■h wm^- iiii ^^^^H ^^H 621.365 I i G55te no. 50-57 cop. 2 Digitized by the Internet Archive in 2013 http://archive.org/details/inputimpedanceso56tang Antenna Laboratory Technical Report No. 56 INPUT IMPEDANCES OF SOME CURVED WIRE ANTENNAS by C. H. Tang Contract AF33 (657) -8460 Project No. 6278, Task No. 40572 June 1962 Sponsored by: AERONAUTICAL SYSTEMS DIVISION Electrical Engineering Research Laboratory Engineering Experiment Station University of Illinois Urbana. Illinois Antenna Laboratory Technical Report No. 56 INPUT IMPEDANCES OF SOME CURVED WIRE ANTENNAS by C. H. Tang Contract AF33 (657)-8460 Project No. 6278, Task No. 40572 June 1962 Sponsored by: AERONAUTICAL SYSTEMS DIVISION Electrical Engineering Research Laboratory Engineering Experiment Station University of Illinois Urbana, Illinois ACKNOWLEDGEMENT The author wishes to express his deep appreciation of the guidance of Professor G. A. Deschamps whose advice and suggestions were always illumi- nating. Many helpful discussions with members of the antenna laboratory are acknowledged. The computer program used in this work was done by Mr . V H. Gonzalez. It is a pleasure to acknowledge the financial support of Wright Air Development Division of the United States Air Force. ABSTRACT The problem of the input impedance of curved wire antennas is formu- lated in terms of an integral equation. A stationary formula is used in evaluating the input impedances of arc antennas and helical antennas of which the cylindrical antenna is a special case. The computational results are presented in graphical form. The impedance characteristics of these curved antennas are discussed. iv CONTENTS Page 1. Introduction 1 2. Integral Equation Formulation for Wire Antennas 4 2.1 Integro-Dif f erential Equation for Thin Wire Antennas 4 2.2 Linearization 9 2.3 Transformation of the Integro-Dilf erential Equation 11 2.4 The Singular Integral Equation and the Approximate Kernel Function 13 2.5 Structures Which Give Closed Cycle Type Kernels 17 3. Variational Formulation of the Input Impedance of Wire Antennas 20 3.1 Derivation of the Variation Function 20 3.2 The First Order Solution and the Assumed Current Distribution 23 3.3 Higher Order Solutions 25 3.4 First Order Solution for Arc Antennas and Helical Antennas 28 4. Computational Results and Discussions 35 4.1 Computational Method 35 4.2 The Input Impedances of Cylindrical Antennas 36 4.3 The Input Impedance of Arc Antennas 44 4.4 Electromagnetic Resonance of Thin Wire Conductors 54 4.5 The Input Impedances of Helical Antennas 59 4.6 Equivalent Helical Antennas 72 5. Summary of the Results and Conclusions 75 Bibliography 78 Appendix A 79 Appendix B 82 Appendix C 89 ILLUSTRATIONS Figure Number Page 1 An Idealized Antenna Structure 5 2 Bent Cylindrical Coordinate System 6 3 A Helical Antenna 8 4 An Arc Antenna 9 5 A Center Driven Idealized Cylindrical Antenna 15 6 The Input Impedances of Cylindrical Antennas (ki = 1.5 to 3.7) 40 7 The Input Impedances of Cylindrical Antennas (ki = 4.7 to 6.9) 41 8 The Input Impedances of Cylindrical Antennas (ki = 7.9 to 9.9) 42 9 The Input Impedances of Arc Antennas 45 10 Resonant Resistances of Arc Antennas 47 11 Antiresonant Resistances of Arc Antennas 48 12 Resonant Lengths and Antiresonant Lengths for Arc Antennas 49 13 Shortenings of Resonant Length and Antiresonant Length of Arc Antennas 50 14 The Ratio of Resonant Wavelength to Arc Length as a Function of C — Comparison Between Englund's Measured Results and That Obtained From the Impedance Graphs . 51 15 Resistive Part of the Input Impedance Along Equi-ki Contours 53 16 The Input Impedances of Helical Antennas C = .5 60 17 The Input Impedances of Helical Antennas C = 0.75 61 18 The Input Impedances of Helical Antennas C = 1 62 19 Resonant Resistance as a Function of Pitch in Wavelength 63 vi ILLUSTRATIONS (Cont'd) Figure Number Page 20 Antiresonant Resistance as a Function of Pitch in Wavelength 21a The Input Impedances of Helical Antennas 21b The Input Impedances of Helical Antennas 22a The Input Impedances of Helical Antennas 22b The Input Impedances of Helical Antennas 23 Resonant Resistance as a Function of Circumference in Wavelength 24 Antiresonant Resistance as a Function of Circumference 71 in Wavelength 25 Equal Resonant Resistance Contours 73 26 Equal Antiresonant Resistance Contours 74 64 C 2 = 0.25 65 C = 0.25 66 C 2 = 0.5 67 C 2 = 0.5 68 tf erence 70 1 1. INTRODUCTION It is of practical importance to know the characteristics of the input impedance of an antenna, as a function of the frequency of the source and the geometry of the antenna structure. As far as wire antennas are concerned, the input impedance of cylindrical antennas has been investigated to a great 1,2,3,4 extent by many authors ; but practically no theoretical results have been obtained in the case of curved wires. The problem to be considered in this report is that of evaluating the input impedance for some particular curve wires . The theoretical model used in formulating the problem is the idealized wire antenna with a 6-source excitation, which is usually called a slice generator. This idealization avoids the problem of the actual transition between the trans- mission line and the radiating part of the antenna. With proper end zone correction, the input impedance computed from the theoretical model becomes the terminal impedance of the transmission line. In this report, only the idealized antenna problem will be considered. The integral equation formulation for the electromagnetic problem is an important technique in antenna analysis, especially for cylindrical antennas. 