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, . A NUMERICAL AND EXPERIMENTAL FACILITY
FOR WIRE ANTENNA ARRAY ANALYSIS
Jeriy M. Lemanczyk, B.Sc.
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A thesis submitted to the Faculty of Graduàte Studies and Research in ,partial fulfillment of the
requirements for the degree of . Master Of Engineering,
, Department of ~lectr;cal Engineering , ,
McGill University (
~Jerzy M. Lemanczyk'
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Abstract
The d~velopment and application of a comprehensive numerical and experimental fatility for the analys~s of ~ire antenna arrays i~
described. Ln the analytical and numerical mOdels, based on a Method of Momen~5 solution of Pocklington's Integral Equation, special attention is given to the evaluation of the equation's kernel in order to imprôve conve'rgence. The tolerance to ill-cond-itjoning of the . ~ --General Impeaance Matrix has been improved throùgh the 'use of the Householder Decomposition Techni~ue, whe~ solving for the antennà ~urrents.·
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A novel eigenanalysi$ is also described ana results fro~ its \ 1
application show a consistent and predictable.behaviour w;th ~espe\t to . the solution and it is speculated, t.h"at the exainination of the char~cteristics of the eigenvalue; alone may give an indication of the prob~ble
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correctness of the final solution before it 1S actually evaluated.1 The rxperimental facil ity and the improvements to its. in~tru::
mentation are described, including the design and construction of'a unique probe-positioning mechanism and boom which allows ~ive. degrees of motional freedom.
-. Jn order ta demonstrate and utilize the effective,ness of ~he ,'. above numerical and experimental techniques, a systematic studywas under- ~.,
taken of two, 'three and four el ement yagi arrays.
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\' Cette thèse raite 1a réa1isation et l'application d'une
-,a~lyse num~rique et 1 périmen~ale de réseaux d'antennes a brins minc~ , Les méthodes de résol tions ana1ytiques et numériques ;~asées sur la .
.'\ '!Iéthode moderne des servent a étudier la solutiion ,de l'équation'
"intégrale de type Po klington. Une attention, ,articuli.~re est accordée
a la formation du n yau afin d'améliorer la co vergencJ de la solution.
1\ Pour. dé~ermin,~r la 'is't.ribution du courant de l'antenne~ la tOléradce
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" dè lq stabilité co Clidonelle de la matrice d s impédances gënéralAsées " '1 /
; a1été améliorée g~ ~~ a'l 'utilisation d'une echnique de décom~osition 1 1 • t J
, Householder. Les ~cteurs et. 1 es va l ~urs pr pres de 1 a ma tri ce e's
imp~dances 9énérali~ées,sont obtenus p~r un nouveau procédé qui étermine
u'ne r'elation cara têristique ,entre les yale rs propres et la solution. 1
, l ' D'où l'on<peut'.co cl,ure que "es valeurs pr pres agissent comme es
indicat'eurs e'n c qu,i/conc'er~e la validité/die la solution. /
Pre~en é ici est une descriptio ~e 1 'appareillage d~ meSure
utilisé dans la ham~re anechoiq,u,e exist nte. Les aménage~entlb qui ont
été apportés con istent dans la con~epti et la réalisation d'un , • 1
'support ro~lant ur ~equel est monté un écanisme qui permet de positioner
l:antenne récept iee l mesure. L'empl c ment de l'antenne réceptrice a. mesure est amov; b e a dés i r étant don é 1 a fl exi bil Hé du mécani sme qui.
"permet un ~::n~:: :ts::~:~:t~:u~':r:~~:~:~:p~~s:':uc~:~~:n::sv::atial.~. expérimental et nu érique en utilis ~ des réseaux d'antennes 'Yagi' ae
deux, trois et qua r~ brins)rayonna ~s d'oD il a été posstbl~ de
dêmo~,trer la 'just. s.'"' du ;y~tême 1 i,
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ACKNOWLEDGEMENTS~
The advice, encouragement and help of a number of people during the çourse of this work must not remain unmentioned. The many roundtable discussions between the author, ~r. S.J. Kubina and C.W. Trueman are
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invaluable in steering a clear way through the maze of numerical, modelling. ,The real~zation of the probe carriage and boom could not have
been possiblE) withoM the ingenuity and,precise work of Messrs. Foldvari, and Tibor of the Electrical Engineering machine shop. The help of Letitia , Muresan in the task of its implementation ln the reactivated anechoic, fac, 1-ity is greatly appreciated as wel1 as her attempts to run the computer program at ,the author's request while he was away. 'The day to day vexat~ons and problems were 1istened to and endured with great patience by S.R. Mishrai it ;s hoped that he will one day recover.
Over everything, the careful and critical eye of the author's ' 'thesis supervisor, Dr. T.J.F. Pavlasek, cou1d be f~lt. Hl~ touch, both light and firm at the same time, allowed freedom of action while prov,jding guidance and inspiration invaluable in the completion of this work. . ,
. The final production of this thesis was accomplished while at the E1ectromagnetics Institute of the Technical University of Oenmark on a Danish State Scholarshi~. Their providing facilities for thi~ is greatly appreciated. The author wishes to especia11y thank Farida Dah1 for her consc~entious typ1ng of the manuscript.
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The thesis work was carried out while on a National Research Coun-cil of Canada 1967 Science Scholarship an9 their support through this scholarship as well as through NRC Grant A-515 1s gratefully acknow1edged.
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TABLE OF CONTENTS
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Page
AB S'TRACT .' ............................................ : .••.....•.•.•.. i l ,
RESUME .....•.•.......•..•..•.•.. 1 ........... 'r' " .... (' ... ~ ............ ; i ACKNOWLEDGEMENTS· ...•.•....••.....•. ' •.•..•.. 7..' •••••••••• - ••••••••••• i i f/
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CHAPTER 1: INTRODUCTION: ...................... ' ..................... 1
CHAPTER II:
2.1.
2,,2.
2.3.
2.4.
2.5.
NUM~~I,CAL MODELLING ... ', ............. ~ ................... 5 J " ,
Inte~ral Equations .•.•.•..•...•...•.••.................. 6 -2.1.1. The EFIE and MFIE ................................ 6 2.1.2. Reduction of the EFIE to Pocklington's
Integral- Equation ................................ 7 ,2.1.3. Hallén's Integral Equation ...................... 10
f.1.4. Application to Arrays of Wire E1ements ....•.•.•. l1 1he Method of f>4oments .••.••.•••...•.•...•.••••..•.••... 12 2.2.1. Basis Functions .•.•..•••.••• ; ••.•..•.•......•.•. 14 2.2:2. Testing Fu'nctions ............... ~._ ............. 16 2.2.3. Galerkin' s Method ............•••.....•.•....•.•. 16 2.2·.4. Collocation and Point Matching ................ , .. 17 Numerical Excitation of Wire J\ntennas .• :.: ............. 17 2.3.1. The Delta Gap ......... : ......................... 18 2.3.2. The Pulse ................•...•••.....•....•.•• :.19 2.3.3. The 14agnetic Frill ...................... : ....... 19 2.3.4. Convergence of the Numerical Model .•.•.•.•••..•. 21 Direct Solution of the GIM .................... ' .......... 22 2.4.1. Gauss Elimination .•..........•.....•.•.....•.... 22 2.4'.2. Crout and Cholesky Decomposition •..••• .! ••••••••• 25 2.4.3. Reduction by Given's Rotations .................. 25 2.4.4. Householder r.ransformations .. ,~ .................. 27' Eiqenanalysis and Matrix Conditioning .•.....•........•. 31
(1 bi'" CHAPTER III: APPLICATION TO WIRE ARRAYS ............ Pt .............. 35
~.l. Method of Moments App1ied to Pocklington's Integral Equation •......•.•.•..•.•..••......•..•.•• ~ .. 35
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3.2. The Excitation Vector ...... t ••••••••••••••••••••••• 40 3.2.1. The Delta Gap Source ........................ 40 3.2.2. Pulse Excitétion ............................ 41 3.2.3. The Magnetic Fri'l .......................... 43
3.3. The Integration 5cheme ............................. 44-3.4. Program Description ................................ 44
CHAPTER IV: THE EXPERIMENTAL FACILITY ..................... , ..... 48
CHAPTER V: RESUL T5 ........................... ' .................... 56 5.1. Comparison with Pub1ished Results .................... 56
5. 1 . 1. Th e D; po 1 e ...... ~ ... ~ ......................... 56 5. l .2. Convergence ·for a Two El ement Ya'gi ..... .- ...... 59 5.1.3. Current'Distribution of a Two Element Yaa; .... 62
5.2. Measured and Calculated Patterns ............... :.~ ... 67 5.2.1. Experiment Descri pt ion ........................ 67 5.2.2. Resu1ts of Vag; Mea~urements and Calculations.70
5.3./1he Eigensolution for Wire Antennas .................. 93 / 5.3.1. Eigenanaly~is of a Dipole ..................... 93
( / ',5.3.2. Eigensolution of Vagi Arrays .................. 95
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1 t~APTER VI: CONCLUS ION ........... ", ............................ l 04
APPENDIX 1: INTEGRAL EQUATIONS ........ -......................... l08 APPENDIX II: NUMERICAL ALGORITHMS ....... J ••••••••••• ~ •••••••••• 110 APPENDIX III: CONDITION NUMBER ...... " ........................... 119 APPENDIX IV: PROGRAM AND SAMPLE INPUT DATA ..................... 122
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APPENDIX V: EIGCe .........•.................................... 133
BIBLIOGRAPHY •.................................... : ." ........... 137 '" (,
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ÇHAPTER 1: l NTRÔDUCT l ON
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The analytical antenna problem i.s lar,gely that <?f'determining its ~ 1 - ~
currents accurately. Once the current distribution is knoW,n, near and fà,r
field radiation patterns as wel1 as the input ~mpedance may be calculated. . /
The antenna problem is fundamentally a boundary value problem. which, start-ing from Maxwe11's Equations, can be expressed an an integral ~quation 1nvolvlng the current distribution of the radiating structure.
The so.lution for currents how~vet, may be avoided in certain clas-""-
ses of problems which are referred ta as being asymptotic. When the s.truc-
ture is small compared to wavelength, low frequencY·asymptotic techn;qu~s
~an be suc~essfully applied (52). Conversely, when the antenna is very larg~ ,. .
compared to wavelength, optical techniques such as the Geometrical Theory . of Diffraction (53) are used. A large region nevertheless remains between the ranges of applicability of the high and low frequency asymptotic methods. In particular, many w!re antenna applications are found in thlS region, sometimes referred, to as the Resonance Region (32). Especially in this region -
- therefore, the integral equations must be solved 'and the current distribution obtained. The availabillty of large scale dlgita1 .... computers has allowed the
development of numerical techniques for the.solution of integral equations and work in recent years has resulted in considerable progres~ in numerica1
Q techniques (9,10, 15, 18,22,54) applicable to the antenna prob1ems. The equations, used to describe wire antennas and antennas in 'gen-"'~
eral predate the electronic ëomputer. Pocklington first formulated the equation for the field of a one dimensional current in 1897 (20). In an âttempt to simplify the solution of Pockl'ington's Equation, the formulation
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of Hallén (23) was put fo~ard in 1938. Methods nat based on integral equa
tions of the current distribution were alsD developed. Predominant among these is the modal method of Séhelkunoff(42) and the resonator theory (42.55)
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) based on the premi,se ;th9t -the field may be express~d as a surnmation of an
infinite number of the antenna's force free oscillation ~r,basic mode~ /None of the'Se methods ,are as di rectly appl i cab le to wi re array arrtennas
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as those based on integral equation formulations a~d none are as amenable
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to computer solution as the integral equation. The most geneval equations. 1 ('\
the Electric and,Magnet~c Fleld Integral Equations. have a1so been long established (43) and it can be shown that the other equations.of'Pock1ington and Hallén.can b~, derivedofrom them.
The process of obtainfng the solution to'an anten~a problem 1S
one of modelling. The integra1 equations are analytical models of the • physical reallty. The choice of which' equations to \.Ise and what assump-
tions to make are questi?ns of modelling. HQwever once the equation i~ formulated, it'needs to be solve'd. Direct analytical solution of the mtegral
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equations is difficult. Th-is,difficulty has caused researchers tq turn to' the digital computer as it provides a proceS~hrough which a solution can be attempted with some degree of,efficacy. But herein lies a second modell;ng process, the modelling of the analytical equation model into a form r
which the computer will accept for manipulation and solution. Measurements provide a Separate avenue to antenna ~roblem solution .
. Hère again though s the questiçn is a form of model1jpg, this time of ~imension. al scaling and the âttempt to simul,ate free space u by 'an anechoic chamber or an open range. The problems encouhtered here'are of physical'scaling ànd . construction. In the end s provi ded that both the ana lytica l /n~meri ca l mode l and the experimental model have been successful in each case, the results obtained from both should be the same. If there is disparity between the two •
, " results s then confidence in one model may then provide insight into the workings of the other and may lead to its improvement.
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The present work addresses the analytical/numerical mOdelling, process and makes use also of the experimental approach through a special ized laboratory facility. While the tHle is then descriptive of the principal go?l of this wQrk. the relationship between numerical and experimental model-
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1ing 'is also elaborated'on as' a necessary guide to the proper application and its interpretation of the numerical modelling procedure~.
Chapter II which follows, is an exposé of the analytical and numer-" ical modelling procedures used for wire antenna array analy~\s. A brief
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introduction of the classical integral equation formulation is fol1owed ~y11 , t,... .... .rr f-
a description of the current' level of development of the numerical mod~lling ., . process known as the Method of Moments. Also presented is a section des-
cr.ibing several numeric~l algorithms. Attention is drawn ta the need for
numerically stable algorithms for the solution of systems of linear ~quatlons.
It is also ~uggested in this chapter that the moment method numerical formu-, la,tion may be treated as a'n. eigenvalue problem. The signifjcance ,of tl)is,
in relating numerical model',l'ing to the corresponding analytical model, will ~ , ~ _ t
be seen inuChapter V. rp,should be ooted that up to the present time, numer-l' ical modelling has beel'Jt'co_nsidered to be merely a method by Whl~ch an analyt-
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ical problem can be sol~ved' on a computer. , " The complete ânalytical and numerical modelling procedure used in ;;
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thïs" work for"modeHing wlre antenna arrays is presented in Chapter III. T.h~:"
method of mOments allows varying degrees of complexity in lts application
but ~"en though more complex mod~ls facil1tate correct solution, the parallel
jncrease in the complexity of the program coding sometime.s causes researchers
,to abstain from their use. ~However, it is shown that ease of programming can
be maintairied_while at the same tim~~ the numer~cal model i~ ~improved~. The
program description composes the last section of the chapter and completes
the'analytical and numerical modelling process. " ,-
Beihg mi ndful that experiments are a necessary part of antenna work,
attention in Chapter J~ is given to ~,description of a càmpact size UHF ane-r \
choie chamber re-equipped and improved for this wqJk. The major improvements/
included impr,ovement of' thé recording apparatus 'and especially the design " '
and construet; çm of a nove 1 probe carri age and boom. Th i s probe ,pos it i oner ,.<'
, .1.' ha~ five degree's of freedom 'and its a(ivantages .are that lt allows probing -,-0." .. ' , of fields at a!1Y point in the èhamber and by Nirtue of its design!l"J:an be
immediately used both for cylindrical and spherical scanning/tech~'(ques for far field patterns derived fr@m near fieHÇmeasurements .. -:/-
.),. -Chapter V ~eal~ with the applic~tA-on of the computing and measur-
ing /ltools" described in the previ'ous three chapters. 0 First the analyt.ical , - ,
and numerical modelling procedures are verified against av~ilable published
results. The ;~nvergent qualities of the approach used are demonstrated
in the solution to a particular arfay problem which in previously publlshed
worK was used as an exa le -to demonst~ate non-convergence associated with
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The program is then systematically appl ied to a
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set of yag,; arrays whose patterns were measured in the anechoi c chamber ~~
descrïbed earl ier';'~'If Chapter IV. , Thé mutually supporting nature of the
calcu19ted and measured result~ is presented. , '
The treatment of th'e numer;cal model as an eigef')'value problem is ,,' re-iterated at the end of Chapter V. The calculation's ,carried out here in-
dicate"that a link exists between the numerical model and the analyticar-. antenna prob lem. The li nk can be shown to be in the comp 1 ex, e i genva 1 ues
which demonstrate a definite and predictable behaviour ;n the solutions
for the radiatiop patterns. It is also sèen that the .. e--~genvalues may, in
their own right, indjcate the correctness of a part5cul-ir mod~l ~eing used.
Thesé two observations resulting from this work suggest that the behavi'our
of the complex eigenvalues is descriptive of the antenna problem like the
pole zero diagramswhich Clèscribe network behaviour. '"
A set of five appendices completes the thesis. They contain de
tai'ls of various aspects of the. analytical and numerical models discussed
in Cha'pters II and III.
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CHAPTER II. NUMERICAL MODELLING
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Wjre antennas and arrays formed by them are commonplace and in
wide use today. The theory required for their analysis' has been known for ,~ . ,
some.-time and is well presented in the classical works on , ,
(6,7,8,42,43). Antennas composed of multiple wires, most sented by the Vagi Ar;ay (45)',,. b~came prevalent after·the
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the subje<::Y' r:. J ~f-';
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War along with the accellerated devélopment of corrvnerc;al radio and wire-
less communication systems. As the transmitting frequencies increased, the
need for wide band di rectiona 1 antennas arose and structures such as log perF , odic antennas were developed. But"just as the antenna designs became more
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complex, so did the analysis of them ;ncrease in d;fficulty.
'" The equations known from the classical theory are a11 applicable -
but there are considerable obstacles to the;r solution., In antenna problems,
as one ;s usually dealing in an unbounded space, the equations are of the in-. 'tegral type. The integrations are normally carried out over the surface of
the antenna structure, but such integrations can be performed analytically
on relatively few simple geometries such as half planes (46) or cy'linders
of infinite extent (47). However with the advent of the computer. numeri
ca-l solutions to these equations can now be attempted. 1
Just which equations shall be used and how they are to be solved - , 11-
with respect to the p.{esent·work is the sub3ect of this chapter. There are
five main se~tions; the first deals'with the equations, usee to model the
antenna system, the second wi th _by whi ch the mode lis then .
---'~p-H~'Tl1:le Il on the computer, the third treats the ~l'l-·important matter
of antenna excitation modellir1g on a .computer whilè the remaining t~o
sections con~;der numerical têchni~es ~er se.
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2.1 Intègra 1 ~Equatifon!s
Any electromagntic field can be represented by a differential'
equation or an integral equation (1~6). Hqw~ver, ,in antenna tproblems, 1 JI,' f"
integral equations are the more popular and there are two reasons for th~s.-
Experience has shown that IIclosed li or internal problems such as
waveglJ\ides, are best solved by differef\tial equations because of known and
fi xed boundary condit ions. Antennas and scatte'r'érs on the other hand, form'
what can be referred to as external problems whose solutions are best form
ulated in terms of integral equations (12). The equations are general
enough tha..t,; as pointed out by Albertsen et al (12) , no approxlmations, are
needed in the formulation of the field, equation although they are still
made,when warranted,to simplify calculations. The most significant approx
imations are introduced in the impl'ementation of the equation ;n a computer
program. - j
Integral' equations-, ~tpecially when" based on a Greenls Function, ".."
have the advantage of reducing the number, of physical dimensions of the
prob1em (16,17) such as reducing a three dimensional problem to a two dim~n
sional one. The question-though is,whether or not a Green's Function may
be defined for the problem at hand or. is an altogether different formula
tion required (44). Antennas though 'are generally loc~ted in free space
and 50 their analysis \'/ith a Green's Function basect integral equation 1S
straightforward .
. ,The seoond reason for! choosing integral equation'representat;ons
over those of other equations is, that they are numerically better be
haved th an their differential counterparts and the problems of underflow
g.ssociated with finite differences are av~ided.
. 2. 1. 1 The EFlY and MFIE h .,. •• ,r
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The two most general integr'al equations used to describe an- l', ~< l ' ~~ .. /
tenna problems are the Electric Field Integral Equation (EFlE) and the ,;~ , , ~Î; t
Magnetic Field Inte~ra1 Equation (",rIE). They- have the;r origins in 1
Maxw!:!l1' s Equations and are shown ~n Append';x 1. Their deri vations
are well establ i she~ (17,19) 'and as ;they are equations using the free
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space Gr~-'-s -FuncYion, they 1end themse1ves q,uite real,iily, ta antenna ~ f ________ -a,nâfys i s.
------ The EFIE and MFIE are not free from approximations, and when
combined with the approximations of numerical mode-lling of the integra-f /
tion surface, ft\has been pointed out (12.13,18,19) that the MFIE does not
give as good a result as Jhe EFIE when/app1ied to electrically, Itth1n" structures such as wi re ~nt'enna5,' . .
