Distribution of current along center-driven cylindrical antenna with variable internal impedance per unit lengt

King, R. W. P.

Wu, T. T.

English

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THE CYLINDRICAL ANTENNA WITH N O N - R E F L E C T I N G  
RESISTIVE LOADING 
By Tai Tsun Wu and  Ronold  W. P. King  
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Prepared u n d e r  G r a n t  No. NsG 579 at 
G o r d o n  Mc Kay L a b o r a t o r y ,  Harvard U n i v e r s i t y  
C a m b r i d g e  , Mas sac husett s 
for 
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION 
. 
https://ntrs.nasa.gov/search.jsp?R=19650019876 2020-03-17T01:57:28+00:00Z
THE CYLINDRICAL ANTENNA WITH NON-REFLECTING 
RESISTIVE WADING 
By Tai Tsun Wu and Ronold W. P. King 
Gordon Mc Kay Laboratory, Harvard University 
Cambridge, Massachusett e 
SUMMARY 
The dietribution of current along a center -driven cylindrical antenna is 
obtained when the material  forming the antenna is resistive. 
kaee ie  coneidered when the impedance per  unit length of the antenna is a 
function of the distance from the end. 
the current is represented by a n  outward traveling wave with no reflected wave. 
The admittance of the antenna and the far field pattern is determined. 
patterns a r e  evaluated for a wide range of lengths. 
by a single major lobe with a very small  minor lobe structure.  
The particular 
A solution is obtained epecifically when 
Field 
These are characterized 
INTRODUCTION 
F o r  some purposes, the directional and broad-band properties of traveling- 
wave antennas a r e  desirable. An example is the traveling-wave V-antenna. 
The f i rs t  work on traveling-wave dipoles was reported by Altschuler ( ref .  1) 
who inserted lumped res i s tors  at  a quarter wavelength from the ends of the 
antenna. Although this location of the res i s tors  is not critical, the traveling- 
wave nature of the current diminishes as the frequency is changed so that the 
lumped re s i s to r s  are no longer at  the maxima of the current.  
In a recent report  (ref. 2) the distribution of current  and the driving-point 
admittance were  determined for a cylindrical antenna with a continuously 
dietributed constant internal impedance per  unit length. It is now proposed to 
investigate the cxlindrjcal antenna with a variable internal impedance per  unit 
length. -- . 
internal impedance for which a pure outward traveling wave exists on an 
antenna of finite length. 
In particular, it is desired to  determine an  axial distribution of the 
THE DIFFERENTIAL EQUATION AND ITS SOLUTION 
The axial component A, ( z ) of the vector potential on the surface of a 
cylindrical antenna that has the internal impedance per  unit length z1 ( z ) 
ca r r i e s  a total axial current I, ( z ) , and is driven at  z = 0 by a delta-function 
generator with emf V e and satisfies the one-dimensional wave equation in the 
form 
, 
0 
is assumed. The internal impedance per  unit if a timerdependence e 
length z1 ( z ) is  expressed as  a function of the axial coordinate z . It is  
given by 
j u t  
i 1 z ( 2 )  = 2TadQz) I J ( Z )  
for a circular tube with constant radius a . In order to vary the impedance 
per  unit length, it i s  assumed that the conductivity IJ on the wall thickness 
d may be functions of location along the antenna. The vector potential on the 
surface of the antenna is 
where 
- j k o r  
e K ( z ,  z ' )  = r 
with 
2 2  (z - z ' )  + a  
(4) 
a 5) 
Since the ratio of vector potential to current  along an  antenna is  
approximately constant, it is possible to  set  
