Current, input impedance, and far field pattern of cylindrical antenna with tapered resistive loadin

Shen, L.-C.

Wu, T.-T.

English

NASA Technical Reports Server

THE CYLJNDRICAL ANTENNA M 
TAPERED RESISTIVE LOADING 
Scientific Report No. 6 
Liing - 
BY 
Chi Shen and Tai Tsun 
IT t 
August 1965 
HATIONAL AERONAUTICS AH0 SPACE AOMI1YtSTRATION 
Preprrsd under Grant No. MsG 679 
6wdm McKay Laboratory, Harvard Univenitr 
Cambridge, Massachusetts 
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https://ntrs.nasa.gov/search.jsp?R=19660026809 2020-03-16T20:29:09+00:00Z
. 
THE CYLINDRICAL ANTENNA WITH TAPERED RESISTIVE UIADING 
Liang-Chi Shen and Tai Tsun Wu 
Scientific Report No. 5 
Augue t 1965 
Prepared under Grant No. NsG 579 at 
Gordon McKay Laboratory, Harvard University 
Cambridge, Massachusetts 
for 
NATIONAL AERONAUTICS AND SPACE ADMINTSTRATION 
. 
b 
THE CYLINDRICAL ANTENNA WITH TAPERED RESISTIVE UIADXG 
Liang-Chi Shen and T a i  Tsun Wu 
Gordon M c K a y  Laboratory, Harvard University 
Cambridge, G s a c h u s e t t s  
SupfMBRp 
The current, the  input impedance and the f a r  f i e l d  pat tern of a cy l indr ica l  
antenna with resistive loading are determined. 
loading along the antenna is a part icular  function multiplied by a constant posi t ive 
parameter a. 
wide range of lengths with several  different  a's ranging from 0 t o  1 and f o r  posi t ive 
integers.  
o r  greater  than 1, the  antenna is non-reflecting. 
The dis t r ibu t ion  of the resistive 
A 
The current on the  antenna and the f i e l d  pat tern are evaluated f o r  a 
They are found not c r i t i c a l l y  dependent on the  parameter a. For a near 
INTRODUCTION 
In a recent paper by Wu and King [l] it is found tha t  i f  the antenna is made of 
resistive material such tha t  the in te rna l  impedance is a par t icu lar  function of the 
posi t ion along the antenna, a pure outward traveling wave e x i s t s  on an antenna of 
f i n i t e  length. They found t h a t  i f  
then the  zero-arder current w i l l  be 
ikz 
I(z)  C(h - I z l )  e 
The constant 'I i n  (1) is determined by 
h 
i k  /- (2-2 ) + a 
e dz ' 
I 
-h 
i 
I n  the  above equations, z is the axial coordinate, z- is the in te rna l  impedance per 
u n i t  length, ro is the i n t r i n s i c  impedance of f r e e  space, h is the half length of 
t he  antenna, k is the free-space wave number, and a is  the radius of the antenna. 
The time dependence is assumed t o  be e -&ut . 
- 2 -  
It is in te res t ing  t o  see how the  t ravel ing wave solut ion is affected when the  
d is t r ibu t ion  of the  r e s i s t i ve  loading is changed from the  value prescribed by (1). 
A change fn t he  internal impedance#involves-two problems. 
finding the new current dietrfbutfon, i.e., solving the  changed d i f f e ren t i a l  equation. 
Secondly, and more inportant€p, is the requirement-that t he  solut ion obtained be 
integrable *the sense tha t  the  f i e l d  pat tern can be obtained numerically without too 
much complication. 
F i r s t ,  is the  task of 
It turns  oat tha t  i f  the internal impedance is changed to  
(4) 
- 4 ~  h - ~ z J  
where a is a posi t ive constant, the  current d i s t r ibu t ion  is the  product of the l i nea r  
decaying t ravel ing wave function (2) and a confluent hypergeometric function. 
and by a new method.; the f i e l d  pattern can be cast in to  a form which is readi ly  
Moreover, 
evaluated numerically. 
It is-clear that-when-a - 0 the current d i s t r ibu t ion  is expected t o  be ident ica l  
with the  zero-order current of an ordinary dipole antenna. When a = 1, the current 
should yield King and Wu's resu l t .  Thus, two references are on hand. 