" The opposite is true wlth regard to "surfacrs" where the MFIE li> ' • ,
has been found.to yield better rèsults. As the work presented here deals
with wire antenna's, exclusion of the MFIE as a choice for the model1ing
equation sèems logical',
It i~ a1so not necessary to apply the EFIE equation ;n its to
tality ta thin wires. Mùch simpler equations such as Pocklington's
Integral Equation and Hallen's -Integral Equation will produce just as good
/results. But as wHl" be seen, the/' equations can pe derived from the ,EFIE'
with assumptions regarding the -w.ire thickness ~nd can th~refore be con
sidered to be special cases of the EFIE.
The reduction of the EFIE t~ Pocklingtan's Integral Equation is
straightforward and it win be useful to present it briefly 50 that the
, relation between the ,,<wo is clear.
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2.1.2 Reduction of. the EFIE to Pocklington's Integr~l Equation
~ ---The EFIE taken from Appèndix l, may be simplified by making use
of the various assu~ptions made 'in antenna prQblems~ namely that the
'medium i s free space and if not,. is at least l ;near and isotropie, Sec
o'ndly, 1t is assumed that the antenn'~elf i5 a perfect conductor.
This then introduces- the bounda~y condition that the tangential
,co'11ponent of the Electric Field ;s zero on\t~e ,surface_ of 'the antenna, or
that
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,Realizing that a11 the sources can be enclosed in a baundary which extends ta infinity, the EFIÉ may be immediately written as
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where n = unit vector normal to the conducting surface' w = 2 il x frequency ~ = permeability of the medium $ = free space Green's Function
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The prime (') denotes the source cQordinates apd the bar on the integral sign its principal-value nature. ..
The equa~ion 2.2 may be further reduced as Maxwell's' Equations and the équivalence principal give
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In ,equation 2.4, v~' represents the operator of the divergence over the surface of the conductor. This then allows the EFIE to be ex· pressed in terms of only'one unknown, J s '
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r,
9
\ The above equation can now be applied to th;n wires. The bas.ic assumption to be made/and shawn to be,vplid'(21) is that for a w're of
circular cross section, where the radius is small cOmpared to wavelen~ih, the cy,rrent which predominates,:;is axially directed. Thus the problem
l,
i~ ,,récjuced to ft single 'dimensional one. With the assumption that t-he cur-. rent i; ax i al. the surface current dens ity may be wri &en as
2~6
'\~f..,:, wi th : \ .~ J~/~';' î being an axially directed unit vector (perpendicular to
section) and ~'being its radius as shown 1~ figure 2.1. , t... the wi re cross , ~.'4 , .' - , : ,,~,
, . ,
(
1 1
Q.
o
Figûre 2.1: The Geometry of the thin wire assumptiOfl.
. , ~==.:~ :.:~i..-,~~~A..~,=""_,*n,.,..,, ""'iUl""!La:œ-..."" ____ _
j
(
1 4
o
.,
,
$ r 1.:
~ " .. ' Î'i
~
i ~
1 i' ~ .. ~
t 1 t
(
, ! 1 1 10
It shoul d De ~j ed ;h~ t s ince the tangJnti a l Ë fi ~ l ds are ,.ro
at the surface of the cot"ctor. the EFIE reduc s to
. 41TJWe: a~· as ~ • Ëinc(x) =/~ ~ r(t t'"k 2 I(x" <1>+ ~ 1(;') 1--4) dt'
, ' 2.7
1. and Mi 11 er (19), i ntegrat ion by parts i s carr; ed
out and assuming tha .the current at the ends of the wire vanishes, equa-
tion 2.7 becomes,
î
equation
Einc(x)
-1 J = 41TjWe: I(x')(t
L
-
.r. 1
• î' k2 <p -, ,
d2
~) dR.dt'
e l ~fùl!ft t s are a 11 cons i dered to be
reduces to
,
-1 l I(X')(k2$ - d2
P J d~' = 41TjwE L ~ dR,d~!
d~' 2.8
pa ra 11 el. the
2.9
Equation 2.9 is immediate1y recognized to be the classical equation whose
kerne1 was first formula"ted prior to the turn of the century (20).
2.1.3 Hallen's Integral Equat10n
The distinct feature of Pock1ington's Equation is that the " boundary conditions are "built-in". This equation'however has been the
( .
source of sorne controversy regarding its convergence (22), the ,alter.nate
proposal being to use Hal1én's Equation first put forward in 1938 (23). A , concise derivation of tflis equation as applied to thin wire is given in
Thiele (11) and is based on the source free vector potential wave
~quation;
\
't
. ~ t
\
1
,
11
, 2.10
.)
App1ying the ~tn!wir approxim~tion and the con traint that
/only, straight parallel w;ire,s are being dealt with and app}y;ng boundary
conditions. the integ aY e1uation for a~ynunetr;cal dipo1;è becomes (24).
+ 9, "2'
J n (Cl cos kx+ C2 sin k (Ix!)) 2.11 9,
2
= intrinsic impedance • 4'
C1't C2."are constants to be deterll]Jined 1 ~ ..... ' • 1
/ from boundary condi' t; ons. (
2.1.4 to Arra s of Wire Elements
mind what has been written until now. i.t would appear
that thé cb<l' e f equation is narrowed to two; these are Pock1ington's ., = and Ha11én's nt gra1 Equations. Although reservations have been expres-
sed by resea che s on the use of Pock1ington~s Equation (24.25).~ its chaice
is s il1 preferred over Hallén's Equation (1.2.11. 15,26)
- l'
as can be seen from the previous section, is
To ex tend this equation to a
three dimen'Siona1 orientation is tedious and one
is 1ed on e m re to tH more general Pocklington Equation shown in 2.8. i -
The appl'cation of Pocklington's Equation is a1so more in line with the
physical situ tion for a number of reasons. In this equation, 1 ;s salved for som arbi rary but known excitation Eine on the left hand side Qf
<
, 2.8 and 2.9, whereas Hallén's Equation is dependent on a delta gap source
~hi~h a thp gh convenient, proves an inadequate model as will be dis
cussed atir on. i
,1 1
1
"
1
c
\ (
<-
1 _
-12 \ '
\
It is later shown in this work how, by proper treatment of the P~ckl;ngton Integral kernel, convergence comparable to that for Hallén's Equation is obtained. And given the above general 'considerations, it i s
------------ , now poss ible to demonstrate ho,w Pockl ington' s Equation can be numer;ca lly imp 1 emented ..
. . 2.2 The Method of Moments
the equaHans' appl icab1e to an'ténnas generally tend to' be of
the integral type and aré deterministic. That is ta say, given g, f,may be determined via application of an integral operâtor
L(f) = 9 2.12
For the antenna, g ... 1S usually the- assumed êxcitation and f is the
response, usually the current or current density on the antenna. Just,hoW the functional equation shown abo~e ;s reduced to t~at of ~ matrix equation (i.e. a set of n linea~ equations in n unknowns) is referred tb as the Method o~Moments and is wel1 described in Harrington (9). An eva1uation
~ of several equationso to which this method is app1ied is givcn in Miller et al (10), - In sunmary the Method of Moments formulation is as follow!'.
, ,Given equation 2.12 and given that the ,operator is 1,i~ear\
an inner product of the two vectors f and 9 can be defined such that o
*
< f.9,> 1
==<g,f>*
< af + 69,h > == a < f,h )' + B < 9th>
< f*, f > > 0 if f f 0
/ = 0 if f ;: 0
By ,linear is meant that the mapping by the operator of f anto 'g is unique and therefore one ta one.
l, .. t~-
_____ 1.. .. ,1-
/1
2.1 i li
,2. 14
2.15
1
( \
(
(
13
where * denotes complex conjugate. -
Thé function f is expanded in a serres of functions within the J
domain of the operâtor L,
L:= ,I Il if n n n
where a and f are referred to as the weighting constants and , n n basis functions respectively. Substituting back into 2.12 ant.,ta-king ,advan~age of linearity yields,
r an L( f rr) = 9 .2. 17 n .
The next step is ta define a set of testing functions. An " inner product w;th these functions yieldsJa~mat~ix equation as shown·below,
, I~ an .~ Wm, -ë ~ ( f n) > = < Wm, 9 ~ 2.18
•
1
The m9tr"ix, comprjsing the elements formed by the inner pro-duct of the testing functions with the operator on the basis functions, is referred to as the General Impedance:Matrix (GIM). The inversion of the GIM yields the anis. ,
It should be noted that ,if the basis and testing funçtions \ form an orthogonal set, then
'''', t·~l. - ..
....
2.19
" ")..,
"
(
o
14
1/
2.2.1 Basis Functions
The choice. of basis functions for the vector f depends
on their relative rates of conv~rgence as well a programmlng
1 effort required. 'Th-us, out o~ /"vlrtually infi ite choice ôf basis fu~etions, only a small number will be of practical se. Evidently,. and as"
noted in practice, better convergence is obtained V/hen the bas,is functlon
resemb1es the actual current distribution to be ca1cu1ated, and as
Thiele (11) reports, may indeed yie1d a belter conditioned (i.e. more
stable) GIM:
Since the basls and testing functions are sel dom orthogon~l~ ,J"
the GIM is always full for integral operators. To have an overlf:.c~oJllplex
basis'function weuld, then praye to be expensive in computation time and ln
the ligh~ çf'what has been mentioned above, not necessarlly convergent.
Basis functions can be categorized into two types, ~orma11y
the entire domain type and the subdomain ,type. Simp1y stafed, the en
tire domain bases are functions f which are defined and different from o n
zero over the entire domain. Common basis functions which span the en-
tire domain are genera1ly of an infinite series such as Fourier Expansions
and Legerdre Polynomia~s. At bést these are difficult to prog do
not lend}.themselves easily to antenna geometries whi,ch are composites of
severa1 st~uctures such as an array of wire antennas.
The subdomain bases however are more easily applied in that
they are defined over a portlon of the geometry and made to be zero every
where else. This great1y simplifies ,integration and reduces computer
t time to evaluate integrals. The priee paid however is slower convergence
as the antenna geometry must be subdlVided in more "subdoma i ns" than the
,.number of terms of an expansion in the entire domain solution (refer to
Hliele (14)). As mer1tioned though, this' drawback is more than made up
by the ability to segment the antenna geometry, allowing for much easier 1
geometr.ica·l modell ing as a 'curved line may now be modelled as a c.hain of
s~rai9ht w;re se~ments. The:basis functions shown in 'figures 2.2 thru
2'.4, a1most universal in their use are the Pulse, Piecewise L inear and -r
Piecewise Sinusoidal. Interpolation formulas can al-so be used in a sub-
doma i n scheme (14).
1 •
~
~ ./ 1 j
,l' 15 . ' t ~ ,
_____ 4
,'0 n
... ,;'
•
SUBDOMAIN n o
Figure 2.2: Pulse basis function. --------
o
- ...-,
SUBOOMAIN n
~ Figure ,2.3: Piecewise linear basis function with equispaced no~es.
",'.
f..,
an = srn(Az) -/ sln( z[] - z)/
sln(A'z) -~
-----~-(:iA
1 ""-
, j -
zn_1 zn Z n.1 l \' - t
:-. v
SUBDOMAIN n- -(;> - "
- , 'J ,f
"
Figure 2.4: Piecew;se sinu~oidal basis, function with equispaced nodes.
\ . 1
,--.".------------ .';;" 'l,"!,
J J
" 1. l 1 '?
-----.---.-. '~. -.J :~
(
-;Y. , '
ç
In the case of the piecewise ljnear and piecewise" sinusoidal
functions as well as the subdomain interpolative for~u1as, the doma;ns
16
.' _ overlap one another. This is shown in figure 2.5 whos~ solution lj shown '
with piecewise linear subdomain basis tunctions.
"-2.2.2
SUBDOMAIN f
Figure 2.5: 1 1 1
etc ...
Solution in terms of piecewise
linear basis functions
Testing Functions
n' • 1 ~"
The t~sting functions follow the same general ,guiâel ines discu's
sed in the pfel,(ious section on basis functions. TITe choice of the\ - , appropriat,e testing funct;on determines the ease of-çomputing and the
- 1" -stability and rate of convergence of the solution: Just wha,t type of
te'sting function is chosen determine.s the -methodology of the solution as 1
1 ;s described immediate1y.
2.2.3 Galerkin's Method
Taking the testing function Wm such that
/ .... '
1 /
/ 2.2ff t
\ .
'. ,
---,-------- ----,----,.
j , ,
(
< .. -~ "
..-:~' :..,.<~
o
~ , 1"-'l,j;'"' ~
~>/~~t
i
le 1
~ ,..
---~-
~
\l 17
\
2 an < f ; -L(f ) > = < f ,g > 2. n m n m
" Note hO,wever, that a~~--4s an integral operator, each lement
of the GIM will require two integrat~ns - a costly proc~dure at
A ~ethpd will be discussed later whereby the advantagetf a Galerk'n
method can bè obtained without exp~~citlY performing t e two integra- , .
tions.. requir:ed. ,
/
2.2.4 Collocation and Point Matching
One method'by which the integration of the inner , ,
be avoide,a is to use :the dirac delta flJnction as a testing
physica-l significance of this is that the boundary'Condition Et n = 0
is only sat.isfied at idiscrete points .1long the antenna whereas Galerkin
method a ttempts \to erl(orce i t al ong the ent ire doma in. ,
The relativ~ ease of programming makes this a very attra~tive
procedure and it has peen shown that good results may be ob ained by use
of ~his t.ype of testi;9 function_~_~5,11 ,12,13,14). Ho~~ver Thie1e (11) po; nts out that the a curacy of \..;tfie result i s not on l.,y de endent on the
;
number of pOints chas n but Qn"their 1 ocatJon as we) 1. n most cases
hO'wever', equispace<L p'~ints provide the use'~ with goo,d r su1ts. C: -. ,..
t I:~'
2.3 Numerica1 Excit~tion of Wire Antennas
. < Considered Jere are tloe commonly useo nun;eri ca 1 nlode 1 s for
th'ë~x<~tati'On of Wire~antennas. ,AU the resolts appearing in ~h~ter V use eite of the mo~el presentedi"tere. The following is a description .; \ 1 0
of these models and ho they are applied.
\\ ;\ , 1
1 1_ -----------------_.--- ~----------;.~ .. ~ - .. >-
*'
l
,
(
("
\ ....
18
2: 3. 1 The De Ha da p \ \
'. If all the constants of Poeklington's Equation are eollected
on one si de' ami the e.9u~-tï on rewr1tt,en 50 ; t coi nci des with the opera tor notation of equatiot~.12, the right hand side, or the excitation side of thè equation becomes
RHS . -Eine ::; Jwt: 2.22
From the previous sections, and applying the method 'of moments, an 1nner product is made,of tQe RHS and the testing f~;tion.
~.);'j\
~wt: 'J WmËinc(X) dx , L
, '
o
But if the field is taken to'be that ge't1~t?a:ted by a delta gap,
then
Ëinc = V <S(x)
, i
The integration is thus avoided leaving simply
~lassical1y, this source representation proves inadequat~ , especially when one reqlJllires convergence in t,he determination of input
• f '
2.24
2 .. 25
/
-, "
< ,; .
1 .1
1" 1
1
. , l '1 1, _.:1 • L"""", .,
1 ,.f
f
'.
(
a4,mittance (11). Ta obtain valid results in such cases, more complex
'mode l sare requi red. 1
2.3.2 The Pulse
c
Keeping the same right hand side and inner product definition of the previous section, a norma1ized voltage excitation is ,defined over a portion of the antenna. In the càse where subdomairi bases are in use,'" the pulse is usually taken to span one such subdomain. As the voltage. V is constant, the exciting field becomes
J '
-1here 6x ;5 the subdomain length and the right after setti~ the, exciting vol,tage ta unit y, becomes '
\
RHS = j~ 1 Wm (x) qx 6x
2.3.3 The Magnet, c ,Frill \
J
hand side,
, \ In rea1ity', wire antennas ar'e nct, isolated in space but are
-
2.26
2.2J
mounted on structures and are fed by transmission ltnes, For exampl • in the case of a c~-axially fed monopole ~ounted on an infinite ground pla~e,\
, q
then by image theory, a'dipole is simulated. It has been shown that the coaxial feed can be successf 11y
-:%~\t)deJJed as pn <;mnular frill of magnetlc current at the feed poi t (27). lts appliêatfon' (11,13,27) has been successful both for near 'a d far field pattern predictions as wel1 as admittance calculations. '
~ . '
, Questions arise as to the actual physical sile of he frill but this poses f~ more of a proble~ than the question of ~ap wi th or s~gment length a~ posed ~ the previous excitation models.
\
.. -~- -
I~
'...-. ' ... 0";: ;-. ~:
t f
1 1
1 ,
1 !.
l ----------- . J
1 -~t .. ~-_,~.4 '..~
\ \,
·.------- - - ....
20
z (
1 .tjJ'
~~--------------------- y
x
Figure 2.6: The Magnetic Frill.
As shown in Tsai (27), the frill derivation is straightforward. " Considering the geometry and paraméters as shown in figure 2.6 and assuming , '
axial 5y~etry, the r directed Ë d~stribution i5,
,
E 1 (r, 1 _) = V, if
r 2r'tn (*J
By image theory and by setting the voltage ted magnetic cur~nt dis~ribution may be qbtained
/ 1
j l-
I <.. ';/
--.----. -' /'
2.28
A
to unit y, the cp 'direc-
. 2.29
, 1 ~l
!
!
l, j'
(
'C
21
. ,
The general wave equation can be reduced quickly by realizing , , ,
jhat with on1y a magnetic current present, the magnetic vector potential ~'is zero 1e~ving ~n th~s casJ ~
2.30 "
where F is the electric vector potenti:a{and is defined to be
\ E 1 ff ejkIR-R'1 F{p)='4~,r 'K'{p') __ ds l
frill ' IR-RII 2.31
surface
It should tie noted however, that the frill has a near and far fiel~ which must be taken into account when radiation patterns are ta be calculated.
2.3.4 Convergence of tpe Numerical Madel
One is now faced with the task of choosing basis and testing functions and using one of the various excitation models. The ultimate goal is ta obtain a convérgent result with as litt1e computationa1
\ . effort as possible. As experience in the application ,of the Method of Moments develops, guidelines to help determine these choices are beginning ta evolve '(10,17 ,29) ..
1
But just as impor~ant as the above choi~s, it is important to avoid unbalanced matrices. The GI~ inversion'process itself may lead to a non-convergent result. As stressed in Webster (15), a judicious choi ce of ~umeri ca 1 a lJlorithms th'én becomes ; mportant and i s a subject of further discussion in this- c~apter.
. ". ,
. ,~ .
i !
1 \
)
",1 \
(
( ! •
,/
22
f,
2.4 Direct Solution of the GIM (General Impedance Matrix)
The Method of the Moments provides a 'means by which an integral equation can be reduced to a set of n linear simultaneous equations in n unknowns. Thus, given the matrix equation
. Ax = b
whete A is the'matrix of coefficients, x the vector of upknowns and b the known "excitation" vector, then,
2.32 a
2.32 b
~ 1 Normally one does not explicitly form the inverted matrix A ,
however, 'the classical methods of solution, based on the algorithms of inversion, must necessarily be subject to the same inaccuracies which
occur when the matrtx, although theoretically nonsingular, behaveslin such a way as to yield inaccurate results. Such matrices are said to
be il1 conditioned. Classical methods a~l re~uire that t~e m~tri1 under manipulation be "well conditioned". 1
.. f 1.
2.4. l Gauss El imination 1 /' Almost all t~e methods of matri,x solution bas~d on ir;t'~ersion
algorithm require reduction of the matrix, into triangular form!. Once/ this is accomplished, the solution is obtained by "back ~u st/itutiOj"' The lNFL * (29) program for thi s al gorithm i s stra~ 9htforwai< and __
.. <Ir ~~ 1 \ 1
* 1
,1
INForma l Language ·1
1
/
, ,
ï
"
.'
(:
~I 1
, 1
easy to comprehend. Let A ~ enxm (where enxm dfinotes an nxm complex space) be
. upper triangular and non singul~r ~nd also let b € en, then x, the \ .... 1'" ... 1 ~,J,
.. .'" ;le'
solution, becomes J'
for i = n, n-1, n-2, ... t 1
2.33
','
where the ais are the e1ements of A. 1
The technique of Gauss !limination is described in many' texts: on numerical analysis (e.g. Stewart '(29)). Basically, the process is the multiplication of the A ~atrix and b vector by e1èmentary lower triangular matrices of the fonJ
and
1 ra: me T . k
,T denotes the transpose
, ( f ..
'''ek = kth cO,lumn of the Ident.ity t1atrix
m\ = a constant needed ~o,~ero the , subdiagonal elements
me/ ~/O'~' (.i '= 1,2, ... ,k) ~-
2.34
2.35
,'"
, ,
Straightforward Gauss El mindtion is not a stable numerical process and more el aborate schemes . based on it are used (15,29), these being 'the a~gor;thm of Gauss Elimi ations with'comp1ete and partial pivoting (Vil. Appendix II for a~ INFL program description of each al-gorithm<appearing in thi,s chapter As the matrices ëalculated from
, FI 1
r.-! ~
f
1 j . l , ,
(
(
24
integr~l equ~tion-s witrincluded Green's functions are usually diagonally dominant, that is ta say that the elements on the diagonal are much
(, 1arger tpan the.remaining elements in the'~atrix, th en it can be safely \ said tha't the matrix is non singular. However, it is suscepti ble ta
i11 conditioning. By il1 conditioning is meant the following:
.. ,J'
Denoting the functio~s of Gauss flimination and subsequent back substitutiOfÎ~,b1 G, ,and realizing that tAe matrix A'lS an approximation to, the true matrix due ta computational pro'cedures and cal1ing this approximation A* and the true matrix A, the difference between G(A*) and G(A) may be quite large. If this is so, then A is said to be il1-co~ditioned.