F 
2 
where !P is the value where the current I, ( a  ) has a maximum. 
With (3) - ( 6 )  it followe that 
may be approximated by 
with the notation 
f ( a ) = m z ( t )  4n i 
0 
where Co =,/m = 120n ohme, this equation become6 
1 2 t k 0  2 - j k o f ( z )  I , ( Z ) = f J  E- 
Except at the driving point z = 0 , the current must satiafy the differential 
e quat ion 
It ie  readily verified by direct substitution in (1 1) that when 
2 
f ( 2 )  = ? 
h -  I 4  
80 that (1 1) becomes 
3 
. 
a solution is 
Note in particular, that a eolution of the form ejklzI does not satiefy the 
e quat ion. 
The expression (14) represents a wave of current  traveling in the direction 
of increasing 
no reflected wave traveling in the opposite direction. 
1. , that ie, from the generator toward both ends. There is 
THE IMPEDANCE AND EXPANSION PARAMETER 
If the current has  the form (14) , it follows that the vector potential i o  given 
by 
The scalar  potential satisfies the Lorentz conditiop 
If the driving voltage is defined by 
4 
voe = 4(+0) - 4  ( - 0 )  = 2 6 ( + 0 )  
it  follows that 
Hence, 
The driving-point admittance is 
- 2n 1 
' 0  - 1 - j / k o h  
The impedance ie  a resietance in aeries with a capacitance 
zo = Ro - J /a Go 
where 
= - -  " O  - 60 *ohms 
RO 2u 
and 
C o = E h .  0 
Note that when ko h > > 1 , R > > l/wCo . 
0 
The parameter is defined in t e rms  of the function 
- j k r 2  
t e ] dzl 
'2 
- j k z  
f ( h - r ' )  e 
0 Q ( z )  = 
(h-z) e 
. 
where r l  - [- , r 2  =,fm . Since I( . )  and A z ( z )  
both have maximum amplitudes at z = 0 , it is desirable to define Q =  Q(0) 
That is  
. 
0 
h - j k r  
dz Z '  - j k z s  e Q =  2 J t l - i ; ) e  r 0 
0 
where ro - /= . Since k o a  < < 1 , and a < < h , no serious .error 1s 
made by setting k z g  4 kro in the exponent in the first  integral and ro  = z )  in 
the second integral. 
h - i 2 k r  h 
0 0 U 
With A = ka , it follows that 
) * (29) 
- j 2kh Q E  2[s inh- l  a - C ( 2 A ,  2 k h ) - j S ( 2 A ,  2 k h ) ] t m ( 1 - e  1 
Specifically, when kh = T/2  , h/a = 75 , &? = 10, ka = 1. 57 /75 = . 021, 
2ka = , 042 
Q g  2 [ 5 . 7 0 - 1 . 6 6 - j 1 . 8 5 ] t j ~  2 ( l t l )  = 8 . 0 8 - j 2 . 4 3  . 
Similarly, for a thin antenna with h/a = 11, 013, or &? = 20, 
ka = 1. 57 / 11, 013 = 1.41 x 
4 \E 2[ 10 .69  - 1.65 - j 1 .851  t j ?  = 18.08 - j 2 . 4 3  . 
In ( 2 9 ) ,  C ( a ,  x )  and S ( a ,  x) a r e  the generalized sine and cosine integrals: 
C ( a , x )  = 
0 0 
1 - 
where W = 
6 
. 
THE DISTRIBUTED RESISTIVE LOADING 
The continuously varying resistive loading of the antenna ia defined by (9) 
with (12). Thus 
i f0 1 15P  
8.n 'iqq = 'iqq- 2 ( 2 )  = 
where the coefficient 15 is in ohms. 
15* ohms ranges from 121 - j 36 ohms to 272 - j 36 ohms. 
with h2= 20, h = 3 2 m  so that ~ ' ( 0 )  - 15*/h = w, = 3 . 9  - j .  5 ohme/m 
is the impedance per  unit length at  the driving point. 
With zi = 1/2n adr 
6 thickness d = 10- , 
for carbon of thickness d = 10 -4 , 
of aluminum or carbon a range from below 3.9 ohms/m a t  2 = 0 to very 
large values as z-h may be conetrudted. 
With 62 = 10 to 20 , the coefficient 
A t  A = 288 m 
In this case a = 6. 54 mm. 
m and zi = 8. 55, 85. 5, and 855ohms/m 
i 
2: = 0 . 6 6 ,  6.6 , and 66 ohms/m for aluminum of 
and 
and m . Thus with thin layere 
THE ELECTROMAGNETIC FIELD 
The far-zone electric field is given by 
r = j o s h 0  A2 r EQ 
where 
j kozt COB 0 -jk r h 
A 2 = 4 r u o r o  -h 
0 0  
e 1 ~ ~ ( z t )  e det . r 
With 
- j  ko14 
I ( z ' )  = C ( h -  l z l )  e (33) 
