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AFCRC-TN.59-779 WW K»K iMSmSii^

/> . ^^ J^ NEW YORK UNIVERSITY ubrary

2 Nj W ^ Institute of Mathematical Sciences ^ ^''^ ?^ ^V y««t j, k x



'^^cccxx*^



Division of Electromagnetic Research



RESEARCH REPORT No. EM-142



Maximal Gains of Antennas



NATHAN NEWMAN and WILHELM MAGNUS



Contract No. AF 19(604)5238
SEPTEMBER, 1 959



ATCRC-TN- •y^-ll^



NEW YORK UNIVERSITY
Institute of Mathematical Sciences
Division of Electromagnetic Research

Research Report No. EM - II42



MAXIMAL GAINS OF ANTENNAS



Nathan Newman and Wiltelm Magnus



'VVj;(;;{i^ ^^*^



Nathan Herman



Wilhelm Magnus



~~\'^ tlr. Werner Gerbes v



Sidney Bopowitz a

Acting Director ^^ Contract Monitor

The research reported in this document has been sponsored
by the Electronics Research Directorate of the Air Force
Cambridge Research Center, Air Research and Development
Command, under Contract No. AF 19(60^)5238.



Requests for additional copies by Agencies of the Department of Defense,

their contractors, and other Government agencies should be directed to the;

ARMED SERVICES TECHinCAL INFORMATION AGENCY
DOCUf-ENTS SERVICE CEOTER
ARLINGTON HALL STATION
ARLINGTON 12, VIRGHIA

Department of Defense contractors must be established for A3TIA services

or have their 'need-to-know' certified by the cognizant military agency

of their project or contract.

All other persons and organizations should apply to the:

U.S. DEPARTMENT OF COMMERCE
OFFICE OF TECHNICAL SEKWCES
WASHINGTON 25, D.C.






Abstract



The gain of an antenna is defined as the ratio of the
radiated energy c 'to the heat loss-V^. For a linear, finite antenna
of length 2J^, this gain can be shown to have a least upper bound B
for all possible current distributions and for all /C. . We determine
B explicitly. For a fixed small value of J^ , the maximal gain is
computed up to an error term of oirier^ . If the gain of an
antenna is defined as the ratio of the energy (^ radiated into a
half-space, to the total heat lessor* similar but less complete
results can be derived. Finally, it is shown that the problem of
maximizing ^ /f does not have a solution. There exists no realizable
current distribution on the antenna which maximizes this quotient.



Table of Contents



Page



1. Introduction. 1

2. Notations and basic formulas 2

3. Integral equations for the current-distribution Hi
U. Behavior of the largest eigenvalue X, (S) for small ^ . 19

5. Direct methods, 25

6. Limit and upper bound for the radiated energy, 29

7. Integral equations for the second problem, JP

8. Behavior of X^(^), the largest eigenvalue of the

second problem, for small ^, 33

9. Bounds and limits for the radiated energy in the

second problem. 35

10. Comment on a certain problem, Ul

References hS



- 1 -

1. Introduction and Suinmary

It is well known and easily proved that we can approximate any-
arbitrary radiation pattern by using an antenna having a simple shape
as, for instance, a spterical antenna, if we can generate any prescribed
distribution of current on the antenna. To do this is a difficult
technical problem which we shall not investigate here. Another fact
which is almost trivial is this: let 0,6 be the angles in a system of
polar coordinates and let

P(0,&)
be a radiation pattern. This meflns that for large distanceG r from
the antenna, a particular component of the electromagnetic field is
of the form

ikr

2 — P(0,e)
r

Then P(0,&) is an analytic function of 0,9 provided that the antenna
is a finite one and that the current distribution is an absolutely
integrable fiinction of the coordinates on the antenna. It follows
from this fact that we cannot produce a radiation pattern P such that P
is identically zero in a cone with a finite opening as, for instance,
in a half space. However, we still can approximate such a non-analj'i.ic
radiation pattern to any degree of accuracy if we are free to choose
the current distribution on the antenna as we wish. Under these
circumstances, it seems impossible to define an " optimal" current
distribution.