Letting
A = A + E
where E = error matrix, Stewart (29) shows that the relative error in inverting A may be magnified by an amount X(A) which'ls referred to as the conditio,[l.,number of A with respect to inversion .
.dç -.....--
2.36
error = relative ~)rror* X(A) 2.37 ... ' ~"
=it-H • 2.38
where " • Il refers to matri ~ not;'ms so that although the matrix is non.singular, it may still be numeritally unstable (30) .
Nevertheles~, for a simple antenna geometry away from resonant regions, Gauss Elimination with partial pivoting is an adequate matrix" " , ~!
solving algorithm. _ • ' I\ •
The choice of partial pivoting over complete plvotlng is one ~
based on economics as the complete pivot requires sear~hing the entire matrix for the largest element while partial pivoting searches out the 1· 1argest element be10w and inc1uding the diagonal eleme~t of the column
. . of the matrix under consideration. In well conditioned problems, partial pivotingOis quit~ sufficient to yield excellent ~esults which are as good as those obtainable by more elaborate procedur~s (15,29,30).
·' (
1",,'
2~
2.4.2 Crout and Cholesky Decomposition
, ,
Two other common methods of square matrix triangularization are the Crout and Cholésky algorithms, both of which are variants QQthe Gauss reduction schemes described in the previous section.
The Crout reduction scheme. w~ich from inspection of Appendix II has the same number of operations as Gaussian Elimination and yields
>P 1 ~ l " # \ ., j
the same res~lt, would seem t~ be of little valu~. Its ad~anfage however, is in the reduction of round off errors wh~ch ~or large matrice~ of poor conditioning may be rather large. This is bècause"the linear products in th~ Crout 'reduction can be accumulated ,in"double precision. To do the same for' Gauss Elimin-ation would require retaining~~if (ref. Appendix II) in double precision as well, thu~ doubling storage requirements.
Should the matrix A be symmetric positive definite, wh.i,ch is never the case with the GIM, then a simp1ification of the Crout algorithm is the Cholesky Algorithm which yields resu1ts the same as the Crout reduction but with only ha~f the number of multiplications as shown in Appendix II.
, 2.4.3 R~duction by Given's R6tations
• This method, used extensively in matri~ e'igen equation solution can be use~ to triangularize a matrix or to reduce it,to a triti~agonal
1
form for which standard algorithms may be applied ,to obtain the eigen-~alue of the system (30). The Given method is based on the concept " of plane rotations otherwise referred to as elementary unitary transformations and are of the following form;
1 • ,,..
r = r.*. = e- jd cose U JJ'
2.39
2.40
, J
i .~
'"
(
----...._-----
,1 26
Its modulus is seen to be unit y: The inverse of the elemen- .
tary unitary transformation·matrix ;s simply its transpose and premulti
plication of a matr;x by U is ana"logous to Ithe subtraction of multiples of
,other rows from the one currently under consideration, in this case the i th
row. The elements are placed with their diagonal coinciding with 'the diagonal
of the matrix to be reduced whilè the, r~~ain;ng two ~leme~ts are plafed
with i and j ~eing the row and column of' the element to be zeroed as
shown below. 4I}~s an example, it is desired to eliminate (3.2) from the , .
matri x A, , ;'~ ft."
x )( x x X
,) " a x x x x ." f[ 1
,,4-.,. , ,·f •
'A = a x x , ~ } ... ~
0 x x a x x
element (3. 2)~
matrix U is constructed as follows,
l 0 a a a
a - e-jacosa -e -jesina a 0 1
U ejBsine = 0 eja~osa a 0 0.41
a a ",,0 0
a a h'" o/,-JQ a
'"'
, . ". ---~ Once again, the complete algorithm is presented i'n Appendix II.· . - ...... _-
o
•
A more efficient method for the reduction of a matrix was put'
forward as early as,1958 by A.S. HousehQlder' (31). Although it.can be
shown (30) that the two method$ are'analytically the same, the number of , '
r ~ 4 ,
.\ /'
-t
\1, 1 .Ii
1
c
... '
~.
••
, 1
27
, 1
operations for a Householder reduction is muc" less as can be seen from
Append~x II and from a programming point of view. much more straight
forward in its apy1ication.
.1 .. '
2.4.4 Householder Transformations
Householder Transformat;o~ , like Given's Reduction and Gauss
'Elimination. is based on the use df orthogonal matrices' (29). For this " scheme an ,elementary matrix, known as a 'Householder Transformation, is
defined to be
U == 1 - 2uuH 2.42 Ji
, -'~ ..... ,~- ...
where 1 = identity matrix . '-..",'"
u = a vector to be defined
H = denotes Hermitian (i.e. uHu = 1)
The matri'x U then has the property of being hermi'tian (UH = U), , H
orthogon~l (U U = 1) and invo1untary (U 2 = 1). For proofs of these . properties, the 'reader is referred ,ta Stewart (29) and Wilkinson (30).
As seen.ear1ier, the reason for using these matrices'is that
they can intraduce ze,:"os in il vector (co1umn of a' matrix) as r.equired.
Therefore, given any vector x, a transfor,mation U may be defined so as to
zer~ a 11 the è 1 ements\ bel ow a chosen one. Thus ; f one w; shEts on 1 y the
first e1ement of a vector to be non zero, the fo11owing e~uat;6n and~proof
of its va1idity can be estab1ished, /
2.42 l'
'-,
. .,
" 1 f J i !
, l' 1 1
\
L
(
.. " ,
~ ..
.
(
~here 0 = constant to be determined -ei = first col~mn 'of identity'matrix 1.
_ - ~ 1'-t 1
Defining the following vector horms,
Il x "2 = f.H: and
- Il x 11 2
- 2 = (II X Il)2
A vector U can be defined such that
u = x + oe,
Define as wel1 a constant n such that
; 1
'/ 1
1
, 1 :
(
(, '
1 !
i
,l l
"'1-... --------
1 1 ___
/ l 28 '1 ._'
/ -------~--./ -- 1 !
, t .. ~ 1\ ,'.. ) ,
/
--------
2.43 '
2~44)
\ ..
, 1
1
)f. • ~
1
~ l 1.;#" 2.45' i ~
f -~.-._.
---==-. ~ t- ,---._--'~- "-1 ......... ---~ - i .t
f '-----=- il'_
.. ~ ~ ,
1 ,.
-- ----, 2.46 ..L-. --, -- ~ ----- i
- Î 1
and rewri~e equation 2.42 as
U 1 _ -1 H = n uu
, . ,
Mu'ltiplying the matrix U, by the vector x yie'lds ..---- !.,
[-• 1 ••
Ux, = x -,",
2.48
• 1
. _. ,
( - .. i
, -.
.' ..... which' ô~ substitution 'Of the vector ~ M:definéd by 2.45, equation 2.48 i~ediately reduces to that of '2. 42't.: .. ," "
This establishes the validity of the definlt'ion and shows - ,-es~entially how ~he.vector is ca'1cu,1ated .. The vector u and c
o can be seen then às.
-----, ---
29
•
2.41
-u. =-X-.'·- i--=2~3.- .... n ._ ... 2.50 _. ___ _ _ .1, 1 ___ _ '
~--~----------~----~~~~~=;~~::::====~~-==--~~----~--~~------------~---'- 1T = oU
1 , 2. 51
-,-. -"---' - The complete algorithm is, ,once agafri'. 'presentecî in Appendix n. " '
The elementary matrices U (these are n x n .matri'x) are rievet explicitl
~ ,-- -:-2-:-52 .--
.. "
H Ua. = a. _ [~ a i] ~
1 1 71' 2.53
/
, .
../ /'
1 •
------....... -----------------------. 30
'- ----:::~. Thus ooly a vectQ,r 'u' and constant ,'a'_need' be stored ins~ead
of a complete matrix.
Tne storage requirements can be minimized using the scneme· ----Cl---- ' , shown belo~ in figure 2~-7..:,"'It can be ~é-n--tl'fa:t-"two_rows Otextra stor-
age wi 11 be requi red. -
;,
.'
Un1 u n2 u .> u n3 nn
, 1T 1 1T2 - .-
~ 1T3~ -..... - - _ ••• 1l , ---.---.. n
1 1
an a22 a33 -----------------... ~ -----a nn·
~ .. i . " ~igure 2.7: Storage scheme for Householder,
o Decompos~tion
-'~
"". --------------- ---------------, ~/ ____________ îhe adva~tage to 'be gained 1s that this scheme is an uncondition-
~ ally stable ,me:thod fo~ solving l inear' systems (29) as there are' no growth
(: \ ,
J'
\ . factors assoCÏ'ated with numerical errors. Thu~f"as antenna problems h.ave .a tendency to~ards 'il1 ;Gonditi~n'ing and car1\theFefore lead toi erroneous
. -ltr.y
reslllts, thfln ":he u~e of a Householder scheme can.be readily justified.
,Practice hàs shown that indeed it does give margina11y better results
than Gauss Elimination for well conditioned problems (30) however, it
tends to ~e much better behaved as the GIM becomes more and more i11 corditioned. Besearchers (14) have shown that this i11·conditioning is not a
"local" phenomenon 'but can spread over a wide range. Just"how one can
p
c
,1 • + e
31
relative stability of a J
Wij~X is one of the subjects in section.
2.5 Ei Conditi on i n J,
f antenna en9ineering problems were more associated with ~/ell
• condition d matrices, there would be little need to discuss sorne of the niceties f numeriéal analysis and routine techniques like Gauss Elimina
tion would be universally applied. But as this ;s not so, theoretical l ,
work in he area o'f matrix inversion and system solution still continues.
Wexler 15) presents the c1assical methods ~hihfmore elaborate te.ch
niques uch as those of Chen (36) and Peters and Wi.1k'Ïnson (37) a11 point o
to the need for a method of matrix solution that in the case of antenna
probl ~Sl would not require inversion of the GIM. " "'
Work by Garbacz (32,35) and more re~en,tJy by ~arring~on (33)
po;'n to a ,method ,through whi,ch solutions may be obtained via the route
·of igenana1ysis. An ~dded featur,e of this route i'nd pointed oU~DYKl in and Mittra (3~),,'and described later. is, that IfnOwing the eigen'-
alues, and eigenvec-tors 1 an insight into the conditiono;ng of the matrix
is also obtained.
1
The eigenvalue prob1em may be stated as follows -{ ,
~, , .- --~f;'~' -...i/- "
. Wheré-- ei
Ae. = À.e: l 1 1
= eige~vector assoçiated with eigenvalue " th' i
(not to be c~nfused,wlth, the; colunm of
the identity matrix 1).
2.54
, 0:' It will now be'shown how ~.é~s.olution to a system of equations '/ ~ ~. .
',can be obtained directly once its~~fgenvalues and" associated eigenvectors
. are knôwn. Thè solution. is obtainè~ by making use of the,p~Qperty that
- the eigenvectors fom an orthogonaf'set (32,33,,3à). ,-This being' the -ease,
,
, t
l ,1
(
, " l - 32
one can expand the v'ectors of the matrix equa.tion 2.32 by defining a set
,of coefficients mi and ni by suitaole inner products: .' ~:;;,
~ ..
mi = < x, e. 1
n· = < b, e. , 1
The expansions are therefore;
n x = r T\ii~i
i=l j'
• n / b = l /
i :f l 1
1
n.e. l 1
>
" > r
r
2.55
2.58
SUbstituting 2.57 and 2.58 into 2.32 and placin~~A' under the
summat i on yie'l ds, ,
n n y Am.e. = r n.e. 2,59 i::: 1 1 1 ;= l 1 1
-,-se- l'
- l- ,,. ~ ;,r Making use of 2.-54. 12.59 can now be wr; tten as ",,, .. '
( , ; '- "
"-:: .. "'" 1 n n t m.À.e. = 1. n .e. 2.60
, 1 l l 1 i=l ' , ;=1
One can now define an inner product of the eigenVéctors and be
cause of the property of orthogonality, the summation may ,be take~};)ut
of the -i nne r product to yi el d,
) .
!. Î
"
11 ~
~I " ~
"
j 1
1 1 /
.1 j"
- , j ,
l " 1
1 . 1
(
/ /
! / .. ,.
1
1
/ ' 1
1
il , • 1
J Ir 1
'.
, . n L j=l
À.m. < e., e. > l 1 J 1
33
n = L
i = 1 2.61 n. <e., e. >
1 J l
The above equation is".defined only when -i=j once again gecause
of the orthogonality principal. There'fiore, when i:::j, the folJowing re
lation holds;
À.m. = n. , 1 ,
Reca 11 ing equation 2.56 and subst,ituting into/2.62 gives,
< b, e. > m. 1 = )..
_f' ' .. At 1 ,
which on substitution into equation 2.57 finally results in 1
n < b, e. > x = r ,
X." e. i = 1 1
"
r Thus it can now be seen that the,' solutig.n
the weigh~ed eigenvectors where the 'weights are given by
< b, e;_ >,
X. 1
/ , )
for the i th eigenvector.
/
, 1 ,
2.62
2.63
2.65
~
\ t
,
l ,
(
"
( "
Tjete weights wou1 d be same as those referred to in the
Har.rj:ny:t.q,1'a'nd 'Garbacz formulatlon of characteristic modes'. Thus, the
fiek:i produced by each eigenvector (an "eigencurrent ll) may be obtained
directly from equati-on 2.64. These"eigencu-rrents produce the "modal"
fields so va1uab1e in scattering problems (35) and useful in antenna
synthesi~ problems (33).
3'4
1t would seem that since the weights vary as l/À., eigencurrents , 1
associated with large eigen~alues have little iri'fluenèe on the result:
Thus the sol ut i on may di spl,ay a converge,nee before a 11 n modes are summed
i'n'equation 2.6;4. Indeéd it will be shown that for simple structures,
not a11 the modes are necessar.y and as will be shown in Chapter V, for
a dipole, a fairly good result ;s obta;ned with only the first mOde. Th~
criterion with whichthe nimpori~nt"omodes are chosen;s the subject of a , . nO,~~l,discovery and ;s presented in Chapter V as wel,l.
,-' Having the eigenvalues available though, ;'t can bè shown that
a condition number X(A) can bj:! obtained and thus an, indiçation of the
relative conditioning of the problem 'at hand. Stewart (29) defines the
condjtion number oT a matrix with respect to inversion as
" , X(A) = Il A 11
2 Il ü_+ ___ 2.66
/ ,
where A+ is the ,r~oore-Penrose ps udo in erse, which for a'" .
sq~are non-s;ng~'ïar matr;x is th~ actual inverse -1. The pseudo inve'JrS,ft -
is well treated in Wilkinson (30),and Peters and Wilkinson (37). And it is
shbwn ;n App!ndix III that equation 2.66 reduces to simply
X(A) ::; Àmax(A) Àmin(A)
or the max to min ratio of the eigenva lues of A.
1
2.67
1
t; l
,1
• " ;:
(
(
(J
•
1 35
/
'CHAPTER III: Application to Wire Arrays
.1 • The computer program which was wri tten for thi s work cons i sts
of a number of subroutines based on the numerical ~lgorithms discu$sed in
the previous chapter. 'The subroutines, bound together by the main pro
gram, allow the user various options in solving the GIM (General Impedance,
Ma~r;x) as well as the choice of exc1tation mode'l (delta gap, pulse or!
magnetic frill). The sections which follow"describe the program construc
tion and organization. , The philos9phy followed in writing the program was, td have a
set of subprograms ~lowing the user to tailor a comprehensive program
l " for his particular problem. For instance, eigenanalysis cé11culation for
the input admittance of a dipole is not necessary. The GIM formulation
follows a scheme first propose~ by Wilton and Butler (25), This sche~e results'in Hallên's EqUation-;'talculatiorfs\avin'g 'convergent qualities
r
which are super;or to those of Pocklington's Equation (22).
3.1 Method of Moments Applied to Pocklington's Integral Equa'tion
Although it has been stated that Hallén 's Eql1ation has superiQr , 1
convergence qualities over Pocklington's Equation, use of the latter i~ , '
preferred because it i5 a more general formulation 9f the antenna prob}em.
The cOlll11only used form of Hallén's Equation as it appears' in Appendix il, 1,-.
assumes a delta gap generator. The Pocklington form,however, allows for
other forms of excitation the user may desire.
Recent work (25) shows that Pocklington's Equation with testing {
functions more complex than dirac d,elta functions achieves convergence. analagous to Ha1lén's Equation with point matching. This,po;nt of view
is .corroborated by Klein and Mittra (34). This may seein a disadvantage
~as it leads to a double integral in the formulation of each GIM element
1 1
~l
(
• 1
as the inner product is
Pocklington's Eq~atibn /
function.
Eine (x) 3.1
. .' Once again, the» unprimed a~dr-prime parameters refer to source
and field points respect.ively. Taking on y the left hand side of (3.1) t:
and using a testing function tm(x) in the pp1ication of the Method of
Moments as described in the previous cha the inner produ'ct yie1ds
becomes
equation
, ,
LHS = . J ti(x) SUBDOMAIN
At .,the
1
1 - j~€ . J RHS =
+ k2] J I(x') G(xtx') dx'
9..
side of equation (3.1)
\ SUBDOMAIN i
Dea1ing ~9W exclusively w)th the left. hand side as (3.2), expansion y:ie1ds .
shown in
t.
- 1
1 LHS = J SUBDOMAIN i
• 1 •
,';/1
+ k~ (' ti(x) l l(x') G(X,X').~X'dx
3.2
3.3
SUBo6MAIN i 9.. 3.4 1 l, \ 1
, - 1 1
1
! : ,6
36
\ \
(
.. 1.1
, 3]
'Le"t. a paramet r fi. now be defined as ... ~.,'.I' "
l'
3.5 \. .
J,I(X')'-~(~,x~') dx'
R.
LHS =
'If
f t.(x)$ Â(~}dx+k2 -,f ti(x)A(x)dx 3.6 1 dx2
SUBDOMA Ni' SUBDOMAIN i
in equation (3.6) may be handled by parts
dA/v\ 1 J 'd ' d LHS = t; x) ~ di t;~x) dx A(x) dx
SUBDOMAIN i SUBDOMAIN i
j' + k2 J ti(~) A(~) dx 3.7
SUBDOMAIN i
• Applying parts again to the first of the two integrals, equa-
(3.7) then becomes, <::> ,
, 1
[ dt.(x) , 1 -.l . A(x)
dx - SUBDOMAIN i
, 1
-, , .
; ( .. • 1
-J d:: t~(x) A(~),dx J + k~ f ti(x) A(x~ dx 3.8 1
SUBOOMAIN ; . SUBOOMAIN i ~-'- - - - -
- "--._- -----
,,'
("
" 1
l'
~-
R'iri ij i"Y'flW"
, ,
38
If now a piecewise linear testing function is ch.psen as shown in figure 2.3 of th'e previo~s chapter,
(
l::.x -t; (x) =
~or eqUiS~aCed riode po;nt~
l ,6x
1 - 6x
~ t;(X) dx2
for
lx - xi 1 6x
::: O'
x < X. 1 '.
,,}' 1
Inserting eqUati~t 3.10
x'·> x. 1
1 1
d311 ' "8 .1 1d" an . lnto <1,' yle s
• ,
3.9
..;.. 3. 10
3.11
, j ,
, 1
~ JI'
'1 , ,1 ~
l ~
{
f '- A(x».- h~ (A(x) A(X'-i\x»}' J
~-~-~ ------/' LHS = t. (x) d~
1 uX
/
-------~
-- --- -X-.+tDC 1
oJ -+: /(2 J l~i(X) A(x) dx
"JI x.-6x! (i"" l
1- '-
3.12
• l , The remai,ning integral in ~.12 may be eva]uated bY.inserting
<fi" the tes t .funct i 01' as de fi ned in equa t i on 3. ~.
.... ,
1
i " x.+6 1
k~ J" tj(x) A(x) dx = 'k'ilxA(x.) , l
Xi -6X
«
. , ..
