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JtlMMJMaDlQJQnrUJllU JL. J
IMHII
L I B R.AR.Y
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UNIVERSITY
Of ILLINOIS
621. 3C5
IJl655\e
no. 3 1 -36
cop
.2
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UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN
APR 1 8 HI76
MAR 2 8 1976
L161 — O-1096
ANTENNA LABORATORY
Technical Report No. 32
THE EFFICIENCY OF EXCITATION OF A
SURFACE WAVE ON A DIELECTRIC CYLINDER
by
James Wilbur Duncan*
25 May 1958
Contract AF33 (616)-3220
Project No. 6(7-4600) Task 40572
Sponsored by:
WRIGHT AIR DEVELOPMENT CENTER
Electrical Engineering Research Laboratory
Engineering Experiment Station
University of Illinois
Urbana, Illinois
♦Submitted in partial fulfillment of the requirements for the degree of
Doctor of Philosophy in Electrical Engineering at the University of Illinois, 1958
~JL6' 3 ACKNOWLEDGMENT
The author wishes to thank Prof. V.H. Rumsey who was his adviser at
the beginning of the research and Dr. E.C. Jordan who was his adviser at
its conclusion. He is indebted to Dr. R.H. DuHamel for several helpful
suggestions and is particularly grateful to Dr. P.E. Mayes of the Antenna
Laboratory for his counsel throughout the entire work.
The help of G. Berryman and K. Rosenberg, who performed most of the
experimental measurements and numerical computations, is sincerely
appreciated.
Digitized by the Internet Archive
in 2013
http://archive.org/details/efficiencyofexci32dunc
iv
CONTENTS
Page
Acknowledgement iii
Abstract vii
List of Symbols viii
1. Introduction 1
2. Mathematical Formulation 5
3. Solution of the Boundary Value Problem 10
4. Singularities of the Integrand — Solution of the Mode Equation 17
4.1 Branch Points 17
-4.2 Poles 19
4,3 Solution of the Mode Equation 28
5. Evaluation of the Contour Integral 30
5.1 The Radiation Field 30
5.2 The Surface Wave Field 38
6. The Power Integrals and Excitation Efficiency 42
6.1 Radiated Power 43
6.2 Surface Wave Power 45
6.3 Excitation Efficiency 48
7 . Experimental Investigation 54
7.1 The Source Representation and Measurement Method 54
7.2 Experimental Equipment 58
7.3 Experimental Results 66
7.4 Elimination of the Ground Plane 75
Conclusions 76
Bibliography 78
Appendix A 80
CONTENTS (CONTINUED)
Page
Appendix B 83
Appendix C 106
Distribution List
vi
ILLUSTRATIONS
Figure
Number Page
1. Infinitely Long Dielectric Rod Excited by a Circular Filament 6
of Magnetic Current
2. Path of Integration in the t, Plane 18
W
3. Graph of the Functions F(X ) = X , . and
1 1 J n (X )
k (£) 1 1
f(I)
ylT 23
4. Graph of X = f (£) and X 2 + £ 2 = R 2
25
5. Distribution of Poles on the Real Axis 27
6. Path of Integration in the f Plane 32
7. Circular Cylindrical Coordinate System (fi,(j),z) and Spherical
Coordinate System (r.C/;, 9) 34
8. Excitation Efficiency as a Function of the Source Dimension
k a 52
9. Excitation Efficiency as a Function of the Source Dimension
V 53
10. View of the Dielectric Rod and Ground Plane 55
11. Representation of a Two Port Junction 56
12. Plot of the Image Circle on the Reflection Coefficient Plane 59
13. Cross Section of the Coaxial Exciter 61
14. Coaxial Exciter Mounted on the Ground Plane 64
15. Diagram of the Measurement Equipment 65
16. Measurement of a Junction with a Lossy Short Circuit 67
17. Arrangement for Measuring Rod Attenuation 68
18. Arrangement for Measuring the Transition Efficiency 70
19. Comparison Between Theoretical and Measured Efficiency 73
20. Comparison Between Theoretical and Measured Efficiency 74
21. Solution of the Mode Equation 83
vii
ABSTRACT
This paper treats the excitation of a transverse magnetic surface
wave on a dielectric cylinder. Specifically, the efficiency with which
a circular filament of magnetic current excites the E mode on an
infinite, dielectric rod is determined. The E mode is the lowest
order, transverse magnetic mode which can propagate on a dielectric rod.
The excitation efficiency of a source is defined as the ratio of the
power converted to the surface wave mode to the total power which is
delivered by the source.
The solution for the field generated by the magnetic current fila-
ment is obtained in the form of an inverse Fourier transform. The inte-
gral is evaluated by considering it as a contour integral in the plane
of the complex propagation constant. The far zone radiation field of
the source is obtained from the asymptotic evaluation of the integral
by means of the saddle point method of integration. The poles of the
integrand yield the TM surface wave modes, which are all circularly
symmetric. By restricting the range of k b, which is the circumference
of the dielectric rod in wavelengths, it is assured that only the E
mode is launched by the source. Curves are presented which show excita-
tion efficiency as a function of k a, the circumferential length of the
filament. It is found that a filament of the proper diameter will
excite the E_ mode with an efficiency of approximately 95 percent.
In the laboratory, a very narrow, annular slot in a large metal
sheet was used to approximate the magnetic current filament. The excita-
tion efficiency of the slot was measured using Deschamps' method for
calibrating a two-port waveguide junction. The measured efficiencies are
in close agreement with the results predicted by the theory.
viii
LIST OF SYMBOLS
77 Constant, 7T = 3.14159.
a Radius of the current filament,
b Radius of the dielectric rod.
€ or £ Permittivity.
(j. Permeability.
€ Relative dielectric constant,
r
(7 Conductivity
C Real variable of integration.
w Angular frequency.
X Free space wavelength,
o
X Guide wavelength.
(fi,(p,z) Circular cylinder coordinates.
(r,
, z). The rod is considered lossless, (J = 0, with
a magnetic permeability n and permittivity £, = € € , where € is the
relative dielectric constant. The medium surrounding the rod and
extending to infinity is free space, with constants n and € . The
electromagnetic field source is a filamentary ring of magnetic current
located at the plane z = 0. The ring is of radius a, where < a < b,
and is infinitesimally small in cross section. The source distribution is
represented as a product of Dirac delta functions in the p and z
coordinates as follows:
K =^5(P-a) o(z) (2.D
where (j) is a unit vector in the (j) direction, The source distribution
~K is independent of (j) (uniform), and is a unit source such that
ff*
a+A o+ />
0da = J j 5(p-a) §(z) • dz dp = 1
a-A o-A
a+A o+ A
£(p-a) dp = 1, and j £(z) dz = 1.
a-A o- A
K has the dimensions of volts per square meter. A harmonic time
dependence of e is selected for the source.
iU
r
02^
The electromagnetic field is a solution of Maxwell s equations.
-iG)t
Written in differential form for e time dependence, we have
VX E = + i'OfjiH - K
(2.2)
VX H = - iCi)€ E.
Taking the curl of the second equation and then substituting the first
relation for VX E yields
_ 2 — —
- V X V X H +0) !i6 H = - iO€ K. (2.3)
The only non-zero component of K is the coefficient of the unit vector
. We may write the component of the vector eq„ (2.3) and obtain
the non-homogenous scalar equation
[- v x vx h] . + coVe ■ H . = - iO>€ £(p-a) $(z),
where the (j) component of the bracketed term is indicated.
The magnetic current filament generates a field having components
H., E , and E , while the components E ., H , and H are equal to zero.
Due to the symmetry of the source, the field is independent of e_1 ^ Z dz - ( 2 ° 6 >
The inverse transform is given by
+oo
H 4> (p,z) = h J h (P>C) e+iCz d ^ (2 - 7)
Assuming that the transform of H. (P,z) exists, we multiply each term
-i£z
of (2.4) by e b and integrate over the infinite range with respect
to z to obtain
A + 1 dh + (k 2 _ ^2 1 ) h = _ lo€ 5(p _ a) (2 8)
dp p dp P
where the source variation £(z) is no longer present since
+co
S(z) e" 1 ^ 2 dz = 1.
9
We may consider (2.8) as an ordinary differential equation in which £
is a parameter constant. The boundary conditions on h (P,£,) are
obtained by taking the Fourier transform of the original boundary
conditions on H. (P,z) and -r-£. The problem is one of solving (2.8)
9 OH
forh(p,£) subject to the transformed boundary conditions, H. (P,z)
is then obtained by use of the inversion integral (2.7) . The integral
expression for H. (P,z) is derived in the following chapter .
10
3. SOLUTION OF THE BOUNDARY VALUE PROBLEM
Consider the form of the non-homogeneous differential equation (2.8),
One could solve (2o8) using the Hankel transform; however, the definition
of the delta function allows one to solve (2.8) in a simpler manner. The
delta function is defined by the relations
a+A
S
a-A
6(pra) dp = 1, and 6(p-a) = for p / a.
Consequently, for all p other than p = a, equation (2.8) reduces to the
homogeneous differential equation
d h 1 dh ,,2 -2 1 x ,_
(3.1)
The delta function 6(p-a) implies a boundary condition which h(p,£)
must satisfy at p = a. Multiplying each term of (2.8) by dp and inte-
grating over the interval 2A from p=a-Atop=a+A, one obtains
dh
dp
a+A
-I a-A
, , !<*>«, + (k » . t «,
h dp
a+A
a+A
i(^) dp = -1U8 J S(p-a) dp
-A a-A
(3.2)
Assuming that h(p,£) is continuous for all p, in the limit as A— »0
and p-»a, Eq„ (3.2) reduces to
= -io>8
(3.3)
p=a
11
We have shown that a continuous h(p,£>) which satisfies the homogeneous
equation (3.1) and whose first derivative is discontinuous by -iwg, at
p = a is a solution of (2.8).
The remaining boundary conditions on h(p,£>) follow from the boundary
conditions imposed onllAp,z) and E (p,z) by Maxwell's equations. Refer-
ring to Fig. 1, we denote the cross sectional area of the rod as region
I and the space outside the rod as region II. Since tangential H is
continuous at a magnetic current discontinuity, we see that Hi(p,z) is
continuous at p = a for all z including the filament position z = 0.
