For a helical antenna, a reduction of the ground plane radius to approxim ately the radius of the helix results in a tran sitio n from forward to backfire radiation [13]. A PIFA attached to a rectangular m etal box (handset) has been analyzed in [15]. The handset and th e an ten n a has been modeled by wire grids and th e MoM has been used to com pute th e antenna characteristics. A null around 0 = 230° is observed in th e E ~ plane radiation p a tte rn due to th e m etal box.
To satisfy th e rapid grow th of th e mobile telephony, an additional frequency band is used (e.g. PCS or DCS) [16]. Since th ere are two different frequency bands allo cated, subscribers who travel over service areas employing different frequency bands need two separate antennas unless a dual-frequency an ten n a is used. Obviously th e later is a b e tte r choice in term s of cost, complexity, and space usage. Recently, several dual-band antennas for handsets have been described [2], [3], [17]-[19]. These include b o th planar and wire antennas.
A dual frequency planar inverted F an tenn a (PIFA) has been described by Liu et. ah in [2]. A sm aller an tenna is inscribed in the original one to get th e higher operating frequency and th e two antennas are excited using separate feeds. A ban d w idth of 7% and 6.25% has been obtained for lower and higher frequency bands, respectively. A capacitively loaded PIFA with capacitive feed has been proposed in [18] for dual frequency operation. The PIFA described in [18] also uses two
differ-eut antennas and separate feeds. A sm aller an ten n a size is achieved a t th e expense of a narrower bandwidth and complex structure. Both the antennas presented in [2] and [18] have been analyzed num erically using FD TD m eth od w ith th e handset m odeled as a rectangular m etal box. The num erical results have been verified using experim ental m easurem ents. E ratuali et. al. has described a dual frequency wire an ten n a consisting of two separate antennas (a monopole and a norm al m ode helix) placed w ith th e monopole along the axis of th e helix. T he monopole and th e helix are designed for using at th e higher frequency and lower frequency band, respectively. Bandwidths obtained for higher and lower frequency are 9.6% and 8%, respectively. T he handset in this work has been m odeled using a wire grid m odel and the antenna has been analyzed using MoM.
U ntil recently, m odeling of handset antennas has not tak en th e proxim ity of th e user into account due to th e com plexity of m odeling a hum an body. However, it is well known th a t an ten na perform ance is affected by th e presence of th e u ser’s head and often also th e hand, which are heterogeneous lossy dielectric m aterials. Recent advances in num erical electrom agnetic m odeling techniques have m ade m odeling of hum an body viable. Several num erical m ethods can and have been used for investiga tions of an tenn a in th e proxim ity of models of human body. T he models of th e head range from a simple homogeneous semi-infinite plane or box [21], homogeneous or layered spheres [22], to anatom ically based heterogeneous models [23], [25]. For h e t erogeneous models, th e m ost suitable are th e FD T D , th e finite volum e tim e dom ain (FV T D ), and finite elem ent (FEM ) m ethod.
Effects of th e user proxim ity on th e an ten n a perform ance have been investigated by several authors [20] - [26] using experimental and numerical techniques. An early
experim ental work on a 600 MHz dipole an ten n a n ext to a hum an body m odel has shown th a t scattering of fields by th e body p ertu rb s th e current distrib u tio n on th e an ten n a [20]. The body of th e user has been m odeled as a rectangular cylindrical plexiglass container filled w ith saline solution. A num erical analysis using th e FD T D technique has been perform ed to investigate th e effects of th e body proxim ity on the antenna resonant frequency, input impedance, e@ciency and far-field radiation pattern [22]. The numerical modeling includes a monopole antenna on a PCS device (a m etal box), a head (m odeled as a sphere of m uscle tissue) and a h an d grasping th e phone (m odeled as a block of muscle tissue). A n tenna characteristics have been investigated at 914 MHz and 1890 MHz and verified by experim ental m easurem ents. T he results show a decrease in resonant frequency of 10% and a decrease in efficiency of 55-57% due to the presence of the head and hand. A considerable amount of distortion is also observed in th e radiation p attern s of th e antenna. D iffraction and scattering from th e head also result in significant cross-polarization. T he presence of th e head results in a shadow effect, m eaning th a t th e m agnitude of th e far field radiation is less in th e direction of th e head com pared to th e other directions (by 2 dB at 915 MHz and by 12 dB at 1890 MHz). Sim ilar tests have been perform ed by other investigators using an anatom ically accurate head model (obtained from m agnetic resonant imaging), a heterogeneous hand m odel and a variety of different an ten n a configurations m ounted on handsets [23]. Antenna perform ance has been investigated at 915 MHz. A decrease in resonant frequency and bandw idth of 32 and 52% has been found, respectively. The num erical analysis, using a F D T D algorithm , agreed well with measurements.