. TBE DIFFGREIUTIAL EQUATION'AND'ITS SOLUTIOEI 
The d i f f e r e n t i a l  equation.to be,solved is the following: 
For z > 0 ,  let  , 
ikz 
I (z)  = A e ' (h - z) -@[2ik(h - z ) ]  
then, from ( 5 )  
- 3 -  
where y - 2ik(h - E). 
t ha t  @(y) = B O(l-a,Z;y). 
convenient t o  define @(y) by a contour in tegra l  as follows: 
A comparison of (7) with equation (6.1.2) of Reference 2 shaws 
Since a is t o  vary from 0 t o  any posit ive number, it is 
8 
a 
@ (y) ’. = . f  ep (e) du 
C 
Subst i tute  (8) into-(7) ,  and the resu l t  is 
One Of the  possible choices of C tomake the r igh t  hand s ide  of (9 )  vanish is Shawn i n  
Fig. 1. 
boundary condition that I(h) = 0.  
Such”a choke  also mak,es O(y) fiaite when y -+ 0 so that it s a t i s f i e s  the  
Thus, the current d i s t r ibu t ion  is forWtlly expressed as 
i k l  I 
I(z) = A e (h -Iz*I> *[2ik(h - 1 ~ 1 ) J  
The constant A can be found from the equation 
It is found t o  be 
4nik VE 1 
A - T COY 2 
Note t h a t  the confluent hypergeometric 
a x  @(a;c;x) = 1 + -- c I! 
and has the  in tegra l ’  representation 
ftmctian is defined as 
2 +a0 x+ 0 . .  
c(c+l) 21 
1 
- 4 -  
a 
Evidently (13) cannot hold f o r  a 0, But i f  'D(a,c;x) is defined i n  terms of a 
d i f fe ren t  contour, say along C in Fig. 1 ra ther  than from 0 t o  1, as follows: 
C 
then i reduces-to 0 (except for a complex constant) for a > 0 and c-a-l > 0.  It c m  
*be shown that (14) is equivalent t o  the  series representation (12) even f o r  a 0 and 
c-a-1 > 0. This fs-accomplished by means of.the-foflowing r e l a t ion  
(15) d -  a -  dx 'P(a-;c;x') = - 4(atl,c+l;x) C 
I 
Repeated use of (15)would f ina l ly  make the f i r s t  argument of CP on the  r igh t  in (15) 
posit ive,  and whenever it is posi t ive i t  is equivalent t o  the  series representation 
(12). The series is then integrated as many times aa the  relation (15) w a s  used. 
This completes the proof. 
Since i n  the present case a = 1-a, it follows tha t  i f  a is a posi t ive integer,  
'D(a,c;x) i s , a  polynomial as-seen from (12). 
EVALUATION OF 'I 
y is determined by* the  follbwing equation: 
h i k  /7 (z-z ) + a 
ikiz' 1(h- lz ' l )#[2i~(h-lz '  l)]yz-z, 2+ a2' dz ' 
-h I. 
Y =  " 
e ik I I (h- I z I ) 4 [ 2ik( h- I z I ) ] 
when z = 0 ,  
dz 
h i k z + a  v
rn 2 eikz(l- E) @[2ik(h- z ) l e  - 0  P (17) 
#(2ikh) 
I f  t h e  integral . representat ion (8) fo r  0 is substi tuted i n t o  (17), and r z + a is  
equated approximately ~ o ' c ,  the result is 
- 5 -  
* .  h 
zWh-ro)u l-u 2 lu 2ltrh(h=t)u lu 
(;I d u d z - * l e  f e  (y) du dz 
0 C T =  
(18) I :ibu I-u a I (3 du 
C 
where ro = m. A change in the order of integrat lop of (18) gives 
T =  1-u a e2ikhu du (19) s (* 
C 
where A = ka(1-u), E - Ich(1-u), aod C and S are tabulated functions [3, Appendix] 
whose defini t ions are as folluus: 
_. t.3' 
/E F sin u + a 1 - c o s u + a  X 
Thus f ina l ly  
Y = 2 s i n h - l h +  y a 
where 
128 i a 2ikhu e [iS(2A,ZH) -C(2A,2H) - $1-e ) ]  du - 
a 2ilchu y =  2 L  
C I(+) e du 
The i n t eg ra l  in (22) can be evaluated numerically. 