this expression becomes 
7 
. 
The integral is readily evaluated with the following formulae: 
ax 
a t b  
(a co8 bx t b einbx)  $saxcoebxd*. e ( 3 5a)' 
e ax {[ a 2 - b 2 ]  0 [ 2ab ] } x e a X c o r b x d x = n  ax---Z c o e b x t  b x - . m  einbx (35b) s a t b  a t b  a t b  
Thus, 
.az I  
f ( h  - e ' )  e coebz'  dz' = 
0 
a t b  {[ a ( h - 2 ' )  t a ~ - b ~ ] ~ ~ ~ b ~ ~   t b  r 1 b ( h - z l )  +,*I e inb  
a t b  
L -I 
- 1 ( a 2  - b 2 )  cos bh t 2ab ( a2 + b 2 )  + a 2  - b 
( a2 t b2 )2 
2 2 2  2 2 With a = - j k , ,  b =  k c o s @ ,  a t b 2 = - k  
it follows that 
sin 0 ,  a 2 - b 2  = - k  ( l t c o e  a), 
0 0 
2 O)coe(kohcosO). - j2coeOein(kohcoe0)]e - jkohsin 2 O t (  1 +cos 
ko2 s in4@ 
(37) 
8 
. 
The electric field is 
- j k  r 
0 j y  hCe 
F (k h ,  0) ( 3 8 4  Eo 2~ r 0 
r -  0 - 
where the vertical field factor is 
F ( k o h ,  0) = 
h s i n  2 Ot(1tcos 2 O)-[j2cosOsin(kohcosO)t(ltcos 2 O)cos(koh cosO)]e  
0 
koh s i n 3 0  
This function vanishes along the axis 0 = 0 and has the value 
- j (koh-  s i n k  h) t ( 1  -coskoh)  F ( k  h , T ' )  P = 0 
k h  
0, 0 
in the equatorial plane, 0 = n/2  . When ko 2 2  h < < 1 , 
F ( koh, 0) = koh s in  0 
which is the same as for any electrically short  antenna. 
imaginary par ts  are 
The real and 
(39) 
FR ( k o h ,  0 )  = 
(1 t cos2 O) ' [1  - cos koh COB ( koh COB O)] - 2 cos 0 sin kohsin ( koh cos 0) - 
koh sin3 ,O 
FI (koh, 0 )  = 
- koh s in  2 0 - 2 cos@ COB koh sin (koh cos@) t (1 +cos 2 0) s in  koh cos (koh cos 0) 
- 
koh sin3 0 
(41b) 
9 
. . 
a When koh = 
l t c o s  2 0 -  2 c o s O a i n ( ~ c o s 9 )  P 
T F R ( T ,  0) = $- Bin3 o 
2 sin2@ t (1 t cos 0) COB(+ cos 0 )  
( 4 W  It - 2  F (-, 0) = 
I 2  + s in30  
Graphe of FR(koh ,  0) , FI (koh ,  0) , and IF(koh,  0)l are in Figs. 1-3 
for a range of values of koh extending from 7r/2 to 50n , 
IF ( koh, 0) 1 has one large maximum that ie located a t  0 = 90° for koh 4 T 
and moves toward 0 = 0 as k h ics increased. 
maximum is near 0 = 1l0.  
lobes a r e  little more than a small ripple on the broad tail of the major  
maximum for all values of k h > 1~ . 
It is seen that 
v i t h  koh = 5011 the 
0 
Of particular interest  is the fact that minor 
0 
CONCLUSION 
The properties of a center-driven cylindrical antenna that is characterized 
by a pure traveling wave of current have been investigated. 
combination of two such antennas into traveling-wave V-antennae is reserved 
for  another paper. 
A etudy of the 
July 1964 
Gordon McKay k b o r a t o r y ,  Harvard University 
Camb r idg e, Ma 8 s a c hus e tt s 
10 
REFERENCES 
1.  Altshuler, E. E. : The Traveling-Wave Linear Antenna. Trans. IRE, 
Vol .M-9,  July 1961, pp. 324-329. 
2. King, R.  W. P . ,  and Wu, Tai Tsun. : The Imperfectly Conducting 
Cylindrical Transmitting Antenna. 
No. 440, Harvard University,. March 1964, 
Cruft Laboratory' Technical Report 
11 
5 .(I 
4.5 
4.0 
3.5 
3.0 
2.5 
2.0 
I .5 
I .o 
.5 
I I I I I I I I 
7T 
m 
OO IO 20 30 40 50 60 70 80 90 0 
FIG. 1 REAL PART OF FAR-FIELD PATTERN OF NON -REFLECTING ANTENNA. 
. 
- I . (  
-1.5 
-2.c 
-2.5 
-3.0 
-2.5 
-4.0 
-4.5 
- 5.0 
' L  
I I I 1 I I 1 I 
10 20 30 40 50 60 70 0 80 90 
FIG. 2 IMAGINARY PART OF FAR-FIELD PATTERN OF NON-REFLECTING ANTENNA. 
1 I I I I I 
6 .O 
5.5 
5 .O 
4.5 
4.0 
3.5 
3 .O 
2.5 
2.0 
I .5 
I .o 
.5 
0 
1.75 
+ 1.50 
1.25 
-a. 
I.-.- 
FIG. 3 MAGNITUDE OF FAR-FIELD PATTERN OF NON-REFLECTING ANTENNA. 
L 
. 
c 
1 
0 


The cylindrical antenna with non-reflecting resistive loading

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