an ten n a and th e u ser’s head using bo th canonical and anatom ically based models. It has been shown th a t b o th box and spherical models yield overestim ated SAR values and box models provide distorted and unreliable results for th e an ten n a far-held p a tte rn . W atanabe et. al. [26] has shown th a t m axim um local SAR is lower for a head model without ear than for one with ear. In both of these cases dielectric covered rectangular m etal box has been used as th e m odel of th e handset. Tinnisw ood et.al. [27] used more accurate C om puter Aided Design (CAD) hies for th e handset m odel and investigated th e effects of wires and circuit boards inside th e telephone on th e a n ten n a perform ance and SAR distribution. It has been observed th a t th e CAD hie and simple box m odel of the handset produce sim ilar SAR distributions w ithin th e head in the region of th e highest EM absorption close to th e antenna.
2.2
B ro a d B a n d P a tc h A n te n n a s
M icrostrip an ten n a technology has been rapidly developing for th e last two decades. It has found applications in a wide variety of microwave system s due to m any advan tages (e.g. low profile ; lightweight, ease of fabrication, low cost) [4]. In addition, these antennas are m echanically rigid, which makes them less susceptible to dam age th a n wire antennas. M icrostrip antennas present an altern ativ e option for cellular tele- phone handset as potentially the amount of radiation absorbed by the user should be minimized. However, th ere are some inherent drawbacks too. A narrow impedance bandwidth is probably th e most significant disadvantage for this ty p e of antenna. A typical im pedance bandw idth for a basic m icrostrip patch elem ent is 1 to 3% [28], compared with a 15% to 20% bandwidth of dipole, slot and horn antennas [4]. Thus, much of the research on microstrip antennas has been devoted to various techniques
for increasing th e im pedance bandw idth. Electrically thick su b strate increases th e bandw idth, b u t th e im pedance locus becomes increasingly inductive [29], [30], m ak ing im pedance m atching difficult. In addition, surface waves are excited th a t lower th e an ten n a efficiency. O ther techniques generally fall into three categories: external im pedance m atching, use of m ultiple resonances, and addition of losses (to sacrifice ef ficiency for bandw idth). Usually, an im provem ent in bandw idth for printed antennas has only been achieved at the expense of an increase in volume or a loss in efficiency
p
i].
E x tern al im pedance m atching is an effective and sim ple m eth o d of w idening the bandw idth because it usually does not require any m odification of th e an ten n a ele m ent itself. Im pedance m atching is typically achieved by adding a m atching circuit to th e feed netw ork, usually on th e same su b strate as th e an ten n a [32], [33]. The m atching netw ork may consists of tuning stubs, quarter-w ave transform er sections, capacitively coupled lines, or active devices. Good results are achieved when the m atching circuits are close to th e an ten n a elem ent. However, care m ust be taken to prevent th e m atching circuit from interfering w ith th e an ten n a radiation p a tte rn . An im pedance bandw idth of about 25% has been obtained by m atching th e in pu t im pedance of a single m icrostrip elem ent [33]. Using transistors in th e m atching n et work, m atching com bined w ith am plification has achieved a bandw idth of 24% and an added signal gain of approxim ately 10 dB [34]. As m entioned earlier, th e in p u t im pedance of a patch becomes increasingly inductive as th e su b strate thickness in creases, so an obvious approach to bandw idth im provem ent is to tu n e this inductance w ith a series capacitor. This technique has been im plem ented in [35], where th e end of the center conductor of a coax feed is formed into a tab that does not contact the
patch element. This arrangement forms a series capacitor which can be controlled by varying the size of the tab and spacing from the patch. Another approach, suggested by H all [36], is to feed th e patch w ith a coax probe in the usual m anner, but w ith a circular or linear gap in the patch conductor around th e feed point. This gap, how ever, m ust be very narrow to obtain sufficient coupling to th e patch, so fabrication is difficult. Im pedance m atching can also be achieved by m odifying th e antenna itself (e.g. by creating slots in the patch e.g. [37]).