- 6 -  
THE FIELD PATTERN 
then the  f i e l d  p s t t e m  i s  essent ia l ly  3 s )  + *-C) 
(h-z)I"(g) + k2(h-z)I(z) + 2iak I (z )  
When z >, 0 ,  (5) gives 
c 
- 
= V (h-z) 6(z) 
where 
The following re la t ions  are known: 
, (24) 
* -  
("(z) (h-z)e dz - -hI'(O)+I(O) (-l+ihy) 
0 
h 
- (z) [ c2 (h-z)+2ic] e i cz  dz 
0 
Clearly I ' (0)  
both s ides  of 
is zero s ince I (z )  is aq even function. 
(24) from 0 t o  h with respect t o  z, t o  get 
The next s tep  is t o  in tegra te  
Bote that the  6-function has been taken care of properly. 
(26) is 
The integrating factor  of 
l+a 1-a -ihc 2 2  (k -5 )h + 2i(ak-c) 
2 2  = e (W) ( k - ~ )  f (k  -i 1 
Thus t h e  solution of (26) is essent ia l ly  
3 5 )  e -ihC (k+C)'* (k-C)l-a - k2 
. 
- 7 -  
It follows with (10) and (11) and the f a c t  that F(*l)  is bounded, t ha t  
l2lChu a 1-u 
al(kh) = -Ah du 
C 
a 
a2(kh) = -Ah 
C 
Usually the f i e l d  pat tern is obtained from a d i r ec t  integration of (23). 
subst i tuted i n t o  (231, t h i s  becomes 
If (10) is 
ikz i C Z  2(r) = A I e (h-z) 0[2ik(h-z)J e de 
0 
- 8 -  
If' the second term is integrated by parts the  r e su l t  is 
- ih(r+k) i2khu - 
c 
It can be show tha t  (29) satisfies the  differential-equat&m W). The procedure 
involves only s t r a igh t  forward subst i tut ion although the  algebra is tedious. The form 
of (29) is inconvenient f o r  numerical evaluation. 
g(c) when a is a posi t ive integer. 
However, it can be used t o  calculate  
.r 
NUMERICAL CLLLCULATI~S 
A. The Field P e t t e r n ,  Oc a c 1 
The electric f i e l d  pat tern is obtained from (281, since the electric f i e l d  is 
1 a 7 [al p,(y) + a2 p2(y)l 
1-Y 
%ot= 
ikhz 
P1(y,lch) = '7 dz e (2ikh - a - ikhz) 
(2 - 0 
- 9 -  
8 
ikhz 
d t  P2(y,kh) = i' e likh(2-e) (1-z) -a(l-z)-(2-z)] 
(2-2) l-O 0 
4-Y - 1 k h Z  
dz a (2-1) e 
(2-2) a 0 
2 ikh 
e -A h I 
FO 2 (1-a) 
B. The Current, 0 a c 1 
From (10) and (28b) the current is expressed in the following form 
Or, as seen from (30b), 
ikh(1- 'E' 
]+( 1-a) e 
(32) 
The numerical integration for the P-functions cannot be done by standard methdds 
This is due to the fact that the integrand has no Taylor's such as Simpson's rule. 
series expansion at one of the integration limits, namely, z = 0. 
integration near such a point is developed in the Appendix and is tried here for 
calculating P-functions. 
A formula for numerical 
C. The Current, when a is a positive integer 
From KUmmer's series (12), it follows that 
ikz 
I(z) = A e (h-z) for a = 1 
- 10 - 
ikr: 
I(z) - A e (h-z)[l - ik(h-z)] f o r  a = 2 
2 2  2 ike I(e) - A e (h-e)[l - Zik(h-z) - + (h-e) ] f o r  a = 3 
And so on. 
D. The Field Pattern,  when a is a posit ive integer 
From ( 2 9 ) ,  t he  electric f i e l d  pattern is  obtained easi ly .  1 
fo r  a = 2 
W+Y)kh 
F(y) - - (6ikh + 12k2h2- 4ik3h3) + (6 - 6e - 24ik4 -12k2h2) 
1+Y (W2 
f(l+Y)& i ( l + Y )  kh + (-24 + 24e + 24ikh) + ,+(24 - 24e ) fo r  a = 3. 
( l e )  (l+Y) 
And so on. 
PERTURBATION OF a ON TEE CURRENT 
For a near 1, the P-functions (31) can be expanded i n t o  powers of (1-a). 