Since tangential E and H are continuous across a dielectric boundary,
we note that H.(p,z) and E (p,z) must be continuous at p = b for all z.
The corresponding conditions on h(p,£) are:
(i) Since UAp,z) is continuous at p = a for all z,
pt) = J° H (p,z) e-^ :
must also be continuous at p = a, hence
»(P^)] a . A = »(P.O] a+A (3.4)
(ii) Similarly, since H,(p,z) is continuous at p = b for all z,
h(P.O] b . A = h(p,U] b+A (3.5)
(iii) From (2.5) we write
l ^ E d> l
12
In region I, where - - b, the permittivity £ = £ g . In region II
where p > b, £ = g . E (p,z) continuous at p = b for all z requires
!5* i
= 8.
?5* i
ep + p H
b-A
b+A
Taking the transform, we obtain
dh 1 ,.
dp + p h
= e
b-A
dh 1 .
ap + P h
b+A
which reduces to
S r [f] - [§] ♦ [| >] C8 r -X) - o
L ^ J b+A L ^ J b-A LH J b+A
(3.6)
by reason of (3.5)
In order to determine H*(p,z) in region II, we must solve Eq. (2.8)
for the corresponding h(p,£>). Inside the dielectric rod, for all p
except p = a, (2.8) becomes
dp
h 1 dh ,,2
2 + p dp + (k l
^ - i) h -
(3.7)
where k, z = CO'
^O&i
Outside the dielectric rod and for all p, (2.8) becomes
dh ldh .2 ,2 1 N .
~2 + p dp + (k - & " "2> h = °
dp ^ ^ p
(3.8)
where k Q 2 =0) fi 0&() .
13
We proceed by writing general solutions h(p,£>) of (3.7) for the
regions - p - a, and a - p - b. The discontinuity at p = a appears
in the boundary condition (3.3). A general solution of (3.8) is written
which yields h(p,£) for the region p - b. The three solutions possess
six arbitrary constants. The constants are determined from the require-
ments that the field be finite at p =0, that the field be regular at
infinity and that it satisfy the radiation condition and the four boundary
conditions (3.3), (3.4), (3.5), and (3.6). Choosing the solution h(p,£)
for region II, the inversion integral (2.7) must then be evaluated to
yield H.(p,z) for p > b.
Equation (3.7) is recognized as a form of Bessel's differential
equation. It has the general solution
h(p,£) = A J^T^p) + P y^-tJ^p)
where A and P are arbitrary constants, J (li p) and Y C\) p) are Bessel
functions of the first and second kind, respectively, and
Consider the case - p - a. Since Hj(p,z) and, therefore, h(p,£)
must be finite at p = Q, we determine that P = 0, since Y (V p) is
unbounded as p— >0.
Hence
h(p,£) = A J x Ci^p) (3,9)
where
6 p £
^=[ kl -^]
2_l/2
M
For the region a - p - b, the Y function must be included, and we have
where a - p - b;
h(p,t,) = B J 1 Cl) 1 P) + C Y ± {]) p) (3.10)
P 1/2
For the region b - p we choose to write the general solution in terms of
Hankel functions of the first and second kind
h(p,;> = d h 1 (1) ay)) + q h 1 (2) g)^)
where
^o - K 2 - tf*
It will be seen later that, in general , "U is complex; therefore we
define the argument ofi) to be — arg l) - IT. As P-=»oo, H ^cp^
(2)
vanishes and H ^c\P} is unbounded for the defined argument ofV ft ;
therefore Q = and
h(p,t,) = D H 1 U) (V o P) (3.11)
where
b ^ p
^o " t k o " tl
- arg-u> Q - 7L
In order to solve for the arbitrary constants A, B, C, and D, we
apply the boundary conditions (3.3), (3.4), (3.5), and (3.6) to the
15
appropriate solutions (3.9), (3.10), and (3.11). Omitting the details,
the following four equations are obtained:
- (v x b) A J x ' (v x a) + (v^) B J^ (v^) + (v^) C Y^ (v^) =-x^ ± b
A J 1 (v i a) - B J^i^a) - C Y (v^a) =
B J^v^b) + C Y 1 (v i b) - D H^V^b) =
> i b) B j' (v i b) + (v i b) C Y^ (v^) - D[£ r (v Q b) H 1 / (v Q b) +
+(6 r -D Vv o b)] =
(3.12)
This system of equations was written in matrix form and solved by
finding the inverse of the coefficient matrix. Details are given in
Appendix A. The constant D was determined to be
J x (va)
D = 1U V (I) (I)
(v x b) J (v ib ) H/ ; (v Q b) -£ r (v o b) J^b) H C ; (v Q b)
(3.13)
Substituting (3.13) into (3.11) we obtain h(p,?,) for the region p- b.
The solution h(p,£>) may then be substituted into the inverse trans-form
(2.7) to yield the integral expression for Hj(P,z) in region II. This
result is . .
icj^a p +0 ° J^a) H^V^) e +i ^ z d£
V P ' Z> ^ (v^J^b) Hl (1) (v b) -S r (v b) J^b) H o a) (v b)
(3.14)
16
where p ^ b ,
v i = L k i - 1 J j V - L k - 1 J •
The evaluation of (3.14) is the subject of Chapter 5.
17
4. SINGULARITIES OF THE INTEGRANT*— SOLUTION OF THE MODE EQUATION
The integral for H . (p, z) is treated as a contour integral in
the complex t, plane where the path of integration is along the real
axis from -co to +co . In order to carry out the integration, it is
first necessary to determine the singularities of the integrand. A
detailed analysis of the singularities is presented in this chapter.
The expression for H. (P,z) is
+i0)€ ia + ^ J^a) Hl (1) (V Q P) e +iCz dC
H.(p,z) = — - , t^t ^p-
211 ^ ( V l b > J o< V l b > *! < V o b > - M V „ b >V V l b >0 ( V b >
-oo 1 ol 1 o rollo o
(4.1)
4.1 Branch Points
Consider the variable v, which appears in the arguments of J and
1 o
I 2 2
J in (4.1). Since v = k - £ , then v is multiple-valued in any
neighborhood of I, = + k . However, if one considers the power series
expansions of J and J , one sees that the integrand is an even function
of v so that the points t, = +: k are not actually branch points. The
I 2 2
Hankel function arguments contain the variable v = k - t, . The
Hankel function has a logarithmic singularity at v =0, and so is
multiple-valued in any neighborhood of t, - + k . The points £ = + k
are, therefqre, branch points of the integrand. We select branch
cuts in the t, plane as shown in Fig. 2. The points + k and + k lie
on the real axis with I k, I > ' k I since € , > € . The branch cuts
I 1| ' I o I 1 o
are defined as t = + k f i In 1.
— o
18
hi
t
r t
E
«
*J(x
i
19
4 2 Poles
Before we investigate the integrand of (4 1) for poles, it will
be advantageous to consider the significance of a pole in the physical
problem. Assume that the point t, = £ is a pole of the integrand.
When (4.1) is evaluated as a contour integral, a residue contribution
at £ = 4 must be included. Thus, Hj(P,z) will have a term of the form
n x it z
Ah/ A; (v p) e ° (4.2)
1 o
where A = 27Ti Residue
*-V
2 2
Since we assumed a lossless dielectric rod by taking k = 03 \jl€ real, we
will expect (4.2) to represent a wave propagating without attenuation,
which implies that t is real. If t is real and It I > |k | ,
o o I o | I o I
then v is pure imaginary and H (v P) reduces to the K. Bessel
function which decays exponentially with increasing argument. Under
these conditions, (4.2) represents a surface wave of amplitude A
having a radial dependence of H (v P) which propagates along the
-i(CJt-£ o z) l °
rod according to e .We shall now determine if there are
values of t, real which cause the denominator of (4.1) to vanish.
Equating the denominator of (4.1) to zero yields
J (v.b) H (1) (v b)
1 o
For convenience of notation, let v b = X, and v b = X ; then (4.3) may
11 o o
be written
J (X,) H (1) (X )
v ol = g o o
1 W r ° Hl (1) (x)
1 o
Recalling that
v l =
k 2 - 1
o
K 2 -
2 2
where k > k > 0,
1 o y
(i) Consider the case of £ real and It, | > Ik I > Ik I
Then
v, and v , and therefore X, and X will be pure imaginary „ Denote
1 o 1 o
X, = ip and X = i£ , where p and £ are real; then (4.4) becomes
1 o
J (ip) H (1) (i£)
(iP) tW- H (1) (X ) H (X )
1 o 00
where H* denotes the derivative of H
o o
21
Using the definition of the Hankel functions, (4.6) may be written
^V _ . £ x W - | W
A l J n (X n ) ~ r o J / (X ) + i Y / (X ) " K *'"
11 o o o o
Since X is real, the left side of (4.7) is always real, while the
right side is obviously complex for real X . Is it possible for the
o
right side to be real for some X ? The right side can be real only
if the argument of the denominator is equal to that of the numerator
by a multiple of nTT.
Let
a = arg [j^) + i Y^)]
P = arg [J o (X o ) + i Y o '(X o )]
Assume |3 = a + nlT,
where n = + 1 , + 2 ,
then
since
a 4. , «\ tan a + tan nff
tan p = tan ( a+nTT) = - — — — — =-= tan a
r 1 - tan a tan nTT
Y (X ) Yj{X n )
tan a = and tan [3
We obtain
J (X ) J ' (X )
o o' o o
Y (X ) Y > (X )
o o o o
J (X ) J / (X )
o o o o
J (X ) Y ' (X ) - Y (X ) J I (X ) = 0. (4.8)
oooooooo
However, the left side of (4.8) is the Wronskian of J , Y , which
o o
is non-vanishing for finite X :
o
W £J ,Y ? = J (X ) Y' (X ) -Y (X )J \x ) = -z§- *
t o OJ oooo oooO 7TX
22
We conclude that the right side of (4„7) must always be complex.