5 ^2 It was first used by Pocklington and later developed by Hall e n . The method has the advantage of generality and is conceptually simple, although rigorous solutions are in general difficult to obtain. A large number of papers have been written on its application to cylindrical antennas using various means of approximation, much less attention was paid to the discussion of curved wire structures 6 ' 7 # With the aid of the integral equation formulation, it can be shown that the input impedance of an antenna can be expressed in such a way that it is 2 stationary with respect to the current distribution in the antenna. It is recognized that the integral equation for the current distribution serves as an Euler equation of the variational problem of the input impedance. The well known Hallen iterative solution to the integral equation has been shown to give a good result for thin antennas, for which the asymptotic solution of the current distribution is known to be sinusoidal. The asymptotic so- lution, when applicable, can be used in the variational expression for the 8 9 input impedance. Storer and Tai have calculated the input impedance of thin cylindrical antennas by such a method. Their results agree well with those obtained by the iteration method. We shall extend these solutions to some curved wire antennas . Those curved wire antennas which we shall discuss are (1) arc antennas — the antenna arms are bent into the shape of circular arcs with a given radius. (2) Helical antennas — the antenna arms are sections of a helix with a given radius and pitch. Cylindrical antennas are special cases of these two classes The common property of being invariant under a one dimensional Abelian group of congruent transformations sets these structures apart. This particular symmetry implies that the kernel of the linearized integral equation has a special form. K(s,s') = K( . I s - s'l ) This is known as a closed cycle type kernel. The above mentioned structures are the only ones which lead to such a kernel. In this work, the computation of input impedances have been performed for both arc antennas and helical antennas . Results are shown in graphical forms . For arc antennas the input impedances are given for various radii 3 of the arc while those of helical antennas are shown as functions of the radius and the pitch of the helix. The computational results show the following general trend: in comparison with the cylindrical antenna, the curved wire antenna has a higher quality factor Q and radiates less with respect to the same input current. In many respects, a curved wire antenna behaves like a corresponding thinner cylindrical antenna. The input impedance graph for the arc antennas also exhibits the extreme phenomena observed by Englund in his measurements for the natural period of linear conductors Based on the computed input impedances of helical antennas the contour lines of equivalent helical antennas are given and they are closely described by the circles for the constant radius of the curvature of the helix. 4 2. INTEGRAL EQUATION FORMULATION FOR WIRE ANTENNAS 2.1 Integro-Dif f erential Equation for Thin Wire Antennas It is well known that in electromagnetic problems one can find the electric field strength E produced by current source J through the use of the vector potential function A -} 2 •- jw e E = [grad div + k ] A (1) with -> 1 ^ * A(x) = — f J(x') G(x,x» ) dv (2) In Equation (1) k is the free space propagation constant. The integration in Equation (2) is performed over the source region v and G(x,x') is a Green's function of the vector wave equation 2 ~b 2 "$ -) V A + k A = -J (3) with appropriate boundary conditions. However, in" a general case it is not possible to find the Green's function for the region outside the antenna, with specified boundary conditions on the antenna surface, therefore we re- duce the antenna problem to the problem of currents in free space and let -jkr(x,x' ) G(x,x')=- 7 pr— (4) ' r(x,x') where r(x,x') is the linear distance between the source point x' and the observation point x. Equations (1) and (2) are the basic relations for the antenna analysis If the current distribution J was given, the field strength E could readily be evaluated. If, on the other hand, the tangential electric field strength is specified on a close surface enclosing the antenna, then Equations (1) and (2) lead to an integro-dif f erential equation for the current distri- bution on the antenna. For convenience, the surface is usually taken on the boundary of the antenna, and we shall discuss only antennas made of perfect conductors . GAP SOURCE Figure 1. An idealized antenna structure Since the tangential electric field vanishes on the antenna surface, the boundary value problem can be formulated in terms of an integro-dif f erential equation with the relation "*t " " 3 1 <5) where E is the tangent electric field deduced from Equation (1) E is the given impressed field which is assumed to be a 6-function The integro-dif f erential equation so formulated will give a unique solution if the support for the current distribution is specified. For a general antenna surface, the vector integro-dif f erential relation leads to two coupled scaler integro-dif f erential equations for two components of the current distribution. However, for thin antennas of unifrom cross- section, the current flow is mostly in the axial direction; therefore we have a single two-dimensional integro-dif f erential equation, instead of two. For the thin wire antennas of circular cross sections, the integro-dif f erential equation can be expressed with the aid of a bent cylindrical coordinate system. Figure 2 describes a bent cylindrical coordinate system. C (s) is a A A given smooth curve, s is the unit tangential vector at s, n is the unit princi- A pal normal vector and b is the unit binomal vector. A point p in the A A n-b plane is described by the polar coordinates (P, (p) and hence any point in the neighborhood of the curve can be described by (P,
, s) in the direction
parallel to axial tangent at s
and
3 2 2/V A -Jkr(s,s •,*>,*>•)
K(S ' S '' W ' ) = [ -3-^ tMS - 5,)1 .C,.',W) (?)
The expression for r (s, s ' ,<£>,<^' ) is known when the geometric configuration
of the antenna is given. For cylindrical antennas, we have
r = [(Z-Z') 2 + 4a 2 sin 2 (&2L) ] 1/2
(8)
where
Z = s
and
a is the radius of the wire
For helical antennas, we can write
r Jr 2 [2- 2 cos 2=2L + tan 2 4; ( £Z£1)
\ o R R
o o
+ 2a 2 [1 + sin ^ sin (
o o
(9)
where
^ is the pitch angle
tan ^ =
27T R
and
R is the radius of the helix
o
p is the pitch of the helix
a - RADIU3 OF I it
GAP SOURCE
Figure 3. A helical antenna
For arc antennas, *\> = 0, and Equation (9) becomes
r = J [2R 2 + 2aR (cos » ^ J Ks')l-9^- r+ k 2 (^ -s') G a (s,s') ds« (6')
Symbolically, it can be written as
E = H(l (37)
where
and
E = E t 1 (s)
lj\±s an integro-dif f erential operator
K l = 4^3T f 2 I ( s '>[w " * 2 (s ' s^jGjs^s') d.<
The kernel K can be shown by the Lorentz Reciprocity Theorem, to have the
symmetry property expressed by the following equality of inner products
-)
I = / M
o v
F o< v ) r . r F o( v >
= f F 1 (u) ( y -^— dv) du + A P 1 (u) ( j -^2— dv)
(B.6)
Applying these formulas, a list of double integrals which are useful
to the derivation of input impedance formulas are given below.
