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L I B RAR.Y
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UNIVERSITY
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621. 365
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no. 50-51
cop. 2
Digitized by the Internet Archive
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Raytheon Company
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Dr c Frank Fu Fang
IBM Research Laboratory
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Dr. A K. Chatterjee
Vice Principal
Birla Engineering College
Pilani, Rajasthan
India
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Head Antenna Department
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Aerospace Corporation
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Dr„ D„ E„ Royal
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Dr u S Das gup t a
Government Engineering College
Jabalpur, M o P
I ndi a
Dr. Richard C „ Becker
10829 Berkshire
Westchester, Illinois
ANTENNA LABORATORY
Technical Report No. 53
A STUDY OF THE NON-UNIFORM CONVERGENCE OF THE INVERSE OF A DOUBLY -
INFINITE MATRIX ASSOCIATED WITH A BOUNDARY VALUE
PROBLEM IN A WAVEGUIDE
by
R. Mittra
Contract AF33 (616) -6079
Project No. 9-Q3-6278) Task 40572
Sponsored by:
AERONAUTICAL SYSTEMS DIVISION
Electrical Engineering Research Laboratory
Engineering Experiment Station
University of Illinois
Urbana, Illinois
ENGINEhKING LIBRARY
ACKNOWLEDGMENT
The author wishes to acknowledge the helpful discussions with his col
leagues in the Antenna Laboratory of the University of Illinois and the
sponsorship of Aeronautical Systems Division, Wright-Patterson Air Force
Base, Ohio, through the research grant AF33 (616) -6079.
CONTENTS
1 . Introduction
2. Development of the Infinite Set of Equations for H-Plane
Bifurcation
3. Solution of Truncated Set of Equations
4. Numerical Calculations of Solutions of Finite and Infinite
Set of Equations
5. Asymptotic Behavior of Higher Order Coefficients
6. The Edge Condition and the Proper Choice of C
7. Further Work
Page
1
3
6
10
14
20
21
1. INTRODUCTION
In formulating electromagnetic boundary value problems we are often led to
an infinite set of equations. In most cases it is not possible to invert the
infinite matrix and we are forced to resort to truncating the above to a finite
size. We then solve the equations for a number of increasing sizes of the
matrix and study the convergence of the solution. If at least the leading
members of the unknown coefficients (which are usually of primary interest)
tend to converge, we feel satisfied and assume that we have obtained a reason-
ably good approximation for the leading coefficients.
It is the purpose of this paper to show in the first instance that for a
particular doubly infinite set of equations associated with the bifurcation
problem in a waveguide, there is a conditional convergence of the solution,
meaning that the solution converges to a different set of answers for every
different choice of the ratio C = P/Q. P and Q are the numbers of equations
from the first and second set, respectively, out of the doubly infinite set
of equations. These equations for the bifurcation problem have been obtained
by Hurd and Gruenberg , who have also presented an exact solution of the
infinite set through the use of calculus of residues , The above set, because
it has a known solution, is particularly suited for our purpose which is to
demonstrate by inverting several finite size matrices that there results a
conditional convergence when P -> oo and Q -> °o. Furthermore, we are also able
to find the ratio of P/Q which yeilds the correct answer in the limit.
In the second part of this paper we present a theoretical basis for
choosing the correct ratio of P/Q when working with a truncated set. With this
choice, when the size of the set is increased indefinitely while keeping the
2
ratio C constant, the solution does converge to the correct answer. It is
shown through the study of the asymptotic behavior of the higher order unknown
coefficients and the application of the edge condition, why only an unique
choice of the ratio would make the solution asymptotically tend to the correct
one and why otherwise an incorrect solution will result.
In a separate paper to be published in the near future we shall apply the
ideas developed here to other problems in a waveguide which cannot be solved
exactly. We will again show that there is non-uniform convergence of the
doubly infinite set and how using the ideas developed here one is able to
choose the correct ratio of P/Q for the problem under consideration.
2. DEVELOPMENT OF THE INFINITE SET OF EQUATIONS FOR II-PLANE BIFURCATION
The infinite set of equations which will be discussed here in connection
with the problem of H-plane bifurcation in a rectangular waveguide have been
derived by Hurd . We shall therefore skip the details and merely outline the
procedure for their derivation.
©
^0
X=b
X = Q
Figure 1. H-Plane Bifurcation of a Rectangular Waveguide
The geometry under consideration is shown in Figure 1. Assume that the
incident wave from the negative z-direction is a TE mode with the electric
vector parallel to the edge of the septum. It can be easily shown that the
only non-zero field components are E , II and H and that they can all be
expressed in terms of a scalar function (ft = E and its partial derivatives.
The problem can be stated in terms of the equation
V xz 4> + ^ 4>= °
(1)
and the following boundary conditions on (ftt
a) (ft and V$ are finite everywhere in the region concerned except at the
edge of the bifuraction at z = 0, x = b where \J (ft becomes infinite.
b) (ft and SJ(ft are continuous in the subregions and at z = 0,
c) (^behaves as an outgoing wave at large (z) apart from the incident field.