Since the left side of (4.7) can vanish, is it possible for the right
side to be zero? For the right side to be zero, J (X ) and Y (X )
o o o o
9
must be zero simultaneously, Watson proves that the positive zeros of
two distinct cylinder functions of the same order are interlaced;
therefore, J (X ) and Y (X ) are never zero simultaneously. The
same holds for -X since J (-X ) = J (X ) and Y (-X ) = Y (X ) .
o o o o o o o o o
Since the right side of (4.7) is always complex and never vanishes,
there are no real values X. and X for which (4,6) can be satisfied.
1 o
We conclude no poles exist for t, real and I £ < Ik
(iii) Consider Z, real and Ik I < |£| / Ik, I ; then y is real,v
is pure imaginary, and therefore X, is real and X is pure imaginary.
1 o
Let X = ib , where b is positive and real, Equation (4.4) becomes
i ( V
J (X,) H (1) (i£) K (£)
= € (-i£) -°-- = 6 e ^TT . (4.9)
j o (x x ) k o (£)
A graph of the functions -X . . and € £ g is presented in Fig. 3.
1 J n (X ) r K n (5 )
J o (X 1 }
The function - X . > is discontinuous each time J, (X, ) vanishes
1 1
l v 1
and so an infinity of branches occur for ever increasing X . It
passes through zero each time J (X, ) has a zero. The function
o 1
K Q (b)
€ £ v (t\ approaches zero for b — »0 and increases positively for
increasing £. It is evident in Fig. 3 that for every finite b, an
infinity of values X exist which satisfy Eq« (4.9). In order to
obtain a finite number of unique solutions of the mode equation,
a second relation between X 1 and £ is needed.
2;j
xT -uP
\
" If
L 1L
1
1
1
1
i
x"
13
C
(0 -^
«
\
s
\
\
\
>
o P
II
^^
fa
LL LL
24
It follows from the definitions
A 2 " 2
, 2 r 2
v = . k - t,
o \o
where £ is identically the same in v and v .
,2
Elimination of ?, yields
2
2 2 2 2 27T
v/ - V / = k_- - k = (J-) (€ - 1)
1 o 1 o A. r
o
2
Multiplying by b yields
(v lb ) 2 - (v o b) 2 = C 2 ^) 2 (€ r -l)
o
o
Si
nee X = i£ , we obtain the second relation between X and % , which is
oqq
X r + i = R (4.10)
where
,27Tb
) JV i = ko bjvr
* HP) €
The graphical solution of the mode Eq. (4.9) is illustrated in Fig. 4,
which includes curves of X as a function of § obtained from Fig. 3,
and the relation (4.10). Note that (4.10) is the equation of a circle
of radius R with center at the origin.
The multiple-valued solutions X for every £ as discussed with
Fig. 3 are evident in Fig. 4. The first branch of X starts at X = 2.405
for t = and approaches X =3.83 asymptotically with increasing |.
X x = 2.405 is the first zero of J^) and X = 3,83 is the first zero
25
7
i
5
A-
i
1
c Mode
r
Solution
(§, x x )
Vi
1
5
2
^^
\
o
C 5
FIGURE 4. GRAPH OF X = f(^) and X 2 + £ 2 = R
26
of J (X ) , The second branch commences at 5.52 and approaches 7,02,
which corresponds to the second zeros of J (X ) and J (X ) , respectively
An infinity of branches is thus established
Consider the case R < 2.405. Since X ± = \| R - £ 2 ^ R, Fig, 3
shows that Eq. (4.9) cannot be satisfied for X < 2.405, that is,,
no solution of the mode equation exists for R < 2.405. This is also
evident in Fig. 4, Physically ; this means that the dielectric rod
waveguide is below cutoff and cannot propagate a surface wave of
the transverse magnetic type.
Referring to Fig. 4, if 2.405 < R <5.52, a single, unique solution
(£, X ) of (4.9) and (4.10) results. Thus, (4.1) has poles at t, = + t, ,
where t, may be determined from the solution (£ , X_ ) „ The poles occur
on the real t, axis in the region | k J / \t,\ • Ik I . We see that when
R = Pr— ) . |€ -1 is restricted to the range 2.405 < R < 5.52, the
dielectric rod propagates a single, surface wave which is the lowest
order, circularly symmetric, transverse magnetic mode. It is known
i£ o Z -i£ o z
as the E mode and propagates as e for z -7 o and e for z < o.
As R increases, the number of solutions (£ , X ) of (4.9) and (4.10)
also increases, and the dielectric rod supports an ever increasing
number of higher order TM modes. The number of poles occurring
between k» and k increases with R, as is shown in Fig. 5. It is
interesting to note the behavior of k , k , and the poles as R increases.
From the definition of v we obtain
o
. k 2 + (|) 2 (4.11)
\| o b
27
H
1
<
v
c
a
4J> J
:
:
\uu\\ wwwww
+
i<
1
)
:
if,
i
'\\\\ \\\\U\\\\\\
CO
i
1
1
»
28
Consider R just greater than 2.405; then X ft* 2.405 and b is a very
small positive quantity. Equation (4.11) shows £ is just greater
than k , that is, the poles + £ first appear in the neighborhood
of the branch points + k with \t,\ ~7 k , Referring to Fig. 4,
as R increases, the s of any solution increases, so that £, corresponding
to (£ , X ) also increases. Thus, the poles + £ move away from the
origin. Each new pair of poles + t, first appear near t, = + k , Now
27T 1 I
k = (r— ) , k, = J6 k , and R = k^b €- - 1, so we may think of k and k
o \ 1 M r 6 \J r ' 1
as proportional to R. The distance from k to k , (k -k ) = k ( J€~-1)
is also proportional to R. As R increases, k and k move away from
the origin, the interval (k -k ) increases, and the number of poles
between k and k increases.
4.3 Solution of the Mode Equation
The object of this work is to determine the efficiency of a magnetic
current filament in exciting the E _ mode on a dielectric rod. For
single mode operation, R must be restricted to the range 2.405 < R < 5.52,
Since R = k b I € -1 , this restriction on R is actually a limitation on
k b and € , the physical constants which describe the dielectric rod
r
as a waveguide. Specific solutions of (4.9) and (4.10) are required
in order to calculate the power transmitted in the surface wave and,
ultimately, efficiency. Equations (4.9) and (4.10) were solved for
the case € = 2.56, a relative dielectric constant typical for
polystyrene. Six solutions (5 , X ) were obtained for selected values
of k b equal to 2.2, 2.6, 3.0, 3.4, 3.8, and 4.2. Graphical solution
of (4.9) and (4.10) as illustrated in Fig. 4 was considered too
29
inaccurate for the purposes of numerical computation and so (4.9)
and (4.10) were solved, utilizing the University of Illinois digital
computer. The solutions (£ , X ± ) are accurate to four decimal places.
Details of the method used to solve the equations on the computer
are given in Appendix B. Table I presents i, X , and the ratio
W\ corresponding to each value of k b. The ratio X |X follows
s ' g| o
from the definition of v .
X |X_ =
g
" M^© 3
(4,12)
TABLE I
V
2.2
2.6
3.0
3.4
3.8
4.2
5
X l
x rx n
g| o
5603
2.6901
.9691
1.2329
3.0043
.9036
1.9353
3.2086
.8403
2.6269
3.3366
.7913
3.2905
3 . 4204
.7560
3 . 9264
3.4788
.7305
30
5, EVALUATION OF THE CONTOUR INTEGRAL
For the purposes of the following discussion, we repeat the integral
solution for H-(p,z) outside the dielectric rod, which is
+iG)&a p J.(va) H (1) (v n p)e +i ^> z
V P ' Z) " IT J -IT) " ' (1) di;
c (v i b) J o ( V» H i < v o b) " W> J i ( V> H o V>
(5.1)
where C is the contour along the real axis with appropriate indentations
o
+ + *
at the branch points - k and poles - b„, as shown in Fig. 2. Expression
o
(5.1) is evaluated by transforming from the £ plane to the complex -r plane
and employing the saddle point method of integration to obtain an asymp-
10,11
totic value of the integral. The saddle point method is applicable
only when certain parameters in the integrand are relatively large. It is
used to obtain an approximation to the far zone radiation field. The sur-
face wave field follows from the residue contribution at the pole. In the
T plane the contour of integration is deformed until it passes through the
saddle point of the integrand. The integrand contains an exponential
function, and when the deformed contour corresponds to the path of w steep-
est descent the exponential function decreases very rapidly away from the
saddle point. The integrand vanishes at the ends of the path, and so the
principal contribution to the integral comes from evaluation along the path
of steepest descent in the very near vicinity of the saddle point.
5.1 The Radiation Field
In order to apply the saddle point method of integration, we introduce
31
the transformation of variable
£> = k sin t
o
where
t = 4« + iT| (5.2)
Now t = sin (£|k ) is a multiple-valued function of £, and the region of
integration in the C plane transforms into a strip in the T plane bound
by two curved lines which correspond to the branch cuts in the £, plane.