A lternatively, th e bandw idth can also be increased by introducing dual or multiple closely-spaced resonances. For microstrip antennas, this can be achieved using stacked m ultiple patches [38], [39], or co-planar m ultiple-resonator elem ents [40], [41], and slot loading (cutting slots into th e patch) [42]. T he stacked patch configuration occupies less surface area than the co-planar patch configuration, and tight coupling is m ore easily achieved. The stacked p atch can also be used in array configurations w ithout th e need for increased elem ent spacing and th e concom itant danger of grating lobes. However, using th e stacked configuration makes fabrication, modification, and addition of com ponents m ore difficult. In practice, th e b o tto m and top patches are very close in size, w ith one slightly larger th a n th e other to resonate at a lower fi-equency. Bandwidths of up to 10% to 20% have been achieved with stacked patches [43]. Using co-planar parasitic patches, a bandw idth of up to 25% has been achieved w ith one center fed patch and four surrounding parasitic patches [43]. This design has some po ten tial drawbacks when compared to th e stacked p atch approach. F irst, to achieve close coupling betw een parasitic patches, very small gaps betw een th e elem ents m ust be used, which makes a fabrication tolerance critical. In addition, since difierent parts of the configuration radiate with different phase and amplitude
at different frequencies, th e p a tte rn and phase center usually change m arkedly over th e frequency ban d of operation, especially for w ider ban d w id th designs. It m ay also be difficult to position coplanar feeds and m atching networks on th e board, since th ere is less space to m ount them . For this reason, th e driven patch is usually fed by coaxial probe, although ap ertu re coupling can be used very conveniently. Slot loading does not increase the size of the antenna, as do the other two methods. Therefore, it is preferable in applications where size is critical. A ban d w id th as wide as 47% obtained w ith slot loading has been reported in [42]. T he resonant frequency and bandw idth can be adjusted by changing th e dep th and w idth of th e notches.
B andw idth im provem ent can also be achieved a t th e expense of efficiency by adding loss into th e an ten n a system . This results, though, in decrease of rad ia tion efficiency. A dding a 6 dB a tte n u ato r in series w ith a m icrostrip an ten n a leads to a minimum of 12 dB return loss over a broad band. The antenna gain is also reduced by 6 dB [43]. Losses can be added externally using a tten u ato rs, d istrib u ted using lossy su b strate m aterials, or added to th e an ten n a directly using chip resistors or other loads. This m ethod of broad-banding is generally undesirable for all an ten n a design, and fu rth er reduces th e already relatively low radiation efficiency of m icrostrip antennas.