4
d)
0.
1/2
e) (j) satisfies the adge condition and hence goes to zero at the edge as r
where r is the distance from the edge. V 0.
It is easily verified that (j) and (f) expressed in the following
equations satisfy the equation (1) and the conditions (a), (c) and (d) . The
expressions are:
_. a z
, 7Tx< °S, . n7Tx n ,_ .
d> = A sin ■ — + S A sin — e (2a)
^A a n=1 n a
d> . = Z B sin —— e ^n (2b)
^B n b
n=l
, £ „ n7Tx ~^ n z ,„ ,
(h = 2 C sin — r- e n (2c)
^C , n jc
n=i
where
A = amplitude of the incident field
a =[( 2V - k 2 ] 1/2
n l a J
P ft «[(^) 2 -k 2 ] V2
n77 2 2 1/2
277
k = '— r-, X = free space wave length.
a ^ (3 and y' s are the mode propagation constants in the three regions.
By applying the continuity conditions at z = 0, and subsequently equating
the Fourier components of the resulting equations in the range < x < b and
5
b < x < c one arrives at the following doubly infinite set of equations after
some manipulations (for details see Hurd):
nTTb
(-) P 2p77 S (A 6' +■ A ) sin
— r— n=l n n
B = ^
a 2 -P 2
n P
n7Tb . 7Tb
^ A sin A sin —
* a - p = a + (3 P-l, ....-oo (3a)
n=l n p 1 p
n7Tb . 7Tb
^ A sin A sin —
2 -2_ _L- = a q = 1, ....oo (3b)
The set of infinite equations (3) has been solved exactly by Hurd. We
shall however^ concentrate on the solution of the truncated set of equations
for various combinations of P and Q where these are the number of equations
from the set (3a) and (3b) respectively. We proceed to do this in the follow-
ing section.
3. SOLUTION OF TRUNCATED SET OF EQUATIONS
Consider the solution of the turncated set of equations
n77b . 7Tb
P+Q A sin — — A sin —
2 n a
o - P
n=l n p
a i + Pp
p = 1, 2, . ... P
(4a)
ntfb . 7Tb
P+Q A sin — — A sn,n ■ —
y, _n ?L_.— a
a - v a. + v
n=l n Y q 1 T q
q = 1, 2, . ... Q
(4b)
In attempting to solve the set of equations (4) we recognize first of all
that the determinant of the equation is of a particular kind which is called a
double alternant. Written explicitly the determinant is
A =
1
i
1
>
y
1
1
a -P >
1
1 .
1
a i-e 2 '
2 r 2
a 3" P 2
a -6
P+Q ^2
1
1
1
)
)
1
1
a -6 '
2 HP
1
)
a P+Q" (3 P
1
a rV
a &i
a p+Q _Y l
a i"V
2 Y Q
a P+Q~ >l 'Q
(5)
It is to be noted that we have grouped the factors sin n7TJj/a with the
unknowns A .
n
It is observed that each element of the determinant is a reciprocal of the
difference of two quantities, a and (3 or -y in general, only one of which, vi;
7
the a, changes as one goes along the columns whereas only (3 or y changes as
one goes down the rows. Hence it' a > a , where r and n are two different
subscripts, then it is obvious that the determinant A -> and that the zero
is a simple one. This is effective to saying that (a - a ) for various
combinations of r and n are the factors in the continued product expansion of
the determinant. In a similar manner we observe that (S - (3 ) , (v - v ) and
r n ' "r 'n
((3 - -y ) are also to be included in the product expansion of the determinant.
We have to make sure, however, that all such factors are included in the
product and that none is repeated. It is also obvious that the denominator of
this expression for A must contain the factors (a - (3 ) and (a - -y ) for various
combinations of the subscripts. The expression for the A developed on the
basis of above arguments, is
M,M-1 P, P-l Q,Q-1 Q,P
n " (a - a ) IT ((3 - {3 ) II (v -y ) II (p - y )
yj , ,» , m n . . r m r n Y m Y n r m T n
y m=(nntl),n=l m=n-f-l , n=l m=n+l , n=l m=l ; n=l
A= (_1) M,P M,Q
n (a -(3 ) n (a -v )
m n , m "n
m= 1 , ( n= 1 m=i , n= 1
where M = P + Q; v is an even or odd integer depending on P and Q and as yet
undertermined, although we shall not really need to find v. This is because
in order to calculate the coefficients A we will have to find only the ratio
r J
of the determinants A /A where A is the determinant obtained by replacing
the rth column of A by the column representing the right hand side of (4).