Now it has been observed long ago by K. Fraenz (and probably by
others) that any discussion of " optimal" current distribution should



- 2 -



include the heat losses as a gauf^e in terms of which the " gain"
(no matter how defined) should be measured. Following this remark,
we shall investigate the ratios

p = total radiated energy to heat losses,
pr^ =» energj"^ radiated into a half space to heat losses
for a straight linear antenna of finite length 2)U

We shall show that, for a given j^ and a given wave number k, both
p and p''*' have well defined maxima. Their dependency on ^ will be
investigated, particularly for small ^ and for jc->oo. Finally,
we shall discuss the problems which arise if we try to find a
maximum for a'= p^'/p.

The maxiFia of p and p* for a given k will be computed as the first
eigenvalues of certain Fredholm integral equations (3.6) and (7.3), respectively.
For this reason, the actual maximum of p will be denoted by X.. (^) and the
maximum of p )^(^) are monotonlcally increasing functions of ^ , They



-3-

satisfy the inequalities

and the limit relations

lim KU) = lim xf(^) « A ^Sn

^-> 00 i-> 00

For small values of k£., the eigenf unctions (i.e., the current distri-
butions) belonging to V and X.. can be expanded in power series in
k^ with coefficients which are power series in kz. However, the
exact range of convergence of these power series is not known. For
large values of kA , practically any current distribution will produce
•' almost " the maximal amount of radiated energy; in particular, this
xiiill be true if we have either one of the functions

cos(n Tt z/A ), i sin(n n z)/£ , (n=l, 2, 3, ...)

as current distributions. For details, see Section ^. Obviously,
no such statement can be true for the current distributions belonging
to \. , unless we take a kx which is so large that X_ is practically
equal to X^ .

Whereas the ratio of radiated energy (either into the whole
space or into a half space) versus heat losses can be maximized by
current distributions which are continuous and even different iable
functions of the coordinate z on the antenna, no such statement can
be true if vie consider the ratio p*/p of the energy radiated into a
half space versus the total radiated energy. This will be shown in
Section 10. If we try to maximize p*/p, either the current distribution



-Ij-



will tend towards a function which is so singular that the usual
arguments leading to an integral equation for this f\inction cannot
be applied, or any sequence of current distributions which produces
a monotonically increasing sequence of values of pVp will be such
that the values of both p-«- and p tend towards zero.

2. Notations and basic formulas.

Consider the time-periodic linear current distribution, I, given
by the function

I = f(z)e

Where f(z) = a(z) +i^(z) } i = f^ ', ^ is the (constant) frequency;
X, y, z and t, Cartesian space coordinates and time respectively.

We make the following assumptions:

f(z) continuous for -jl < z 0.

3) >f , the heat losses.



1) The Pete nninat io n of Cj the Radiated Energy

(^ UcR
Let M = .2-— f(s)ds

k ^

where R = /x +y +(z-s) and k(wave number) = — j

c the speed of light,

I-et M = (0, 0, M), Then the vector i"otential N of the electromagnetic
field generated by this current will be given by

Then the electric field E and magnetic field H will be given, respectively,

by

E =

CO



E = — curl curl H, and H = curl N



The following four steps are an outline for the procedure for finding
the energy radiated per second;
(i) Consider the time-dependent Poynting vector

S(t) = 2ReE X Ret . (x : vector product)

Let a be the time-average of S(t) so that

" ' tSoo 1 J 2(He^ X Ret) dt



(ii) Let S be the radial component of S.



Then



S = - St + 2 So + - S-
n r 1 r 2 r 3



/' 2 6 2 ^

where r » yx + y + z and S = (S,, S-, S^),



(iii) Let A be a large sphere with radius p, and center at the origin.



-6-

Form the surface integral



S dA - 8
I n p p

A
P

(iv) Take -_>qq C " C, the radiated energy.