84
>2!
1. / jsin u G(u) — du dv = / u sin u G(u) du + / (2i-u) sin u G(u) du
rr r 1 r 2i
2. / / cos u G(u) — du dv =/ u cos u G(u) du + / (2i-u) cos u G(u) du
u v
3. / /
U V
sin u G(u) — du dv = / [ cos v - cos (2i-v) ] G(v ) dv
/ cos u G(v) — du dv = / [sin (2i-v) - sin v] G(v) dv
| sin v G(v) — du dv =
/ cos v G(v) - du dv = 2(i-u) cos v G(v) dv
7. i / sin v G(u) — du dv =
| cos v G(u) - du dv =1 sin u G(u) du + sin (2i-u) G(u) du
f " i
/ | u sin u G(u) — du dv =/ u sin u G(u) du 4
ff
U V
). u cos u G(u) ~ du dv = I u" cos u G(u) du + (2i-u) u cos u G(u) du
85
11. I I u sin u G < v ) — du dv = J [sin (2i -v)- (2i-v) cos (2i-v)-sinv+v cos v ]G(v)dv
12. I I u cos u G(v) - dudv =] [ cos (2i -v)+(2i-v)sin(2i-v)-cos v-v sin v]G(v) dv
| u sin v G(v) - du dv =
l! V
14. / u cos v G(v) — du dv = |
u v
.5. I] u sin v G(u) — du dv =
ff x r 1 f
6. j u cos v G(u) — du dv = I u sin u G(u) du + I u sin (2i-u) G(u) du
u J v ' ^o J I
v sin u G(u) — ■ du dv =
io / / v cos u G(u) — du dv =
I I v sin u G(v) - du dv =
v cos u G(v) — du dv =
86
| v sin v G(v) - du dv =
U V
s v G(v) - du dv =
23. . ' v sin v G(u) — du dv = | [sin u - u cos u]G(u)du +
u v o
2i
[sin(2i-u)-(2i-u) cos(2i-u) ] G(u) du
\ } v cos v G(u) — du dv =
■IP
I u sin u G(u) — du dv = | u sin u G(u) du + \ u (2i-u)sin u G(u) du
I j u cos u G(u) - du dv = J u cos u G(u)du + / u (2i-u)cos u G(u) du
■IP
: // u sin u G(v) ^ du dv = I [ 2 (21 -v)sin(2i-v)- (4i 2 -4i v+v -2) cos (2i-v)
2
■2v sin v + (v -2)cos v] G(v) dv
ffi
j ;u cos u G(v) - du dv = I [2(2i-v) cos (2i-v) + (4i -4iv+v -2)sin(2i-v)
Jo
2
-2v cos v -(v -2) sin v] G(v) dv
87
2 1
29. / j u sin v G(v) — du dv =
u v
\U-
1 '13 2 2 3
| u"cos v G(v) — du dv = I -[81 -12i v+6iv -2v ] cos v G(v) dv
31. I / u sin v G(u) — du dv =
21
32. | I u" cos v G(u) ^ du dv = j u sin u G(u) du + u sin(2i-u)G(u) du
J J u cos v G(u) — du dv = I
u v
2£
2 1 13 I 1 3
/ / v sin u G(u) — du dv = I — u sin u G(u) du + / — (2i-u) sin u G(u)du
fr f r i
I I 2 1 I 1 3 J 1 3
34. / / v cos u G(u) - du dv = — u cos u G(u)du +i — (2i-u) cos u G(u) du
u v ^o " 1
[35. j j v sin u G(v) — du dv = I v [cos v-cos (2i-v) ] G(v) dv
u^v Jo
36. / / v cos u G(v) - du dv = v [sin (2i-v)-sin v] G(v) dv
2 1
v sin v G(v) — du dv =
KK
38. / / v cos v G(v) — du dv = J 2v (i-v) cos v G(v) dv
f 2 1
39. J v sin v G(u) — du dv =
; | v cos v G(u) — du dv = I [ 2u cos u + (u -2) sin u] G(u) du
u v
21
2 2
[2(2i-u)cos(2i-u) + (4i -4iu+u -2) sin(2i-u)
G(u) du
89
APPENDIX C
COMPUTATIONAL PROCEDURES
The numerical computations were carried out on ILLIAC, the digital
computer of the University of Illinois . The computation involved two parts.
(a) The integration of the set of integrals v f v , and v . They
can be written as
L 2L
f ™ / \ cos T ^ x ) ^ I t, ( ^ cos ~r(x)
1 F (x) — — — r dx + I F. (x) =^7— -
I ll r(x) J i2 r(x)
Jr, T.
dx
[
_ . . sin r(x) ,
F. n (x) — _. ; dx +
ll r(x)
2L
/„ , . sin r
F. (x) _.
i2 r(x
(x)
1,2,3.
where F 's and F 's are given by the Equations (86) through (91) and
r(x) is given by Equation (93) for the helical antennas and Equation (95)
for the arc antennas .
The integration was performed by using Simpson's rule. The set of
functions F. .'s were first evaluated by the auxiliary routines; the re-
sults then entered the integration routine with the assigned number of
intervals for the integration « This number was determined by the convergence
of the result^, which will be explained later a
(b) The algebraic operations for the impedance formula.
After the set of integrals v , v and v are evaluated, the set of
1' 2 3
complex numbers
. + ■ j T .
i i
1,2,3
90
entered the first order impedance formula as follows
Z U> = j30 L_J 2 m a ± jp
(sin 2 L) v„ - (L sin 2L) v. + (L 2 cos 2 L) v, Y + j5
«J ^ 1
aY + (36 . p\ - a
Y 2 ♦ 6 2 V + 6 2
where
• = -sotVa + Vs - 2a 2 T 2 J
M* 1 V T lV! !t, 2 !l
\ = CT sin 2 L - (7 L sin 2L + (J L cos L
2 2 2
6 = t sin L - t L sin 2L + t L cos L
The computer program was prepared for computing the impedance of the
helical antennas for a given set of parameters C , C , L and £1. These
1 2i
parameters are defined as
C = kR = 27TR A
loo
C 2
=
kb =
P/>
L
=
ki
ft
_
2 in
2i_
[where
|
R is the radius of the helix
o
i is the half length of the antenna
a is the thickness of the antenna wire
91
and
b = p/277
where
p is the pitch of the helix
The arc antennas and the cylindrical antennas are special cases of the
helical antennas . For the arc antennas
c 2 = o
and for the cylindrical antennas
C l = *»> C 2 = °
The program was coded in such a way that it could compute automatically
the impedances for several values of L. The published results of the input
impedances for cylindrical antennas were used as guides in checking the
computation program.