They are defined by the equations
C sin ^ coshTj =1 C sin ^ cosh T\ = ~1
1 L ^ I L
(^ cos + sinh 7] = ^-2 > L cos ^ sinh r\ = ™-^ ^ ( 5 . 3)
o o
The contour of integration C in the ^ plane, which is defined as Im %, =
with necessary detours, transforms to the path C in the t plane, as shown
in Fig. 6. It follows from the definition Im £ = which corresponds to
k cos 4* sinh 71 = in the t plane while maintaining C within the curved
o ' 1
strip. The direction of C follows from consideration of the real part
of £ along C_, where Re £ = k sin 4 1 coshTh The branch points £ = - k
U o v o
transform to the points t = - 7T/2 while the images of the poles - fe are
the points T = +7T/2 - i cosh C /k and -f = -7T 2 + i cosh £» /k .
o ' o / o o ' o' o
Substituting (5.2) into the expression for v yields v = Jk 2 - C 2 -
- k cos 1" „ The choice of sign here is determined by the previous
definition of arg v in the £> plane. On the contour C_ , %, is real and
o U
when |£>| > k I we obtain v = j v I g ± *• V 2 . Since we chose to
define < arg v < 77", the negative sign is excluded and we have
V = V n e+ 1 ■ In ord e^ that corresponding values of f on C yield
32
33
the same argument for v the positive sign is required;, thus we establish
v n = + k cos t. (5.4)
o
2 Y 2 + 2
The function v, becomes v = k - fe = - k , £ - sin t. Since the
1 1 \| 1 \j^r
integrand of (5.1) is an even function of v , either sign may be used
and we choose
2
- sin t = k w(t) (5.5)
o \J r o
where for convenience of notation we let w, a function of t, represent
the radical, w = £ - sin 2 T. Substitution of (5.2), (5.4), and (5,5)
into (5.1) with d£ = k cos t dT yields the transformed integral in the t
o
plane which is
ico£ a n J, (k a w) H (1) (k p cos t) e lk ° Z sinT cos T dr
rr/^\ 1 lO 1 O
H . (P , Z) = T-T pj
v 27Tb U w J (k b w) H, (k b cos t)- £ cos t J (k bw) Bt> (kb cost)
Do lo r louo
C l
(5.6)
If p is large and k p cos T =f 0, the Hankel function may be replaced with
its asymptotic representation, Then the Hankel function and exponential
in (5.6) combine in the form
H l (1) (k p cos T)e ik ° Z Sin T ~ e" 1 ?^ 2 > e ik o (P cos T + z sin T >
o r \ 7Tk p cos t /
(5.7)
34
Consider the spherical coordinate system illustrated in Fig. 7, where the
FIGURE 7 CIRCULAR CYLINDRICAL COORDINATE SYSTEM (p,0,z) AND SPHERICAL
COORDINATE SYSTEM (r,0,9)
polar angle Q is measured from the plane z = 0. In this coordinate system
we note that p = r cos 9 and z = r sin 9. Substituting for p and z in the
right side of (5.7) we obtain
H n (1) (k p cosT)
1 o
37T
ik z sinT ^ ft -i-£- p 2 ;
r cos 6 cos f-
Lit k
V2
ik Q r cos(-f-0)
(5.8)
We substitute (5.8) into (5.6) to obtain the following approximation for
35
for H. (r,9) which holds for r large and k r cos 9 # „
i/i
iooe a 37T I „ , ^
H*0 for -7T+0<4 J <©
and
7]<0 for < 4< < IT J (5.14)
The shaded area of Fig. 6 is the region of convergence and we see that the
integrand of (5.9) does vanish on C as T) approaches - oo .
The relation of the path of steepest descent and the poler n will
now be considered. Evaluation of (5.9) along C yields the total field
s
in the region outside the dielectric rod. Since the saddle point is
equal to the polar angle 0, as increases towards 7T/2, the steepest
descent path shifts to the right in Fig. 6. Ultimately, a residue term
must be included because C will cross to the right of the pole T«. There-
s
fore, the total field outside the rod, which is obtained as takes on
values ^0 < tt/2 s may be considered as the sum of the residue at the pole
and the steepest descent contribution at the saddle point, that is, the
surface wave and the radiation field .
Due to the rapid decay of the exponential term away from the saddle
point, it is only necessary to integrate over a short segment of C in the
s
near vicinity of Y= 6.
37
We proceed by setting
-T - 9 = a e" 1 *
.7T
dX = do- e~ 17 ^
(5.15)
where a is the distance along C measured from the saddle point. Consider-
s
ing \ f - -9| small, we may approximate
2 2 7T
cos (T- ©) ,* 1 - {rf ~ 2 B) - = 1 - "Y e _i 2 (5.16)
Except for the exponential term, the integrand of (5.9) is slowly varying
about T 1 = @ and may be assumed as constant. We substitute (5.15) and (5.16)
into (5.9) to obtain
V r ' S) = «b
L. [_|_] F(e) e lk o
L 7T k r J
+A
ik^r
k ro^
o
dcr,
(5.17)
Since r is very large, the following approximation is valid
J e -V- d(r « / e -V- do . = [|i]
*6
(5.18)
Substituting (5.18) into (5.17) and noting that W£ = k £
1 or
the far zone radiation field, which is
^o
, we obtain
yr,6) =
. OJ 7Tb r
\^o
(5.19)
where
F(0)
J (k a w)
1 o
wl(k bw)R (k b cos 0)-£ cos J, (k bw) H (k b cos ©)
Oolo r loOo
IH
and
S - sin 9
r
It follows from Maxwell s equations that
E e (r ' e) = -"-S-> p(9) I
(5.20)
5.2 The Surface Wave Field
The surface wave propagating in the positive z direction is obtained
by evaluating the residue of (5.1) at the pole ?> . Denote the denominator
of (5.1) by G(£) so that (5.1) may be written
yp.«> - w
f^ (1) iCz
ico^a J^a) H^ ' (v Q p) e lb
27T
G (&)
d£
(5.21)
where
G(t) = (v 1 b) J (v a li) H 1 (1) (v () b) -£ r (v Q b) J 1 (v 1 b) H (1) (v b)
The integral (5.21) is the solution for the total field H,(p,z) in Region II
where p > b. Let H (Pi>z) represent the surface wave portion of the field
in Region II. It is given by
H/X (P> z ) = 27ri Residue
(1)
j^a Jj^a) H x v (v Q p) e
277
G(&)
(5.22)
t-6.
where z > 0.
The pole £ follows from the solution (I , X ) of the mode equation G(£) = 0,
Since G'(& o ) =^ G(&)
£= 0, C is a simple pole and the residue of
& »&,
of the integrand at £ is given by
Res
iut.. J i( 5 V H (1) ,i^)eV
1 b b . (5.23)
0' + t ( T> + &r ( r^ W K o (l) +
1 1
+ (6 - 1) J, (X.) K. (I) > (5.24)
r 1 1 1 /
Using the relation H (i £ £) = - ~ K (§^), it follows from (5.22),
(5.23), and (5.24) that
H0 II (P,z) =A n K x (£^) e^o z (5.25)
where p > b, z >
^ (=> j, r ( 4 >+( r )] J (X l' *>< e >*<6,-» WV 6) }
with
and
& b = J ( V ) 2 + 4 2
R 2 = Xl 2 + | 2 .
It is interesting to note that the surface wave solution (5,25)
exist since £> is real. This situation results from the simplifying
40
assumption that the dielectric rod is lossless. If the finite loss of the
rod is included, the constant k 2 = d) 2 u £,,(1 + i^]?") is complex since
i^oz (5 = 27)
H z > = A n j^) J i (x i V e
41
We obtain EAp,z) in Regions I and II from E = — - — v y H, hence
P — iCoL
?)H I £
V (p ' z) = ( I3 l ) ^f = fe^ V (p ' z) (5 - 28)
E p II(p,z) = ( 4^ V 1 (p,z) (529)
The field components (5.25), (5,27), and (5.28), and (5.29) will be used
to calculate the surface wa.e power in Chapter 6.
42
6. THE POWER INTEGRALS AND EXCITATION EFFICIENCY
In the previous section, the surface wave field and radiation
field generated by the magnetic current source were determined. It
is now possible to calculate the corresponding powers and obtain the
efficiency of the source. The efficiency with which the source
delivers power to the surface wave is called the excitation efficiency
of the source. Denoting efficiency by the symbol T , it is defined as
S
m _ W_
; - T
w
s
where W is the surface wave power
T
and W is the total power delivered by the source „
12
Goubau has proved that, for lossless surface waveguides, the
radiation and surface wave fields are orthogonal with regard to power
considerations. In this case, the total energy delivered by the source
is equal to the sum of the surface wave power and the radiated power.
Since we are considering a lossless dielectric rod, the orthogonality
condition holds and y is given by
w s + w*
g
where W is the surface wave power
and W is the radiated power.
In order to determine T , we proceed with the calculation of W and W .
(6.1)
(6.2)
43
6,1 Radiated Power
The radiated power W is obtained by integrating the time average
Poynting vector over the surface of a large sphere of radius r„
W =J jp" av ■ da = j | Re ¥ X H* • da (6.3)
2 .
where da = r cos 9 d 9 d
[ A n j^yll V< J o 2(x i ) + J i 2 (x i> - ^ VV V x iY («.io)
tor region II where p > b, W 11 is given by
27T p=o>
w " = / / i ' 2 ReE P n V I%ai (6 - u >
o p=b
Substituting (5.29) and (5.25) into (6,11) yields
p=oo
f i --&*ii r p»k a '*&*
p=b
(6.12)
he definite integral in (6.12) is also given by McLachlan, 14
btain W 11 as
w " ■ <£f> A „ 2 4U 2 <6> - k x 2 (I) + I k o (?) Kl j
le constant A^ is defined in Eq. (5.25), We may now substitute
5 »10), (6.13), and the value of A into (6.7) to obtain W S . After
(6.13)
considerable manipulation of constants one obtains the expression
for the total surface wave power which is
47
where
n x \
SI o
o
K, (I)
Nr
r 1 x 1
(6.14)
" ( 4 )2+1 l r a X ! n
3— ■jJ (X 1 )K 1 (6) + [€ r (^) + (^J o (X 1 )K o (e) +
[K o 2 (e)- Kl 2 (e) + f k o <£) ^«a/
(€ r - 1) j i (x 1 ) 1^ (I)
2 V
J i
where D , D , and D are the lengths defined in Fig. 12. The origin
of the reflection coefficient plane is the point 0, the iconocenter
is , and the center of the image circle is C. Formula (7.2) was
used to calculate the efficiency for all of the experimental measurements.
Before the results are presented, a brief description of the experimental
apparatus and coaxial exciter will be given.
7.2 Experimental Equipment
A two inch diameter polystyrene rod was selected for the experimental
work. This choice followed from two main considerations. First, a two
inch rod can stand on end without additional support. This method of
vertical mounting avoids the usual problem of how to support a surface
wave structure without disturbing the field. Second , it is easier
to fabricate annular slots for a large rod than for a small rod.