We carry out step (i) as follows:
Writing

t = e"^*(| curl curl 7^), f! = e"^*(curl t)

and letting

- curl curl ^ = EL+ iEp » curf !t = Tt + iiL

we have

E - e-^*(f^ - i^2^ and t - e'^^it^ + d^) ,

so that

2Re!^ X Re^ = 2^ ^ x Ttj^ cos^cot + ^^x'^ sin^cot

+ 2(^^ X "^^ + t^ X t^) sin cot cos cot



Theni

/o



2(ReE X ReH) dt



sin coT cos coT



(, , sin co^^cos COT ) ^^ ^ ^



+ (1 . sin COT cos COT) ^ ^ ^
COT 2 2



Letting T -►>



(overhead bar indicates complex
conjugate)



S = Re J (i curl curl M) x ( curl T^)

Gi

Step (ii). In this section, subscripts x, y, z means partial differentiation
with respect to these variables.



curl M = (M , M , 0) ,
y' X '



— curl curl M = —

CO CO



K , M , -(M + M )
xz ' yz' XX yy



and



S = Re



(- curl curl fi) x (curl M)



. Re j^- i Y.



J



_ (M + M ), l (M + M ), 1 M + !? M
X XX yy y XX yy ' x xz y yz



so that



S = Re.
n



Z/^t;



- - U^ 1 (M +M ) + ^ ?? (M +M ) + -(M M + M M J
CO r X XX yy r y xx yy r x xz y yz



yy



y y2



We now compute the derivatives occuring in S ^



X R ' y R » ^z - ^ '



, ikR, ikx ikR / ikRv iky ikR / xkRx ik(z -s) ikR
(e )^=-^e ; (e ) = - ^ e ; (e )„ = -As '- e j



'x R



'y R



R



ikR ..^ .,
/e V ^ikR /ikx X n



ikR .,c
^e >> _ ^ikR



R^ R^



.IkR



(V~) - e^C



ikR/ik(z-s) (z-s)



R'



)



-8-



ilcR iiV -^^-^ -^ '-2



(V->xx- ^



3iK 2 k^ 2 ^ 3 ^2 ^ ik 1



^^ ikR/ 3ik 2 k^ "

R yy



R R-



,e^^N ikR/ 3ik 2 k"^ 2 ^ 3 2 _^ ik 1



R-



R'



ikR



(VV.-""(-f ^'-='-g >• *^^ \ >

Z 1 (M +M ) = y ^ . ^ "^y ^ P(a) ^(bT + ^ P(b) TCbJ
r y XX yy r r ^ '

2 2 2

f \ \z = ^ ^^^^ "^^ - -^ \^^^ ^^^

^ \ tt = 4^ P(a) PTbT - ^ P.Ca) HbJ
r y yz r r J.

but the siun of the terms on the left hand side of these equations

constitutes the bracket in 3 • Adding the right hand sides of these

n

equations we get

.(x^.yW)(x^V^) p(^) ^^ ^ ,^£^ ,p(^),2 _ z(xfi£) p^(^) 7^



Letting x = r sin 9 cos

y = r sin 9 sin
z = r cos

this Stan becomes

r^ sin^ © P(a) TTbT + 2r sin^ |P(b)|^ - r^sin^© cos f) P^(a) KbT

so that



S = Re ; - i
n 1 CO



r^sin^ P(a) Kb) + 2r sin^e |P(b)|^- r^sin^e cos 52^P^(a) TTbT



Step (iii) Substituting spherical coordinates in the definition fore ^nci
letting p = r gives



-ID-



,2n



K'])



r'^ sin © S d^^ d©

n ^



o



J ReU T^slr?^' cos )?} P^(a) P^b)) d0 dS

'0 0^ /

11 Re (- ^ 2Ain^ |F(b)|^ld$2( d9




Step (iv)



Let U^(3,R) = -k^e^Cs)
U,^(s,R) = -3ike^'*f(s)
U^(s,R) = 3e^f(s)