Since for those integrals involving Cos r(x)/r(x), the integrand gives
a large value when x = 0, the integral
/
, . cos r(x)
F._(x) dx
r(x)
was split into
A L
f F (x) COS r(x) dx + f Ffa) cos r(x)
J iX Too / xl TOO
dx
92
where the value A together with the number of points in the Simpson's integra-
tion rule were determined by the stability of the results. A fixed number
n = 20 for the integration interval was used to determine the stable value for
A- The computation results for a series of integrals indicate that A = 0.25
was an optimum value . The value A = 0o25 was then used in the splitting of
the above mentioned integrals „ To determine the number of the integration
intervals the integrals were evaluated with a set of different number n.
After analyzing these results,, we chose the value n = 30. For this combi-
nation of values of A and n, the impedance for the cylindrical antennas were
compared with that obtained by Tai, for i2 = 10 and i2 = 15. The comparison
showed that in the range L < 6, the difference between these two results
is 3 ^ at the most .
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Ann Arbor Michigan
idio Corporation of America
ZA. Laboratories Division
ttn: Technical Library
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Antenna Section
rawer 37
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)00 Woodly Avenue
in Nuys, California
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Attn: Librarian (Antenna Lab)
State College, Pennsylvania
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P. 0. Box 1828
Clearwater, Florida
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Attn: Librarian (Antenna Lab)
Stanford, California
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ttn: Librarian (Antenna Lab)
moga Park, California
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Attn: Librarian (Antenna Lab)
Menlo Park, California
md Corporation
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700 Main Street
inta Monica, California
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itn: Librarian (Antenna Lab)
3999 Ventura Blvd.
llabasas, California
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389 Washington Street
jwton, Massachusetts
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oplied Research & Development
Division
:tn: Librarian (Antenna Lab)
irmingdale, New York
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Attn: Librarian (M/F Antenna &
Microwave Lab)
100 First Street
Waltham 54, Massachusetts
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Attn: Librarian (Antenna Lab)
P. 0. Box 188
Mountain View, California
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Attn: Librarian (Antenna Section)
2 Aerial Way
Syosset, New York
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Temco Aircraft Division
Attn: Librarian (Antenna Lab)
Garland, Texas
mders Associates
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IS Canal Street
ishua, New Hampshire
imthwest Research Institute
:tn: Librarian (Antenna Lab)
)00 Culebra Road
m Antonio, Texas
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Attn: Librarian (Antenna Lab)
6000 Lemmon Ave.
Dallas 9, Texas
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Attn: Librarian (Antenna Lab)
355 Providence Highway
Westwood, Massachusetts
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Head Antenna Department
Physical Science Laboratory
University Park, New Mexico
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Attn: ASAD - Library
Wright-Patterson Air Force Base
Ohio
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Whippany Laboratory
Whippany, New Jersey
Attn: Technical Reports Librarian
Room 2A-165
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Attn: Dr. A. G. McNish
Washington 25, D. C.
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Box 95085
Los Angeles 45, California
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10829 Berkshire
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Raytheon Company
Surface Radar and Navigation
Operations
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IBM Research Laboratory
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Collins Radio Corporation
Antenna Section
Dallas. Texas
Dr. A. K. Chatterjee
Vice Principal & Head of the Department
of Research
Birla Institute of Technology
P. 0. Mesra
District-Ranchi (Bihar) India
ANTENNA LABORATORY
TECHNICAL REPORTS AND MEMORANDA ISSUED
i rntract AF33<616)-310
Synthesis cf Aperture Antennas,, 111 ' Technical Report No, 1„ C»ToAo Johnk,
itober, 1954„*
i Synthesis Method .for Broad-band Antenna Impedance Matching Networks,"
' tcfanical R eport No. 2, Nicholas ¥aru, 1 February 1955,,* AD 61049 „
'be Assymmetrically Excited Spherical Antenna,," Technical Report No. 3 ,
M>ert C. Hansen,, 30 April 1955 „*
\nalysis of an Airborne Homing System,," Technical Report. No u 4, Paul E„
lyes, 1 June 1955 (CONFIDENTIAL) .
'oupling of Antenna Elements to a Circular Surface Waveguides," Technical
[port No, 5, H„ Eo King and I. H. DuHamel,, 30 June 1955 ,*
Vxially Excited Surface Wave Antennas /° Techni cal Rep ort NOo 7 9 Do E Royal,
! October 1955 o* ™_ _«=__«_=_______
'oming Antennas for the F-86F Aircraft (450-2500 mc)'° ' Techni cal Report No 8,
] Eo Mayes,, Ro F„ Hyneman,, and R Co Becker, 20 February 1957, (CONFIDENTIAL),
'round Screen Pattern Range/ Technic al Memorandum No. 1 ,-, Roger R» Trapp„
1 July 1955 o*
C atract A F33 (616)) -3220
Vffective Permeability of Spheroidal Shells/" Technical Report No 9, Eo J.