[t is fairly straightforward to see that
A r = A A(a r ^ - a ± ) (7)
'here A(a -> - a, ) is the determinant A with a replaced by - cl . We can
' r 1 r 1
therefore obtain through the ase of (6) and (7) an expression for A in the
product form, which is
sin (■ — -) ACa^ -> - a 1 )
A - — — — "Str- = A r — r—
r , 77b. A
sm ( ■ — )
P (a - P ) Q (a - v ) r-1 (-a_ -a ) M (a + a )
= A n * ?\ n . r \ • n ., - 1 " n n 1
_ (-a n -(3 ) (-CL-.-V ) _ ( a -a ) (a ■
n=l 1 n n=l 1 "n n=l r n n=l n
a )
r
P+Q P (a r~ p n ) Q (a r" V (1) (
r n
^n
n
(9)
We also quote below for comparison and reference, the result arrived at
by Hurd as a solution of the infinite set of equations. His expression is
n (1) (-a ljtt ) n (a JL ,|3) Jl (a pY )
J1 U) (a r a);ri ( Wl ,p) n*(-a 1 ,y)
x exp [ (2a/70 1 - a + bin(a/b) + cin(a/c) 1 ]
(10)
where for instance
00 *: d7T
n (cj a) = n (a - <*>)(=-) e p "
p=l P P?7
and the superscript (1) implies as before that the factor corresponding to
p ; = 1 is to be omitted.
10
4. NUMERICAL CALCULATIONS OF SOLUTIONS OF FINITE AND INFINITE SET OF EQUATIONS
In this section we shall present and compare the numerical values of the
reflection coefficient R for certain choices of b/c, b/a (note these fix c/a) v
etc. We shall see that even when (P + Q) is very large the expression (9) for
a reflection coefficient yields different values for different ratios of P/Q.
We shall then compare the answer for a particular choice of P/Q and see that
it indeed converges to the exact value of R calculated through the use of (10).
For the purposes of numerical calculation: we shall consider the particular
case when all the mode propagation constants excepting a are real and a is
purely imaginary. This is done merely for the ease of numerical work, and the
conclusions regarding the convergence phenomenon reached in this case will
still be applicable to the general case when the above condition is not true.
For this case it is obvious from (9) that R=e i.e. I r| = 1. The
angle = argument R is given by
= arg R
S tan F
n=l K n
It - 16
1j tan —
n=l ^n
P+Q
3 tan"
n=l
1 6
(11)
where a = j6
But since (3
2 2 / 2 9
(nTT/b) - k and 6 = ^ k (7T/a) , one can write
.-16 • -l
tan p— = sin
b5_
77
(n - — )
After similarly expressing the other arctans in terms of arc sines, one
can rewrite (11) as
11
a& b6 c6
e P ; Q . -i -? ; , -i -j l . -i i
- = 1] sin — S sin - S sin
2
n=l v/V - 1 n=l /2 7b.2 n=l / 2 ,c N 2
/n -(-) /n - (-)
(12)
For the purposes of numerical calculation (12) is more suitable than (8)
A similar expression for the exact 9 has been derived from (10) and is given
in the following
| (1 . ^ in a _ c in a 5
7T a b a c 2 1'
where
■^1.0, -8,(^,0,
S (u ; v 0) = 2 [sin
N m , 2 K 2 ^ 1 / 2 n
n=N (n - b )
Calculations were done for 9/2 for the following of dimensions of the
waveguide bifurcation
a = 2.286 b/a = 0.313 hence c/a = 0.687 b/c = 0.456
and for \ = 3 cms: The results for 9, calculated using (11) are shown in the
following table for various combinations for P and Q and are calculated to the
nearest degree. The 9 values are compared with 9 calculated from (13).
It is seen that the values of 9 converge to different numbers for different
values of P/Q but that the value of 9 is not too sensitive for slight changes
in P/Q as may be seen from the last four lines of the table in which 9 is seen
to agree with 9 . It is also observed that there is a considerable deviation
from the correct value for some choices of P/Q. The correct choice for P/Q
TABLE 1
12
p
Q
P/Q
(radians)
(calculated)
8 (radians)
ex
(exact)
10
10
1
-42°
20
20
1
-42°
20
10
2
-18°
30
15
2
-18°
-50°
8
17
0.471
-50°
10
22
0.455
-50°
20
44
0.455
-50°
10
20
0.5
-50°
13
seems to be in the vicinity of P/Q = 0.48 and to find it more accurately
through numerical means one has to calculate the results to a higher degree of
accuracy. In the following section we give an explanation for the behavior of
the results displayed in Table 1 and a theoretical basis for t^e correct choice
of P/Q. It will be shown that the criterion for the proper choice of P/Q is
based on the asymptotic behavior of the higher order mode coefficients A } for
(P + Q) large. We develop the necessary formulas for the study of the asymp-
totic behaviors in the following section.
14
5. ASYMPTOTIC BEHAVIOR OF HIGHER ORDER COEFFICIENTS
In this section starting from (8) we shall develop an asymptotic expres-
sion for A /A for large P and Q.