3; 13
\
1 ' , 1
J :' 1
1
(: ','
Also noting that
t.(x.-llx) = t.(x.+~x) = 0 l l ,1 l
equation 3.12 further simpl ifies to' ;,/
Il / A(x.+~x) 2A(x.)'+ A(x.+l\x)
LHS = ~.....:.1 ____ .,..:1 __ ~1 -- + k2 ôxA( x. )~ llx _< • l
.- , f'
39
3.14
3.15
• 1
The in.tegral on the right hand side as shown in 3.3 is n'ot sim
plified here asH is dependent on the excitation model used. l.t will be , \
seen that the integral'will a,t times have to be evaluated numerically _as
in the case 'where a magnetic fri11 excitation model is used~
Recal1 ing however that the finite di fference approximation of the
second deriyative of a function B~x) at a point x is given by
" . L B(x) = B(x+l\x) 2B(x) + B(x-lIx) dx 2 l\x2
3. 16
/
/ " / ,
Substituting the above approximation directly into equat-ion 3.1 1
and using a dirac del,ta testing ,function; the left hand side immediately
reduces to
1 _,
3. 17
Thus the left hand side as shown in equation (3.15) t which forms
a 1lIM entry, /can be rewritten in the .form shown in (3.17). It" ctm be / • '1
, noted that fn the case of (3.15), app1ying a piecewise 1 inear testing'
function to explicit differentiation corresponds ta the finite dirference 1 .. ~ ~
approximation as used in equation (3.17) above.
, +
" 1
i
f JI , ,
I~C 1 i 1
1 / , /
/
/ ,
40
~ Thus there is achieve~ the programrning ease of point matching wtth the convergence associat'ed with piecewise 1 inear' testing functions
/i~h Pocklington's -Equation.
.'
3.Z The Excitation Vector \,
1 .' ~ ~~ 4. ~,
The right hand side of Pocklington's Equation is used ',*ith", ~ ,1~
an excitation model which in the Method of ~1oments yields a vecto! ~ ...... ~
called the excitation vector and provides a rlght hand side for the , 1
solution of the GIM. Three different types of frequently used excita-I
tion will be considered here .
. __ 3.2.1 The' Delta·GCllrSource-
The 5implest source model is that of the delta gap where
3.18
. ,
This is analogous ta one point integration and"there will onJy 'bé"one ent~y to the excitationvector 'with the remalning elements .. be,ing ......
Il
zero.
- jtxE: 'J t;(X) f.i,nc(x) dx"= - j:: Vô(xP " 3.19
SUBDOMAIN i
The ea se ,of. programmi ng i s offset by the poor convergent qua lit i es . of the above scheme"evén when ~ppli~ed to as simple a problem as a half
~ _"./.-.... wa ve--'a,'pô f~::'''''-'' , ~ .~p~ ~~~~ .. ~ .... '
___ " 1 _____________ -.
j
! '
- '1
1 ..
41
~ 3:2.2 Pulse Excitation .!
" r' ,
An improvement on the delta gap model is the pulse excitation as shown below in figure 3.1.: J~
\
.--__ .... ______ V
AX
SUBDDMAIN I_1
x 1.'
------------ ~_________ SUBDOMAIN. -.- --- - - _. ___ 1
Figure 3.1: Pulse excitation wi~h applicabl~ domains of integration.
Applying- the piecewise 1 inear testing function in the inner /- -
product with this model yields three non-zero integrals over tner, domain of the prob1em fbr each excitation. These are
I lné . , ïnc . ti_l(x) E (x) dx == ~ b.X E
SUBDOMAIN i -1 .
3.20
J inc . - _ , ~t i ( x) E ( x) . dx
S~BDOMAIN ; "
3.21 '
"./' """," ,-
'J . t i+1(x) Einc(X),dX = l f:.~ ~inc .SUBDOMAIN i+1
3.22
• i 1
l
1 i " 1 -
el 1 J j
1.
/
(
/
~- -" r
L
"
'JI," -
r t'
Norma li z i n9 the input voltage ta unit Y 9 ives
v == 2Ax Eine = 1
or -------.~
Denoting the excitat,ion vector as b, it ;5 seen ta be
,
.: ,
b. 2 = a 1-
-- ~ ~j
b. l jw,,; - -·w 1-
b. jtil = _ E:
1 26X
bi +1 = b. l l-I
bi +2 '= a
b ~ a n
>- t
, '
42
\
3.23
~ 3.24
3.25
•
~,. .. .... 't-' J ~/':
, ~;~
l '
4 f j
l ',<
;1' ,
. j 1 ' /
"
••
......
43
3.2.3 The Magnetic Frill
For the case of magnetic frill excitation, no simplification
'!cor~esponding to the pulse or delta gap cases can be achieved. It
......
must also be noted that the integration over the subdomain including the
source point ,must be done fi nely in order to co,rrectly extract the
"peaky" nature of the fri1l . . The fields of the frill are calculated at the node points and
the program used is essentially the one by Tsai (27). Their values are then used in the i,Jltegration scheme wi,th the piecewise 'linear testirîg
, --function. <"Seen J>elow in figure 3.2 vecJors a1'ter performi ng the i nner
lized to one volt excitation.
, \ " . t,
; /
/
\
.;.0
'1. ....-.,. ... ~~"..~ r J
,
is a comparison of the excitation ,
produc~. All the voltag~s ére norma-
-1·0
'\\ -'\ .' , \_._~
-delta --pulse . __ 2- frllr",
• 4~J
Figure 3.2: Comparison of'imaginary parts of excitation vectors r Q ". with ~ = '1
/::'x 1 1
" The 1\1;i gure 3" 2 shows that the pulse exci t: t i,on" ~ode 1 i s a good
appr~ximation to the magnetic f i11 at a fraction of the complexity and
cast of the la\tter. T.hus in ma y cases, using the piecewise linear t'est o \ Gi ,
scheme, use ofl rela,tively simple pu.lse model gives as good a result as the magneti c frill r '
i /
-/
, ~ "
1
(
/
, "
c'
.. -'\ ~"r
7 -
~ • é).. - '::~
. " 3.3 Integration Scheme
44 -l
. '
.. ",
; The integrations are performed by a Gauss Quadrature 'scheme (39) "
whicQ essentia11y approximates the integral as a,finite sum
:, ,J ~
b, + 1
-J f (x ) -~ d~ = ~ f ~ ( t) dt
a " -1
'. where -F(t)'= ; (b2a t + ~;a )
and the integr~l is approximatQf by
Ob
J f(X)'dX zl b2a 'I a \
,
W.r:. 1 1
,'-
3.26
3.27
3-.28
Wi and ti a-r-e the' Gauss- weighJ;s and the node points respectivelyo
for the, integratioQ. ~anQ are ~vai1able i.n most texts dea,ling with numerical, . in'tegration (39).
3.4 program Description ,
:'J' •
~ As mentioned previouslYt the program cons;sts of a ma,;n progra!Û whi~h i~puts data, outputs resu1ts and ca11s the vari9us subprograms as
, requi red or as requested by the user. The De~era 1 program st ucture mây be seen in figure 3.3 below~ 'A,short description of each sub outine
then follows immedlately. - A m~6re' detaileâ description o.f th pfog;am to~ether with examp1es of input data are to be found in -App ndHclV.
,>
,
o ' '
, , , ,
J
1
\. (
'"
..
•
1 1
,1
..
•
~ -,,~ )
, ~' " " .
.j'
. ~ 45 , .
" ,
1
0 GIM & EXCITATION: , , GQUADl ,
FUNC1 0 1 MAGFR
\ ~ 1
-, -'j a
EIGENANALY5 15: , ,r , ,
DI RECT SOLUTION: MODES 1
.1 - MAIN PROGRAM -'EIGSLN - -, .. CSOLVE - " CHLDR 1 .
1 . , "
1 ~ ... "- -. . ( 1 l' ;.,
c , -
ANTENNA PATTERN: (" PATTRN , EFLD
. ,'-
. ~
Figure 3.3: The program may be tailored to suit the requirements of the user. :
GQUA~, f:
,
FUNC 1:'
MAGFR : \
1 . , , ,..Lot- '
The main subroUtines are as follows:
This is the main integr~tion program wh~ch for the GIM generat i on a 11 ows th-e pr:ogrammer up to fi ve poi nt' Gauss Quadrature integration ;'
Called by GQUA01, is -only used when the,GIM is be/ng genera
t~d ~nd 1s the 'program needed to evaluate the kernel ,o~ ~ Pocklingtonls Integral Equation~with a ~iecewise lJnear
'.' "basis functi.on as well as employing the finite difference ,.' .... 'scheme tô simul ate piecewi se l inear testing .
. ,"
, Should a more sophisticated excitation be required, this . ' program will calculate an excitation field du~ to a fri1l of magnetic curren,t (after Tsa; (27)) placed at any. point along the'aritenna. The 1nner radius of the frill . . is taken to be that of the antenna itself whil~ the c~oice
-/ ... / ·r ,
'OÇ'
\ "
~ --- ,. ~ ,<
,1
\
jJ 1
, -l ,
1 . i ,
\~ " •
~. ---------....,.., -...-, --.,....-;-. ".~, .~: ."--:-:--:------~-----_. "'~ . -":. .. .... 1 . --_._~~-+
(
CSOLVE:
CHLDR:
MODES:
EIGSLN:
()
'c'
~ .. ~ ... --~----~------ -----------
46
'of outer radius is any multiple of it greater than onè and
is spec;fied by the user.
This program is used to solve the matrix equation directly to obta ii(.t;he .p~rent dens it; es al ong the wi ris. The a 1-
-~.. \~
gorithm used is based' on tbe Househo~der scheme out1ined in Chapter II. The program to decompose the matrix makes use of the program CHLDR.
Ca 11 ed by CSO program 'decomposes a g~~J~ra lrm x ~ atrix into upper triangular form using t~e'H6usehol r
Transformation 'Scheme of Unitary Ref1ector Matrices.
Should an e;gen solution be preferroed. MODES ~ày be used instead of CSOLVE. This progra~~finds the-eigenvalues and ei~nvectors of the General IImpedance Matrix. Calculation of the matrix conditioning is also carried out. The
1
present version makes use of the IMSL routine EIGCC (ref. 1
Appendix V) to calculate the eigenfunctions. As this " program ;s ;nef{;cient, it is s gqested that an eigen- ,-
.... -! 1
package may be written by making use of the routine CHLDR tà' decompose the matrix. The eigenvalueS ofl,-an antenna problem . .
-, tend to hav,e great variation and only ~hose with the smallest absolute value which are associated with the prin~pal
, ,
mOQes are.of interest. A reduction via the Householdér technique ta tridiagona1 form iSI easily accomplished and on1y a subprogram for the QR algbr.ithm (29,30) wpuld then
1 "
need to be written. 1 i
, 1 "',
Once.th~ eigenyalues and'eigênfunctions are known, the charactJristic currents'themse1ves or their combinations mJy ~ obtained as wel1 is 'the radiation patterns for eacJ~,éase. Each eigenfunction i5 weighted according ;'1 " .' to thé, 'procedure out 1 i ned ~ in Chapter IL
1 .--
> ' , "
1
, ,1 ,
,
, î 1 1 1 "
1
~1
(
( 1
4
PATTRN:
(
All 'the far field patterns are gener~ted from a current vector supplied to it by EIGSLN or CSOLVE as well as
".~
47
geometrical data. The pattern subprogram does the bookkeeping of the vector quantities in th~ far field radiation
EFLD:
~
pattern equation. The program makes use of EFLD ta ca1-culate the field at a~point-in space.
1
The field at a point (e,~) is calculated by this -program. The routine calculates any specified E or H plane pattern or bath .
. After coding 'the program, the numerical modelling' process is now ready for application.
aside to be r~turned to in Chapter V.
At this point, however, it is set ~
In the next chaPter a description is given of the anechoic facility and what contributions_ were made to it in the course of this work. This anechoic facility was th en used ta make the measurements a150 presented' in Chapter V.
1
~
\
v
1 l /
! 1 1
1 f
1 1
!
/ 1
. ~ , 1 1 1 1 1 1
\ 1 1
l , , 1
1
•
, , 1
, 1
1
~ ,
! 1 , 1
1 1
1
; 1 1
" 1 1
; 1
1 /-
1
i 1 •
"
Cf ,
48,
" , CHAPTER IV. THE EXPERIMENTAL FACILITY
Prior to th advent of the now prevalent numerical techniques ,
for antenna analysis, virtually the on1y tools avai1able to the antenna engine~r were either pen or anechoic ranges (56). The numerical techniques however, bring ith them thei),own problems as the designer is
now faced with the pr Ib1em of having c~nfidence in the calculated values of antenna currents, nfar and far field patterns as well as input impedance. Should numer;crl~onvergence occur, th en one can be reasonahly assured that this \esurt ;s numerically correct. But questions arise about the choice of th~ numerica1 modelling procedUre used and the numerical .. algorithms en1ploY~d for their solution.
It shouJ~~be~ noted that nat on1y the numerical model can be ~ , fI"
open to questions. As1can be se en from Chapter II, the antenna prob-lem's analytical mode1 1 may a1so be questioned as to its va1idity.
Thus the solution o~ a~ antenna pr~blem is subject to the ~pproximations of two modelling proce~ures, the analytical modelling of the physical \ p~oblem an~ the numeri ~l mod~11in9 of·the analysis. The only recourse i s to revert to the an choi c chambe'r, and i ts own set of mode 11 i n9 groblems,ta see whether wh t is calculated agrees with measured results.
The fol1owin is a summary description of the measurement facility and sorne of tne lmpr9vements made, ta make possible the measure~ ments presented in Chapter V,giving an experiment~l verification of the analytical and numerical modelling procedure described in Chapters II
, '
and II I. - ,
The anèchoic'room operating in the UHF region in the Department of Electrical Engineering at Mé:Gill University was buHt in 1967 and is documented in Kubina {l}. It has been used extensively ta verify numeric-
'" ' al modelling procedures used for the study of complex antenna configura-tions ranging from basic geometries (e.g. spheres and monopoles) to
!
!'
49
, stru.ctures such as aircraft (1,2,3.4,5) .. located on the roof of a buil-ding, it was desig~ed as a temp07ary structure and ,as can be seen from' \ ~igures 4.1 and 4.2, weather creates its own completely different set of problems.
As th i s work co'rrmenced however, on'e of the goa 1 s wa s ta re-l
habilitate the strûcture and ta ~~tend the versati1ity and capabilities of this UHF chamber. Since orîginally the facility was conceived as a tempç'raryone, it was necessary ta first rearrange the wiring and ,jnstrumentation. A sche~tic of the measurement system can be seen in figure 4.3. The' installation, of permanent cables' and junction boxes for the chamber's centrally located turntables as well as the fabrication of a'position control panel (seen in figure'4.4) for the turntable was required. A polar pl~tter was also acquired and incorporated into the
measurement _séheme. \ ' 1 . .
\
Figur.e 4.1: UHF anechoic chamber at McGill University in. the bright sun.of summer. Access door to the chamber can be seén open.
, '
(
(
50
~ ~ 1
-... -.-'---..;,-~ -
Figore' 4.2: UHF chamber and the Montreal winter. Access d9or, closed.
probe test rotary joint
omphtude & phase
14----------i SO,UrCé ~
r measuring J : equipment
" 1 . "----r-~--' r--
polar piotter
1 1
1 1 ___ .:.J
1 1 , l-i JJprocessor ,
CON TROLROOM
----------1 OC MOTOR &. SELSYNS
JeT. BOX 1 CHAMBER 1 L---------JCT. BOX 1
POSITION CONTROL
, 'Figure ~ Schematic of the UHF measurement 'faci11ty. /" ._. , 1
1
200 MHz --"3 JO GHz 1 __ _
" ,i "
,.;
~ J ~ , t ! 1 L_ ---
1 1
\
'1
·1 ,1
1
(
(
/
1
ri.··· Il, - ,- ~,""
51
, "
Figure 4.4: Instrumentation showing network analyzer, vector voltmeter and turnstable posftion'control panel.
An important improvement was the design and construction of a prpbe arriage and boom. Due ta the cy1indrica1 structure of the chamber, having he antenna under test ;n the centre of th~ chamber ànd acting as tran mitter, multipath specular ref1ectians are minimized as al1 radiation-ir o·nnalli-inciden-t o~ the walls·,· The requirement was ta be able to probe warious portions of the field in .three cf;mensiQns for differènt ~olarizat{ons as well ~s ta allow rotation of the model for simple az;mutha(pl'ots. The aluminum and. steel carriage assembly is shown in par-
/
tial completion in figure 4.5. This structure may be "hidden" with the use of microwave absorber. The boom supporting the probe antenna needed to Me rigid but obviously could nct be made of metal. Phenolic tubing was tried but not being o'f sufficient stiffness, was replaced by a bamboo rad, Furthermore, ta minimize interaction, the boom itself was coated
"
-.. ' .,'
•• kt
JI
" .. "
. ,
" ,
, ' 1 \
! ~
, ,
1 ,
1
t
(
\
"
Figure 4.5:
Figure 4.6:
, f'~,
PrO,be carriage assembly. "
Schematic showing the five degrees of freedom . \
of the probe carriage and boom.
52
..
1
(
1 (
53
with a lossy carbon-based compound. The five degrees ~ freedom of
the ,assembly as shown in figure 4.6, allow the experimenter to measure a cross sectional area of ~TIr2 as seen in figures 4.7 and 4.8. If the
model is th en rotated, this corresponds to a rotation of the measurement plane. Thus measurements can be made for 2n steradians of solid
."
.. ' ,
an~le. The typ~ of probe used to d~te and adequate with respect to l directivity and ,polarizationl'discriniination for E fields, is ~he half ! wave dipo]e with a balun tuned to the resonant frequency. ln fact t~--'arrangeme~t Ms' been s~own ov~r th~~ate / {fOr far fleld a~~measurements (6,7). However, as one j of the-~the carriage and boom arrangement is to allow probing . J
l of near fields, a further project outside the scope of the present work, !
would be the development of probes for the measurement of near E and H fields.
The mechanical work described above constitutes the first st~p of improvements designed to make the UHF facility more productive
_~. • 1
and permane~t. The second stage consists of combining the measurement and display equipment with a Motorola MC6800 based microprocessor system (ref. figure 4.3) currently associated with a 12 mm anechoic room mea- ) surement facility in the same labor'atory. This wiil'enable bulk storage
, " of data enabling transfer to thé University's éomputer centre for the purpose of producing fi~ld maps as we11 as pattern integration to deter-
, 1 1 if
mine antenna gain and eventua11y to perform near-far field transformations.
J
/ !
J , l r
{
~ ,i
1 f .1 , !
i l
The physical operation ~f th;'carriage, now manual, is désigned for eventual automation. This will enable, with _the use of the above mentioned microprocessor based data acquisition system and process contr011er, the scanning of the lield, allowing E-and H plane .measurements
l l ~' .'
f
'IIi thout reori ent i n9 the mod~' on the mount i n9 ~o 1 "m~ -...
1
1 1
.' "
.', 1. '
2·44m
)·05m'
~ 1
WEATHER
COVE~AGE AREA Op...PROBE---
MODEl MOUNTING COlUMN
COVER ---~
, TURNTABlE DRIVE & SElSYNS
Figure- 4.7: Secti,on throu~h the UHF anechoic chamber. The shaded area depicts the principal probe' coverage area.\
.------ -- ---------~-
- 54
"
i , 1 ! ,
)
h 1 J.
: '
~ "
~.1 ' i
( ...
'1 (-'c
oo. ___ ~_1 _0
55
~ Area covere
~ Effective ârea of coverage due to'model rotation.' _________
,----::>UPPORT COLUMN
PRINCIPAL MEASUREMEN PLANE ---------<il~
ABSORBER PANEL
TURNTABLE
RAISEO FLOOR ,
7·32m
....----+------7·32m ----------t '---.nOORWAY &.
PROBE CARRIAGE LOCATION
, Figure 4.8: UHF anechoic chamber - plan view.
/
,1 Il
,
,. .
(
o
;'1
56
<:l
eH"PTER JJ-~ RÉSUL TS
T~e previous three chapters describ~ the analytical, numerical and experimental basis for the study of wire antenna arrays. This chapter presents the results of their application. The results are those of measure-
1
ments made in the anechoic chamber and calculations for a number of yagi array configurations of up to four elements. A systematic eigenanqlysis was
1
also carried out,from ~hich a novel relationship between numerica1 mode1-ling and the real world app'ears to emerge and is presented here.
, , In order to show the validity of the numerical program, ca1cu1a-
tions are first presented -fora set of simple dipoles and their curren,t distributions'. These are compared with the measured values availab.1e
~ ,
in the literature (40). As described previous1y, by using a particular treatment of the kernel. the calculations for a two-e1ement array can be made to converge when using Pocklington's Equation even in those cases
/
which in prev;ous work (26) proved to be non-convergent. / The results sUbstantiating this convergence are presented herê.
\ -
5.1 l'Comparisons with P~b{i shed Res'u'Üs
, 5. 1'. 1 T~e Oi po le
The basic initial problem fo~ a wire antenna program is the dipole and this is presented in figures 5.1 through 5.3. The values calcu1ated by the program written for. this work are in good agreement with the;' accepted ê\nalytic;al and experimenta1 values. - The input admittance cijry.efor a dipo1e as a function of the length is a1so ca1cUlatFd. Once, aga\~'n, excellent agreement between, available established curvesfand the one pr'-sented in figure 5.4 can be seen. 1
1
"
(
(
liA
n,
\ .. ./
0·10
0.05 •
o 2 4 10 1
molV
Figure ~.l! Current d;~tribut;on,on a half wave dipole. o Calculated "_, Measured MACK (40)
.~ ·z \
-\
liA
o molV
, f
•
. ." o 1 z
57
,
Figure s.2i Current distribution 6n a three quarter wave dipole. 1 MACK (40) .. ; .0 Calculated - Measured
~---
1 1
1
, "
r ..