V^(s,R) = -k^s e^^f(s)
V|^(s,R) - -3iks e^(s)
V^(s,R) = 3se^''^f(s)

W^CsjR) = ik e^(s)
and W^(s,R) = -e^(s)



Then P(a)




V



\U) =



and P(b)



n=3 Ij^ R



n
n



ds



^l



W
— ds

r"



-11-



Pnd



P(a) TTb)



S 5



^i



V7



m



Jjin



-/



ds — ds



m = 2,3



min (j + m) = 5', where j = 3 and

P^(a) PTbT



m



V

— ds
n,m I r" ;

4 -



W

-i2ds

r"



P(b) KbT = 21



min(k + m) = 5



min (j + j') = U .



n
m



3,U,5
2,3



2,3

2,3



I«t Er = ^1 "" ^^2 "■ ^-^



where /U /2ir



i «ii...,3c



Q^ = I I Rel i r^sin% cos {i^ P^(a) TTbJ dji^ d9



o 'o
/n /27T



«2 =



>i I



CO



o \



Re I - ^ 2r^3±T?9' \ P(b) |^ ] d0 d&
T^s±t?e P(a) PTbTld?^ d&



then li™ C = li^ q . li^ q + 1^ q,
r->oo *-r r->oo 1 r->oo 2 r->oo j



We note tiie following statements:
As r ->co



-12-



A) R = r-scos» + o(-)

r

ikR ikr -iks cos » °^r^
e = e e ® i

-HcR -ikr iks cos © r

e =? e e e



I



n
C) lira — = 1

r"



Itm



.i >0



R'



D) Ifj^



-iks cos ©^/ s ,
e i(.s; ds



de



y



Letting s = u in the first bracketed integral and s = v in the second
bracketed integral, we get

^-iku cos 9^(^) ^^ j ^-ik cos a ^(^^ ^^

^-ik cose(u-v)^(^)^^^^^^






-/ -/



cos



r(u-v)k cos 9] (a(u)d(v)+p(u)p(v)j
-sinr(u-v)k cos e fa(u)p(v)-c(v)p(u)J



du dT ,



-Ill-



Substituting back and changing order of integration we get



£



f^^






[a(u)c(v) +p(u)R(v)J



CO



'-i -I



sin^e



cos (u-v)k cos BJ



a(u)p(v)-a(v)B(u) J sin"^© sin (u-v)k cos » dB



dudr



dudv



We note that
/n/2



/



sin-^Cr) co3(S^ cosy) dy = y2v. - \i>^



/Tt/2
and I sin^Cr) sin(i^ cosy) dy = ^ ''-^






where J and H represent the Bessel and Struve functions respectively.
It is clear that

I sin-^Cy) 3in(5^ cosy) dy = - I sin-^Cy) sin(^ cosy) dy



and



V2



sin'^(y) cos(S^ cosy) dy = I sin-^(y) cos(^ cosy) dy
V2 ^



Therefore
/It



sin^e cosr(u-v)k cos» d« = 2/2ii -^^



(k(u-v))



(k(u-vjj^



7?



and



/n

sin-^» sinj(u-v)k cose de = ,



-15-



So tl^t wa get for the radiated energy,



CO



.:^ ^^



/ rk(u-v)l / X

llfz J (a(u)a(v)+p(u)p(v)) du



dv



2) Computation of £ , the energy radiated to upper half space «

The four-step procedure outlined for computing c is repeated
identically with one exception. In step (iii) instead of forming the
surface integral with A , a sphere of radius p and center at the origin,
we now form the surface integral with A , the half of this sphere lying
above the x,y plane. In carrying out this computation foro this
means only the difference of taking the limits of integration with
respect to & to be ^ to n, instead of to •^.