Let us rewrite (8) as
a' = n
r n=l
p (1 - pr> q (1 - ^ p+q
n
n
(i + -i-)
(i)
(1 + p-) (1 + — )
r n "n
(1 -_)
(14)
where
r7Tb
A' = A
r r
7Tb
Since
n£
b
n77
c
and
n77
for large n,
it is convenient to introduce some additional factors in the numerator and
denominator of (14) and rearrange it as
1
n
n=l
a
1 - *r>
n
(1 - — )
(1 )
p -' V n ' P+ Q (r)
a b , a c
n
x < n
n=1 n * %
(i + g-)
r n
n
n=l
f~> P + Q (1 + f*
— n ■ -2-
n=l
(1
a a a b a c
a ) n/i--f-> V 1 '
r n=l n=l
77
P+Q
n (i
n=l
(1
P+Q
-) n (i
p
n (i +
:1
a x b
Q
) n (i
n=l
15
Now notice first of all that the factors inside the first two curly
brackets tend asymptotically to constants as P and Q are increased indefinitely
for a given r. This is because the factors like
=> 1 for large n
a
a - Q -i>
and similar reasoning holds for other factors appearing inside the curly
brackets.
It will therefore be sufficient to study the behavior of the ratio of the
products F, where
a b a c
p _£_ Q _£_
n (1--I-) n a --L-) V
P = ^— 55L-. (i --JL-) (16)
a a r
P+Q -T-
n (i L_)
T n
n=l
md of a similar ratio of products with - a replacing a
Let us rewrite (16) as
a b a c
f ' = " (1 ~- P n£l (1 - ~r- ) Q nSl (1 - -r- > (P + Q) r
a a a b/7T a c/7T
a b/77 P+Q -|- P r Q r
(p+q) r n (i - — _ >
n=1 " (17)
2
nd then use the following representation (see Magnus page 2) of l/r(z+l)
here T(x) is the Gamma function of argument x.
16
lim m Z n (1 + -) (18)
T(z+1) m -> oo _ n
to recast (16) into its asymptotic form for large P and Q. We derive using
(18) in (17) and letting P/Q = C(constant)
a a a a
-£_ r oo 77
a b/77 a c/77
X (1 + ^) r (1 + C) r (20)
The next step is to study the limit of F as r and hence a becomes very
large We have from Stirling's formula,
z
r (z) ~ -TJo for a lar g e z ( 21 )
z i/2
Using (21) in (20) and letting a -» r77/a we find
-.rb.rb/a re re/ a
F _* K -1 / • 4?2 (1 + F> rb/a "(l - C) rC/a
(r) r r 3/2 C
for large r ? P and Q,
(22)
17
where K is a function with a limited upper bound lor arbitrarily large r. II
is to be noted that we have used the fact that (1 - (a a/7T)/r) /sin a a has the
r / r
limit r7T/a as a -> rTT/a,
Writing
r ,rb rc.rb/a + rc/i
• = ( — + — )
,e obtain from (22)
^ (i + i) rb/a (i + o rc/i
^T 7 * a + !) rb/a (i + :V C/1
b c
for P/Q = C, P -> oo , Q -> co and r large.
Equation (23) gives the desired asymptotic behavior of F. It is not
ifficult to show that the ratio of products
P+Q —
n (i +-1— ) .
n=l
(23)
a--?-)
a b a c
p JL ' q JL
n a ■+-£-) na + -f)
n=l n=l
= G, say,
is the asymptotic behavior
lim G . i . ra * ^ ra + ^ (1 + I,-i^ (1 + c,-i^
a a a a C
Q-*«> IT r(1 + IT*
W (1--4-) (24)
18
and hence tends to a constant as r becomes large. We can sum all this up and
arrive at the limit
K x (1 + i) rb/a (1 + O rC/a
D lim A r ~ "372 * — crb/a „ b,rc/a f ° r large T > (25)
P -»oo r (1 + — ) (1 + -)
Q->*> be
where K is a constant.
Suppose now that we pick a C which is different than the ratio of b/c and
let C > b/c. Calling X = b/c, we can write (25) after some manipulation as
K l
lim A ' ^
P -*0o r
Q -> oo
%$% ■ <§>* rc/a '« i«- '■ < 26 >
Without loss of generality we can let X (= b/c) > 1, and show that
C X + C X X > X X + X X C, since C > X > 1
and hence
(£) X > ( I_±_C) or (i-i-C) X > 1, (28)
P -»«w r
Q -> 00
where "H < 1 .
This shows that under this condition the higher order coefficients have an
exponential decay with r unless T| = l which happens when C = X.
19
Similarly it may be shown that if C < X > 1,
/ 1 m r
lim A ~ b for large r
p _>co r r °/ 2
Q -* °°
where b > 1, i.e.. that in this case the coefficient A has an exponential
growth with r unless h = 1, which happens again when C = X.
When C = X, then the coefficients A have an algebraic behavior for large
3/2
r and go to zero as 1/r
This completes the study of the asymptotic behavior of the higher order
coefficients for different ranges of C/X with X > 1. In summarizing we note;
(i) A has an exponential growth for large r when X > C and an expo-
r
nential decay for C > X.
(ii) A has an algebraic behavior and is 0(l/r ) for large r when
C = X.
In the next section we appeal to the edge condition given by condition
(e) in Section 2 and show how the proper choice for C = P/Q is to be made.