1-a fyz 1 
pl(0,Y) J I e [12y - iyz - 1 + (1-a) 1 dz 
0 
and subs t i t u t e  t h i s  i n to  (33). It becomes 
(33) 
- 11 - 
i Y  
where 
Pl0(y) = 1 ? [f2y - lye - 11 dz = e - 2 
0 
me follawing re la t ion  has been used i n  the above integration: 
1 I In 2 dz\e )dz = (e - 1 ) l n  x + c(0,yx) - is(0,yX) d iYz iYx 
0 
Similarly,  we expand P3(0,y) but s d y  its f i r s t  term is neq4d as can be seen from 
(32) 1 -fya 1 -iye 
In order t o  get f i r s t  order perturbation t e w ,  we have t o  expand the coeff ic ient  A. 
ikh 
@(2ikh) = 9 = 1 { l-(l-a) [C(O,Zlrh) - iS(0,Zkh) + s i n  kh]] 1-a 
2ikh 
1 1 1 - e  co'(2ikh) = - - 2ikh [l + 2ikh 
- 12 - 
2nikve 1 
Let, A = A. + then 
2ifi-3 , 
(fkh-l)[C(0,2kh) - iS(0,Pkh)l + e 
(ikh - 1) Aa = Ao(l-a) 
Let, I(t) = I , ( t )  + Ia(z), then 
Note that (34a) is exactly what has been obtained by Wu and King [ref. 1, eq. 211. 
When a = 0, the expansions are as follows 
where * 
fyz 
e (2iy - iyz) dz 9 eiy - (eiy- 1) 
iY 
Pl0(Y) = j 2-2 
0 
- 13 - 
where 
ikh 
sin k(h- I z  1) i(, (2) 3 - e * o  k 
ikh 
Ka(z) = -e  lsin k(h-121) + S(O,2k(h-lzl)) sin k(h-121) 
k \  
1 
i - iC(O,Sk(h-lzJ)) COS k(h-ltl) 
Sikh 
1 
J 
2ikh e -1 - a -. ' [ S (0,2kh)+C (0,2kh) ]e +C (0,2kh) -S (0,2kh) 2ikh ( W i k h )  = 2 i h
2 ikh 
e [sin2kh+l-cosZlrh-S(0.2kh)-C(0.2kh)]+[l-cas2kh-sin2kh-C(O.2kh)+S(0,2kh) + 
(2ikh)2 
A A o + a A a  
- 14 - 
s i n  k(h- I z I) 
cos kh I0(z) AoKo = - 
COY 
Ia(z) = AaKo + AoKo + AoKa (35b) 
Mote that (35a) is exactly what has been obtained by King, [ref.  3, eq. 11-18.5). 
The f i r s t  order perturbation terms are i n  terms of tabulated functions; thust 
hand calculation of t he  current dis t r ibut ion is possible for  a e i the r  near 1 or  near 
zero. The following t ab le  shows the comparison of the current obtained by numerical 
integrat ion (carried out by computet) of the exact formula (32), denoted i n  the  tab le  
as IA and #A, with tha t  obtained by perturbation formulas (34) and (35), denoted as 
I and eB. B 
z - 
h 0 ,oo 
1.000 I A  
1.000 I B  
0.000 
@A 
0 . 000 
@B 
a = 0.75, kh = 2n 
0.25 0.50 0.75 
0.818 0.589 0.371 
0 . 812 0.588 0.343 
1 . 564 3.129 4.658 
1.559 3.120 4.651 
c 
a = 0.25,  kh = 2a 
z - 
h 0.00 
1.000 
1.000 
0.000 
IA 
IB 
#A 
@B 0.000 
0.25 0.50 0.75 
1.450 0.754 1 . 260 
1.230 0.782 1.170 
1.561 3.132 4 . 661 
1.570 3.141 4.710 
EFFICIENCY 
To ca lcu la te  t he  efficiency, the power l o s t  on t he  r e s i s t i v e  antenna is compared w i t 1  
t he  power radiated. 
Over a l a rge  spherical  surface i n  the  f a r  f i e l d  ra ther  than by calculating the  input 
The radiated power is obtained by integrat ing the Popnting vector 
impedance which is only of zero-order accuracy in  the present theory. 