Table III lists the wavelength X , frequency f , the ratio X (X , and
X for a two inch polystyrene rod, assuming € =2.56.
Using Deschamps 7 method, one obtains the efficiency of the entire
transition between the input reference plane and the output reference
plane, that is, between the measuring probe and the short circuit
termination. Since we are interested in measuring the dissipative
59
Unit Circle
FIGURE 12 . PLOT OF THE IMAGE CIRCLE ON THE REFLECTION
COEFFICIENT PLANE
TABLE III
k b
o
X
o
f
g/ o
X
g
centimeters
megacycles
sec
centimeters
2 2
7 25
4132
0.969
7.03
2,6
6.14
4884
0.904
5.55
3.0
5.32
5635
0.840
4.47
3.4
4.69
6387
0.791
3.71
3.8
4.20
7138
0.756
3.17
4.2
3.80
7889
0.730
2.78
attenuation of just the annular slot, it is essential that the rest of
the transition shall introduce only negligible attenuation. For the
present we shall ignore the dielectric rod loss since it is very small.
It follows, then, that a low loss exciter must be constructed to
illuminate the annular slot. The annular slot presents a very low
conductance and capacitive susceptance to the exciting waveguide.
Therefore, the feed waveguide must be a low impedance line with some
means of tuning out the capacitive susceptance of the slot at the
ground plane position. Figure 13 shows a cross sectional view of the
low impedance coaxial line which was constructed for this purpose.
The inner diameter of the outer wall of the coax was 17/8 inches
and the inner conductor diameter was 1 l/4 inches, which yields a
24 ohm line. A two wavelength tapered section transformed the 24 ohm
line to a standard 50 ohm, type N connector. Two polystyrene rings
centered the inner conductor within the cylinder. The ring located
at the tapered section was one-half wavelength long, so that no impedance
61
w
CP
J.
c
c
,3
h
w
QJ
1
•J
<
62
discontinuity would result. The ring at the ground plane position
was a quarter-wavelength transformer which would match the 24 ohm line
to a 9.5 ohm resistance load. A circular tuning disc or washer was
placed on the end of the center conductor to provide a series inductive
reactance to cancel the capacitive susceptance of the slot. The circular
disc which formed the inner boundary of the annular slot was fixed to
the tuning washer and the coax center conductor by a special mounting
screw., Since the neighborhood of the annular slot and tuning washer is
a resonant cavity, all parts were silver-plated to minimize losses,.
A family of discs and rings were fabricated to permit varying the
annular slot radius from 1/2 inch to 7/8 inch while maintaining the
slot width constant at 1/8 inch. The 1/8 inch slot width corresponds
to .067 X and .076 X at k b equal to 3.4 and 3.8, respectively,
o o o
Recalling that the radius of the dielectric rod is one inch, it is
convenient to express the slot radius in the normalized from a/b;
then the source dimension k a for any k b is given by k b (a/b)„ Six
o o o '
slots were constructed for the measurements. Table IV gives the
normalized slot radius a/b, and the corresponding dimension k a at
the two values of k b which were used for the measurements,
o
A slot 1/8 inch wide and two inches long was milled in the side
wall of the exciter so that the standing wave ratio could be measured.
After a particular slot was mounted on the exciter, various tuning
washers were tested until one was found which reduced the VSWR in the
exciter to less than two when the dielectric rod was terminated with
a matched load. The image circle was determined from measurements
63
TABLE IV
Normalized Slot o o
Radius a/b when k b=3.4 when k b^3.8
o o
0.50 1.70 1.90
0,625 2.12 2.38
0.687 2.34 2.61
0.75 2.55 2.85
0.812 2.76 3.08
0.875 2.98 3.32
made in a coaxial slotted line which was connected directly to the
exciter through a type N elbow connector. A view of the slotted line,
coaxial exciter , and the underside of the ground plane is given in
Fig. 14. A block diagram of the waveguide apparatus used for the
measurements is given in Fig. 15. When Desehamps' procedure was
carried out, the dielectric rod was terminated with a six inch
diameter plate, machined from 1/16 inch brass and silver plated. The ground
plane, which may be seen in Fig. 10, was 60 inches square. For most
of the measurements the dielectric rod was 40 centimeters long,
although in some instances, the length was increased to 134 centimeters.
Finally, it should be mentioned that an external probe was constructed
to measure the standing wave ratio of the field along the dielectric
rod. It is shown schematically in Fig. 15. It was used to measure
the guide wavelength, X „ when the rod was operated at k b equal
to 3.4 and 3.8. In each case, the measured guide wavelength corresponded
exactly to the value predicted in Table III. A matched load was made
from four tapered lengths of 300 ohm resistance cord which were
64
66
mounted symmetrically on the rod. The matched load is visible in
Fig 10 The VSWR on the dielectric rod was less than 1.1 when the
rod was terminated with this load,
7 3 Experimental Results
When efficiency was measured as shown in Fig. 11, the result obtained
was not precisely the excitation efficiency of the annular slot; instead
it was the efficiency of the entire transition between the measuring
probe and the short circuit plate. Although the losses in the system
were small, the measured efficiency was reduced somewhat by dielectric
rod attenuation and by the loss between the measuring probe and the
ground plane. We shall discuss these two effects separately.
First, consider the dielectric rod attenuation which caused the
two port network to be terminated in a lossy short circuit. The lossy
rod acts as a reflection coefficient transformer, so that the magnitude
of the reflection coefficient at the ground plane, IfL is given by
h
? "''" (7.3)
where a is the attenuation constant of the wave on the dielectric
rod and z is the length of rod between the ground plane and the short
ircuit plate. It is assumed, of course, that a perfect short terminates
the rod yielding a reflection coefficient of unity at that position. We
wish to determine the efficiency of the slot while excluding the effect
of the lossy rod. Consider Fig. 16(a), which shows [s'] , the four
Pole to be measured; it is located between the slotted line and the
lossy short circuit. Figure 16(b) is an alternate representation of
67
Lossy Short
51. %*
I-
(a)
Pev-fec.+ Shovl-
(b)
FIGURE 16. MEASUREMENT OF A JUNCTION WITH A LOSSY SHORT CIRCUIT
Fig. 16(a) in which the reflection coefficient transformer I P|has
been associated with [s» ] to form the fictitious four pole [s] which
is terminated with a perfect short, when one measures the four pole,
assuming a perfect short, the composite four pole [s] is obtained.
Assuming that | P |is known, the parameters of [S» ] are given by ]
17
S »
11
11
S '
22
_22
12
"12
Referring to Eq. (7.2), it is evident that the efficienty,^ f , of
[S> ] is related to T of [S] by
T« =
T
(7.4)
(7.5)
is the slot efficiency which would be measured if the dielectric
rod were lossless, and Eq. (7.5) gives the relation between the measured
6!'
p
efficiency T» and T ' .
In order to apply (7.5) to the experimental measurements,! 1 I was
obtained by measuring the efficiency of the same source with two
different lengths of dielectric rod. The distance from the ground
plane to the short circuit plate was 40 centimeters and 134 centimeters
for the two measurements. Referring to Fig. 17, let T and f represent
this efficiencies which would be measured when the short was located
at the specified positions.
GjirOund Plane.
Die-le+vic Rod — 7
",40m
-.94n
Cireu i+
FIGURE 17, ARRANGEMENT FOR MEASURING ROD ATTENUATION
From (7,3) and (7.5) it follows that
£-•*» (7.6)
( o
where z is equal to 0,94 meters,,
Substituting the measured values of T and HP into (7.6), one
obtains the value of the attenuation constant a. Having determined a,
it follows from (7,5) that
T» = Te 2aZ (7.7)
where z is now 0,40 meters, '"P ' may be thought of as the measured
efficiency referred to the ground plane position. Equation (7,7)
defines the adjustment which is applied to the measured efficiency, T,
to account for the dielectric rod attenuation. The value of a was
measured when k b was equal to 3.4 and 3.8, The results are
o
- ,0692,
k b = 3o4
,0716
k b = 3.8
(7.8)
o o
Before (7.7) is applied to the experimental measurements, the attenuation
between the measuring probe and the ground plane will be discussed.
The attenuation of the transition between the measuring probe
and the ground plane results from losses in the type N connector
and the coaxial exciter. The loss is, of course, a function of the
standing wave ratio in the transition. When a particular slot is being
measured , the VSWR changes with the short circuit position because the
junction is lossy. Therefore, the transition loss is different for
70
each position of the short, and it is impracticable to correct every
standing wave measurement performed. One may, however, establish the
maximum loss, or, correspondingly,, the minimum efficiency of the
transition. This was accomplished by mounting a two inch diameter,
circular waveguide perpendicular to the ground plane and concentric to
the annular slot, and measuring the efficiency by Deschamps' method.
The circular guide was fitted with a movable shorting plunger. The
annular slot excites the TM __ mode in the guide. The schematic
representation of the two port junction is shown in Fig. 18.
I Detector j
Qround Plane
Circular
Waveguide
FIGURE 18. ARRANGEMENT FOR MEASURING THE TRANS I ST ION
EFFICIENCY
71
Assuming that the circular guide was lossless, the measured
efficiency, T, is the efficiency of the transition from the slotted
line to the annular slot. It is now possible to adjust the measured
slot efficiency to account for both the transition loss and the
dielectric rod attenuation. Let T q denote the corrected slot efficiency;
it is given by
2az
T
TL_ Tel
T T~ TT
(7.9)
where Tis the efficiency measured with the lossy rod present, and
T T is the transition efficiency. It should be noted that T provides
a maximum correction to T since, in most cases, the transition efficiency
will be greater than the value Xp- This results because standing wave
ratios are higher when "^ is measured than when T is determined. T
was measured when k Q b was equal to 3.4 and 3.8. The results are
T
0.939,
T
kb=3,4
o
= 0.934
(7.10)
b = 3.8
Substituting (7.8) and (7.10) into (7.9) yields
* 1.125 T, T
k b = 3.4
o
= 1.132T
k b = 3.8
o
(7.11)
The measured efficiencies were corrected according to Eqs. (7.11).
tote that the correction factor is approximate. It was obtained
Ln order to fix the order of magnitude of the system losses. One
lay see, however, that the excitation efficiency of a slot is somewhat
72
greater than the measured value.