20
6 . THE EDGE CONDITION AND THE PROPER CHOICE OF C
The condition at the edge of septum requires that the field potential X = b/c. the expression
for d(f) A/3z I at the edge z = has a bounded sum and hence violates the edge
condition. Also when C < X, A 7 has an exponential growth and hence when sub-
stituted in (29) j makes d A/9z go to infinity in a much stronger manner than
z as z -^ 0. It is only when C = X and A / is 0(l/n ) for large n that
-1/2
the sum of the series is 0(z ) as z -> as may be shown by following a method
due to Hurd . It may be pointed out that in the exact solution obtained by
/ 3/2
Hurd A is 0(l/n ) fcr large n, as it of course must,
n
The conclusion is then that the proper choice of C = P/Q is C = b/c = X;
only when such a choice is made and, P and Q are increased indif initely, that
the solution converges to the one which satisfies the edge condition, and thus
yields an unique answer satisfying the physical condition.
This section will be concluded with one further remark which concerns the
coefficients A , m = 1.2.... etc., m finite. From (15), (16) and (19) it is
nr
clear that even when P ->oo and Q -» oo the limits of A A etc. are dependent or
C, i.e. j on the choice of the ratio of P and Q. Numerical calculations for A
for various values of C have been made using the above equations and the results
agree with those presented in Table 1. Only the choice of C - b/c yields the
correct answer for A although the result for the leading coefficient A is not
very sensitive in the vicinity of the correct choice of C.
21
7. FURTHER WORK
Further work has been done along this line in connection with the step
discontinuity problem in a waveguide. Although the exact solution of the
step problem is not possible, the correct choice of P/Q can be made on the
basis of asymptotic behavior of the higher ortjer coefficients. A detailed
discussion on this problem will be the subject of a later report.
ANTENNA LABORATORY
TECHNICAL REPORTS AND MEMORANDA ISSUED
Contr act AF33 (616 ) -310
"Synthesis of Aperture Antennas/' Technical Report No. 1, C.T.A. Johnk,
October, 1954.*
"A Synthesis Method for Broad-band Antenna Impedance Matching Networks,"
Technical Report No. 2, Nicholas Yaru, 1 February 1955.*
"The Asymmetrically Excited Spherical Antenna," Technical Report No . 3,
Robert C. Hansen, 30 April 1955.*
"Analysis of an Airborne Homing System," Technical Report No. 4, Paul E.
Mayes, 1 June 1955 (CONFIDENTIAL).
"Coupling of Antenna Elements to a Circular Surface Waveguide," Technical
Rep ort N o. 5, H. E. King and R. H. DuHamel, 30 June 1955.*
"Axially Excited Surface Wave Antennas," Technical Report No. 7, D. E. Royal,
10 October 1955. *
"Homing Antennas for the F-86F Aircraft (450-2500mc), "' Technical Report No. 8,
P. E. Mayes, R, F. Hyneman, and R. C. Becker, 20 February 1957, (CONFIDENTIAL)
"Ground Screen Pattern Range," Technical Memor andum N o. 1, Roger R. Trapp,
30 July 1955.*
Contract AF33 (616) -3220
"Effective Permeability of Spheroidal Shells," Technica l Report No. 9, E. J.
Scott and R. H DuHamel, 16 April 1956.
An Analytical Study of Spaced Loop ADF Antenna Systems," Technical Report
No. 10, D. G. Berry and J. B. Kreer, 10 May 1956.
A Technique for Controlling the Radiation from Dielectric Rod Waveguides,"
technical Report No, 11, J. W. Duncan and R. H. DuHamel, 15 July 1956.*
Directional Characteristics of a U-Shaped Slot Antenna,"' Technical Report
ko. 12, Richard C Becker, 30 September 1956.**
|
Impedance of Ferrite Loop Antennas," Technical Report No, 13, V. H. Rumsey
nd W. L Weeks, 15 October 1956. -- — —
Closely Spaced Transverse Slots in Rectangular Waveguide," Technical Repor t
EL_Ai> Rich ard F. Hyneman, 20 December 1956.
"Distributed Coupling to Surface Wave Antennas, " Technical Report No. 15,
Ralph Richard Hodges, Jr., 5 January 1957.
"The Characteristic Impedance of the Fin Antenna of Infinite Length," Technical
Report No, 16, Robert L. Carrel, 15 January 1957.
"On the Estimation of Ferrite Loop Antenna Impedance," Technical Report No. 17,
Walter L e Weeks, 10 April 1957.*
"A Note Concerning a Mechanical Scanning System for a Flush Mounted Line Source
Antenna," Technical Report No. 18, Walter L. Weeks, 20 April 1957.
"Broadband Logarithmically Periodic Antenna Structures," Technical Report No. 3 ,
R. H. DuHamel and D„ E. Isbell, 1 May 1957.
"Frequency Independent Antennas," Technical Report N o. 20, V. H. Rumsey, 25
October 1957.
"The Equiangular Spiral Antenna," Technical Report No, 21, J. D. Dyson, 15
September 1957.
"Experimental Investigation of the Conical Spiral Antenna," Technical Report
No^_22, R. L„ Carrel, 25 May 1957.**
"Coupling between a Parallel Plate Waveguide and a Surface Waveguide," Technic;
Report No. 23, E. J. Scott, 10 August 1957.