2.3
P B G S tr u c tu r e s for A n te n n a A p p lic a tio n s
Photonic ban d gap (PB G ) structures are a class of com posite periodic stru ctu res th a t exhibit transm ission (pass) and reflection (stop) bands in th e ir frequency response. These bands in a photonic stru ctu re occur due to th e constructive and destructive interference of th e electrom agnetic waves [44]. P B G stru ctu res can be m ade of di
electric only [45], or of m etallic elem ents em bedded in a dielectric m aterial [46]. Two and three dimensional arrangements of the PBG crystals have been explored [47]. Recently, there has been increasing interest in PBG engineering for nse at microwave and millimeter-wave frequencies. Numerous engineering applications of these stru c tu res have been described at microwave frequencies, such as m icrostrip filters [48], high power components [49], and m agnetic conducting surfaces [50]. A n application of PBG structures in antenna systems is also receiving interest. Kesler et. uZ [52] has described th e application of a finite thickness slab of 2D, all dielectric P B G m a terial as a planar reflector for a dipole antenna. A slab of woodpile PBG m aterial has been exam ined as a planar, all-dielectric reflector for a dipole/m onopole an ten n a [53]. A ntenna characteristics such as th e field p a tte rn and input im pedance have been obtained using full wave analysis.
P B G stru ctu res have been analyzed num erically by eith er calculating th e disper sion diagram (/? — ui) [55], [56], or calculating the transm ission and reflection coef ficients [57], [58]. Due to th e com plexity of PB G stru ctu res, th eir characterization can be achieved only w ith full wave analysis. N um erical techniques such as plane wave expansion m ethod [59], integral equation m eth o d [60], finite elem ent m ethod [61], finite difference b o th in frequency dom ain [62], [63], and tim e dom ain [52], [64] have proved effective for different structures. Some of these structures have also been m odeled analytically. Plihal et. al. [56] has described a m eth o d of calculating th e dispersion diagram of a 2D dielectric PB G stru ctu re. T he stru ctu re in [56] consists of infinitely long, parallel, dielectric rods w ith circular cross section em bedded in a different dielectric m aterial. A position-dependent dielectric constant and th e plane wave approximation are used to cedculate the dispersion diagram of the structure.
Recently, a new claas of 2D planar PBG structure has emerged [55}, [50], [65]. The trad itio n al 2D structures are uniform and extend to infinity in th e third dim ension, whereas this new class of 2D PB G structures is finite in th e th ird dimension. The 2D planar PB G structures can prevent electrom agnetic wave propagation along the surface of th e structures and serve as high im pedance surfaces w ithin the stop band [55], [65], [66]. The PB G stru ctu re presented in [65], and shown in Fig. 2.1, is an array of small metal sheets (square or hexagon) connected to a continuous metal sheet using th in m etal wires. The space betw een th e top and b o tto m m etal layer is filled w ith a dielectric m aterial. This structure has been m odeled using a “L G ” parallel resonant circuit to predict th e operating frequency. A finite elem ent m ethod has been used to calculate th e dispersion diagram and has been verified by m easurem ents. A nother PB G stru ctu re, described in [50], and shown in Fig. 2.2, consists of a square m etal pad w ith four narrow connecting branches and supported by a continuous ground plane. A dielectric m aterial fills th e space betw een th e to p m etal layer and the ground plane. Transm ission properties of th e stru ctu re have been calculated using th e FD TD m ethod. A grounded periodic dielectric su b strate w ith square la ttic e and finite height as shown in Fig. 2.3 has been analyzed in [55]. T he effect of th e height of th e su b strate on th e band diagram has been determ ined using th e FD T D m ethod.
The high impedance property of 2D planar PBG structures such as those men tioned above makes them suitable candidates for ground planes for different antennas, whenever surface waves deteriorate anten n a performance [51], [54] [66], [67]. Q ian et. al. [51] have analyzed a simple m icrostrip patch antenna placed in a photonic band gap stru ctu re described in [65]. It has been shown th a t th e radiation p a tte rn of the patch antenna in the L'-plane becomes narrower due to the suppression of the wave
w ± g X TO P V IEW
T
i
I
r ~ i T '
FRO N T V IEWFigure 2.1: PB G stru ctu re consists of square array of square m etal plates w ith short ing pins [65].
TOP VIEW
I'
'_______________________________________FRONT VIEW
Figure 2.2: PB G stru ctu re consists of array of square m etal plates w ith connecting branches [50].