( , .
I~
10'
• j
-~
, .........
"
•
-20 -10
liA
o mo/V
... '
. .
. \ ~o
Figure 5.3: 'Currènt ~istribution on a full wave dipole. . 0 Calcul ated _ Measured
/' l
1 /1
1 1
l , 1
.~ --, , ,,"
01$
-G
--- 8 .\
\
.--, 1
" ... •
..
58
, . " ,,-
,r -
ln
figur~ ~.4: I~put~admittance.as.a·funct;on Of~ipole length~
..
i !
:
~ / \
r .. '
L' \
...
'.'
, (:~ \ .
l '-
1 -
1
\ 59
,
5.1. 2 for a Two Element Va
The two element yagi is not a configuration in widesprea'd -prac
tical use but it is a useful staJ'ting point in the study of antenna model
l~ng pr~bllèms. In the case of the configuration sh"o~n i,~ fig'ur~ 5.5,'
ft has been shown (26) that for a=O.oon, 1=O.5À and d=O.15À, that a~ con
vergent r.esul't will not De obtained by using Pock,lington's Integral Equa
tion. ,T,hê~efore, .this is a val~test of t~e convergence qualities of the
I\rograffi-'as outline~ in detail in Chapter III.
. As can be seen from figures 5.5 and 5.7, convergence is attained
with as fewas eighteen unknowns on the driven element. The excitation model
used was a simple pulse normalized to one volt. À..1-hese figures were ca1su!. '
lated with the driven and parasitic elements 'having the same number of cur
rent uAknowns. However, convergence can still be obtained even when using
fewer segments on the parasitic element. To show this,'calculations were
made for a similar two element array with a=O.007022~" R.=O.5À and d=O.25À .
"
,~, Figure 5.5: l' ,
.:>
1
\ '
, 0
A two e lement yagi, ,~rray.
r , , Ji
•
...
n \;,
\' .1
t ., ,\ .
f, 1
1 .1 , 1
'1
,
r-
(
n .18 .. • DRIVEN
10 • PARASITE
Il 1 mo./V
5
o 0125
I/~
, , .Figu.re 5:6; Curnents on a two element yaqi with 1"'0.5.>.,
a=O.'oon, d=O.15À and where n is the rtumber of segments OA each element.
10 -
III 01 cerir. of drlving
gnlenna. 5
o-0
\
..
005
10,51/(nl
p
'f
().1 ...... 1"'"
:JI' - " ~/i 1.
Figure 5.7: Convergenc,e of the driving point .current of a two element yagi with 1=O.5À, a=O.OOl À, d=O.15'>' as a function of the number oJ antenna segments n.
\
i'
,
!
60
1
1
k
" I
r' i .
(
\ l
... '"
(,
j
61
This particular configuration was chosen since the results for the input
admittance can be compared to published values (41) which were obtained
by the more comp1icated Bubanov-Galerkin Methoç. The resu1ts can be seen
/; n- ~igure 5.8 where ccnvergence ; s shawn as a functi on of both the number
of unknowns on the dr,i ven as we 11 as the 'paras it;c el ements. It can be
seen that convergence is attained with fewer e1ements on the parasiti,e ~
that Wt' the driver. This is important when th-e economics of numerical
modelltng need to be considered. \
o
5·0
1
IY. 1 , ln 45 -r'
ma/V ORrVEN ELEMENT
0 ~ segments· , . - 8 "
• 10 .,
~ il
{,·O 0·1 0·125 0·167, 025
1/( no. of porasite segments 1 \
Fi ure 5.8: Conver en ce of a two element
the number of segments on the parasite ant~nna.
a = O.007022)}, t = O.5À, d = 0.25>". . ,
• l'il"-
//
.- ----~---~~-
- , ., J;
J 1
1 r .. ,
1 .
(
. ' ~ i
l
"
62
," .§.l.h3':,.,;r Current Distributions of a Two Element Vag;
1 • In this section, the ca1culated current a;stribution for ,sey-
eral two e1ement arrays are presented and when possible, compared with
e~perimental1y determined values of Ma~k (40). Four different configura
tions are ca1culated as shown in table 5.1~
Figure Antenna Antenna Calculated Currents No. Lengths Sepa~tion Dri y'en Parasite
i{À) d(X) Re lm Re lm
5.9 0.5 O. 125 1 -~ "'1'
0.250 1·:..·' V . O. 500
''-''jJ: 1
5.10 0.5 0.125 1 ,
0.250 / Q.500 1
5.11 0.5 O. 125 ;; 0.250
, 1
0.500 1 .
5.12 0.5 0.125 , / 0.250 1 0.500 f 1 , .
5.13 1 • 1.0 0.125 1 1 ..... --
0.250 { 1 0.500 1 1
, 5.14 1.0 0.125 1 O.~50 /
\ 0.500 /0'
- 5.15 1.0 0.125 1 "
0.250 \
1 0.500 1
Table 5.1: Current calcu1ations on two Element Vagi Arrays
\ A11 Element Rad;i = 0.007022À \
1
1
(
( (
"li.
o 5
maN
LEGENO:
d caL Oî25 0
0250 D
G500 A
10
fi3
meas.ll.Ol • • ••
Figure 5.9: Currents on a two element yagi. Refer to Table 5.1. and
) -
p.
'"
'ln. ,J, , .)
025
020
015
. -".
0-10
005
____ J-----...... ~--''"----~ 0,0 0/
-10
Figure 5.10:
1
Currents on a two elemen~yag;. Refer to and '.Fi gure 5.9 for 1 egenr ' '
1 1
Table 5.1
/
> •
"
-,
1
(
. "
l.-
,
Il). 64
().2S 1
1 j
1
\
020 1
'\ 1
0.15
010 \
0.05 1
o
malV
Figure 5.11: Currents on a two el and Figure 5.9 for l
agi. Refer to Table 5.1
I/A
-~ 0.15
010 ..
0.05 ..
A 0
5
InG/V
/
\
/, ,1
/ 1
,1 J ,
/. i; !~,
,
;
l ,1
Figure 5.12: Currents. or a two element iagi.f, Refer to Table .1 and Fig ej 5.9 for legend. l
)} 1
1
, 1
, 1
l'
1 1
,
i, , , :
",,'
'f: , 1/ J
1
1 -1
, '.
1 (
_ ... --_ ..... _""'._~--~-- -
\ 1
1
_ J i
. '
~ ~
...,.. *
,.-..
-2·0
Figure
... ,TM __ • __ ' ____ _
\ \
\
lIÀ
0.5
0·4
Im{1} 03 R~
10·2
~.
-1·5 - ~·o -0·5 , 0 0.5 1·0
ma/V Current distribution on~a two element yagi. Refer to Table 5.1 and Figure 5.9, for lege~. J,
'J, •
• ~~-~-~-------J.--_ ....... t-1_"
1.5
~
..-.
lu C'\
<'lJ1
•
1
(
./
04
03
02
0·1
o o 01 02 04
- -- - -- --- - ----~ ma/y
, ~-
Figure 5.14: Currents on a two element yagi. Refer to Table 5.1 and Figure 5.9 for legend.
0·3 0-2 01
1/),
o moIV
" ..... ~. ---
o·, 0-2 OJ
Fi gure 5. 1 5: Currents on a two el ement yagi. Refer to Table 5. l --- and'figure5.9forlegend. kw' "
"1 \
66
,
1 • -l ~
).;'" j
1
1 ! ~
67 1
excitation used,when the elements are bO~h a half wavelength, , ·was a pul se normal ized to a gap voltage of one volt an the case where
--~~
f,
R,=1.OÀ. the excitation was a magnetic fri". Use of t e magnetic fril1 al-
allowed to be~_us'èd thus justifying the dded expense of ~ -' ,
a more co plicated excitation model. When both element were half wave-, l engths , he number of unknowns used was seventeen on t e driven el ement ~
on the p.arasit;c while with'the use of a frill excitation when
of one wavelength, t~become .twentY-fi~e and eleven
"
. 5.2 Measured and Calculated Patterns
'\ . It has been stressed that one objecti))e( of this work ;s to estab-
1 ;sh an i.ntegrated approach for wire antenna array study combining carcu
lations and measure]llents. In--this section, measurements are presented
on a variety of yagi antenna arrays with their accompanying calculated
results. All the measurements were carried out in the facility described
~n Chapt1r IV.
5.2.1 E~periment Description 1
~he nature of this work demanded that a number of, wire configu,a
tions be measured. This called for a flexible physical modelling system'
allowing for an easy,change of one or more of physical pa'rameters of the
antenna configuration being studied. A systematic set of measurements
was carried out for a set of two and three element arrays.
1 It shoyld be noted that the small size of the chamber with re
spéct;to frequency will not allow accurate far fielud determ;nation for
arrays of more than a few e1e~ents. The method by which the arrays are
modelled is shawn in figure 5.16 by making use of an imaging ground plane. l'
The actual elem~nts used and their mounting in the anechoic· chamber are 1
shown in figures 5.17 and 5.18.
,-
- - "'\
i' ;
/ l,
(
"
"
(
J
ground Plane
mounting 'disG
__________ driven -anœM~"'---L.
paraSile\
o 1
NI--OSM(SMAJ flanged connector
ground plane
column
1 figure,S.16: Ph~sical model used fdr radiatjon pattern
measurement of y'agi an:tenna arrays.
'.
68
- --'-l~~-
( / l
(':
J
Figure 5.17: Array elements. The ,driven elemert can be seen attached to the mounting disc while two passiveelements stand on their flanged bases.
Fi9uré 5.18: A three e1ement array set up on the ~round p,lane in the anecho; c chambe r .
69
, -
i
j
(
'.
i, ~
i ~ ft
t J" t' ," ( ~:.; .
\ C' 1
~ i~
\l
1 1
.... . , -----------. ----- ' ------------~
----------------- . 70
-------------- . The use of a ground plane has the advantage-tnat-fee~bles and
- the antenna mounting co1umn can b'ë "hidden ll from the "free space" ~ _____________ -
chamber. It does however impose l imitat-;ons. by its finite s ize. on the
number of elements that can accurate1y be measured. But the ground~p1ane ..
a1so allows, as seen ;n figures 5.16 through 5.18, easy feeding ollenergy
ta the driven element as this can now be, done coaxia1ly, obv;ating the need for a balun. ~,"
As shawn, the passive elements consist of rigid brass rods,at
tached to base plates. These can be fabricated ea,sily and an array mày
be simu1ated by simply taping these elements to the ground plane at the,
des;red locations using concfucting aluminum tape. This procedure can be
seen quite cl early in figure 5.18.
. The driver was made 0.08 m long giving an electric length of
O.25À at 937.5 MHz. The w;re stock used to fabricate a11 e1ements was
pf 0.0625" (_ 1.5 nm) diameter giving an:..~Jectrica1 radius for calculation
purposes of 0.00248À. At the frequency mentioned, the probe could o~ly
be located about "eight wavelengt~s'from the centre of the chamber and
the, infinite ground plane had a diameter of just under four wavelengths, --' '.
Even sa, excellent agreement, between measurements and' calcu1ations was ... , - , ,..
obtained, as will be seen, for tWD and three element arrays. ).,/'
A ha lf wa ve di po le with a ba 1 un tun4~~ to the resonant frequency
was used as a measuring antenna and despite its directivity and being ~ , f'
almost ;n the near field of the arrays, it provided good results.
5.2.2' Re'sultS of Vagi Measurements and Calculations \
1
A set of systematic measurements and calculations are presented
for two and three element yagi arrays. Measurement of four element'arrays was also attempted and sorne results -are included. Measurements on five
element arrays praved unsucces5ful because the addecJ. element increased
j,
the di rectivi ty of the array such that its equiva lent aperture' became 50
hlarge that the measuring antenna was well in the near 'field making d'i rect ... _~~!.eld measurement impo'ss;ble. The assumption that the 1.2 m diameter -
ground plane (- 4À) is of inf;nite extent alsQ contributed ta the diJcrep-. ~ '-l, l
ancies betw~en calculated and measured results as the number of elem~nts increased.
----------- ---
., ,
----
, ,
! ~, f ~ l , f-
I \
l
, -,
~-----
71
The presentation of the two element array results begins with
a table (5.2) of the cases measured .and ca.lculated for on p~ge 73, whi1e;-~ ..
_ the three element array presentation begins in the same manner on page 80.
For the cases wi th the two el ement arrays. agreement between .t',... ..
meas'urèments and cal-culations is within one'dB. in.dicating that the two
models, numerical anéJ experimenta,l, are here mutua1>ly supporting one
another. The same good agreement is also evident for the measurements and
éalculations presented for the three elerrrent arll"ays.
Although these systematic measurements were made to ',verify the
operation of the chamber as desc'ribed in Chapter IV, parametric curves , . , may be obtained from the accompanying calculations. For these a rrays
th en , curves of input admittance as a function of_the-ir various geometries -~ ,
are presented. Prior ta thi.s_though-;.a~ook at the four element results
wou~d be useful as "re';,;-h~re that a "breakdown" of the model; ing proce-_:..". ------- 1
dures in their mutually supportive role occurs. Nejther is "wrong"; it' ."
is simply the case that the assumptions used for both ryJodels are either
inapplicable or are no longer the same. Thus the model being calculated
is not the same 'as that being measured. Added to this is that the probe
is almost in the near field, if such a "border" exists. and thus the
results as seen in figure 5.25 are obtained in the presence of near
field components not significant in the far field radiation pattern.
The ,instrumentation of the anechoic chamber howeyer allows the simultan'
eous measurement of amplitude as well a's phase thus allowing near field
far field transformation techniques to be considered for future work in
the chamber but autside the scope of the wo'rk at hand. However, it can
be sa id that the probe carr; age des i gn out 1; ned in Chapter IV, l ends i tse lf
very wel1 ta the cyl indrical and spherical scann'ing pr6cedures required \
for the near field far field transformation (48, 49, 50).
The measurements themselves are_~rror in their own dght-becaus-e-of- model misalignment and imperfect flatness of the ground plane
dise. Due to the nature of the experiméntal Iset up. the actual H plane
could not be measured as can be seen from figure 5.l9 below. This has no
effect on a non-directive antenna, but poses a growing problem as direc
tivity is increased. Thus the measut:'ements themselves in -the cases
!......f -
(
l
'>", l \
72 ., •
/,
1
array bèing measured~
'j ~-.. - ~-------- L
~~--_ ....... ....:.._-- actuol H-plane
Fig. 5.19: Schematic of,error introduced by probe orientation for
H-plane radiation pattern measurements.
of the two and three el,ement arrays are thought to be onlyw; thin one-' . . dB of the actual values for all the reasons mentioned above. This ac-
curacy is achieved only with great care and is al1 that can be expec-• 1
ted of this,particular measurement facility at this time.
\-
The set of systematic calculations allowed the construction of
parametric graphs for the input admittance of the various arrays and they , , <
are presented onrthe fo-llowing pages in figures 5.26 thru 5.28., The be- , " 1 ~
haviour of these cur~es becomes rather int~iguing when considering those
for/ the three element array in filgure 5.28. It is seen how in effect a "timily" of curves have been drawn al10wing for a crude-interpretation-----_j
" ,,'
,f-I
1
t 'predict, values of input admittance for spacings not calculated for. ',I~--~ - l i uch as dl = O.1350À and d2 = 0.375>'. These curves suggest that t
the ~pplication of "synoptic" techniques (51) may be' us.eful, not only in pattern prediction, but as importantly"-'fur input admittance as well •
''l:I!J~ r
However, this is out,side the scope of the present work.
______ .~ 1-1--[ _0 '_"-_'
t.
','
• :- . 1 t 1 _
'" !
! ! -!'
·c
o ,
, ,.
o • . . Il • ' .. " ., .
"
l'
SI ~
... '~ ..... J .. ,
0'0" . ~ \
"' ~
~r. ;.
r,"
0,
. • • ~ a ' ,
' "
"
\
1 • 1
. ,
. , ),
" ~r~
.'
"( 1
. \
• J
d
Figure 5.20: Geometry and definition of parameters fo}? Tabl 1
and Figures- 22a through 221.
, . ~
o •
figure No. ~1 p.) R.2
().) d( X) . 0
'5.21 'a 0.5 0.5 0.125
b - 0.250 , .. ,
c 0.375.
~ 0.500 . 1 e 0.625 ,
1 .;.. fI', - 0.750'
~.21 9 1 0.5 0.75 '. 0.125
j..
.:.' h ~
.j.. 0.250 J.-
i .
0.375 " , 1
0' j . 0.500 , t
k .. -r~'" 0.625
R. '0.750 .,
~ Table 5.2: Two el ement yag i array cases measured
. '" , . and ca1cu1ated . .. ,
f' l't.
rF , j;. \ ,
73
5.2
1
\
\
j
! \ ,
t 1 §
J. • Î i,
. r
1
ri . J
1
'\
-------,~_ .. ----
.---" ,(!
'"
"'.
0 0
(dB)
(,
" .'
-, w10 --
T:'_
,',
'\ '0
_ ~ ..... ~ ........ "fon ....... ~--...,.. .. -~"""'" ..... -l"~. -_ -~ . ..,... ·"'Il ..... ,,~~...,. ..... ~ ........ ;t, .... - ..... , ... , ... ....,...-._
~
, ~.
"1;1
!
, '. ,~ .. ,\ ..... ,
" , , . . ~ .. " -.. ., ..... :~
" "
.. \
,1 . / ,. ~,
"\'
, ' .. "
.... ,/
,
,- '
" -:;
/ - Measured
• Calculated
, // ,; ,
. ,
\>' j . . ~ . ,-.
"\ ~
\ , -
1 1
1 1
1 '[
1 "
/
..
-' rJ> --
0 ... ---
(dB)
-10
'- j
, .
')
Fi'gure 5.21a:-Two element array. Figure 5.21b: Two element array. '1=O.5À"2=0.5À,d=O.25À. ,- 11.=0. 5À, 12=0. 5À,d=0 .125L ~
v-
:tP'1.". sr' u1 • ....,. ........ <. .... ,.. .. ~ ...... ____ .--~~ ... ~~ _ ...
g fer_ M • -~_.~ .. ..... -":'\~ ..... ~,. _6"nnt· ... -,.L...a:uJIII!iLtil ... Ll ....... e.t .. b ... & .. ~ ... ~ .. ~~
.-
"
,"
. ,,-.--\
~
" "-J +:>0
~~,,: .. ,. ~ ...,
~i i lb: _"" .... t-n-_~:.""_~ ..... ~ ... ~/_--
,-.,
~
(/)
'- o
(dB) L
-10
~l 1
~.:;'
~ , '"
,
l l
" 1
'" 1
--.-.-,...",~_ .... ~~,,~"""""""'" ~r .... ~~~'t:>~~)-'""-~ç.~~"""f?'F~~ii"~%-"'''''Ç'~ ..... ,';1" ~~
1
"
.,4, ~ • -'1" "'
::':. l 1
~ \1
"._"
"
-Measured
• Calculated
.----
!
.. """- ---- .
o
(dB)
-10
.(/)
• •
l ,
." .
"
,
,:
... -.- ,
'. .. -. .;" . . ;
/
'\.- /.' - .\
! \
\
, •• ~ •• 6 , _ t : \
\ ... \. . ' , .. ... . ~ ._6 .-_ .. ~~ ... - ... . "'1.' ....... , .... .
, . -. '; . \
• " ~ 4 I: l 1 , 'j . " l
'. l' /,' ~-. • J."!'
'. /,' , / 1
, / / , ' ,
" /,'
Fi~ure 5.21c: Two element ayray. . 11:0.5À~12=O.5À,d=~.375À.
.. Figure 5.21d: Two element array . 11=O.5À,1 2=O.5À,d=O.5À.
, '{ ~
..... ..te ;. " ."t'j'''; = ... : '1 .e" .... " d es ..,_~ •
"
" (JI
-) -~,.
\
~ \
, .
'0.
~,
~
/1'
cl
"
(/) -, - Mealiured
(dB}
o
-1y 1 in. ~v ~ r
4
, "
.'
,1
."'"' .. "
:-_-~_ .. _- ---~----_ .......... -
.i" .\
"\ . .\
./
'"
Figure 5.21e: Two element array. 1,=O.5À,1 2=O.5À,d=O:625À.
---
• Ca;yulated
-----.
~
f/) ~ "
o~ Il
o : ,t,
' .. ..1
(dB) 1 "
"
". , " ./. >'" -' /-.
, ,.~ ~ ",-' .. - -- '\
.- ~ .. , ~ ." . \
"\.,,
~
.'
. \ 1 .
-, \ _._. ____ J
..
-1 . ~.
/
.:.- ..
, .
Figure 5.21f: Two element array. ',=O.5À,1 2=O.5À,d=O.75À.
~ \
....... -- .. 1 ., f
- -~"-"''''.'-~ .. ~_ ~ ~ ... .,.,... .. ~~_4~ro,!(" " ........ ~ ...