ifbtt and Ro Ho DuHamel „ 16 April 1956 „
'n Analytical Study of Spaced Loop ADF Antenna Systems," Technical Report
jo 10. Do Go Berry and J» B„ Kreer, 10 May 1956, AD 98615
':' Technique for Controlling the Radiation from Dielectric Rod Waveguides/ 8
I chnical Report No, 11, J, ff„ Duncan and R„ Ho DuHamel,, 15 July 1956 c *
'irectional Characteristics of a U-Shaped Slot Antenna , " Technical Report
j'^^12,, Richard Co Becker, 30 September 1956 »**
Ifflpedaiace of Ferrite Loop Antennas/ Technical Repo rt NOo 13, V. H. Rumsey
aid Wo Lo Weeks, 15 October 1956 „ AD 119780
'jlosely Spaced Transverse Slots in Rectangular Waveguide/ Technical Report
Ko 14, Richard F. Hyneman, 20 December 1956 „
Dist'ibuted Coupling to Surface Wave Antennas/' Te chnical Re port No., 15,
;alph Richard Hodges, Jr.,, 5 January 1957,
'The Characteristic Impedance of the Fin Antenna of Infinite Length," Technical
teport No. 16 , Robert L. Carrel, 15 January 1957 „*
'On the Estimation of Ferrite Loop Antenna Impedance," Technical Report No. 17 ,
'alter L. Weeks, 10 April 1957.* AD 143989
A Note Concerning a Mechanical Scanning System for a Flush Mounted Line Source
,ntenna," Techni cal Report No. 18 ,, Walter L„ Weeks,, 20 April 1957 „
Broadband Logarithmically Periodic Antenna Structures," Technical Report No. 19 ,
U H« DuHamel and D. E, Isbell, 1 May 1957. AD 140734
'Frequency Independent Antennas,' Technical Rep ort No. 20 , V. Ho Rumsey, 25
ictober 1957.
'The Equiangular Spiral Antenna,,'" Technical Report No. 21, J. D. Dyson, 15
eptember 1957, AD 145019
'Experimental Investigation of the Conical Spiral Antenna, " Technical Report
to. 22, R„ L. Carrel, 25 May 1957.** AD 144021
'Coupling between a Parallel Plate Waveguide and a Surface Waveguide," Technical
teport No. 23., E. J. Scott, 10 August 1957.
'Launching Efficiency of Wires and Slots for a Dielectric Rod Waveguide,"
ecfanical Report No. 24, J. W» Duncan and R„ H, DuHamel, August 1957.
■'The Characteristic Impedance of an Infinite Biconical Antenna of Arbitrary
Toss Section," Technical Report No. 25, Robert L. Carrel, August 1957.
i'Cavity-Backed Slot Antennas,," Technical Report No. 26, R„ J. Tector, 30
'ctober 1957.
I' Coupled Waveguide Excitation of Traveling Wave Slot Antennas," Technical
Report No. 27, W. L. Weeks, 1 December 1957.
Thase Velocities in Rectangular Waveguide Partially Filled with Dielectric,"
technical Report No. 28, W. L. Weeks, 20 December 1957.
Measuring the Capacitance per Unit Length of Biconical Structures of Arbitrary
i'ross Section/ Technical Report No. 29, J. D. Dyson, 10 January 1958.
j'Non-Planar Logarithmically Periodic Antenna Structure," Technical Report No. 30 ,
! 'v E. Isbell, 20 February 1958. AD 156203
! Electromagnetic Fields in Rectangular Slots/" Technical, Report No. 31 , N. J.
i:uhn and P. E. Mast, 10 March 1958.
The Efficiency of Excitation of a Surface Wave on a Dielectric Cylinder,'
echnical Report No, 32,, J. W Duncan, 25 May 1958 „
A Unidirectional Equiangular Spiral Antenna" 1 Technic a l Re port No. 33„
, Do Dyson, 10 July 1958 . AD 201138
Dielectric Coated Spberiodal Radiators," 1 Technical Report No. 34 , W„ L„
eeks, 12 September 1958 . AD 204547
A Theoretical Study of the Equiangular Spiral Antenna," Technical Report
o. 35, P. E. Mast, 12 September 1958. AD 204548
ontract AF33(616)-6079
Use of Coupled Waveguides in a Traveling Wave Scanning Antenna," Technical
eport No. 36, R. Ho MacPhie, 30 April 1959. AD 215558
On the Solution of a Class of Wiener-Hopf Integral Equations in Finite and
nfinite Ranges,," Technical Report No. 37,, Raj Mittra, 15 May 1959 .
Prolate Spheroidal Wave Functions for Electromagnetic Theory/' Technical
eport No, 38 , W. L. Weeks, 5 June 1959 „
Log Periodic Dipole Arrays," Technical Report .No. 39,, Do E. Isbell, 1 June 1959 .
D 220651
A Study of the Coma-Corrected Zoned Mirror by Diffraction Theory," Technical
eport No. 40 ,, So Dasgupta and Y. To Lo, 17 July 1959.
The Radiation Pattern of a Dipole on a Finite Dielectric Sheet," Technical
eport No. 41 , K, Go Balmain, 1 August 1959.
the Finite Range Wiener-Hopf Integral Equation and a Boundary Value Problem
n a Waveguide," Technical Report No. 42, Raj Mittra,, 1 October 1959 .
Impedance Properties of Complementary Multiterminal Planar Stnactwes, "
schnical Report No. 43 ,, G. A. Deschamps, 11 November 1959 „
On the Synthesis of Strip Sources," Technical Report No. 44 , Raj Mittra,
December 1959.
Numerical Analysis of the Eigenvalue Problem of Waves in Cylindrical Waveguides,'
,echnical Report No, 45 ,, Co H. Tang and Y„ T. Lo, 11 March I960.
New Circularly Polarized Frequency Independent Antennas with Conical Beam or
.unidirectional Patterns," Technica l Report No. 46, J. D. Dyson and P„ E. Mayes ;
June I960. AD 241321
Logarithmically Periodic Resonant -V Arrays,," Technical Report No. 47 , P. E. Mayes ,
j ad Ro L. Carrel, 15 July 1960. AD 246302
Study of Chromatic Aberration of a Coma-Corrected Zoned Mirror," Technical
port Noo 48, Y, To Lo, June 1960
valuation of Cross-Correlation Methods in the Utilization of Antenna Systems,"
chni cal Report No. 49 , R. H„ MacPhie, 25 January 1961
ynthesis of Antenna Product Patterns Obtained from a Single Array," Technical
!port NOo 50 , Ro Ho MacPhie, 25 January 1961 „
)n the Solution of a Class of Dual Integral Equations," Technical Report No, 51 ,
Mittra s 1 October 1961 . AD 264557
inalysis and Design of the Log-Periodic Dipole Antenna/* Technical Report No, 52 ,
>bert Lo Carrel, 1 October 1961 .* AD 264558
I Study of the Non-Uniform Convergence of the Inverse of a Doubly-Infinite
itrix Associated with a Boundary Value Problem in a Waveguide," Technical Report
). 53, R„ Mittra, 1 October 1961 . AD 264556
Copies available for a three-week loan period,
* Copies no longer available .