Six slots were measured in the laboratory; the results are presented
in Table V, which includes the measured efficiency °f and the corrected
efficiency f for each source k a„ Measurements were performed at
frequencies of 6387 megacycles/sec and 7138 megacycles/sec which
result in k b equal to 3.4 and 3.8, respectively. The data of Table V
have been plotted in Figs. 19 and 20 for comparison with the theoretical
curves of efficiency. The very close agreement between the experimental
points and the theoretical curve is evident. In all cases the measured
TABLE V
k b k a Measured Corrected
Efficiency Efficiency
T Tc
3 4 1.70 0.44 0.495
2.12 0.65 0.73
2.34 0.76 0.855
2.55 0.835 0.94
2.76 0.77 0.867
2.98 0.65 0.732
3 8 1.90 0.48 0.544
2.38 0.775 0.88
2.61 0.85 0.96
2.85 0.77 872
3.08 0,63 0.714
f 3.32 0.44 0.498
efficiency, which included the system losses, was within 10 percent of
the efficiency predicted by theory. The experimental measurements
verify that an excitation efficiency of approximately 95 percent may
be obtained from an annular slot of dimension k a = 2.6.
o
73
!
^X
/
<
X
\
X
\
v.
X
X
1 X
5-°
>>
o
K
c
>>
•H
>>
o
O
rt
\
0)
5h
O
\
\
01
Si
Q)
cS
fc
A
1
S
X
1
M
<
«S
u
w
C\l
o
&
Q
H
«3
B
(0
s
rvi
oa
*)
8
CVi
1
<
*
j
CVJ
<
KH
% 6 I
O
* §
CVJ
H
55
°p
N ^O IO i; (O
/CouQi-oij-j-g
CM -
74
o
fO
'
/
>
\
8 ^
>>
o
S3
•H 0)
«H -H
o
«H O
W -H
(1)
-P
d) Vi
o
ft 3
03
0)
Sh
u
1
Xi CD
o
\
1 *
o
1
c\>
tf
7
<0
1")
€
10
>i
^
o
IP
cv
o
fo
E
w
O
It*
p
3
&
(V o b)
Denote the coefficient matrix of (A s l) by j~A. . "1 and let the cof actor
A. . be designated X. .. The solution of (A.l) is
of
Det
[ A iJ
x ll
X 2!
X 31
X 41
X 12
X 22
X 32
X 42
*13
X 23
X 33
X 43
\4
X 24
X 34
X 44
- ia)€ b
(A, 2)
which reduces to
HI
•Mb
Det
[ A u]
r~ -1
\l
X 12
\3
\4
Evaluating Det ["A. .1 , we obtain
DGt [ A ij] = *L1 (A 22 *23 A 44 + ^23 A»4 A 42>
- *11 (A 32 ^3 A 44 + ^2 ^4 A 43 )
-\2 (A 21 ^3 A 44 " ^1 ^4 A 43 )
+ ^3 (A 21 ^2 A 44 " *2l ^4 A 42 )
(A3)
(A, 4)
since A = - A L2 and Ag = - \ , (A. 4) reduces to
DGt [ A ij] " (A 13 ^1 " *ll A 23 ) (A 32 A 44 " ^4 A 42>'
(A. 5)
We note that
(A L3 *21 " *L1 A 23 ) = (V l b) [ J l (V i a) Y l (V i a) " Y l (V i a) J l (V l a) J
= (V l b) ^aT
(A. 6)
since the bracketed term is the Wronskian of J (v a) , Y (v a) which is
| equal to [2/7T (v a)]. Substituting (A ,6) into (A, 5) we obtain
2b
Det [ A ij] ■ 15 (A 32 A 44 " *34 A 42>
(A.7)
The cof actor X is given by
From (A, 3), (A. 7) and (A. 8) we have
iJJ ff (A 32 A 44 " ^4 A 42>
Consider
82
14 = " Aai (A 32 A 43 -^a^a>- < A - 8 >
• * h ^ * A 21 ( A 32 A 43 - A 42 A 33 )
D = " 1Cj€ l b Bet FA. .1 = + laC l* 2 , A A T" — — ■ < A ' 9 >
L ijJ = (A „ A_ - A_ A )
(A 32 A 43 " A 42 ^3 ) = (V l b) LVV* T l ( V> " \ V ]
= (v i b) ^bT = l (A ' 10)
because of Wronskian relation. Utilizing Bessel function recurrence
relations we determine that
(A 32 A 44 " ^4 A 42> = (V l b) J o (V l b) *i™ (V o b) " V\> b) J l (v l b)H o (1) < V o b)
(A.1D
Substituting (A. 10) and (A, 11) into (A, 9) yields
+ ±cj€ a J (v a)
>- — i_L_i (A. 12)
(v ib ) J o ( Vl b) H x ( >(v b) - € r (v o b) J l(Vl b) H/ (v o b)
83
APPENDIX B
The graphical solution of Eqs . (4„9) and (4.10) is illustrated in
Fig. 4 of Chapter 4. Equations (4.9) and (4.10) are
J Q (X) K Q (£)
T[ao ' £ r (e> K]T¥)
=
(4.9)
r 2 = (k n b) 2 (s - i) = r 2
u r
(4 JO)
where we omit the subscript 1 of X for convenience of notation.
Referring to Fig, 4, we see that the intersection of the circle and
the first branch of X = f(£) is the unique solution of (4.9) and (4.10)
Consider the magnified view at the point of intersection of the curves,
as shown in Fig. 21.
FIGURE 21. SOLUTION OF THE MODE EQUATION
84
Denote points on the circle corresponding to values of % by X, where
X = sj R 2 - i 2 from (4.10). The points X, which define the curve with
positive slope, are solutions of (4.9), which may be written in the form
J (X)
- X j7W-V = ° (B.l)
k (6)
where v = £ (£) _. /gv is a constant for any selected £ and value of %.
» r K (5) r
We see that the points X are simply real-roots of the equation
J (X)
- X ( v - y = 0. Let the point of intersection, which is the exact
solution of (4.9) and (4.10), have the coordinates % and X . The basic
s s
method for determining the point of intersection using the digital
computer is illustrated in Fig. 21. It is evident that for any value
£ less than § , X' > X, and when I is greater than I , x'
4 X x in 10
4 X 2 in 12
16
17
18
19
20
21
22
23
24
25
26
27
28
JO
(2)
50
16L
26
115 F
JO
(2)
J4
4 F
50
18 L
26
59 F
92
975 F
L5
(3)
LO
(5)
40
(6)
50
523 F
7J
523 F
40
(7)
50
(6)
7J
(7)
40
(7)
L5
(1)
10
2 F
40
(6)
50
(6)
7J
(6)
40
(6)
L5
(7)
LO
(6)
waste
enter HI
waste
print X^/4
4 spaces
4 <£,-!) in (6)
(kb/8) 2 in (7)
(^[(kb)^ (£-!)] m (7)
16 in (6)
1 n2
(— >* [(kb) 2 (^-1) - e. 2 ] in A
90
50
28 L
29
26
149 F
50
(2)
30
00
2 F
JO
(2)
31
J4
4 F
50
31 L
32
26
59 F
L5
(1)
33
LO
(9)
<=
36
39 L
34
L5
(1)
L4
(10)
35
40
(1)
50
11 L
36
K5
F
42
11 L
37
50
12 L
K5
F
38
42
12 L
enter Rl , leave r-r X . ' in A
16 l
clear Q
form i X, '
4 i
waste
print — X. '
i. | + .088
test: (^ - -^ )
advance £.
i. + .01
l
in (1)
advance (500) address
advance (510) address
22
7 L
transfer control to 7 L
39
L5
42
(11)
11 L
reset (500)
91
40 L5
(12)
42
12 L
reset (510)
41 59
LO
(2)
523 F
— in A, clear Q
test: . _
4.0 - kb
8
42 < 36
L
OF
F
43 (1) 00
F
00
F
44 (2) 00
F
00
F
45 (3) 40
F
00
140 000 000 000 J
46 (4) 00
F
00
F
47 (5) 00
F
00
250 000 000 000 J
48 (6) 00
F
00
F
49 (7) 00
F
00
F
50 (3) 00
F
00
22 000 000 000 J
51 (9) 00
F
00
F
52 GO) 00
F
00
2 500
000 000 J
92
(11)
00
F
00
500
F
(12)
00
F
00
510
F
Auxiliary Subroutine
40
K5
18
F
F
rescue X. /4
42
8
L
plant link
L5
18
F
50
2
F
enter V8 , leave J Q (X) • 5
50
2
L
-19
and J (X) -2 in7.
26
158
F
L5
18
F
10
2
F
form X/16, store in 19.
40
19
F
50
6
F
79
19
F
50
(2)
66
7
F
1 J o (x) 1
f ° rm ' 16 X J]L (X) " 16 T
S5
F
L0
(4)
JO
(2)
waste
22
F
•19
93
TABLE B.l
275
1375
.6711
.6730
1400
.6725
.6725 -
.0001, interpolating to obtain — £ (£) ( r for each £, until inter-
section was detected by the method shown in Fig. 21. When intersection
was detected, the computer printed out %, X, and X' on each side of
crossover in the following form:
k b/8
o
K
hi
4 i+1
K
I x
4 *i+l
ix'.