. ~.:.-_...:o:..._......;.. ___ ~~)~. ~ "'. ..... 'b, pe, 1 'ie" • • t -nz •
/
------
........ 0'\
.-
--
; :,
, r·
<1;
~,
~ \
\ \
t,
"
<1>
O-r:-:-
(dB)
-10
i ... 1
• . . ~_._ ..
. Figure 5.219: Two element array., 11=O~5À,12:0.75À,d=~.125À.
- Measured
• Calculated
, ,
"
Ir
\"-."
-.
q;
o r .~ .' .- ..
(dB) - .1
- ..
-10 " .': . ~ . ....... . ~ ..... :: "
.~
"--------.-------------------
. . . .
Figure 5.21h: Two element array . 11=O.5À,'2=O.75À,d=O.25À.
•
E '.'
:--J '-J
~ "'~""~-'-'....v.)t' 'C.!!i!Jti(~~~~~:lw!:::-' '"-'~ ~ ... ~
.')
fê+n - • N 1
-"'\
J' .--'."
~ .\
"
'"
" .,
..
- \
"'''' .. "'-." .... ,, ...... , ,
- - \
~
'-(/>
01 ...
(dB)
...
-10
- Measured o
Ca 1 cul ated •
.. __ ~(dBJ
~.
~
~ , .. .....;., j
•
-' . --
"
--- '---"
-10 -1 -~
l .L:'-.. .,.. ,- -- -' , .
,-. .,/
,-
..-.. -1
~
-,
J
\ ..?
, -. r. · - ~ .. _~_:!\~ -----• • • • --
I":,,
ti9~re 5.21;: Two 'element.array, , 11=O.5~,12=O,75~,d=O.375À.
",
"
",
",
..
..- ."
• • • • ---
Fi gure 5-,21 j: Two el ement array. 11=O.5À,12=O.75À,d=O.5À.
.. ......., co
~._-;-- --,:':,~..:_-...
""""", .. ,
\
\ ..--..., \
"r :, \
1
, ,
(dB)
'.
~ 'ta ~
t!4 .. ' ...... "1.'..J' ~\ .«
'(/)
o· ---
\
\ • • • • \ ~.
, \
\
.'
.
"'" \
- ...... ~ ..... .....
~ a ... ..
...... -.... oO ...... _ ..
....... -.....
.. t' Figure 5.21k:' Two-element array.-\ ',=O.5À, 12=O.7~À,d=Q~.625À.
'/
t.1eas'ured
• Cël.lculated
r
.. . '
(dB)
o
-10
\
'~ l' ,
• •
Figure 5.211: Two element array . '1=Œ.5À"2~O.75À,d=O.75À.
és-=.-: ~- ---..-.. ....... ,--~ .... .., .. .!".,.\~ ........... "'~ .• ) •• ~~.~ ..... ~~~_ ......... ~~~ ,
f M' "etn~,~ __ ,~.
~
---------
..... \D
-
,t
1
. , ,
.'
0·75 À ) 0·5 À
Figure 5.22:
, . Geometry an~ definition of parameters for table 5.3 and
,figures 23, a through p.
. 5.23 a
b .,
c-
d
5.23 e
f
9 h
. 5.23 m n
o p
0.0625
0.1250
0.1875
0.2500
/ .
d2(>.)
0.125 0.250
:.-> 0.375 0.500
0.125 0.250 0.375 0.500
0.125 0.250
~
0.375 0.500
0.125 . 0.250
0.375 0.500
. ""
,~ Tàb1e 5.3: "I:hree element yagi array cases measured and'
cah: 1ated.
. , ,
- . j
80
~
, ,~
,. 1
" .~
l ~#, ): i 1 :~
r i
, l ,
f ,l j ., j f
/'~
.0
' ..
\ '
_!, Measured
• ,Ca' cu, ated
\', '.
\ ,- \".~ .. . ,. '. '
. \. ;. .-
.r
_ ........ .. , " ......
.... ,
• • "
, .
;
: .......... .
..
Figure S.23a: Three element array. dl=O.06~5À,d2=O.'25À~
.------
(dB)
,.-... 1
(/>
0' r ~'
-10
..... ''--" ~'
" .~ "
.. ... '.
"".... .. ,\
~)\"""" , , ': .è " ." "\
", \
.'
.. " \
" " "\ : ",~,' \ \ - \ . i
"
\ _:' - :~. -~ - ' \ \':. ,': '" ,
" , .. " .-"
, ' 1 , f
, "
'\ \ l j . L:.;:;:.~
.~': -: . .' ~~ .' r . .-:---r. ,1 .' .
. ./ J//"~"-/ 1 1
,/ /' '/ .':: '.
'/ ' ._ /_-'~' •• f, ---,......... . .. ,,' \ ' .
\ _., '.
Figure 5.23b: Three element array. d,=O.Os25À,d2=O.25À.
------------------:-:::~...--~-~ - ..... -- ... '"' ...... ~,.· .... ~~~-. .. ,,~'iJ!',J. .. ~ .. 2u-rl.eP .1 TJi"n,é:'e"6>-""'~- "--, v t .. :_
b' b li? fbWm~ Il." tbs<rt 'reb .. ;;", t -
-
ex> -.J
<,
. ,'--'"
~ ..•
'.'
i\ r ~
'.
...
o
(dB')
:--
-10
?
_.".
. :.:~:-- \ -',::' 'l
. 1 ': .:.:'.:/
"
,,' c
.'''''
.<:.: " ',- '~,>,/~, ": ~~:.--.\. . ' -~ ... \- .-....
: ;(
. ,'
: .. :~:::~\ .... -
..-.--_ .. "
. ... ...
i\.· .. ·· '.:~~~:
J'i 1./ .. ,:~.: :.
• ; 1 ... ~ ...
~~.//' .:.;.: .. ;~:.' "
~.' :' .. / ..... ::.:: " .. ' .. "A .', , _,_--~ , • _ " _,' .J
--_.----. ,,' '.' " '
, .\ .... . ,' If'-: -~ --,,"-
4
. Figure 5.23c: Three element array. dT =-0 • 0625). ,d2=O. 375;\.,
..
- Measured
• Calculated
..
,-... .
"'-"'"
(/>
~1 --....
.' . .
(dB) 4 .-' ',. ~ . .-
"
-10 \ .. . ~. . ". \
4 ..... :a"', .... ~
"
, , . , . : ".
/ ./'
Fipure 5.23d: Thrëe element array . d1=o.0625À,d2=o.5À.
.. ... -.......
~ ~ ... " .. -- •• 0 ••
.: ~ -.. "-. .. -. ' .
~
.... -""· ...... -~~~lf ''&R?' .. 5wi?}t'",1 ~t .~*'"~ _____ --,..- - ..--1-4_ ..........
le /
-~~-- ~~\
'''1. ..
co N
\ \ \
!," ':-J .0 ,,~~;"'1F" 'F*"I'"""eu;:'ljjiIfkG"l +i""~-';::~.-"'~". ,,,..?",,,," ~ __________________________________ _
~.-.
'" " /')
r,""
~
"
.; "'~ , "
(dB)
-'1 -10
'~-, ~'
'!f
"id>
••
. - . ... ~.
..-
,-
Figure 5.23e: Three\~lement array. dl=O.1~5À,d2=O.1~5À.
. '\ /'
-fr'easured
• Calculated
. .. -~ -... .. ~
.....
".
.;
~ ... \ -~~ .... . ,
\
~ = "::-)/ tei± ;"},~._~
-~~..!I"" :É4W""~_·;...,,. .... r
,-.
-1- \
fi)
0' ---
(dB)
\ - 0 { -1
•
" . . -
k
\ .- ..
;
,/
::.' .-
\
.'
.... ': ,
. /" /
.... \ ." .of
.' .'
.' ,.'
~:: -~~~:--~.-...... -... -...... .. -.. ---
......... -.
: ~- ....
-....... ..
- . ..... ",
Figure 5~3f: Three element array. ·-dl=o.125À,d2=~·25À.
1
•
, ,"
co w
-
1.. .tili 'dI'tJ""'~: ,-" '("~",
'1
J , ~ \
;:
, '
~
"
.;.
~
~
"<!
r-'
o
(dB)
. -10
0"""
l-"
.~
• .. .. " ..
,.~ ....
lit
_.,
.:
...... , . ',>. " .. , \ \ .' , " \
- \ \.
~
- Measured '
• Ca 1 cùl ated
:. ~~
? ...
.-" ........
- " ': -. .._~ .. ----
-----}
: .. :-. .\ ····Î'
. ,\:: .... /
,-_~ .. : ./'",1
,_e .: .... .- ••• ' :~/.. • ",.... / -,., /. .. . . .. -' . ./, .:.' ~.:-. ~
"
" '.
"
Figure 5.23g: Three elemént arraY. d1=O.125À,d2=O,315À.
"
.. ~ . . t '
, \
lit
~
tt--. , "
C/J
D 1· ~
~ --, .. ~ - ..... . _- --... '. . (dB) .' '-
, .... ~.
-te
Fi gure 5. 23h: Three' el ement array. dl~O.125~,d2~O.5À:
.. '
.. ~ .. .... ..... ... .........
.....
-- ....
~
f"' --ièsb ' 'C --~~'~/iiiii~'"'''\'~'''''''''"''~--'~ j;te Mt -"Eiee li' 1è ~-......~ '--
:'\ .... - '" -.!
./-
co ~
"
Il
~
----
l~
,
k. ,," • .,...\!fi~~
:""""1 1-
'-
.,.
..
-----....---
r
'"
'\
~
\
'P, Or
-- ~ .. ~~
-- --:-... _-;-
(dB)
-10 \
-) .. ~ .
\
'--
.t-~ , .. \
.' - ~easured
- • Ca leul ated
. \
"
',-.-'
.......... .' .. ,--
. , '.
;1/.: ~'~-::~ ...... . . \ .: " ... -. . . .. : . -._- -.. .' .. .- _.: .. ' .", . . -: :' .- \: '-' .. :.\ 1· : ,\; ..,."
~ .. ; .'- l' \
' 1 '/ .1 .:, .... ",' .
"- ,1.'
~'
J
, "
'~
Figure 5.23i. Three elernent J~ray. dl,=O. ~~Z3_,d2::=1L1251_=--,- - - --.-' - -- \ .
"
-r--~
'\ \
"
(dB)
\
'"
(/) 1 .... 1
,........" ,
o~ , .
-10
/ ....... "
, "
"
" ... " y
---... .~.
" .. " .-.-,,\ -:\ \ \ ':1 . .~ ~ ~
. l :
\, ..... " " ' .. ---_.
"'-, :.\ 1
.,-.... ,1
'"
~ ... t -.. '! ... ~
..l':· .. ·\" ' v ...... _- ..
" .\1 ,: . . , ' , .' . / :,./\ ,/,
, > ./,' :,":'\':: " . ___ . ..~,;// ,.Ô ' '. '.:,
_ .~, .~.". r ~ .. , .. :' , .. \:' \.
• 1
.. , .
Figure 5.23j: Three el ement array.\ dl~O .1875:\ ,di",O.25À. \
'} \
~
\ \ .\
~ w..". ri ... .' ?tH œ' . \ " kA 'H ... ·TI &* . _~ •• "' .• _ _ . • iIl4lt.....-.~~-,.-- ~
c. '
,
CP U"1
-"
........... :
~--
\ r
\i
•
.. " !"
, \
~.
• •
:{
.. ~,,"'"
::f, ;~41.
:'''\''''- " ... 0_',
0i ~ ,(/J' •
===---- f'~ 1 ~
(dB)~-, ,~~
1 ,!
-10
" , ... - 1
v
~" . , ;. t '\ ~ •
.Q
, r . . . . . . . '. :".:';~'.:. /":':", '. , , L .. ~.~ .. " .. ~ " .. . .. -
. '" ........ ' '.
..... ".'
..... \":::'t~: :.:.: .. ".::. ::::.~~.~ .. ~~ ... 0- ._
..... ~~.J . . . ~.:~:::.::.
,
, " - Measured
~
• Calculated
(1
.. 'O. ~
, 1 ~; . .:~"~ -/
.. :-: .. ~ .j .' . ,- .....
o
~
,,' •
................ .# ........ -., .
/ If "::',
'. " .. _ .. : JO
," - .. . .......
- .... '" .. . .. .. - -... ..
l,; : .... < -/ '
.. ')' ... "',
.. .. .. .
.. . . ..
'f-•
Figure 5.23k: Three element array. d1=O .. 1875À,di=O.375À.
/1 "--
..
"
'< '
,~.
. \
,,1'.
o
• (dB) "
~
't
(/J
~ " . '. . . ' .. - "' ..
''--'--. -',---...
lé ........... ----~ ;-~
,"
\ '\
-10 " "."" \. '\'" \.
" '
Figure
/ /
\
\ L-I
!
,.
.'
. \'
5.231: Three element array. dl=O.~875À:d2=O,5~ .
-. ~.
..
--'-"'I--"~--",,;, ....... ii,4i r&SIIUR"é: u,·? •• ,.nit'!tGi :.i**. Sot ............. ~..-.
" . ~
-----
J
. ..
co ,Ol
....
.. -----
, . \
"'7--1:-:'
-~ \
.... "'E!4''4+fI"fi "'Ji ;~ ........ '1!"~.."..., ...... __ , ........ ~_ ""~
r 1
~ , ,
;.
• l
, 1 "
....
1 !--'" '} fi
.----~._---.
(/> o~
(dB)
-10
..
...
"
"
...
r • 1. ,.-;
,."/ -.,..~ ......
/
~
:
.' . . . '
~ ... Il.. • .:~
. .. 4.-· ......... ~ .. ". .; .. . ........... \
......... 1 • ::~:~::~::J
• . ... 1 .. . .. _ .... :
· .' • . . ..... -.
~~
'.
'.
~-.)
Figure 5.23m: Three element array. d1=O.25À,d2=O.125À.
~ 1)
-1
- Measured
• Calculated
,al. ..,.. ~~"""""""'"Jôo."-I""t&._
;. ~
po
~
1· : 1.
t/>
O~ . "
(dB) ..
.. .. ,
' .... ::. ;: .. 1
~
\". -.
#./ ,,"
.,(:. ""
i'" ,,' • '. '.
-'-
,J\
. "'v' - .-.
"1"
. · •... , ,,: ..
" ... " ...... o· .oo .. '. . .' .
\ .
.', ., .' ..... . ""'."
\~\ .. <,~:""
.. .. ,"oo ...... : .. · .... -...... -~ '. "".. . ...
" - .. ~ ..
...... · "
~ ". . ' ..
~ .. -Fi gure 5,. 2?n: Three e 1 ement-array ~
, .. di=o.25À,d2=O.25À.-
~
. " f' (
.~~\ -t
' .
ex> "'-J
'1
<>
/
'--.
1
i
'1 1 ..
./
o
(dB)
-10
ct>
... \
. '.:, .' \ .. "' ....... , . .... ] . ' .', .' ',' '.,
. ~\l". '. J
...
• • 11 •• fi .. ... . ...... ' :'':-.
. ,
"
"
~--:..'---- .' .~ .~
.. "" 'II··· .... ,,".\
.. '. .. '.
..... :" .. · .. .. : ... 1: · .. · . .,
.~. # -.-... ",
._". .. "
,. 1
Figure 5.230: Three element array: dl =0. 25À,d~=0 .3751. ..
,li
.'
. .\
.. -.... .. , ....
"t
- Me,asured
Calculated
\ .
:, .. j 1 ............... -
',.
1 c
.'
,
.. ... p
"
o ct>
, (dB)
'\. 1> + '
-10
-
""
Figure 5.23p: Three element array. d1=0.25l,d2=O.5\.
'~
. ' ..
/\ ~ ~ fi ~'M.·t",",~ot>~:;' ,....,.~~ ~,
•
~
, co co
...
\ . ..--.,.
'"
----~
.'
li
" ·75 5À
d=·06. 5À ,,.125
'"
"!'
'"
-,. Figure ~.24: Four element yaqi array.
rr·~-····· .:- .. "'----------~.- ..
J>
ri>
Dt • ----
•
(dB) -5
-10
... ., ....
- f1ea s u red ( ..
• Ca 1 cu 1 a ted
~ __ ................. -~~I;:;;~,f."i._ ......
.-
~
•
OJ \0
'"t
, J
.. ~
.... ,'"' .........
"la'
• L
.,
'"
. -"
(dB) .
. --,-
. ,
-·75 À ( ·4 À 5À
- 1-1 .
1 , .. ,
. ,
'f 't!'
·25À .125~ ·125:\ .
\
Figure 5.25: Four element'yagi array .
-"
~
_,,,J, .('--J;'ll;~'l'i..~t,
C/J
Dt --
-5 ~
. - .. ---- ..... -..... ...... ~
-101
" • •
• .. •
~ Measured
• Ca'lculated
. .
/ ...
y ",
'\
• " . .. . ': ..
.: .-. , •
\ •
• • . , • , . •
/; 1
1 1
1 1
1 "fi
1 ....... ---
•
-
~~ ..... -
. '. '.
::-
~~
l(>
c,
10 o
}'"
;--; 1
. \
,
~ r
la-
1 1 1 ,
•
• ~ j- ~
.. •
-,
moN
"
. '111'; ..
,1;''''
/ t'\ '-~~-:'J...
J
,
10 i
[Ons, W ",
5 ~ r", 0 002L8).
-- Rèal - ltnoginary
, 0 1
0.25 0.5, - 0.75
dl}.
)1. '" ~,
~5 1 ,
/----...
-'0
Figure 5.26: Input admittance vs. Separation of a two element yag; arra'y. (Computed results)
...
1 ,. .. -. ~- ..... _-,,~~ ......... ~ ... _~_'!>._\/j:!'~-i:~"O-'"
\. pl'
"
"-.
~ "
tILP ~S~ D
10' , d \
, 1/ ~
1 5
r,. 0-00248),
-R.a. -- Imognc.y
moIV 0
\ ~ ,. 0.25 05 . • 0.75
d/i
-5 i
--'0
Fi~ure 5.27: Input admittance vs. Separation of a two element yagi array. (Computed results}
\0 .....
1 ~
f
1
( • ,
(
~-~-- -
" \
<S fJJ 05~. 0-75>'
W 10 dl dL
5 , Real
mo/V o 0·25
-5
-10
..
-15
-20
t
of ./l.,~
().5
d,=02500>' d,=Q.1875 X d, = 0·1250 >.
d, =' 0 0625 >.
r 7 0·0021.8 >.
o
lmoginory o
V
d,:Or2500 ).. , '
d,':O. 1875 ').. "!'-.'
dl = 0·0625).
, ~
Figure 5.28: Input admittance as a functionJ'of element separations on a three eleme~t yagi array.
.,., - ---
....
92
( \'
(
,
'r
1 •
5.3 The Eigensol~tion for Wlre Antennas
The dipole, despite its ~implicity, ~erves as a good introguction to concepts of eigenanalysis as presented earlier in Chapter II. 'The eigenanalysis differs from the establisned analyses of Garbacz
(32,35) and Hanrington (33) as Garbacz applies the theory to scattering ,matrices, and while Harrington works with the GIM. cursory inspection
93
will show the difference between·that analysis and the one presented here.
In the formulation presented in this text, the eigenvalues and eigenfunc-,
tions are taken directly from the GIM as stated bX equation 2~54. The . matrix is not divided intfr i~s real and imaginary parts and the eigen
equation restated in terms or' them as is the èommon pracflc-e ...... {32,)}) a'nd the significance of tmls is shown latèr on when applicat'ion to ~u1tip1e wire arrays is discussed. •
,"
5.3.1 Eigenanalysis of a dipole
The dipole highlights well the concept of flcharacteristic"
currents or ',',modal fi currents which are essentially the currents
associated with each individual eigenvalue and eigenfunction via the '1 • '
analysis be~inning with the equation 2.55 of Chapter III. For the
1 2 3
'. Figure,5.29: Eigenfunctions of a halfwave dipole,~each normalized
" to unity.
)
1
1 , j
.... I.....i ...
(
...
(
1
1 l"
! , 1
"
r
-- Solution with 1 mode
,. ... ..
a ~~------~----------~ o - 5 J
mo/V tO
t \ ! .1 I-f,.
Il Figure 5.30: Ejgensolution .current magnitud,e on half wave dipole.
(r=O.007022À)
:lOO
• 150
100
~IA)
50 0-
o o 0,5 .
r.0.007022
37 w-a(nowns
'.0
/"antenne lengtl," >.