00
a#
000
000
00
1
I###
0N
■
I
T
w
ba
TOP VIEWa - B00I 0 0 » I
- — — I L ■ ■-■i t
X FRO N T VIEWFigure 2.3: G rounded dielectric m aterial w ith square lattice and finite height [55].
propagation along th e surface. This increases th e an ten n a gain. An im provem ent of th e an ten na bandw idth has also been reported. T he use of th e 2D p lan ar PB G stru ctu re as a ground plane for vertical m onopole and horizontal wire antennas im proves th eir radiation p attern s and increases th eir gain due to th e suppression of th e surface waves [65]. Similarly, an aperture-coupled p atch an ten n a on a PB G stru c tu re described in [50], shows a significant im provem ent in anten n a perform ance. T h e fi n ite height grounded periodic dielectric, described in [55] (Fig. 2.3) has been used as a su b strate for m icrostrip phased arrays [67]. It has been shown th a t th e scan blindness of th e phased arrays can be elim inated by using this PB G stru ctu re. A m icrostrip p atch an ten n a w ith a finite ground plane, integrated w ith P B G su b strate is rep o rted in [54]. A significant reduction of back radiation has been observed due to th e reduction of edge currents [54]. T he antennas in [51], [54], [66], and [67] have been modeled with the FDTD technique due to its capability of analyzing complex
structures. The results have often been verified by measurements.
2 ,4
C o n c lu d in g R em a rk s
The review of present state of knowledge on handset antennas has indicated th a t dual ban d antennas are likely be the next generation antennas for handsets. However, th ere is a paucity of reports on th e ir design, and an apparent lack of an ten na designs w ith a single feed. Furtherm ore, only few designs described in the lite ra tu re for b o th single and double band operation have been analyzed in a realistic environm ent th a t includes a handset and a realistic m odel of th e hum an head. T hus, one of th e aims of this research is to design dual ban d antennas for handsets and analyze th eir perform ance in th e proxim ity of an anatom ically based hum an head model.
M icrostrip antennas, due to th e com pact geom etry and lower energy deposition in th e u ser’s body, are an a ttra ctiv e alternative to th e oth er an ten n a configurations for handsets. As discussed, the developm ent of microstrip antennas for handheld devises poses m any challenges, including technological lim itations (e.g. bandw idth and efficiency) and physical constraints (e.g. size). Moreover, most of th e broad- banding techniques for m icrostrip antennas, presented in th e literatu re, achieve wider bandw idth at th e cost of increased volume and degraded perform ance.
A nother new recent m ethod of im proving th e an ten n a perform ance (e.g. gain, effi- ciency, bandwidth) is, as briefiy reviewed, by the use of PBG materials for suppression of surface waves. Integration of 2D planar P E G stru ctu re as a high-im pedance sur face w ith an ten n a system s is relatively new. A shortage of reports on th e ir analytical m odeling is evident, Therefore, as a p a rt of this research com pact broadband patch
antennas w ith th e 2D planar PB G stru ctu res are analyzed and th e ir applications for handsets and other wireless mobile antennas are explored.
For th e analysis of antennas in complex environm ents such as proxim ity of a hum an body and m ounting on a ground plane com parable in size to th e wavelength, th e m ost suitable m ethods are FD T D , FV T D , TLM and FEM . T im e dom ain techniques offer an advantage of com plete characterization of perform ance in th e required range in one sim ulation. T he FD TD has been m ost often used to analyze th e handset antennas in the proximity of an anatomically based head model (composed of voxels) and complex PB G structures. Thus due to its capability of analyzing com plex stru ctu res, robustness, and oth er inherent advantages, th e FD TD m eth o d has been selected for num erical analysis in this research.