The diss ipated power can be found d i r ec t ly  from the  d i f f e r e n t i a l  equation ( 5 )  as 
w i l l  be shown later. However, i n  general, the  radiated power cannot be obtained except 
by numerical integration. The d iss ipa ted  power Ph on the antenna Ss 
h 
Ph = 2 II(z) l 2  [Re  zi(z)]  dz 
0 
4nk V:and since 
6 (2) (7 d2 + k2+ 2icrk, (IR+ iII) = dz h-e 5,  ( - i Y )  
dz 
(37) 
(38) 
(39) 
where subscr ip ts  R and I denote respectively the real and imaginary parts of the quant i ty  
subscr ipted and - 4nk V:
v. = - 
From (38) and (39) 
- 16 - 
2 d2 - 
I, % IR + II 7 II + k21II2 = V h ( V I k +  VRII) 6(z) 
dz dz 
When (41) I s  subst i tuted in to  (36), it becomes 
h Let  
d2 
I + IT 7 II]dz Q =  IE[I,-?[ d2 R
dz dz 0 
Then subs t i tu te  (38) and (399 -fnto (43). fie resalt is 
The second in tegra l  of (44) vanishes so t ha t  (42) becomes 
Usually 1 is taken t o  be real, so t h a t  .. 
(43) 
where Ro is the  real par t  of the  input impedance. 
of t h e  zero-order input impedance takes in to  account only the  ohmic loss  on t he  antenna. 
Thus it is seen tha t  t he  real par t  
- 17 - . 
The power radiated Pr is 
2n n 2 1  
I dri: 2 2  = s in  8 d8 IE(8,r)l ro = - 81 'r 2c0 
0 0 
V e h  
'h 2 
'r 
[Re AcP(2ikh)l 0 - 
1 
- I  
8n 
0 
Using the  re la t ions  (11) and (28b) the above expression is put i n t o  f i n a l  form a8 
f ollars : 
r 
0- 
where al(kh) and a,(&) are given i n  (30). 
D 
&r 
'h+ 'r 
Efficiency = 
In the  par t icu lar  case a = 1, i t  can be shown tha t  ( 4 5 )  reduces t o  
1 
vhere F is given i n  (38b) of Reference 1 [See a l so  Ref. 51. It can a l so  be shown tha t  
t he  diss ipated parer approaches zero as a + 0. The r e su l t s  of the  numerical calculation 
agree with the above statements. 
- 18 - 
The efficiency has been calculated fo r  a - 1 with kh ranging from r /2  t o  40r 
(Fig. 10) curd f o r  kh = 2% with a varying from 0 t o  1 (Fig. 11). 
of Y is chosen t o  be 1 Y (a-1) I i n  one curve and Ya [Ref . 3, Table XI-20.11 i n  the other. 
It is expected that should IY(a)l be used, which could be obtained from (22) by numer- 
ical integration, t he  curve would be between these two curves. 
I n  Fig. 11 the value 
CONCLUSION 
The f i e l d  pat tern of the dipole antenna with tapered r e s i s t i ve  loading is not 
c r i t i c a l l y  dependent on t he  parameter a as long aa a s tays  between 0 and 1. 
brightens the  aspect of the  usefulnees of the  r e s i s t i ve  antenna aa a broadband direct iona 
This 
communication device. An experirneutal study of the  r e s i s t i v e  antenna is in progress. 
Reeults w i l l  be reported i n  another report. 
ACKNOWLEDGEMENT 
The authors wish t o  acknowledge with grat i tude the constant encouragement given 
by Professor R. W. P. King. They are a lso  indebted t o  him fo r  t he  correction of the 
manuscript and several valuable suggestions t o  improve it. 
REFERENCES 
1. T. T. Wu and R. W. P. King, "The Cylindrical Antenna with Nonreflecting Resistive 
Loading," IEEE Transactions on Antennas and Propagation, AP-13, No. 3, 369-373 
May 1965. 
2. Erdelyi, Bigher Transcendental Functions, Vol. I, McGraw-Hill (1953). 
3. 
4. 
5 .  
King, Ronold W. P., Theory of Linear Antennas, Efarvard University Press (1956). 
Hildebrand, Introduction t o  Numerical Analysis, McGraw-Hi l l  (1956). 
L. C. Shen and R. W. P. King, Correction t o  The Cylindrical Antenna with Nonreflec- 
t i n g  Resistive Loading, by Wu and King, t o  be published in IEEE Transactions on 
Antennas and Propagation, Noveqlber 1965. 