1
~ , Figure 5.31: Condition number of the GIM vs. dipole length. (r=O.007022À)
"
, -
/' -
-,-"'-----
f
(
95
case of the half wave dipole, the eigenfunctions form an onal set of even ~nd odd harmonies as stetched in figur each
- sueh eigenfunction an eigencurrent may be dete1"minr' It will b.e shown
that nct al1 the modes are required to reach a convergent result. Indeed,
in the case of the dipo1e, on1y one mode is required to lleld a result and the ca1culation with more modes cannot be'said to' im~ove the solution
as shown in figure 5.30. Calculation with tao may modes can therefore 1ntroduce "numerical noise" into the SO,lut10n. The criteria to determine
which and.how many modes to use are easily determined by the eigenanalysis
formulation put forward in this wor,k and will be discussed in the next
section .. It had been mentioned that having the eigenvalues available.'
an indication of the GIM's conditioning'may be obtained as shown in Appendix'III. Mittra and Klein (34) show that this will vary as the prob-
,1em ' s parameters are ~hanged. Hansen and Rydahl (14, a1so show this in
re.1 a t i on to resonance phenomena and poi nt out tha t the cond it i on number
is a reliab1e method for determining resonant situations. In figure 5.31
can be seen the condition number of the GIM for a dipole as a funct10n of length w1th the number of unknowns rema1ning constant. The plot shows . how the condition number varies as the antenna is brought through its resonant 1ength at just less than 0.5~and once aga1n at a length of just
less than 0.5 "li and once again at a length about double this. It should f
be noted that the first resonant peak coïncides exactly with the point at which the imaginary part of the antenna's input admittance goes'to
zero (refer ta figure 5.4 on page 58 ). The second n,se however could
not have been predicted 50 easily from the antenna1s input adm)ttance a1-
though a resonanee occuring at twice a resonant frequency is to be expec~ed.
q.3.2 Ei2ensolution of Vagi Arrays
In each case ~f the yagi configurations which were ca1culated
and measured for secticn 5.2, the eigensolutions for currents were also obtained. In addition the eigensolutions were also determined for four
and five element arrays. An,examination of these results showed a consis{~nt and definile behaviour of the eigenvalues irr relation~hip ta the
..
/
,
. ,
-..-..........~----
--------------------------------
{
l'
solution. This behaviour leads to a tentative conclusion that certain ,identifiable eigenvalues are sufficient ta determine the complete far field solution and that in sorne cases, even only a subset of these ca~ yiel,~ a good approximation of the solution. This behavioyr re-iterates
\
the useful ness 0 f obset'vi ng the cha racteri s ti cs of the é i gneva lues as ra
96
direct indication o~ the behaviour of the antenna itself. As this is a moment metho~ program, an n x n GIM will be gen
erated for the case of n unknowns. Thus there wlll be n eigenvalues and n correspo,nding eigenfunctions. Each 'ei'genvalue-eigenfunction pair accounts for one of the antenna's characteristic currents, the sUlllTlation of which forms the total curren~ from which the far field radiation pat-
, 0
o tern may be obtained. On the 'pages that follow are presenteEamPles, , and plots of individual modes for three arrays. lt is noted hat c~ri~\~rgence with a finite number of modes occurs quickly and suppo ts the e~rlier statement that not all n modes are necessary. But it is now asserted that not only are they'not required, but that they should not be included at all as "numerical noise" will be introduced into the calculations.
It ;s necessary therefore tO,state prec;sely which modes define . the ,radiating behaviour of the antenna. From t~e case of the dipole in
the previous section, it ;s seen that the "principal" mode is associated with the eigen~alue with the lowest modulus. The sam] appears to be true for the arrays studied herethoweve~ the nature of the significant eigen-
, /
values can be stated even more precisely. In retrospect, it was noted when studYing the numerical results,
that convergence was obtained with but a ,few modes 'and it was generàl1y tr~e that the lowest modulus eigenvalue made correspondingly" the gr~est contribution to the antenna currents and radiation pattern. Further signi~ ficance of the eigenvalue modul; will be discussed later, however, it became clear that after a few modes were summed, the result was not essentially altered with the addition of further modes. It was found t~at there was an exact 1:1 correspondence between "contributing" modes and the nature of their eigenvalues. ONLY EIGENVALUES WITH POSITIVE REAL PART~ PRODUCE CURRENTS THAT CONTRIBUTE TO THE SOLUTION. Thus from figures 5.32 through 5.38, it can be seen 'that for the two and three element arrays, two and
three modes are requ;r~d for their res~ctive com~lete solutions. However a correlatjon should not be draw~een the number of elements
p ( . • 1 1
---...r.....- ___ _
(
First Mode: Second Mode \ .
Figure 5.32: Individual mode solutions of far field H-plane pattern. Two"element array,11=O.5À,12=O.5À,d=O.~25,r=O.00248À.
t 0 '·0 • • • 1 MODE
• 0 o 2 MODES ·Î • 0 • • • • • 0 •
)
0
• • 05 0
0 • • (, .' (l •
• • 0 0 • 0 0 • • .. • •
" . 0
o 90 180
( 0
Figure 5.33: Convergence of mode solution for the array of Fig.,5.3.2. Convergence attained with two modes. . ,
1
(
~ 1" ' !i
/' ,,'
(
(
l't If
lb
-~ cf A
First Mode Second Mode , f
Figure 5.34: lndividual mode solutions for far field H-plane pattern. Two element array,11=O.5À,1~O.75A,d=O.125À,r=O.00248À.
1·0 • • • • •
IE.I I~.IINK os
•
o o
• • o • • •• .. . •
,
• 1 MODE
• 2MODES
90 ~
• • • • • .00
• • . , ...
180
" J
Figure 5.35: Convergence of mode solu~ian for-the array of F~."5.34., Convergence attained with two modes. ' ~
98
~
l'
, .
(
( .
First Mode
rp --1
Third Mode
"
(
./ ;
99
Second Mode
Figu~e 5.36: Individual Mode . solutions for the
far field H-plane pattern. Three element array as in Fig. 5.22 with
,d,sO.0625À,d2=O.t25À,r;O.00248À. , .
q 1
\ \ \
,.
,~
\
~ j '1
\
·1
1
1
/" " ___ .... ""',._. _ ..... -.... _._ .. _ ... -_o.-.-_II\j~ .. ~ ... ~""'~~lIIi'liiIIi-_ •• I~IIiI.iIIII~ .... rao;-.~-.IIIIIIII--____ «' ... - ... -.... ' _ ..... _, __ •• _______ • _____ ~~-. ... ~_ ~~-----; _______ P ... ' A r ~ _l''U' ~m wU:WU%lddwn mi _~ III ---'''-' ... ~ .... - - -'-' ---_. -
'-.
Fig u re 5. 38,:
1
()'6 ° ' 1;,' ° "I~-,!I •
... pC -01 ° • v
•• • :. • 03 • • " •
• • • • e 0, • • • • • 0 • • • • • 0 • 0
0 JO 180
• Plot' showing ~hè"error' between the convergent solution with three lTlodes o d tne two other sol utions with one and, two modes 're pectïvely fref: Fig. 5.37 above).
". '
..
1
1
l'
1
1
(
.. 101
and the number'of modes required as calculation of four and fiv,Ie element
al'Tays al so showed results that were determ; ned by as few ·as three modes.
The only criterion whi'èh avoids an} confusion or ambiguity i5 the
sign of the real part of the eiget\Yalues., Thus for a half wave dlpole,
eige"nalySis shows'only one eigenvalue with a posi.t;v~ real part and
only that one mode is required to form the convergent' solutlon.
How quick"1y a solution converges is to be seen in figures 5.33,
5.35 and 5.37. In figure 5.38, the difference between the normalized pat-~ ~
terns is plotted as an error and it can be se~n that although t~ree modes • 1 "
define the complete solution, two modes already provi,pe a good approxima-•
tion. '. 1
The wor~s' "complete solutionll.J1annot be over-emphasize~ ~s it is
held here as stated above, ei..,genva1ues ,w{th negative real parts will only
adq "~oise" to the solution iflncluded in its calculation. This can be '\ ""'(' .,
shown by the fol1 9w,in g brief mathem~tical argumen1 .' . Assume that an n x 'n matn X' has been gederated and that . ' ~. ~ ~
eigenva 1 ues and n eigenfuncti ons are known. Each ei gencurrent or
tion of 1;hem satisfies the equation ..-.
/'
= V. l
the n
summa; :,
J
where V.=À.I. !
5.2 l l l
/ / . '
T~is ;s seen to be no more than a statement of the eigenvalue . /
problem forml11a"ted in Chapter II. However V. is.an lIexcitation" and
Ii' a current. For the current to radiate P/~wfr, 'the following relation ./
'must be true //
5.3
• or
> 0 ~, 5.4
, .
!
l ~
i ----------..... ---_., -, I~I---------~-~--,---~ -'--.- --_ ..... - =-=W::::=, ,al, ~~. ._ •
\. ".
, "
· t i
r 1 1 i r ~ ",......~ ... - ----~- .. t-- - '---
102
As the eigenv~lues are from the beginning complex,
1.
À,' = a. + J'b , , ;
Substituting the expression of 5.5 back into equatio'n 5.4 gives
that for energy ta be radiated
'" ' 1
This· leads to the obvious conclusion.that ai' the real part of
5.5
5.6
the eigenvalue Ài' must be positive if power is to be radiated. This fact has been borne Qut by the computer calculations made on wire arrays. There
is no reason to suggest however, that this concept cannat be appl ied uni
versally in moment method problems. This therefore provides a highly Â
,efficient me,thod of solution and, as can be seen. is independent of the
the excitation itself as the eigenvalues and vectors are only dependent
on GIM. Then, for any excitation vector, solution for the a.etual current---
distribution is very efficient, results are obtainçd without matrix inver-
sion and exactly how many modes are required is now precisely determined.
The formulation of the GIM always warrants discussion and this
was done in Chapters Il lnd 'III. Improper formulation with two few un
k,nowns or a poor integration scheme could yi eld very poor results. It was
observed hOl'iever, that perhaps there is a correlation between a matrix
'".that will yi~ld a convergent result and the moduli of its eigenvalues. It
was noted that for calculations of arrays as ~erfonned in this work, a non
convergent result was obtaine<;l in s'orne cases (made eventua lly convergent
~ith an increase in unknowns and higher order integration). In these cases
however.' when compareq to the convergent results, even though the number , , ,
of modes required to define the solution did not differ, it was found that
eigenvalues with modul i smaller than sorne moduli of .the "s ignificant" modes
were observed. These did not contribute to the solution but the conclusion
-"..,0-_---_ .. - --
/
1
" -( ;
1
1
(
1
" 103
-to be drawn ~hat if a "non significant" eigenvalue falls within the
circle whose radius is defined by the "s ignificant ll eigenvalu~ further
from the origin t th en the GIM can be said to be poorly formulated (as op-
'posed to illconditioned) and ~inv~lid result can be a-chieved. The valid
ity of this statement though is sUbject to further study and no definite
conclusion is drawn here albeit an i nterest i ng observation is seen \
fi gure 5. 39 .
...
INCORRECT CORRECT
lm lm
Re
Pi gure 5.39:" Schema tic of e i genfu'n'c't ion 1 oca·t i o~ for correct
and' incorrect sol uti on.
in
Re
The behavioui of the eigenvalues b~ings forth interesting questions~
Based on t~e formulation of the GIM t should there exist at all eigenvalues
with negative real parts? What does this say about the numerical model or'
the analytical ,one given that the resul ts obtai ned are correct? What
tolerances, such as presented in Figure 5.39, can be allowed these eigen--
values? These are the ques~ions raised by the current work and further study
of them should lead to a better understanding of the application of
analytical and numerical modelli-ng 'procedures to antenna problems.
\ , , 1
1 / ~ •
-,
(
J
t ) ,
CHAPTER'" VI: CONCLUSION
A maln goal of. this work b'~s been to es'tablish a numerièal and
experimenta1 faci~ity for the study or wire arrays. The complexities of
modelling antenna problems on a computer reqlJire careful application of
numerical techniques t~ the analytical mod~ls which describe the actua1 , ,
antenna problem. S,ince, as was seen in the Introduction, the analytical
models for wire antennas are well established, the major portior) of Chap
ter II was devoted to the numerical modelling procedure. The method of ,
moments was used as wire antenna problems are usually at frequencies at
which the.currents must be determined from an integral equation formula
tion for accurate field and input impedance prediction. However, it is
known that the application of moment methods doès not guarantee a correct }
and convergent result. Thus by the end of Chapter III, three, important
104
modificati ons have been presented whi<:h ,improve upon the usua l appl i cation
of the method'of moment
The method of moments generates a system of simultaneous linear , equations which when solved, gives a scHution ~the currents on the wire
elements. By the very nature of the integral(equations however, the resul
tant matrix can tend to be singular, especially if the problem has resonant
qualities. Thus inversion of the matrix requires algorithms more numeric
ally stable than the conventional Gauss Èlimination. An algorii.hm based
on a Householder Decomposition has been described here and was incorporated
into tne computer program. Kaving addressed the problem of numerical stabil
ity, it should be realized that successful inversion does not necessarily
mean that a convergent resu}; has been obtained. It is shown in Chapter III
how improved convergence of the numerical result can be obtained. The sec
ond order differential of the free space Green's function in the kernel 1
of Pocklington's Integral Equation is treated as a finite difference approxi-1 1
" '
,
( .
105.
mation. When piecewise ,linear basis and test functions are used, the. resultant matrix has better convergènce.qualities than those obta{ned using
, l'
point matching and the same' basis function while maintâining the ease of programming of the point match procedure. ·The convergent qualïties
, are demonstrated later in Chapter V where a convergent result is seen to te obtained for an array geometry which has been used in the literature as'an example of poor conv'ergence qualities of Pocklington's Integr~l Equation, when solved using the method of moments. ~
The thi rd modffication to the standard numerical scheme was to allo~ the numerical model to be treated as an eigenvalue problem. Once the , , eigenvalues and eigenfunctions are known,the currents on the antennas may be calculated from themand the problem of matrix inversion is" avoided. Recent work (33) has shown that the problem of synthesis may be addressed by an ei genana lys i s approach. The d ifference in the ei genva lue scheme pre,.s·ent-ed here and those available from current literature is that the problem is not formulated here to guarantee real eigenvalues; the eigenvalues of tne GIM can thus be complex and display'the behaviour shown in Chapter V. . . .
Part of the work involved the design and construction of a novel 1
probe carriage and boom for the UHF anechoic chamber . The des ign a 11 ows . . five degrees of freedom in motion, and three dim~nsional probing Qf flelds in the chamber is now possible. Of major consequence is that the probe carriage and boom lends itself to n~ar-field meas·urements. 'The cyl indrical shape of the chamber and the probe carriage design as shown in Chapter IV. are ideally suited to both cylindrical and spherical scanning, techniques (48,49,50) used in near-field far-field transformations.
It was seen in, Chapter V that the goal of establishing a numerical , and experimental wire antenna analysis facility was successfully.att~ined.
The validity of the program is shown through the usual comparison of the calculated results with available pUfrlished experimental values. The superior convergence qualities of the numerical approach used are then demonstrated. as previously mentioned, on an antenna configuration which, when
< •
us;ng the same analytical model, yielded a non convergent result. Using the program and the anechoic chamber, a 'systematic set of
,calculat.ions and measurements was made on two and three element yagi arrays. Excellent agreement is to be seen in tr.e results for these cases. However. calculations and measurements of four element arrays show the l1mitations
t_
- ~-- --- ~ - -----.. ~-~-
• 1
1.
(
(
"
106
of the finite size of the anechoic facility and clearly demonstrate that near to far-field transformation techniqu~s should be used in this facility wh~n the mOde~size exceeds a particular'limit.
An elgenanalysis was âlso performed in al1 the case~ for which èaléulations were made. It is clearly dernonstrated in the results presen-
\ , . ted in Chapter V that the elgenanalysis provides a consistent and predic-table pattern of behaviour. It is seen that not'a"· eigenvalues are necessary for the solution of the currents. However, it ,lS also concluded that while some eigenvalues may not be "necessary, sorne should ,not be used at all as they will only add "numerical noise" to the solution. It became , clear that only those eigenv~lues with posltive real parts contribute to
, the solution. Ei.genvalues with negative real parts should ,not be used in the solution. It is also' shown that in sorne caseSi only a subset of the eigenvalues with posltive real parts is sufficient ta provide a good approximation to the final result.
From the results it is thought'that a link may exist between the phyJical an'tenna and the numerical model through the behaviourlof the eigen-
e-
value. No definite conclusions are drawn at this time, but th1S notion is further strenghtened by the closing paragraphs ·of Ch~pte~ V. Here it was seen that an incorrect solution was associated with an "improper" position- ,'1 ~
ing of the eigenvalues in the complex plane. It was'observed that if an eigenvalue with a'negative real part fell into the circle about the origin
1
whose radius is given by the largest eigenvalue modulus of'those eigenvalues wit~ ppsitive real parts, an incorrect solution was obtained. And,even though a correct solution was 'always obtained, the actual significance to , the numerical model of eigenvalues with negative real parts remains unanswered. , ;
The work would nat be complete,howeve~ without also stating its
limitations and suggesting directions for future work. The limitations of the anechoic facility are a function of its,finite sizef the se cannat be avoided. Butl work in this chamber should be directed along tWQ avenues which would no longer place upon it the limitations of its size. Work should be initiated to develop a set of wide band directive probes for both E and H field measurements. The use of directive probes naturally improves the signal-to-noise ratio, however, they do disturb the field and interact with the model under test. Thus the second avenue would take thi~ into account as it is proposed that near~f;eld far-field measurement techniques
)
•
(~
should be implemented in this anechoic chamber which is ideally sûited to this purpose.
107'
The computer program as it stands can be further developed and improved upon. Currently it is maintained that for an active element, eighteen segment~ per half wavelength are required for a convergent'result while ten are more than sufficien't for passive' elements. Thus," for an array size allowing-for one hundred segments, the largest array,cons;st'
'of half wavelength elements with but one dnven antenna.woul ave elements. The use of a more sophisticated integratlon scheme or even just usi,ng more.... points in the Gauss Quadrature, could reduce the number of seg
rments required. It is seen that the eigenvalue program is not memory efficient at this time and development of an eigenvalue package specifically designed for this application could reduce memory requirements 'significa'ntly. by at least a factor of two,as the library routine EIGCC was available only in double precision. There is no reason ta suggest thatjwith the work described above, the number of segments could not be 'extended to an order of 250
, /"
<l.which,at the same segment density, would salve-for a yagi array of twenty-, , ,
four half-wavele~h È!lements. There, is thus an upper bound on the complex-ity of the antennas whic~, coulcf b,e considered. ,
Finally.r work on the implications of the eigenanalysis should be continued especially with régard to the final points presented in Chapter V. This 'is significant since further deve 1 opment in the unders tandi ng of the, relationship between the eigenvalues and a correct ~olution may result in a, deepër appreciation of the manner in which the method of moments should be a,pplied to the antenna program._ There is also no reason to believ,e tha't the' results and observations are solely applicable to wir~ antenna arrays and could therefore be applied just as well to moment method solutions of other antenna forms. Such an extensio~ in the understanding of the eigenvalues ' , significance as well as its application to other antenna geametries would' probably be the most important and immediately fruitful extension of the work presented in this thesis.
i l . 1
(
,.
'/
/
, 1
APPENDIX 1: Intégral Equations
" Presented h~re are the integral equati.ons referred ta in this work:
é ,
Ha~lén's Integral Equation:
l '
(
1 ~ ,1
J I{-z') G(z.z') dz' = - * (Cl,coskz +"C2 s.~nklzl) , -~ / .,
Pocklington's Integral Equation;
l ,
Electric Field Integral Equation'.(EFIE):
'.'
Ë(x) '= TËinc(.x) 4: f (jw~(~' x H)$ - (n 1 x ,Ë) x'V'</>
s "
(nI • È) V,I,cP) ds!
" • Pli _ 9
" '
, 1
108
'.
, '
(
':"
t , ,
,~
(
" ,
! /
t
, , / -
t Ma2hetic Field Integrai Equation (MFIE),
l '
H(X) = TRinc(x) + 4: f (jW€(~1 X Ë) ~ + (ni x H) XV'$
s
, /
1
, f~ •
, 1
, ---
, 1
109
, )
fr
. ! i
" .
i
f
l
('
/
(
APP~NnTX JI: NUMERICAL ALGORITHMS
The algorithms presentedin th/is appendix all deal with transforming the General Impedance Matrix ,into upper trïangular form and are those referred to in the text. They are presented in an INFORMAL (INFL)';'language which gives the basic FORTRAN structure,but,
, ,
enab]es one to quickly grasp the mechanics of the procedure. Specifie rules for INF,L are féw and simple enough so that discussion of them may be dispensed with as they will become selfevident. "
Before presentation of the algorithm~ howe~er, a summary of ' notation would be useful.
110
i ) Capital Ro~an letters (e.g. A, Bt C ... ) refer to matrice~., ~ ~,
• i i ) Lower case Roman letters refer t'o vectors. As matrices
/ II> 4 ,
are comprisà! of ve'ctors, the vector ai (\Ii 11 therefore b,e the i th vector of matr-ix A.'
iii) LoweY" case Greek refer to individual elements of a vector.
.pre~ented.
. aij 'is the jth element of a ve~tor ai i s the' i,th vector of matrîx A~ Or is fh~ i~ jth element of matrix A.
wh; ch in turn directly a ..
lJ
This convention will be employed. throughout the algorithms
, \
, . , .
.