Antenna Laboratory
Technical Report No. 57
POLYGONAL SPIRAL ANTENNAS
by
C. H. Tang
and
0. L. McClelland
Contract AF33(657)-8460
Project No. 6278, Task No. 40572
June 1962
Sponsored by:
AERONAUTICAL SYSTEMS DIVISION
Electrical Engineering Research Laboratory
Engineering Experiment Station
University of Illinois
Urbana, Illinois
ds ds f (50
o iya " '
i ' i
V 2 = V ol = 1 J J^ i o (3) 4 l (fl,) C (S ^ !) dS dS? = V
10
S l S l
2 2
V ll=iS S il(S> 1 l (s, >^a
pq k U sj P
s) i (s T ) /V (s,s f ) ds' ds = v
q v a qp
(60)
and the A. 's are determined by
5z.
(n)
= i = 0,1,2
6A.
> -> —j
(61)
Equation (61) gives a set of equations as follows
SA i (o)
o p p
2 A v .
o p pi
2 S A A v
o o p q pq
i. (o) =
l
(62)
i = 0,1,2, n
From Equations (62) and (59) we have
:. (n) fsA i (cjl i
in I o p p i
L. (o) = j30 S A v
o p pi
27
(63)
M
These are n + 1 equations for the set of variational parameters
i = 0,1,2, n. To solve these equations, we first express the feeding
condition
n
Z (n) i A i (o) = JL
o p p I o
(64)
in
where V is the given voltage difference at the input terminals, and thus
Equation (63) becomes
n V i. (o)
S A v . = * =
o p pi j30 I
In matrix form it can be written as
vA =
J30 I
(65)
where
10
ol
11
In
nO
nl
is a symmetric matrix and
28
A =
the solution of A is
i (o)
o
i 1 (o)
i (o)
n
T V =-1t
(66)
substituting Equation (66) into Equation (64), we obtain Z.
(n)
^^O;^
(67)
This reduces to the form (55) in the case of two parameters.
3.4 First Order Solution for Arc Antennas and Helical Antennas
In this section we shall apply the first order formula to some symmetri-
cally curved wire antennas of half lenth i , and curved wire antennas which we
will consider are those which possess the closed cycle type kernel, namely,
arc antennas and helical antennas, with cylindrical antennas as special cases
The set of trial functions for the current distribution will be
i (s) = sin k (i- I s I )
o
and
i (s) = k Q- I s I ) cos k a- I s I )
29
For these antennas, the kernel f unction \_ _ (s, s ' ) becomes
G ( I s-s' I ) (68)
f/ds-s'l) = (-4 + k 2 C^ ■£•)}
This particular property can be used to simplify the set of double integrals
V V 2 and V
We first convert these integrals into the following forms.
1 * (~di (s) di (s») ~|
V I = k J J Mb 57T- + k f (S > S,) l o (B)1 o (B, -2j V 8 ' 8 '* dS ' dS
-i -i (69)
i ' Pdi (s) di^s') -I
*2 = k J J - "di ds^" + k f(S ^ S,) V S)i l (s,)
-t -1 *- -i
i i
| ai^s; a^ts"; 9 |
G (s.s 1 ) ds' ds
a
(70)
i r r I di i (s) di i (s,) 2 1
*J J Q"5S ^ + kf(s,s.) i 1 (s)i i (s'J
where
-jkr(s.s')
(71)
G )
r cos kv + cos k(2i-u)] (82)
ds ds ' 2
di (s) di (s*) k 3 R 2
— -: — — ; = -— - (2i-u+v)[sin kv + sin k(2i-u)] + — [cos k(2i-u) + cos kv
us as 4 ^
(83)
di (s) di (s>) 4 3
— ^ — — — = |- (2i-u+v)(2i-u-v)[cos kv - cos k(2i-u)] - — (2i-u-v)
k 3 k 2
[sin k(2i-u)-sin kv ] - — - (2i-u+v) [sin k(2i-u) + sin kv ] + — [cos kv +
cos k(2i-u) ] (84)
32
In u-v plane, double integrals v , v , and v can thus be written as single
integrals (see Appendix B) .
If we normalize the length in radians, i.e. let
L = ki
and with x = ku = kv as new variables, then the set of integrals v , v and
v Q become
2L
v i=J F n (x) dx+ S
F_(x) dx + | F 12 (x) G(x) dx
2L
v 2 = f F 2i (x) G(x) dx + P F 22 (X)G(X) dX (85)
2L
v 3 = P F 31 (x) G(x) dx + j P F 32 (x) G(x) dx
o L
where
F n (x) = [x cos (2L-x) - 2(L-x) cos x] [l-f(x)] + [2 sin x - sin (2L-x) ][ 1+f (x)
(86)
F 12 (x) = [l-f(x)] (2L-x) cos (2L-x) + [l+f(x)] sin (2L-x) (87)
F 21 (x) = J|x-L-(L-|) f(x) cos (2i-x) + [l-f(x)]
(| Lx ~2j sin ( 2L -x) - (x -xL-1) sin x + [ (3x-2L) + x f(x)] cos x
(88)
33
F 22 (x) = (L-|) [3 + f(x)] cos (2L-x) - I*- -2Lx + 2L 2 ~) [l-f(x)] sin (2L-x)
(89)
F 31 (x) = [(l-f(x)) (-|-+ Lx 2 " L 2 x +| - L ) + x| cos (2L-x)
+ x 2 - 2Lx + (1+f (x)(-Lx+L -|) sin (2L-x)
+ j-2x 2 + L(l-f(x))x + (l+L 2 )(l+f (x))J sin x
+ [(l-f(x)) (^|- + L 2 x + x - | L 3 ^ + 2x -2lJ
J -x+2L
-x + 4Lx - 4L 2 + -| + -| f (x) J sin (2L-x) (91)
F 32 (x) = [(l-f(x)) f|- -Lx 2 + 2L 2 x - | - | L 3 + L J
and where
-jr(x)
G(x) = -
r(x)
(90)
f(x) =\[ Cl 2 cosf + C 2 2 j (92)
/ N 2 ~| 1/2
7(x) = kr(u) = j^ 2 sin 2 (§-) M 4c/ (|J + 6 2 J (93)
with
6 = ka
34
C = kR
1 o
c 2 = kb
V^
2 —2
+ C 2
For arc antennas C =
f(x) = cos £- (94)
r(x) = |4C 1 2 sin 2 (-£-\ + e 2 l (95)
and for cylindrical antennas C > °o
f(x) = 1
r 2 2i 1/2
r(x) = |^x + € J (96)
The curvature of a helix can be expressed as
R
K = °
2 2
R + b
o
and the normalized radius of curvature of the helix is thus
^2 2
C l + C 2
C = 1 2 (97)
C l
I The Equation (97) will be useful in describing some equivelent helical antennas
■
j which will be explained later.