I 
- 19 - 
APPENDIX I 
B Modified Simpron'r Bula 
In order t o  calculate  numerically the  in t eg ra l  of t he  form I xa F(r) dx,  where 
0 
P(x) CUI ba expanded in a Taylor series around x = 0 ,  a method is developad which ir 
vrry ahdlar  to tha calebratad Siaaproa'r rula. 
Lot f(x) ba a p o l y n d a l  of dagrer 2 which coincidar with F(x) at pointr 
x e x1 x2. Thrn 
0 
2 
k=O 
f(x) 1 Lk(d F ( 5 )  (A-1) 
whrra 
(x-xo) (x-xl) 
L2(x) - (x2-x*) (x,-x,) 
See [Ref. 4, p. 711. 
while S is a new variable.  It follows tha t  
Le t  xl- x = x2- xl = h, x = 2mh + hS where m ie a constant 
0 
1 
Lo(X) = y (S-l)(S-2) 
al(x) = -S(S-2) 
1 
2 a,(x) = - S(S-1) 
Hence, 
dx = h dS 
2 
xa f(x) dx = ha (2mtS)' f(2mh+hS) h dS I x2 
X 0 
0 
= hl*[a f(2mh) + alf(2mh+h) + a2f(2mM.2h)] 
0 
where 
- 20 - 
= i 1 (2m+SIa (S-1)(8-2) dS 
aO 
0 
2 
= - / (2*s)a S(S-2) ds al 
0 
2 
0 
The coef f ic ien ts  a can be calculated easi ly .  Thus, after repeated use of the  above 
f ormala, 
+ b2(k-1) f (2kh) ] 
where 
bl (k) = 4 (2k) 2+o( 2W3ta) -4( 2 kt2) 2+o( 2k-1-a) 
b2(k) = -(2k)2+0(3+a+4k)- 12(a+3) (k+l) (2l~+2)~*+(2k+4)~*(4k+S-a) 
h = B/2N 
(A-2) 
If F(x) is approximated by f(x) ,  i t  follows t h a t  the in t eg ra l  t h a t  involves P(x) 
should be given approximately by (A-2). 
b (k) involve two or three  nearly equal numbers tha t  are subtracted from one another, 
t h e  e r r o r  would be la rge  i f  they were calculated by a computer. 
modified Simpson's r u l e  only at points near t he  s ingular i ty  and t o  use Simpson's ru l e  
elsewhere. 
Note tha t  s ince when k is large the  coeff ic ients  
n 
It is s a f e  to  use the  
- 21 - 
APPENDIX I1 
The figures sham are the result of numerical caluclations. The fie14 pattern 
is normalized to the maximum field and the current is referred to the btivaog point. 
FIG. 1 THE CONTOUR IN THE 
DEFINITION OF @ C y )  
I I I I I I 1 
P 20" 30" . 40" Q 50" 60" 70' 80" 90" 
FIG 2 FIELD PATTERN, k h - T  
SO0 70° EOo 900 -- -- 
8 
FIG, 3 FIELD PATTERN, k h  = 2 7  
o v  I I I I I I I I 
IO" 20" 30" 40" 50" 60" 70" 80' 90" e 
FIG 4 FIELD PATTERN, k h  =67r 
. 
1 
- 
- 
I 
a -0.25 a - 0.25 0.25 
I I I I I I 
70" 80" 90° 
FIG. 6 FIELD PATTERN, k h = 2 0 T  
’ -4.0 
- 3.5 
1.6 - 
4.4 - 
/ 
. ... 
FIG 7 CURRENT OISTRIEUTION, kh *IT 
as - 
04 - 
0 01 0 2  03 0 4  0 8  0 6  0 1  08 OD 10 
L - 
h 
FIG B CURRENT OISTRIEUTION, hh.277 
FIG. 3 CURRENT DiSTR18UT13N, kh-611 
I I I I I 1 
n/2 = 2 a  6n Ion 20r 403 01 I 
kh 
FIG. 10 EFFICIENCY, a-1 , ka -0,021 
I I I I I I I I I 
01 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 
c! 1 
0 
a 
FIG. 11 EFFICIENCY, kh =2r, ka =0.021 


The Cylindrical Antenna with Tapered Resistive Loading Scientific Report No. 5

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