We therefore get

d /I



C* _ 2Tik

^ " CO



3



\v^






|a(u)a(v)+ p(u)p(v)| sin^e cos (u-v)k cQs& dB



dudv



CO



r-l



|a(u).e(v)-a(v)(3(u)J 1 sin^& sin (u-v)k cogelcB



du dv



Substituting from above we have therefore the energy radiated to upper
half space,

j^3^Jk(u-v)]



^-fb^^"^^^'^^^"°^^^'^^M'"



dv .



-16-



3) De-finition of)f, the heat loss
We take the heat loss to be

C f(s) TTsT ds

where C is a constant depending on the carrier (antenna) of the current.
'iVe arbitrarily set C = 1 so that we can write

W = J p(s) + P^(s)] ds

-i

Having defined tlie functionals ^(f), ^(f) and^(f), we now pose the
following three problems:



I, Find p.'i), where p(^) = ""f fe}



T*/



II, Find p*(£), where p*(je) = "^"".^^fT



max



(f)




and ni, Find cTCi), where tfU) - ^



3, Integral equations for the current-distribution.

The radiated energy £and the heat loss Vt are functicr^ls of
the current-distribution f(s) =a(s) + ip(s). If we write

9 r -1

(3.1) £(f) = Y V ""Wl t(u)a(v) + ?(u)p(v)l dudv ,

; ; k(u-Y)]3/^ ^ J



(3.2) H(f) - I p(u)+p2(^)j dy ^

-X



where



(3.3) Y - I (2n)3/2k3 ^



17 -



C(f)
Then the problem of maximizing ^^^ over the class of functions,

0(3) + ip(s) = f(s) 6L^(-/,/)(s real), is the same as maximizing 6(f)

subject to the condition^lCf ) constant, over the same class of functions,

Setting the first variations equal to zero yields the equations



(3.1i)



(3.5)



I J



3/2



[k(u-v)]



J






[k(u-v)]



Tn



a(u) du = Xa(v)



i



J / Jk(u-v)l
[k(u-v)J^/2



We note that both a and p satisfy the equation
(3.6) 7^(v) =



Y y^^ no/o i2S(u) du



/.



[k(u-v)J3/2
If in (3.6) we let u = £x and v = h ^^ ge't

[kX(x-y)]^/^



(3.7) Xf(y)



where f(x) = 5^(^x) arrl f(y) = ^{h) •



We note that



/OO



^-00






dv "



3L/|e^^(k2-32), |s|k



k3'^







If we let



^00



e^^^ 0(v) dv » g(s)



-00



-18-



then



'OO



^(u)



7H



-itu /.\ ,.
e g(t) dt



/-OO



Therefore operating on both sides of (3,6) with



/OO



>



IV s



dv



/



00



we get



r



Xg(s) = ^ ^



k^ 2(/2n



-X



4



ius/, 2 2



e"'"^(k^-sO, Is! k



V




e"^^" g(t) dt



du



Noting that I e^^^^"'^^du = 2



This equation can be written



sin [X(t -s)]



t -3



(3.8) xg(s) =-^ —
k^ /2n



^00



-00



sin [^(t-s)]



t-s






g(t) dt



If X 4 C> thj3 equation becomes

(^ sin [^(t-3)] 2 2
(3.9) Xg(s) = ^-i- i—:^ i (k^-s^) g(t) dt



Letting t = kx and s * ky gives

A sin



(3.10) Xg(ky) = -1.



^ i\ k(x-y)



M(x-y)] 2

^ -^ (1 - y^) g(kx) dx



- 19 -



Letting ' ^



OSSI— = h(x) and ^^i^L. = h(y)



2



vCx^ v4^



^ .■ .,, , sin ?^ A" ''1/2''^'

and noting that — p — ~ ^ "3 — -i /?



this eq\iation becomes

(3.11) Xh(y) = [ U '^y^ ^ ^Wp"* /-y"^ i/^ h(x) dx .

We note that to every eigenvalue of (3.6), (3.7) and (3.11) there
corresponds at least one even or one odd eigenfunction.