"Launching Efficiency of Wires and Slots for a Dielectric Rod Waveguide,"
Technical Report No. 24, J. W. Duncan and R, H, DuHamel, August 1957.
"The Characteristic Impedance of an Infinite Biconical Antenna of Arbitrary
Cross Section," Tec hnical Report No . 25, Robert L. Carrel, August 1957.
"Cavity -Backed Slot Antennas," Technical Report No. 26, R. J. Tector, 30
October 1957.
"Coupled Waveguide Excitation of Traveling Wave Slot Antennas," Technical
Report No. 27, W. L, Weeks, 1 December 1957.
"Phase Velocities in Rectangular Waveguide Partially Filled with Dielectric,"
T e chn i c al Rep o r t^ No . _28_, W. L. Weeks, 20 December 1957.
"Measuring the Capacitance per Unit Length of Biconical Structures of Arbitral-;
Cross Section," Technical Report No. 29, J. D. Dyson, 10 January 1958.
"Non-Planar Logarithmically Periodic Antenna Structure," Technical Report No. '. .
D. W. Isbell, 20 February 1958.
"Electromagnetic Fields in Rectangular Slots," Technical Report No. 31, N. J.
Kuhn and P. E. Mast, 10 March 1958.
"The Efficiency of Excitation of a Surface Wave on a Dielectric Cylinder,"
Technical Report No. 32, J. W„ Duncan, 25 May 1958.
"A Unidirectional Equiangular Spiral Antenna," Technic al Report No. 33,
J. D. Dyson, 10 July 1958.
"Dielectric Coated Spheroidal Radiators," Technical Report No. 34, W. L.
Weeks, 12 September 1958
"A Theoretical Study of the Equiangular Spiral Antenna," Technical Report
No. 35. P. E. Mast, 12 September 1958.
Contract AF33 1616) -6079
"Use of Coupled Waveguides in a Traveling Wave Scanning Antenna, " Technical
RiP^LL-!^— 3 -^ R - H - MacPhie, 30 April 1959.
"On the Solution of a Class of Wiener-Hopf Integral Equations in Finite and
Infinite Ranges/"' Technical Report No. 37, Raj Mittra, 15 May 1959.
"Prolate Spheroidal Wave Functions for Electromagnetic Theory," Technical
Report No . 38, W L„ Weeks, 5 June 1959.
log Periodic Dipole Arrays," Technical Report No. 39, D. E„ Isbell, 1 June 1959.
"A Study of the Coma-Corrected Zoned Mirror by Diffraction Theory," Technical
Report No„ 40, S. Dasgupta and Y. T Lo, 17 July 1959 .
'The Radiation Pattern of a Dipole on a Finite Dielectric Sheet," Technical
R gPP2:t_Ng.° _ . 1L K - G - Balmain, 1 August 1959.
'The Finite Range Wiener-Hopf Integral Equation and a Boundary Value Problem
in a Waveguide," Technical Report No. 42, Raj Mittra, 1 October 1959.
Impedance Properties of Complementary Multiterminal Planar Structures,"
Techni cal Report No. 43, G. A. Deschamps, 11 November 1959.
"On the Synthesis of Strip Sources," Technical Report No. 44, Raj Mittra,
4 December 1959.
Numerical Analysis of the Eigenvalue Problem of Waves in Cylindrical Waveguides,'
Technical Report No. 45, C„ H Tang and Y. T. Lo, 11 March 1960.
('New Circularly Polarized Frequency Independent Antennas with Conical Beam or
Omnidirectional Patterns," Technical Report_No. _46 J) J D. Dyson and P. E. Mayes,
June 1960.
Logarithmically Periodic Resonant-V Arrays," Technical Report No. 47, P. E. Mayes
md R. L, Carrel, 15 July 1960.
|A Study of Chromatic Aberration of a Coma-Corrected Zoned Mirror," Technical
eport No. 48, Y. T. Lo.
"Evaluation of Cross-Correlation Methods in the Utilization of Antenna Systems,
Techn ical Report No. 49, R ff„ MacPhie, 25 January 1961.
I
"Synthesis of Antenna Product Patterns Obtained from a Single Array," Technical
Report No. 50, R„ H. MacPhie.
* Copies available for a three-week loan period.
** Copies no longer available.
AF 33(616)-6079
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Cambridge, Massachusetts
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1200 Duke Street
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Attn: Antenna Section, Code 523
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Walt Whitman Road
Melville, L, I,, N Y,
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Box 552 (Antenna Section)
Lansdale., Pennsylvania
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(Antenna Section)
3412 Century Blvd.
Inglewood, California
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Attn: Librarian (Antenna Section)
299 Atlantic Ave Q
Boston 10, Massachusetts
Amphenol-Borg Electronic Corporatioi
Attn: Librarian (Antenna Section)
2801 S, 25th Avenue
Broadview,, Illinois
Bell Aircraft Corporation
Attn: Technical Library
(Antenna Section)
Buffalo 5, New York
Bendix Radio Division of
Bendix Aviation Corporation
Attn: Technical Library
(For Dept 462-4)
Baltimore 4, Maryland
Boeing Airplane Company
Aero Space Division
Attn: Technical Library
M/F Antenna & Radomes Unit
Seattle, Washington
Boeing Airplane Company
Attn: Technical Library
M F Antenna Systems Staff Un:
Wichita, Kansas
Chance Vought Aircraft Inc.