35
4. COMPUTATIONAL RESULTS AND DISCUSSIONS
4.1 Computational Method
The numerical computations were carried out on ILLIAC, the digital com-
puter of the University of Illinois. Some of the procedures used in computing
the integrals and performing the algebraic operations of the impedance formula
are described in Appendix C.
The computer program was prepared for computing the input impedances of
helical antennas for a given set of parameters C , C , L and Q, . The first
1 ■£
three parameters are the normalized physical dimensions of a helix, as de-
fined previously.
C, = kR
1 o
id £1 is defined as
c 2 = kb
L = k£
fl = 2in 21
For arc antennas, C = 0. and for cylindrical antennas C = 0, C = °°.
The computations for the input impedance of cylindrical antennas have
been performed for
£2=8, 8.5, 9, 9.5, 10, 10.5, 11, 11.5, 12, 12.5, 13, 13.5, 14, 14.5, 15.
While the parameter Q, was held constant (£2 = 10) for the impedance computation
for the arc antennas and the helical antennas . In case of the arc antennas,
a series of computations were performed for the parameter C in the range
0.5 < C < 3.5
36
If the dimensions were such as to make the antenna arms meet, the computation
outline would not be valid thus the largest values of L used in the compu-
tation were limited by the relation
L < 7TC 1
for a given value of C .
For helical antennas, the impedance computations were performed for
the values of C and C in the following ranges .
0.3 < C < 3
0.3 < C < 2
4.2 The Input Impedances of Cylindrical Antennas
Although our main purpose was to evaluate the input impedances of curved
wire antennas, particularly the arc antennas and the helical antennas, the
impedance formula derived in Chapter 3 also gives the input impedances of
cylindrical antennas. This will be discussed first for the following reasons.
(a) A cylindrical antenna is a special case of curved antennas. For
curved wire antennas of small curvature, their input impedances approach that
of cylindrical antennas. Furthermore, when ki is small, the input impedance
of all wire antennas should be practically the same. As ki increases, the
effect of the curvature begins to appear. The input impedance of a cylind-
rical antenna is therefore useful in extracting the effect of the curvature
in the input impedance of a curved wire antenna.
(b) It is the nature of the approximation of the problem that it
jives a better result for structures of relatively small curvature. Yet,
\ve have no way of determining what exactly is the upper bound of the curvature
37
for which the approximation is still valid, there is no explicit condition
for limiting values of C and C in the impedance formula. Hence in inves-
tigating the effect of curvature on the input impedances of these wire antennas,
it is important for the purpose of comparison to compute the input impedances
of the corresponding cylindrical antennas with the same method.
(c) The computing program can be checked by allowing the comparison
of our results with published, experimentally confirmed data for the cylindrical
antenna case.
(d) It will be shown later that in many respects the curved wire antenna
behaves like a corresponding thinner cylindrical antenna.
In case of cylindrical antennas
f(x) = 1
The set of functions F. . 's are simplified
' (x) = 4 sin x - 2 sin (2L - x)
F (x) = 2 sin (2L - x)
F (x) = 2(X-L) cos (2L - x) + 2(2x - L) cos x
F (x) = 2(L - x) cos (2L - x)
(98)
' 31 (x) = x cos (2L-x) + (x 2 - 4Lx + I? - 1) sin (2L-x) + (-2x + 2L +2) sin x
+ (2x-L) cos x
F 32 (x) = (2L-x) cos (2L ~ x ) + (-x 2 + 4Lx " 4Ij2 + D sin (2L _x)
38
For the thin wires ka « 1, i » a the following approximation is valid
P e ' Jr(x ' €) dx .f_dx_ P i - e -J r <^°
J r(x,e) J r(x,e) J r(x,o)
)
- dx
hen the set of integrals v , v , and v for cylindrical antennas can be
9
xpressed in terms of known integrals, as given by Tai .
^ (102)
/£-
1/r
e
by the equation of continuity € can be written as
e
877600
< I'/A I' >
where u> is a natural frequency .
Equating the Equations (101) and (102), we obtain the formula
„ 2 < 1 1 IJC I ' > -rr
For thin wires, the double integrals in Equation (103) can be written as
jf = < I' /|l£ I' > =] |j [I'(s)] 2 |(v (s,s«) ds' ds
J J
JTm = < I /AC m I > = I [I (s)] 2 /\ m (s,s') ds' ds
56
The approximation is justified for thin wires due to the large contribution
of the integrand at the neighborhood of s = s'.
To evaluate resonant wavelength and the quality factor Q, we assume
I(s) =1 sin ~ s = I i(s)
where 2i is the antenna length. From Equation (103), we have
1/2 1/2
i' \K i
and the quality factor Q associated with the nth natural mode can be expressed
n
s,-ft--?[] va -?ar.ff.] v>
n n n n
when R is the radiation resistance referred to a current antinode
n
For cylindrical antennas, we have
where
^■IF ^ - . ± (2n„
JL ° - i 2t [A + s. (2„„)
m 2 n77 l
A = 2n7T |j + log 2 - l|
57
and s. is a sine integral. For arc antennas, we write
and
j-ir^i 1 ^ . , v 2 . , ^ -l 1 .