We will denote tte kernels of (3.6), (3.7) and (3.II) by K, K^ , K^
respectively . We note that the set of eigenvalues of all three equations
is the same. The kernels are all symmetric sind analytic in their arguments
for >ji< QD . Therefore for any /^ be fixed and



p(r',s) =<



9^l(^',s) , |s| K(u,v) p(r ,u) p(/",v) du dv .

'X -I



I Si' /i'

K(u,v) a(^',u) 0a',v) du dv =X3_(X')



U. Behavior of the largest eigenvalue \At) for smallj' .

Rewrite equation (3.7) in the form

fl J-/JkZ(x-y)]

(U.l) X f(y) = Y -^^^ TTT? f(x) dx

y^ [k^(x-y)]3/2



-22-



where



Let






K. =V-^^






= Y^ ^Jk(x.y)p"



n=o



where O- = (-1)
2n



. (2n+2}I ■ 12?^



^TTT J



K, is an entire function of k zr\d jL for bounded x and y.



Let



m



and



Djx,y)=



A



^-1 "-1 ^-1



Kj^(x^,x^) K^(x^,X2).
Kj^(x2,x^) K-j^Cx^jX^).



.K^(x^,x^)
.ir^(x2,x^)



^(V^^ K^(x^,X2).



Vx ,x )



dx, dx^. . . .dx



• • • • 1



K^(x,y) K^(x, x^),
^]^(x^>y) 1^(x^,x^).



.K^(x,xJ
.K^(x,,xJ



'Ki(x^,y) k;^(x^,x^) ^^(V^m)



dx,dXr,. . . .dx
X c m



Consider the disciominant of IL

nl ~n



D(|a) = 1 +



D (£) tx"



D(h) is an entire transcendental function in |j. and JC . The zeros of D(|j.) are
the reciprocals of the eigenvalues of (l;.l)i.e. -j-i — = n. (£) , where

11. (j^) are the zeros of D(|x) for any given £ .



-23-



For any fixed J? , if [^^(/q) is a simple zero of D(ia), we can
find a small enough circle in the [i plane around P-j^(X ) since the
spectrum is discrete, such that
f



and



Ti^



/



where a.(J? ) is the residue of rrrr at [i^iX^),



We observe that since ^1 — y ^"^ m l ^^^ analytic functions of X,

if we can find a neis;hborhood c(K ) and a corresponding neighborhood
e'TiJ-.O^ )j such that for any ^ in t{£^) all |J..(^) j 4 i are excluded

from £'|M..(/ ) then a. (£.) and a.(/C)n..(£) are analytic functions of

in e(^ ), hence ij..(J?) is an snalj'tic function in e(J?-)'

For the kernel K^ we have

D3(£) =-y^[(k/)^ ^^^2 6U ^ j^

In general D (Jt) are even power series in kJL where the lowest order
term in D (X) has the povrer 2n -2,



-2h-



Vfe are now able to demonstrate the anal^aicity of IL {Z) in the
neighborhood oT £ = 0.

If in D(n) we set /^= we get D(ii) = 1 - ?r^ p.. This has only

the zero n^ = v. . /ill other m - becone infinite. We now show that there

raust be a neighborhood of /= in which \i.{)^) - [i.{Z) > t > C, t a
constant for all i ^ 1,

We note that

1) V-A^) are continuous for ^ 4 0.

2) n^(£) = min[n(/)]

3)



H^C'^) =



mini ti(|!)lti(;^) > |a^(J?) •, for those .^ where ^^(-^)






1^



(^)



L



is sinple

, for £■ where \u{^) is
multiple



For i 4 1 all n^(^) and particularly n (/?) tend to infinity as /-> 0,



and since

there is a neighborhood of Ji= where ij.^(je) - |j^(J^) > 6 > 0. 6, a constant.

Therefore m-j^(a) is an analytic function of JZ. in this neipihborhood.

From the knowledf^e of the regular analyticity of [lA^) in the
neighborhood of zero, we find that



^



We



i


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