THRU: BU AER Representative
Attn: Technical Library
M/F Antenna Section
P, 0. Box 5907
Dallas 22, Texas
Collins Radio Company
Attn: Technical Library (Antenna
Section)
Cedar Rapids, Iowa
Convair
Ft. Worth Division
Attn: Technical Library (Antenna
Section)
Grants Lane
Fort Worth, Texas
Convair
Attn: Technical Library (Antenna
Secti on)
P.O. Box 1950
San Diego 12, California
Dalmo Victor Company
Attn technical Library (Antenna
Seci
1515 industrial Way
Belmont,. California
Goodyear Aircraft Corporation
Attn- Technical Library
M F Dept 474
1210 Massilon Road
Akron 15, Ohio
Dome & Margolin, Inc.
Attn Technical Library (Antenna
Section)
30 Sylvester Street
Westbury, L. I , N_ Y,
Granger Associates
Attn: Technical Library
Antenna Section
974 Commercial Street
Palo Alto, California
Dynatronics Inc .
Attn Technical Library (Antenna
Section)
Orlando, Florida
Grumman Aircraft Engineering Corp
Attn; Technical Library
(M/F Avionics Engineering)
Bethpage, New York
Electronic Communications, Inc.
Research Division
Attn Technical Library
1830 York Road
Timonium, Maryland
The Hallicraf ters Company
Attn: Technical Library (Antenna
Section)
4401 W Fifth Avenue
Chicago 24, Illinois
Fairchild Engine & Airplane Corporation
Fairchild Aircraft & Missiles Division
\ttn Technical Library (Antenna
Section)
iagerstown 10 Maryland
jeorgia Institute of Technology
Engineering Experiment Station
ittn Technical Library
M, F Electronics Division
.tlanta 13, Georgia
reneral Electric Company
lectronics Laboratory
ttn; Technical Library
lectronics Park
yracuse, New York
eneral Electronic Labs,, Inc,
ttn: Technical Library (Antenna
Section)
13 Ames Street
ambridge 42 Massachusetts
jsneral Precision Lab,, Division of
General Precision Inc
ittn: Technical Library (Antenna
Section;
k Bedford Road
I easantville,, New York
Hoffman Laboratories Inc.
Attn: Technical Library (Antenna
Section)
Los Angeles 7, California
John Hopkins University
Applied Physics Laboratory
8621 Georgia Avenue
Silver Springs, Maryland
Hughes Aircraft Corporation
Attn: Technical Library (Antenna
Section)
Florence & Teal Street
Culver City^, California
University of Illinois
Attn; Technical Library (Dept of
Electrical Engineering)
Urbana, Illinois
ITT Laboratories
Attn: Technical Library (Antenna
Section)
500 Washington Avenue
Nutley 10, New Jersey
Lincoln Laboratories
Massachusetts Institute of Technology
Attn; Document Room
P, 0. Box 73
Lexington 73 Massachusetts
University of Michigan
Radiation Laboratory
Willow Run
201 Catherine Street
Ann Arbor, Michigan
Litton Industries
Attn: Technical Library (Antenna
Section)
4900 Calvert Road
College Park,, Maryland
Mitre Corporation
Attn: Technical Library (M/F Ele<|
tronic Warfare Dept. D-21)
Middlesex Turnpike
Bedford, Massachusetts
Lockheed Missile & Space Division
Attn: Technical Library (M/F Dept-
58-40, Plant 1, Bldg. 130)
Sunnyvale, California
New Mexico State University
Attn: Technical Library (M/F
Antenna Dept)
University Park, New Mexico
The Martin Company
Attn; Technical Library (Antenna
Section)
P, C Box 179
Denver 1, Colorado
North American Aviation Inc.
Attn: Technical Library (M/F
Engineering Dept.)
4300 E. Fifth Avenue
Columbus 16, Ohio
The Martin Company
Attn: Technical Library (Antenna
Section)
Baltimore 3, Maryland
The Martin Company
Attn: Technical Library (M/F
Microwave Laboratory)
Box 5837
Orlando, Florida
W, L„ Maxson Corporation
Attn: Technical Library (Antenna
Section)
460 West 34th Street
New York 1,, New York
McDonnell Aircraft Corporation
Attn: Technical Library (Antenna
Section)
Box 516
St c Louis 66, Missouri
Mel par 9 Inc
Attn: Technical Library (Antenna
Section)
3000 Arlington Blvd
Falls Church, Virginia
North American Aviation Inc.
Attn: Technical Library
(M/F Dept 56)
International Airport
Los Angeles, California
Northrop Aircraft, Inc.