R o ° R o
2 3
R Q o R Q (nvr)
For n = even
i i 3
R (n7T)
for n = odd
3
OI--^
i
(106)
3 „2 . ,2
R (n77)
o
Therefore, it is seen that in comparison to cylindrical antennas (ki)res
becomes larger for arc antennas and helical antennas while (ki)antires becomes
smaller for curved structures . Since
. A - s. (2n77) .2
(ki)res = SE X (n . [1+| 1 ] (107)
9 \ I A + s (2n7r) 3 2-1
2 \| A + s. (2n7T) 3 ^ R 2 ^
n = 1,3,5.
58
and
(kOant:
/A - s.(2n7T)
1
17 I 2
A + s. (2n7T)
1
3 2
A Ro n77
■ 2,4,6
(108)
They therefore agree with that indicated in Figure 9 for kR > 2.5. The
approximation for deriving Equation (103) holds only for relatively small
curvatures .
For the quality factor Q, we have, From Equation (105)
'res ft
= -TpVtA - s (2n.7T)][A + s (2 n7 r) ] 1 1
K n W X X L a R 2 mr J
(109)
n = 1,3,5
and
wJ^ - s i
(2ni7)][A + s i (2nir) ] 1 + 3
[■
16 i
2
A Ro n 7T-
(110)
n = 2,4,6
Q is larger for curved structures while Q is larger for cylindrical
res ant ires
antennas- since R at resonance decreases as the curvative is increased at a
7 n
2 2
faster rate than the decrement due to the term £ /A R Q nj[ . At antiresonance
R increases as the curvature is increased and its order is also higher than
the increment due the term — .£ /A R n7T •
The results obtained from this analysis agree qualitatively with those
.obtained from the input impedance graphs. Quantitatively speaking, the dis-
i
crepancy between these two results exist even for the cylindrical antenna
59
case. This,porha.(.o.. is due to the different assumption on the current distr]
bution,.
4.5 The Input Impedances of Helical Antennas
For helical antennas, input impedances were calculated as functions of
C and C with £1 = 10, where
C = kR , R is the radius of the helix
1 o> o
p
C = kb, b = ~— } p is the pitch of the helix
Results are presented in the form of circular graphs. Figures 16, 17, and
18 give the input impedances of helical antennas of constant C (C = 0.5,
C = 0.75 and C =1), while Figure 21a and 21b give the input impedances
of constant pitch helical antennas for which C = 0.25 and Figure 22 is
that for C = 0.5.
When the size of the helix is fixed and the pitch is varied, the cir-
cular graph of the input impedance enlarges as we decrease the pitch. The
input impedance of the helical antennas approaches that of the arc antenna
for the small pitch and approaches that of the cylindrical antenna for the
large pitch as expected. The range between these two limits is a function
of C , namely the size of the helix in wavelengths. The range is larger
for smaller C and is narrowed down to zero for large values of C . The
! rate of this convergence is shown in Figure 19 and Figure 20. Figure 19
is the variation of the resonant resistance as a function of C and with
C as a parameter while Figure 20 indicates that of the antiresonant re-
sistance .
60
800
600
400-
200
-200
-400
•600
C.= THE CIRCUMFERENCE IN WAVELENGTH
C 2 =THE PITCH IN WAVELENGTH
Figure 16. The Input Impedances of Helical Antennas C = 0.5
6]
CO
X
o
1200
800
400
-400 -
-800
-1200
Figure 17. The Input Impedances of Helical Antennas
C = 0.75
62
800
•800-
kl
00 (OHMS)
C, ***'*
Cz= p/X
Figure 18. The Input Impedances of Helical Antennas c 1 = 1
RESONANT RESISTANCE IN OHMS
64
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o
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o
CO
CO
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+->
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0)
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m
CO
—
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5
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c8
(1)
SIAIHO Nl 30NV1SIS3U 1NVN0S3U - I1NV
65
1000
0.75 — Ci
1000
2000 \ , (OHMS)
G=»d/X
Figure 21a. The Input Impedances of Helical Antennas
C = 0.25
2
66
kl
1000
500
1500 (OHMS) 2000
■500
Cl = 7rd/X
C 2 = P/X
Figure 21b. The Input Impedance of Helical Antennas C^
0.25
67
800
400
400
1600
R
(OHMS!
Figure 22a. The Input Impedances of Helical Antennas
C 2 = 0.5
68
800
R
(OHM)
Figure 22b. The Input Impedances of Helical Antennas C ? = °- 5
69
The characteristics of the input impedance as a function of the antenna
arm length is exhibited by the equi-ki contours. Generally speaking, when the
antenna arm length is not long, say ki < 2.1, the characteristic of the equi-
ki contours behaves very much like that for the arc antenna; the reactance
is almost constant while the resistance increases monotonically as the pitch
is increased. For larger ki , the characteristic deviates both from that of
the arc antenna and that of the cylindrical antenna. When the pitch of the
helix is held constant and the radius of the helix is increased, the circu-
lar graph of the input impedance is first enlarged from that of the cylindrical
antenna. It reaches a certain maximum and then starts to shrink to the
input impedance of a cylindrical antenna again. This is because of the fact
that helices of zero radius and infinite radius both become cylindrical
antennas. For example, Figures 21a and 21b show the impedance for a con-
stant pitch helical antenna for which C = 0.25; in Figure 21a the circular
graph becomes larger for smaller values of C and it reaches the maximum at
about C = 0.75. For even smaller values of C the circular graph shrinks
and approaches to the impedance of the cylindrical antenna. Figures 22a
and 22b give the impedance for C = 0.5.
The convergence of the input impedance of helical antennas toward that
of cylindrical antennas is shown in Figures 23 and 24. It is seen that
there is a minimum in (/2) and a maximum in (Py ) for each given
'^ res ' antires
C . The locations of these extrema are shifted toward larger C as we increase
(a) The location of minimums of (/P,) are at
V
c l
0.4
0.625
0.825
1.05
1.125
C 2
0.25
0.5
0.75
1
1.05
1.25
V C 2
1.6
1.25
1.1
0.9
1.325
1.5
0.883
70
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a
V.
u
a?
0)
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s
3
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