C h a p ter 3
M o d e ls and M e th o d s
Due to th e complex n atu re of th e antennas and their environm ents considered in this work, num erical modeling is necessary. Besides, a com plex stru ctu re such as P B G needs to be investigated using full wave analysis to get a b e tte r understanding of its behavior. As m entioned earlier, th e FD TD m ethod can be readily applied to th e complex geom etry of inhomogeneous dielectrics and m etals. It is also suitable for obtaining th e d a ta of interest in a broad frequency range in one com putational run. Therefore, it is well suited to th e analysis of th e different antennas an d th eir environm ent studied in this work. This chapter briefly addresses th e issues related to th e accuracy of th e FD TD modeling. A pplication of num erical m ethods to analyzing different antennas and PB G structures is described in m ore detail. The han d set and hum an head models are presented. Methods for th e verification of SAR d eterm in atio n and measurement of antenna properties are also outlined.
3.1
F in ite -D iffe r e n c e T o m e -D o m a in M e th o d
T he finite-difFerence tim e-dom ain m ethod is a popular num erical electrom agnetic technique that is based on solving the differential forms of Maxwell^s equations [69]. T he FD TD m eth o d is a tim e-dom ain technique, which requires th e discretization of b o th tim e and space. The method iterates through tim e for a given num ber of tim e steps, and a t each tim e step, th e field com ponents are u p d a ted using relatively simple linear equations.
The FDTD method, introduced to electromagnetics by Yee in 1976 [68], is a direct solution of M axwell’s differential equations. Two of th e differential equations are [7]:
V x Ë = (3.1)
V X Ë = M - — (3.2)
where È and H are th e electric and m agnetic field stren g th , J and M are th e electric and m agnetic current density, and D and B are th e electric and m agnetic flux density [7]. All of th e field variables are functions of space and tim e, / ( x , y, z, t). To solve the differential equations numericcdly, space and tim e are discretized and differentiation is perform ed using finite-difFerence approxim ations. A com m only used finite-difFerence approxim ation is th e central hnite-difference formula, shown below for differentiation in th e x direction:
^ 0 - / (a: - */, . .
aæ ^ A z ^ ^ ^
where A x is an increm ent in th e x direction. It can be shown th a t applying th e central finite-difference form ula to (3.1) and (3.2) produces a set of six scalar equations in which th e field values at each point in tim e and space are related only to field values at neighboring points [69]. Thus, th e solution of each field value does not depend on th e form ulation of large m atrices, which is a great advantage of th e FD TD m ethod. However, Ë and H m ust be evaluated at all nodes in th e com putational space. This is accomplished by iteratin g through space along a regular grid. A unit cell of th e grid, known as Yee cell, is shown in Fig. 3.1. As shown in Fig. 3.1, Ë and H field com ponents are evaluated at different locations on th e grid. T hey are also evaluated a t a ltern ate tim e steps. To ensure th e num erical stability of th e FD TD m ethod, th e tim e stepping increm ent, A t , m ust be less th a n th e C ourant lim it [70]:
1 1 1
Ay^ Az'^ (3.4)
where fi and e are th e perm eability and p erm ittiv ity of th e m edium , respectively, and S' < 1 is th e stability coefficient.
For sufficient accuracy of th e results, spatial increm ents in th e Yee grid (A x , A y , A z ) must be small in wavelengths (A/10 — A/20), and a sufficient number of tim e steps m ust be chosen [69]. Even sm aller grid or special algorithm s have to be used to rep resent highly non-uniform fields. A ccuracy also depends on th e proper application
AX AZ ! ^ : U ] 1 ! --- — — — _ _ J k r " ' 1 1 i
:
r
111 J--
1 --- — —^ _ . ^ --- ----c> H component -► E component AYFigure 3.1: Yee cell in th e FD TD m ethod.
A bsorbing boundary conditions (ABCs) have been form ulated which sim ulate th e effect of an unbounded region. Different types of ABCs are used for various applica tions, as th eir advantages and lim itations are dependent on th e ir use. One of th e m ost popular and advantageous ABCs is th e perfectly m atched layer (PM L) introduced in 1994 by Berenger [71]. PM Ls can provide a reflection sm aller th a n -60 to -llO d B for an a rb itrary angle of incidence and even when th e boundary is placed no m ore th a n 5 cells from th e surface of the scatterer.