----------_--:.-----_. -- --- -, - -------------".. • « P. ,-
i
--j
'L --....,.
(.
• F. r ... _ ,
o
n.1. GAUSS ELIMINATION ~
Given that the matrix A is of dimension n x n and of full rank, i.e. non-singu1ar, the following a1gorithm reduces it to , tri angu1 ar. form. The multip1iers ).Iik-required for this may be in the
1 )
lower part of A which becomes zeroed.
FOR k = l, 2, ••. , (n-1)
a'k 1 ) ,1 Ct i k + ).Ii k ,= Cl
kk (i=k+l), (k+2) ,
(i=(k+l), (k+2),
(j=(k+l), (k+2),
; UI n.r r Ft
n 3' Number of multi pl i c~t i on::: ""3
•• 1
... ,
... ,
... ,
upper
stated
n)
n)
n)
, ,
111
--- ...
'\
1 ---
\t '
1
l ' !
1
l' ,
/
o
.... et .. t
*/ IL 2.
/ \
. . "" GAUSS ELIMINATION WITH ~ARTIAL PIVOTING
A if Dg dimension n x ,n and of full rank . •
l ) FOR k :i l, 2, ... , (n-l)
1) FIND Pk è': k suèh-that la kl;: max..!la'k l :; 2 k} Pk' ,1
2) IFa k -=.(), .... 1 Pk
STOP, MATR'! X l'S· SJNGULAR
)} CLkJ -+-+ Cl (j = k, (k+ 1 ) , ... , n)
Pkk ~ o'k
4) • l ( i = k+ 1) • (k+2) , n) CL +- ~ '" --- ... , ik i k. 0kk -
5} CL • .... a, ' - ~ik CL kj ( i = (k+ 1 )., k+2), ... , n) lJ, lJ
(j = (k+ 1), ,(K+2). , ... .,. n} "',
,
Number' o~ multiplication ~ ,n;
, "
/
1 •
'II' PM _Mt, ••
112
..
. \ ,
-7-= ;:b' _'!o..
L
.C!
! '\
(,
"
\
[ _r
-I 1."3., GAUSS EL H·1I NA TI ON WITH COMPLETE PIVOT I NG
1 )
-"
,d
A is of dimension n x n and of full ran'k.
FOR k = 1, 2, ... , (n-1 )
1) FIND Pk' Yk ;:: k such that
IClp ykl k . )
2) IF a Yk = 0 Pk
3) ak · +-+-_ (l j J Pk
4) a'k +-+- Cl. 'Yk 1 1 ,
= ma x {I Cl,. : i, j ~ k} lJ
STOP - MATRIX IS SINGULAR-
(J' = k, (k+ l ) , ... , n)
( ; = 1 , 2, •• -t nj
( i = ek+1), ( k+2) , .•. , n)
, . ( i = ( kt l )" k+2), .... , n)
(j = (k+ 1, (k+.2), ... , n)
1 3
N~mber of multiplication: ;
, .
113
•
1
- .. - ---- -~--~-7- .--------..... - .. -
., .. /t
t
l
. '
1 1 1 -; ~
'C~
-,
.-1
• 1
/Ô
II.4. CROUT DISPOSITION
1 )
,\
I.;'e
\ \ ;\ 1.
A is a matrix of dimension of full rank.
~OR k = l', 2, . n ... , 0
.~
k-l l ) (li k +- >-; k = CY. ik - L ÀiR, \JR.k
R.=l !
k-l 2) (lkj +- 1-\j = Àkk
-1 ( CY.kj L À k'R. \) R.j )
R.=1
\
,V
n3 Number of mul tipl icat;on:::: 3-
\
.'
1
.. ~
•
- 1 (; =k, . " . , n)
(j = ( kt l ) , .. " ,
o
114, ~
n)
•
, ' ' ...
, t
l
\
1 ~
1 f t f
i (. ,
( II.~. CHOLESKY DECOMPOSITION
A is a positive definite matrix of order n x n and ful1 rank.
1) FOR k = l, 2, , .. , n
1 ) FOR i = l, 2, ••. , k - l
e ~l( k-l = À" ok' - 1. "l'J' "kJ')
11 l j=l
-,
- 3 Number of multiplication ~ n6
, ,
115
o
..
1
-, "1
: .
. . 116
<' 1I.6. GIVEN'S ROTATIONS
A ;5 a matrix of dimension n x n of full rank.
l ) FOR k = l , 2, " " . , (n-l)
~ 1) FOR (k+ l ) t ( k+2} , = .". ,. n
"
1 ) X' = I,CLk+1,k +CLh'.;· l 1 ~ 'I~ • -; J,
" 2 ) IF,xi = O. "
y = and cr = 0, go, ta 5
3) y = CLk+l,k
Xi
4) CL ik cr =-x· 1
5) CL k+1,k +- X ...
6) a. k .... 0 . 1 ,
~ 7) FOR j = (k+l)', ... , n . .
1 ) 1'1 = a k 1 . y + CL.. a + ,J lJ. j' ., ,
2) \.1 = -CLk l . cr + CL •• Y + ,J lJ , l ~ . . 3}' CL k+ 1 ,j ... n ,
_ '~f.
(i' 4 )1 CL ..
lJ ... II
(';
,.
-- ~-,-- ," - - -"
(
8)
(
J
/ FOR j = l, 2, ••• t n
1 )
2)
3)
4)
n ;: (x. k+l Y + (X .. cr J,
Jl 1
~ = - cxj , k+ l a + aj~
çxj,k+l + n
(x. • J .1
+ ~
"
_ la n 3 Num~er of multipli~ation ---r--
if A ;s symmetric
..
_ 4n 3 - -3-
117
-..
---~----~~~~----~--------, , 1 ... • •
l 1 l,
1
(
c ,
,. zr t ,
118
II.7. HOUSEHOLDER DECQMPOSlfION
"
A is a matrix of dimension n x n of full rank.
1) FOR k = 1, 2, ... , (n-1)
,
, . 2)
. \
-
1 ) n = ma x ,{ 1 Ct i k 1 î = k, (k + 1 ), .... n }
2) IF n = 0
1) Ctn+ 1, k = 0
2) STEP k
\ ,
, 8) FOR j = (k+ 1), (k+2), ... , n
, ~ 1) 1 =:=-L 'J'kCL"
11 k i =k 1 1 J '
2) Ct., -+- Ct,. - 'TV,' k lJ " lJ
N~m~er of multipl ication ::: \ i ~3
p r d tUt r l
(i :' k, ( k+ 1 ), ... , n)
( * denotes comp l ex conj uga te) .
,
(~ = k,(k+l), ••• , n)
-..,
, 1 ..
, "
1
,1
(
1 ~
. '
(-, , "
/
APPENDIX III. CONDITION NUMBER
The condition number with respect to inversion X(A) has been " defined as
where the Il''11 denotes mé\tr'ix norms and'A+ is the'pseudo inverse of A (30,37). The matrjx Amay be decomposed as follows
where if A is a general m x n matrix of full rank, Um x n
119
A. 3.1
A.3.2
and Vn ,x.m are both unitary matrices and En ~ n is a diagonal matrix. The ' matrix two~norm is defined as
"
r' "
A.3.3
so the s~bstitution of the decomposition yields l ,
A.3.4 -:
1 :
", A.3.5 !
1
. l •• ____ r._= ____ 1T_F __ ~X ____ M_· __ j_e_mt __ l _________________ J'1_'1_, _______ . __ . ________ ._r~i ______ .-I.---'d~~
(
"
, t
(
120
\ A.3.6
As eigenvalues and ~atrix norms ~re ,not, altered under unitary ~transformations (29), using V as the unitary transforming matrix yields
l,
- , \
The eigenvalues of A are in fact the eigenval~es of [ (29), the condition number may now be written as
1;
, ,
A.3.7
A.3.B
A.3. g,
The two-norm is d.efined as the lat'gest eigenvalue of the matrix. 1
, 'Thus,
x(A) = ÀmaX(A) x Àmax(t) A.3.l0
" The eigenvalues pre'sented in the decomposition are in fact the eigenvalues of r'and the eigenvalues,of A~ simplj the inverse of those of A, 'hence ; t i 5 ev; dent by the decomp,os'it; on that
'-..
, ,
, .. "1 .. , ,. "U
" , l ,
. ,
i
, ~
1 --------'~.P----__ • ____ ,_) .... .".w •••••• lF ••••• l., ••• _ •••• .-I1._ •••••• 21.-••• t •• I_.'.F_1f_t"_'_I_I~! _______ .r_' _____ r_·_' ______ .. _m __ , __ , ----.~J
\
,(: )f and , otheY"because "~;,, ,
'" ,
, ,
(;
! ",
, that as the eigenvalues of one are the inverse of the ris a di agona l matri x, the (i na l result i s that
X(A) = Àmax(A) Àmin (A),_
r 1
,l,
(
" .
..
1,
121
A.3.12
'. , \ ,
. ,-
, 1
\
---------n_~I_' ________ . __ I.J w-_______ .~_: ______ '._P~_. __ -'_. __ .~_-*~Ml~ l' 1
·c
•
122
APPENDIX IV: PROGRAM AND SAMPLE INPUT DATA
re,l evant In this appendix are presented the ma~n
subroutines written for the numerical main program embodies
computer program and of this work. The
in Chapter 111.
portion the GIM formulation descr~bed
The subroutines show the application of the Householder Decompos i ti on Each sub-scheme as well as the impl~entation'of the eigenanalysis.
routine has prev.iously been described Fo11owing the listings will
; n Chapter IlL ~
be. found two examples of input data organization as an aid to future users of this p~ogram.
. ~
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Input Examp le 1: z - 1 r-
- 2 c
'0·375 À 0.25~~
f '\ x , J 0·25À
0·375 À ,
-
~ ~
,Figure A.Iy.l: Two element yagi array with the driven element located along the z axis and both elements located' syrrmetrically about the x axis. The 'radi u-s of artenna (l) i s ·0.008" while the radius' of antenna (2) is 0.002L
t. ___ ~
T-aking twelve segments for antenna (l) and eighteen - ,
segments for ahtenna (2) and 'using a pulse 'tation while requesting
direct solution to~therwith the calculat'on ~f he,H-plane radia,tion
patter!l every -five degrees results in, the llowing, put data: "
l' , : 1
1 1 1
1 l " 1 ., , .
j. Il " "' " , .. l' " ,1 .. •
1
1 5 2.
r 1 .~
, ,
1
, 1 , 1
0.15
a·su , -0.002.
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, /. 1 • ,
, 1 1
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132
Il ,
lnput Exampl e 2:
For this -exampl e the same array configuration as thé
one in example one will be used. Howe~er th;s ti. the excitation
wi 11 be by a ma gnet i c fri 11 whose ou ter rad i us wi 11 ~e set to 1.5
times the antenna1s radius. Only the èigensolution with the first
three· modes is requested as· well as bath the E and H-plane radiation
patterns calculated every five de~rees. The input ;s as follows:
6.008 ,
..
1 ""J ....... "" :II 'J\ , -'1. '. 'lit _.
~ S 3
1 ~
, • • •• 1
, o.'u l O· $/. o. 002.,
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1 Il. 1
i .. 1 1
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.J ..... _ ........ .
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l h
t ! 1
133
(
APPENDIX V: EIGCC
The IMSl (Internatio'nal Mathem~t;cal Statistical librairy)
\
routine 'tIGpC was used to determine the complex eigenvaluës ~nd eigen-1 vectors of the Gerr~ral Impedance Matrix. The pages that follow are from ~e IMSL manual describing the implementation of the routine.
J ~
. , i
,"
1.
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1 " r
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l "-"\ i , .
r 1 • n
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134
, C SU~ROUTINE EIGCe (A,NdAdJOA,W,ZdZ,WK-,IEfH EICCOOlO C ;' . '" , E-I CC 0 0 2 0 e-EIGeC---------S/O-----LIBRARY 1---------------------------------------EICCOU)0 è c c c C c C 1 CI C C C c
~I c C c C c c c C C C C
, C C C C C c e e c C C t C C C C C C C C C C C C C C c, c c c C c
FUNCTI ON
USAGE PARAMETERS A
PRECISION
N lA
1 JOB
,1
w
z
IZ
WK , •.
IER
RECO. IMSL ROUTTN(S LANGUAGE
EICCOIHO - TO CALCULATE Etr.ENVALUES AND 10PTIONALLY) EICC0050
EIGENVECTORCj OF A COI~PLF.~ GENER6.L MATRIX. EICC0060 - CALL EIGee' (A,NtIA .rJOB,w.z.rI.~l(dERI EICC(1070 - THE INPUT CO~PLEX GENE~AL MATQIX OF OROER N EICC0080
wHO~E EIGENVALUES AND Elr.ENVEeTO~S ARE EICC0090 TO BE COMPUT:O. INPUT A IS DESrROYED IF EICCOIOO 1.J08 15 EaUAL TO 0 OR 1. EleCOllO
- THE ORDEP OF THE MAT~rx A AND MATRIX Z. EICC0120 - THE RO" DIMENSTON OF THE MATRIX A IN THe EICC0130
CALLING PROGRAM. lA MUST BE G~EATER THAN OR E1CC(ll~O EQUAL T~ N. EleC0150
- INPuT OPTION PAPA~ETE~. WHf.N EICco1bO I..1OB Il 0, eO"'PUT€ EIGE'NVALUES ONLY EICC0170 I~OB • }. CO~PUTE EIGENVALUES AND EIGEN- EIC~OldO
VECTQRS. E1CC0190 1.J0& • 2. CO~PUTE EIGF.NVALUES. EIGENVECTOR~ EICC0200
AND PERFORMANCE INDEX. / EleCOZIO 1.J08 • 3. CQ'-'PUTE P.E~FOR~ANCE INOEJ( ONLY. 1 E lCC0220 IF THE PERFOQ"'HICE INOEX 15 COP04PUTED. IT IS EICCO~JO RETU~NEQ I~ WKIl). THE ~OUTINES HAVE EICC0240 PERFORMEO (t/FLL. SATISF'ACTO~ILY' POORLYI IF EIeco;:so WK(l! IS (LESS THAN 1. ~ET\oIEEN 1 AND 100. E1CC0200 GREATER THAN 100). fICC0270
- THE OUTPUT COt-lPLEX VECTOQ OF LENGTH N.. E lCCll2tSO CONT~INING THE EIGENVALU€$.OF A. EICC0290
- T~E OUTPUT N ~y N CDMPLEX ~ATRIX CONTAINING E1CCOJllO 'THE ElGF.:NVECTORS OF A. ElCC0310 THE EIGENVECTOR IN COLUMN J OF Z CORRES- EICCOJ~O PONDS TO THE EIGENVALUE ~(J). EICC0330 IF IJOB Il O. Z \5 NOT U~ED. EICC0340
~ THE HO~ DIMENSTON~OF THE MATRIX Z IN THE EIcch350' CAL1.ING P~DGqAM. IZ MU')T BE G,.:lEATER TH AN EICC03bO OR EOUAL TO hl IF IJOB IS NOT EOUAL TO iERO. E.IecoY7o
wORK A~EA. THE LENGTH OF W~ DEPENDS EICCOJ80 ON THE VALUE OF t.JOB. WH~N EICCOJ90 IJOB • O. THE' LE~GTH OF' .K IS AT LEIIST 2"1. EICCO .. I)O I~OtJ Il 1. TI-I~ LE"IGTH OF' wl( IS AT LEAST ZN. EleCO'dO 1.J08 • 2. THE LE"IGTH OF wK IS AT LEAST ErCC0420
2 (N*N.Nl • , E leCD~JO [JOB ,. ,3, THE' I.ENGTH OF .,1( IS A.T LEAST 1. e: lCC0440
- EPHOR PARA~ETE~ Eleco-so TERMINAL f.RROR EICCO~60
1ER • 121i.~t INDICATt:S Tt'4AT Euune OR HRI-;I2CEICCo'Io.70 fAILED TO r.ONVERG( O~ EIGENVAI.WE J. EICco~dO
EIGENVALUES J.l.J·2 ••••• N HAVE ~EEN EICCO-~O COMPUTED CORQf.CTLY. EICCO~OO THE PEQFORMANCE IND~X IS SET TO 1000.0. (ICC0510
WAHNING ERROR IInT ... FIX) \ EICCOS20 1ER • 6tH l''IC'ICtTES IJOR IS LESS THAN 0 OR EICCllSJO
IJOB IS GRfATEH THAN 3. IJOt; SET Ta 1. EICCO~ioO 1ER. 67. IN'3--.!CATES IJOR 1S NOT EQUAL TO EICCO::iSO
ZERO, ANO/!Z IS LESS Tt-'AN THE OROEM OF E lCC0560 Jo4ATRIX A. IJOA 1S SET ro ZERO. EICCO~70
- SINGL.E/DQ.USLE ,- EICCOS~O - EBALAc.~RatKttFHESSe,ELRHICtELRH2C,UEPTST EICCOSqo - FORT~AN EICCOb04
c------~--~-------------~---------------------------~-------------------EICCOblO , n ' , 1 • \
-... ________ 7 'p.."d ?tns:f ...... HltI'Z:fl'qpDr ... " hll'''' ·tfiàt4+' .. Wt,Mttt'f...,.c;;Ml;Ie • .,..".....'.1A ..... e .p ln' [l;' ... » .... -~~-
1
!
, .
(
135
CALL EIGCC(A,N, IA,IJOB,W,Z,IZ,WK,IER)
Purpose
EIGCe computas eigenvalues and (option.lly) aiaenvectors of a genera1 complex IllAtrix. It can alao compute a performance ladex.
Algorlthm
EIGCe ca11s ~SL routine EBALAC ta balance th. matrixl Then, EHESSe li called to reduce the matrix .to Hessenberg form. ELRHlC 18 called if bnly eigenvalues are ta 'be computt;d. Othervise, ELRll2C and EBBCKC are called td comput. elgenv.lu.s and elgenvectora, and to baclttransform ei&en,vectors.
The performance index 1a defined as follows:
p •
yhen the IllAX 19 taken over the j dgenv.lues W j and assocl.ted elgenvec tors ,j. 'EPS spec if iu
the re1atlve precision of floatins point arithmetic. When P 1's less chan l, che F,erformance of the routines ls considered ta be excellent in the senSe that the residuala A,-v, are AS small as can be expected. When P il between 1 and 100 che performance 15 gaod. When P 1s greater than )00 the performance 1. considered ,poor.
The performance index \las firs't developed and used by the EISPACK project at Argonne National Labo ra tory.
See reference: Wilkinson, J. H., The Algebraic E1genvalue Problem, Oxford, Clarendon Presa; 1965.
Smith, B. T., Boyle, J. M., Garboy, B. S., Ikebe, 1., Klema, V. C., and!'!oler, C. B., liatrix E1gensystem Rout:.ines, Springer-Verlag, 1974.
1
Programming Notes
1. 2. 3.
A, 1s preserted when IJOB-2 or 3. In a11 ocher cases A 15 dest r.Jyed. The eigenvalues are unordared and the eigenvectors are not normal1,ed. When UOB-3 (1.e •• ta computa a performance index only) the e1senvalues, W, and elgenvectors. Z. are assumed to be 1nput.
Example
1 DUiL'lSION 1.(4,4),\01(4) ,Z(4.4),WK(40) COMPLEX A, loi. Z, ZN
In.put:
N • 4 IA • 4
1 IZ 4 IJOB - 2
A • 1 [
5.0+9.01 fi 3.0+3.01
2.0+2.01 1.0+1.01
S.0+5.01 6.0+10.01 3: 0+3.01 2.-0+2.01
, ~
-6.0-6.01 ' -S.0-5.01 ï -1.0+3.0i -3.0-3.01
-7.0-7.01] -6.0-6. 01 -S.O-S.Ol
0.0+4.01
)
, ,1 '
... ~. __ ~ .. ~ ____ ... , __________________ ... _-----"----------... i_ .. __ ~_I1""'.""; _' ..... _:~ ..... ,,·r"'~."'!J ...... "'! .• ~, ~'''i!!!''''!fi:'In~'''!!-~>'';'
1
( ,
( -,
'.
CALL EIGCCCA,N,IA,IJOB,W,Z.IZ,WK,IER) C NORMALIZE EICENVECTORS
DO 10 J-l,N ZN-Z(l,J) DO lOI-l,oN
Z(I,J)-Z(I,J) IZN 10 CONnNU! "
Output:
1ER • W
z •
WK(l)<l.O
",
o ' / - /f (2.0+6.01, 4.0+8.01,
;,; 1
~ 0+7.01, 1. 0+5. 01)
[
1.0+0.01 2.0+0.01
/.1. 0+<).01 1.0+0.01
1. 0+0. ai 1.0+0.0i 1.0+0.01 0.0+0.01
(performance index)
"
t
1.()-K).Oi 1. ()-K). 01 0.0+0.01 1.0+0.01
(e1gel'lva~ue. )
1.~.01]/' 0.5-+0.01 0.5-+0. 01 0.5-+0.01
!
"
" .'
#
:1 - / /
"
136
,>
J
/
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137
(
/ BI BLIÙGRAPHY
/
/ "
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