4 l
— Y>
4 A i+1
Table B.2 presents the numerical results for each of the six values of
k b. Table I was constructed from these data,
o
TABLE B.2
14005
.67251
.67252
14007
.67252
.67252
325
30820
.75107
.75107
30822
.75108
.75106
375
48380
.80214
.80215
48382
.80214
.80213
425
65672
.83415
.83415
65675
.83415
.83413
475
82260
.85510
.85512
82262
.85510
.85510
525
98157
.86969
.86972
98160
.86969
.86969
97
The procedure for calculating X and X" was similar to that used in
the first program. The only essential difference was the addition of
routine II to perform the interpolation. The tape format and order codes
written for the main routine and auxiliary subroutine are as follows:
Tape Format
Routine N12
Routine PI 6
Routine Rl
prerset parameter for HI
Routine HI
pre-set parameter for II
Routine II
Routine V8
Main Routine
Auxiliary Subroutine 1
Stop - Transfer Control to Main Routine
98
Main Routine
50
21 F
50
L
26
200 F
JO
(1)
50
54 F
50
2 L
26
200 F
26
75 L
-3-50 190 F
50
4 L
26
200 F
JO
(1)
50
40 F
50
6 L
26
200 F
JO
(1)
52
45 F
50
8 L
26
200 F
read in 12 fractions, locations 21
through 32
waste
read 12 fractions into 54 through 65
read 6 fractions into 190 through 195
waste
read parameter constants into 40 through 44
waste
-39
read integer a * 2 into 45
gg
10
11
12
13
14
15
16
17
18
19
20
21
L5
42
92
92
92
JO
L5
J4
50
26
L5
40
L5
50
50
26
00
00
00
40
L5
40 10 F
L5 42 F
40 12 F
45 F
17 L
707 F
131 F
515 F
(1)
43 F
3 F
13 L
239 F
40 F
(2)
(2)
4 F
16 L
338 F
F
F
10 000 000 000 J
(3)
41 F
insert a into II entry
number shift
carriage return and line feed
delay
waste
enter PI 6, print kb/8
puts initial £ into (2)
current £ in A
enter II, interpolates to leave
Ti Y inA
store interpolate in (3)
X /4 in 10
X 2 /4 in 12
100
22
23
24
25
26
27
28
29
30
31
32
33
34
50
21 L
26
304 F
40
(4)
L5
(2)
10
2 F
L4
44 F
40
(5)
50
43 F
7J
43 F
40
(6)
50
(6)
7J
(7)
40
(6)
L5
(5)
10
2 F
40
(8)
50
(8)
7J
(8)
40
(8)
41
F
L5
(6)
L0
(8)
50
32 L
26
295 F
50
(1)
00
2 F
40
(9)
enter HI, leave root X. /4 in A
l
store X. /4 in (4)
forms £ . /4 from t
l t
store £;/4 in (5)
(kb/8) in (6)
(— ) 2 [(kb) 2 (£ f l)] in (6)
§./16 in (8)
(i/16) 2 in (8)
clear A and location
( T6" )2 t< kb > 2 <£f D - ^ 2 ] in A
enter Rl , leave — X. J in A
16 i
clear Q
X^/4 in (9)
101
35
36
37
38
39
40
41
42
43
44
45
46
47
48
L5
(4)
L0
(9)
-32
41 L
L5
(5)
40
(10)
L5
(4)
40
(ID
L5
(9)
40
(12)
L5
(2)
L4
(13)
40
(2)
22
15 L
92
131 F
92
515 F
L5
(10)
J4
5 F
50
43 L
26
239 F
92
975 F
JO
(1)
L5
(ID
J4
5 F
50
46 L
26
239 F
92
975 F
JO
(1)
test: X./4 - X. /4
i i
transfer to print
§ /4 in (10)
X./4 in (11)
X. /4 in (12)
advance 6 by .0001
carriage return, line feed, delay
print £ . /4
four spaces
waste
print X./4
four spaces
waste
102
49
50
51
52
53
54
55
56
57
58
60
61
L5
(12)
J4
5 F
50
49 L
26
239 F
92
131 F
92
515 F
L5
(5)
J4
5 F
50
52 L
26
239 F
92
975 F
JO
(1)
L5
(4)
J4
5 F
50
55 L
26
239 F
92
975 F
JO
(1)
L5
(9)
J4
5 F
50
58 L
26
239 F
59
(1)
LO
43 F
36
6 L
OF
F
00
F
print X /4
carriage return and line feed
delay
print i ±+1 /4
waste
print X. , /4
l+l
print X. + 1 /4
in A, clear Q
test:
4.0 - kb
8
transfer control to read in new parameters
103
62 (1) 00 F 00 F
63 (2) 00 F 00 F
64 (3) 00 F 00 F
65 (4) 00 F 00 F
66 (5) 00 F 00 F
67 (6) 00 F 00 F
68 (7) 00 F 00 390 000 000 000 J
69 (8) 00 F 00 F
70 (9) 00 F 00 F
71 (10) 00 F 00 F
72 (11) 00 F 00 F
73 (12) 00 F 00 F
74 (13) 00 F 00 100 000 000 J
75 50 91 F
50 75 L
76 26 200 F
-*— 26 4 L
read 6 fractions into 91 through 96
40
18 F
K5
F
42
8 L
L5
18 F
50
2 F
50
2 L
26
389 F
Auxilliary Subroutine
rescue X. /4
plant link
X/4 in A
enter V8<
J (X)
■J X (X)
2~ 19 in 6
2- 19 in 7
104
L5
18 F
10
2 F
40
19 F
50
6 F
79
19 F
50
(1)
66
7 F
S5
F
L0
(3)
JO
(1)
-< 22
F
form XA6, store in 19 F
TS V x > ' 2 " 19 in A
clear Q
X J o (x)
16 ^(Xj.)
in A
X J
program tape listed the function values, — £ (I) T - , g x , required by
lb r K \b)
the interpolation routine. It also included constants telling the
computer where the table was stored, the initial value of b, X /4, X /4,
1 2i
and k b/8. The parameter tape as it was read into ILLIAC is as follows:
o
+14217 205
+14363 079
+14509 095
+14655 256
+14801 554
+14947 990
+45745 8333
+45902 4109
+46059 0048
+46215 6141
+46372 2411
+46528 8834N
+05055 518
+05089 839
+05216 642
+05343 995
+05471 883
+05600 291
105
+35300 6528
+35455 7627
+35610 9008
+35766 0675
+35921 2618
+36076 4848N
+24685 093
+24837 399
+24989 766
+25142 182
+25294 660
+25447 191N
+55638 3502
+55795 7729
+55953 2083
+56110 6549
+56268 1103
+56425 5788N
+ 555+ 66+ 68+275+ HfN
+ 23+74+76+ 325+25N+N
+93+79+81+ 375+ 25N+N
+ 62+82+84+ 425+5N+N
+ 285+84+86+475+75N+N
+92+86+88+525+75N+100N
106
APPENDIX C
The integral (6.6) which was needed in order to compute the
excitation efficiency is
7T/2
cos 9 F (9) d9 (6.6)
where
F(9) =
J, (k a w)
1 o
w J (k bw) H. (1) (k b cos 9) - € cos 9 J (k bw) H (1) (k b cos 9)
ool o r looo
and
w =
€ - sin 2
r
The program which was written to determine N utilized ILLIAC
library routine E2, which computes an approximation to the integra]
b
7— f (x) dx by Simpson' s rule. This routine requires that the
user supply tabulated values of the integrand at an even number of
equally spaced intervals. The range of integration, which was zero
to 88 degrees, was subdivided into 44 intervals of two degrees each.
For any k a and k b, the integrand was computed at the 45 points
o o
0, 2°, 4°, 6°, ... , 86°, 88°. Since the integrand contains F(9) ,
which involves a division, it was necessary to scale the numerator
whenever it exceeded the denominator in order to maintain | F(9) | < 1;
otherwise the capacity of the ILLIAC registers would be exceeded. This
107
was done by multiplying the numerator by 1/2 as often as needed until
all 45 values of F(0) were within range. Consequently, the computed
integral was scaled by an integer number of multiplications by the
factor 1/2. The scaled integral, I, which was computed was equal to
88 £_
n 90 2 p+l
I = |8^ J <2> C ° S 9 I F(9) I
90 2
where p is the integer scale factor.
R
It follows from (6.6) and (Col) that N is obtained from I by
For a particular choice of k b, the integral (CI) was calculated
o
for k a equal to 0.2, 0.4, 0.6, ... until k a took on the value k b;
o o o
then k b was changed to a new value, and the process was repeated. The
o
scaled quantities k b/8 and k a/8 were used in the computations. The
o o
results were printed in four columns which listed, respectively,
k b/8, k a/8, p, and I. Relation (C.2) was then used to compute the
o o
(C2)
N R
corresponding value of N . Table CI presents the results as printed out
by ILLIAC for k b/8 equal to 0.425, that is, k b equal to 3.4.
o o
The program which was written to compute the integral (C.l) was
comprised of 143 order pairs. Due to its length, the order code will
not be given in detail. However, it should be mentioned that library
routines V8 (Bessel functions), T5 (sine-cosine), Rl (square root),
P16 (print routine), and E2 (integration) were utilized by the main
program.
108
TABLE CI.
k b/8
o
k a/8
o
P
I
.425
.025
1
.0304
.425
.050
1
.1136
.425
.075
1
.2280
.425
.100
2
.1722
.425
.125
2
.2167
.425
.150
3
.1184
.425
.175
3
.1142
.425
.200
3
.0974
.425
.225
2
.1450
.425
.250
2
.0913
.425
.275
1
.0914
.425
.300
1
.0334
.425
.325
1
.0149
.425
.350
1
.0310
.425
.375
1
.0682
.425
.400
1
,1097
.425
.425
1
.1403
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Wright-Patterson Air Force Base, Ohio
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Director
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Belmar, New Jersey
VTTN: Technical Document Center
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?hief
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Department of the Navy
VTTN: Aer-EL-931
:hief
lureau of Ordnance
tepartment of the Navy
TTN: Mr. C.H. Jackson, Code Re 9a
ashington 25, D.C.
ommander
q. A.F. Cambridge Research Center
ir Research and Development Command
iaurence G. Hanscom Field
edford, Massachusetts
TTN: CRRD, R.E. Hiatt
Commander
Air Force Missile Test Center
Patrick Air Force Base, Florida
ATTN: Technical Library
Director
Ballistics Research Lab.
Aberdeen Proving Ground, Maryland
ATTN: Ballistics Measurement Lab.
Office of the Chief Signal Officer
ATTN: SIGNET- 5
ENG. & Technical Division
Washington 25, D.C.