T he accurate m odeling of m etal and dielectric objects is an im p o rtan t issue in the FD TD m ethod, particularly if th e FD TD grid does not conform to the shape of the object, or if th e object is small com pared to th e size of th e cell. A tw o-dim ensional exam ple of th e later situation is shown in Fig. 3.2a. In this geometry, two m etal sheets are modeled whose thickness is less than the height of the Yee cell. Conven- tional FD TD u p d ate equations for this geom etry either ignore th e m etal sheets or assum e th a t they occupy th e entire volume of th eir cells. E ith er approach results in
a considerable error. A finer spatial discretization m ay be used, as shown in F ig .3.2b, b u t th is m ay place im practical dem ands on th e available com puter m em ory or com p u tatio n al tim e. Im provem ents can be m ade to th e conventional FD TD algorithm th a t increases th e m odeling accuracy while m aintaining th e com putational efficiency. These im provem ents include subcell gridding, using nonuniform grids, and subcell m odeling [72]. In th e sub cell gridding m ethod, shown in F ig.3.2c, a fine grid is used around th e small features, while a regular coarse grid is used in th e rem aining region. N onuniform grids, as shown in Fig.3.2d, are m ade gradually denser near th e small features. Subcell m odeling is often used when it is im practical to decrease th e mesh size to th e dim ensions of th e object, even if sub cell gridding or nonuniform gridding is used. In sub cell m odeling th e u p d ate equations are modified near th e object of in terest ra th e r th a n modifying th e grid. The new u p d ate equations are typically based on th e integral forms of M axwell’s equations th a t take into account assum ptions of field behavior in th e region of interest. For exam ple, m odeling of th in wires with radii greater th a n zero and less th a n one cell w idth requires m odifying Yee’s stan d ard FD T D u p d ate equations in th e cells adjacent to th e wires [68]. Sub cell m odeling is also used to m odel wires w ith circular cross section and diam eter no t equal to the m ultiple of th e cell w idth. Modeling of m etal sheet edges is another exam ple of subcell modeling. A special algorithm is also required to accurately m odel th e field singularity at the edges.
A more detailed description of the different aspects of the modeling is discussed in th e subsequent sections for different applications considered in this work. For modeling different structures and antennas studied in this work, an in house developed FD TD is used. T he accuracy of th e code has been verified previously for a wide range
air
mel
(a) (b)
(c) (d)
Figure 3.2: FD TD m odeling of coplanar lines using (a) a coarse grid, (b) a fine grid, (c) a subcell grid, and (d) a nonuniform grid.
of electrom agnetic problem s [25], [73]-[75j.
3.2
C o m p u ta tio n a l M e th o d
3.2.1 A n ten n as on H an dsets
T he Yee-cell, rectangular grid [68] and th e total-held form ulation [69] are used in FD T D code. T he com putational space is tru n cated w ith perfectly m atch ed layers (PM L) placed 4 to 5 cells away from th e nearest surface of th e objects. P aram eters of th e PM L are selected to ensure boundary rehections below a desired lim it w ith a m inim um num ber of layers. For th e discretization of th e com putational space, appropriate m esh sizes are used to represent antennas and oth er stru ctu res accurately and sim ultaneously lim it com putational resources. B oth uniform and nonuniform grids are used.
A special algorithm is used to handle dielectric objects w ith shapes a n d /o r voxels th a t do not coincide w ith th e FD TD mesh. This algorithm considers th e held con tin u ity conditions and th e integral form of Maxwell’s equations in a sub-cell regime. Using fast logic integration, th e weighted hux averages are com puted and look-up tables of th e dielectric constant and conductivity are assem bled for each held com po nent separately. This algorithm increases th e accuracy of com putations, p articu larly when, for exam ple, th e hum an head m odel is not aligned w ith th e coordinate system . In th e cases where th e m etal surfaces do not coincide w ith th e m esh, an o th er special- ized algorithm is used that allows for accurate treatment of helds near these surfaces [76].