NORAIR Division
Attn: Technical Library
(M/F Dept 2135)
Hawthorne, California
Ohio State University Research
Foundation
Attn: Technical Library
(M/F Antenna Laboratory)
1314 Kinnear Road
Columbus 12. Ohio
Philco Corporation
Government & Industrial Division
Attn: Technical Library
(M/F Antenna Section)
4700 Wissachickon Avenue
Philadelphia 44, Pennsylvania
iladio Corporation of America
ICA Laboratories Division
An: I ■ > hni ca] Library
Antenna Section)
>, i ncel rn New Jersey
tadiation Jnc„
Vttn Technical Library (M/F)
Antenna Section
)rawer 37
lei bourne, Florida
tadioplane Company
ittn Librarian (M/F Aerospace Lab)
1000 Woodly Avenue
'an Nuys, California
H R B Si ng i ir Corpor * I
Attn.; Librarian 'Antenna Lab)
Pennsy] vania
Sperry Microwave Electronics Company
Attn; Librarian (Antenna Lab)
P„ 0. Box 1828
Clearwater., Florida
Sperry Gyroscope Company
Attn: Librarian (Antenna Lab)
Great Neck^ L„ I,, New York
Stanford Electronic Laboratory
Attn: Librarian (Antenna Lab)
Stanford, California
:amo-Wooldridge Corporation
>ttn, Librarian (Antenna Lab)
onoga Park., California
Stanford Research Institute
Attn: Librarian (Antenna Lab)
Menlo Park, California
and Corporation
.ttn: Librarian (Antenna Lab)
700 Main Street
anta Monica, California
antec Corporation
ttn; Librarian (Antenna Lab)
3999 Ventura Blvd.
alabasas, California
aytheon Electronics Corporation
ttn: Librarian (Antenna Lab)
089 Washington Street
ewton, Massachusetts
epublic Aviation Corporation
pplied Research & Development
j Division
ttn Librarian (Antenna Lab)
arroingdale. New York
Sylvania Electronic System
Attn,: Librarian (M/F Antenna &
Microwave Lab)
100 First Street
Waltham 54, Massachusetts
Sylvania Electronic System
Attns Librarian (Antenna Lab)
P 0„ Box 188
Mountain View,, California
Technical Research Group
Attns Librarian (Antenna Section)
2 Aerial Way
Syosset, New York
Ling Temco Aircraft Corporation
Temco Aircraft Division
Attn; Librarian (Antenna Lab)
Garland, Texas
anders Associates
ttn: Librarian (Antenna Lab)
|5 Canal Street
ashua ? New Hampshire
Texas Instruments,, Inc„
Attn: Librarian (Antenna Lab)
6000 Lemmon Ave„
Dallas 9, Texas
)uthwest Research Institute
ttn: Librarian (Antenna Lab)
)00 Culebra Road
Jin Antonio, Texas
A „ S Thomas I nc „
Attn: Librarian (Antenna Lab)
355 Providence Highway
Westwood, Massachusetts
Westinghouse Electric Corporation
Air Arms Division
Attn Librarian (Antenna Lab)
P„ : Box 746
Wheeler Laboratories
Attn: Librarian (Antenna Lab)
Box 561
Smithtown, New York
Electrical Engineering Research
Laboratory
University of Texas
Box 8026, Univ„ Station
Austin, Texas
University of Michigan Research
Institute
Electronic Defense Group
Attn: Dr„ J „ A M, Lyons
Ann Arbor, Michigan
Dr„ Harry Letaw, Jr„
Raytheon Company
Surface Radar and Navigation
Operations
State Road West
Wayland, Massachusetts
Dr„ Frank Fu Fang
IBM Research Laboratory
Poughkeepsie, New York
Mr. Dwight Isbell
1422 11th West
Seattle 99, Washington
Dr. A. K„ Chatterjee
Vice Principal
Birla Engineering College
Pilani, Rajasthan
India
New Mexico State University
Head Antenna Department
Physical Science Laboratory
University Park, New Mexico
Bell Telephone Laboratories, Inc„
Whippany Laboratory
Whippany, New Jersey
Attn: Technical Reports Librarian
Room 2A-165
Robert C c Hansen
Aerospace Corporation
Box 95085
Los Angeles 45, California
Dr c D„ E„ Royal
Ramo-Wooldridge, a division of
Thompson Ramo Wooldridge Inc„
8433 Fall brook Avenue
Canoga Park, California
Dr S c Dasgupta
Government Engineering College
Jabalpur, M„P
India
Dr Richard C „ Becker
10829 Berkshire
Westchester., Illinois
Antenna Laboratory
Technical Report No. 55
AN INVESTIGATION OF THE NEAR FIELDS ON THE CONICAL
EQUIANGULAR SPIRAL ANTENNA
by
0. L. McClelland
Contract AF33 (657)-8460
Project No. 6278, Task No. 40572
May 1962
Sponsored by:
AERONAUTICAL SYSTEMS DIVISION
Electrical Engineering Research Laboratory
Engineering Experiment Station
University of Illinois
Urbana, Illinois