Commander
Rome Air Development Center
ATTN: RCERA-1 D. Mather
Griffiss Air Force Base
Rome, N.Y.
Airborne Instruments Lab., Inc.
ATTN: Dr. E.G. Fubini
Antenna Section
160 Old Country Road
Mineola, New York
M/F Contract AF33(616)-2143
Andrew Alford Consulting Engrs .
ATTN: Dr, A. Alford
299 Atlantic Ave.
Boston 10, Massachusetts
M/F Contract AF33(038)-23700
Bell Aircraft Corporation
Attn: Mr. J.D, Shantz
Buffalo 5, New York
M/F Contract W-33(038)-l4l69
Chief
Bureau of Ships
Department of the Navy
ATTN: Code 838D, L.E. Shoemaker
Washington 25, D.C.
Director
Naval Research Laboratory
ATTN: Dr. J.I. Bohnert
Anacostia
Washington 25, D.C.
page 2
DISTRIBUTION LIST (CONTINUED)
National Bureau of Standards
Department of Commerce
ATTN: Dr. A,G. McHish
Washington 25, D.C.
Director
U.S. Navy Electronics Lab.
Point Loma
San Diego 52, California
Commander
USA White Sands Signal Agency
White Sands Proving Command
ATTN: SIGWS-FC-02
White Sands, N.M.
Consolidated Vultee Aircraft Corp,
Fort Worth Division
ATTN: CR. Curnutt
Fort Worth, Texas
M/F Contract AF33(038)-21117
Dome & Margolin
29 New York Ave.
Westbury
Long Islai.d, New York
M/F Contract AF33( 616) -2037
Douglas Aircraft Company, Inc.
Long Beach Plant
ATTN: J.C. Buckwalter
Long Beach 1, California
M/F Contract AF33( 600) -25669
Boeing Airplane Company
VTTN: F. Bushman
7755 Marginal Way
Seattle, Washington
i/F Contract AF33(038)-31096
'hance-Vought Aircraft Division
Inited Aircraft Corporation
iTTN: Mr. F.N. Kickerman
'HRU: BuAer Representative
•alias, Texas
onsolidated-Vultee Aircraft Corp,
TTN: Mr. R.E. Honer
.0. Box 1950
ian Diego 12, California
/F Contract AF33( 600) -26530
Grumman Aircraft Engineering Corp.
ATTN: J.S. Erickson,
Asst . Chief, Avionics Dept .
Bethpage
Long Island, New York
M/F Contract NOa(s) 51-118
Hallicrafters Corporation
ATTN: Norman Foot
440 W. 5th Avenue
Chicago, Illinois
M/F Contract AF33(600)-26117
Hoffman Laboratories, Inc.
ATTN: S. Varian
Los Angeles, California
M/F Contract AF33<600)-17529
Hughes Aircraft Corporation
Division of Hughes Tool Company
ATTN: D. Adcock
Florence Avenue at Teale
Culver City, California
M/F Contract AF33(600)-27615
Illinois, University of
Head, Department of Electrical Engineering
ATTN: Dr. E.C. Jordan
Urbana, Illinois
Johns Hopkins University
Radiation Laboratory
ATTN: Librarian
1315 St. Paul Street
Baltimore 2, Maryland
M/F Contract AF(33)616)-68
Glenn L. Martin Company
Baltimore 3, Maryland
M/F Contract AF33( 600) -21703
McDonnell Aircraft Corporation
ATTN: Engineering Library
Lambert Municipal Airport
St. Louis 21, Missouri
M/F Contract AF33(600)-3743
Michigan, University of
Aeronautical Research Center
ATTN: Dr. L. Cutrona
Willow Run Airport
Ypsilanti , Michigan
M/F Contract AF33(038)-21573
page 3
DISTRIBUTION LIST (CONTINUED)
Massachusetts Institute of Tech.
ATTN: Prof. H.J. Zimmermann
Research Lab. of Electronics
Cambridge, Massachusetts
M/F Contract AF33(616)-2107
North American Aviation, Inc.
Aerophysics Laboratory
ATTN: Dr. J. A. Marsh
12214 Lakewood Boulevard
Downey, California
M/F Contract AF33( 038) 18319
North American Aviation, Inc.
Los Angeles International Airport
ATTN: Mr. Dave Mason
Engineering Data Section
Los Angeles 45, California
M/F Contract AF33(038)18319
Northrop Aircraft, Incorporated
ATTN: Northrop Library
Dept. 2135
Hawthorne, California
M/F Contract AF33(600<)-22313
Radioplane Company
Van Nuys , California
M/F Contract AF33(600)-23893
Lockheed Aircraft Corporation
ATTN: C.L. Johnson
P.O. Box 55
Burbank, California
M/F N0a(s)-52-763
Raytheon Manufacturing Company
ATTN: Robert Borts
Wayland Laboratory, Wayland, Mass.
Republic Aviation Corporation
ATTN: Engineering Library
Farmingdale
Long Island, New York
M/F Contract AF33(038)-14810
Sperry Gyroscope Company
ATTN: Mr, B. Berkowtiz
Great Neck
Long Island, New York
M/F Contract AF33(038)-14524
Temco Aircraft Corp,
ATTN: Mr. George Cramer
Garland, Texas
Contract AF33(600)21714
Farnsworth Electronics Co,
ATTN: George Giffin
Ft. Wayne, Indiana
Marked: For Con; AF33( 600) -25523
North American Aviation, Inc.
4300 E. Fifth Ave.
Columbus , Ohio
ATTN: Mr. James D. Leonard
Contract N0a(s) 54-323
Westinghouse Electric Corporation
Air Arm Division
ATTN: Mr. P.D. Newhouser
Development Engineering
Friendship Airport, Maryland
Contract AF33(600)-27852
Ohio State Univ. Research Foundation
ATTN: Dr. T.C. Tice
310 Administration Bldg
Ohio State University
Columbus 10, Ohio
M/F Contract AF33(616)-3353
Air Force Development Field
Representative
ATTN: Capt. Carl B, Ausfahl, Code 1010
Naval Research Laboratory
Washington 25, D.C.
Chief of Naval Research
Department of the Navy
ATTN: Mr, Harry Harrison
Code 427, Room 2604
Bldg, T-3
Washington 25, D.C.
Beech Aircraft Corporation
ATTN: Chief Engineer
6600 E. Central Avenue
Wichita 1, Kansas
M/F Contract AF33(600)-20910
page 4
DISTRIBUTION LIST (CONTINUED)
Land-Air, Incorporated
Cheyenne Division
ATTN: Mr. R.J. Klessig
Chief Engineer
Cheyenne, Wyoming
M/F Contract AF33(600)-22964
Director, National Security Agency
RADE 1GM, ATTN: Lt . Manning
Washington 25, D.C.
Melpar, Inc.
3000 Arlington Blvd.
Falls Church, Virginia
ATTN: K.S. Kelleher
Naval Air Missile Test Center
Point Mugu, California
ATTN: Antenna Section
Fairchild Engine & Airplane Corp.
Fairchild Airplane Division
ATTN: L, Fahnestock
Hagerstown, Maryland
M/F Contract AF33(038)-18499
Federal Telecommunications Lab.
ATTN: Mr. A. Kandoian
500 Washington Avenue
Nutley 10, New Jersey
M/F Contract AF33(616)-3071
Ryan Aeronautical Company
Lindbergh Drive
San Diego 12, California
M/F Contract W-33(038)-ac-21370
Republic Aviation Corporation
ATTN: Mr. Thatcher
Hicksville, Long Island, New York
M/F Contract AF18( 600) -1602
General Electric Co.
French Road
Utica, New York
ATTN: Mr. Grimm, LMEED
M/F Contract AF33(600)-30632
D.E. Royal
The Ramo-Wooldridge Corp
Communications Division
P.O. Box 45444
Airport Station
Los Angeles 45, California
Motorola, Inc.
Defense Systems Lab.
ATTN: Mr. A.W. Boekelheide
3102 N. 56th Street
Phoenix , Arizona
M/F Contract NOa(s)-53-492-c
Stanford Research Institute
Southern California Laboratories
ATTN: Document Librarian
820 Mission Street
South Pasadena, California
Contract AF19(604)-1296
Prof. J.R. Whinnery
Dept . of Electrical Engineering
University of California
Berkeley, California
Professor Morris Kline
Mathematics Research Group
New York University
45 Astor Place
New York, N.Y.
Prof. A. A. Oliner
Microwave Research Institute
Polytechnic Institute of Brooklyn
55 Johnson Street - Third Floor
Brooklyn, New York
Dr. C.H. Papas
Dept. of Electrical Engineering
California Institute of Technology
Pasadena, California
Electronics Research Laboratory
Stanford University
Stanford, California
ATTN; Dr. F.E. Terman
page 5
DISTRIBUTION LIST (CONTINUED)
Radio Corporation of America
R.C.A. Laboratories Division
Princeton, New Jersey
Electrical Engineering Res. Lab.
University of Texas
Box 8026, University Station
Austin, Texas
Dr. Robert Hansen
8356 Chase Avenue
Los Angeles 45, California
Technical Library
Bell Telephone Laborat-drxes
463 West St.
New York 14, N,Y.
Dr . R . E . Beam
Microwave Laboratory
Northwestern University
Evanston, Illinois
Department of Electrical Engineering
Cornell University
Ithaca, New York
ATTN: Dr. H.G. Booker
Applied Physics Laboratory
Johns Hopkins University
8621 Georgia Avenue
Silver Spring, Maryland
Exchange and Gift Division
The Library of Congress
Washington 25, D.C.
Ennis Kuhlman
c/o Mc Donnell Aircraft
P.O. Box 516
Lambert Municipal Airport
St. Louis 21, Mo.
Physical Science Lab.
New Mexico College of A and MA
State College, New Mexico
ATTN: R. Dressel
Technical Reports Collection
303 A, Pierce Hall
Harvard University
Cambridge 38, Mass.
ATTN: Mrs. E.L. Hufschmidt, Librarian
Dr. R.H. DuHamel
Collins Radio Company
Cedar Rapids, Iowa
Dr . R . F . Hyneman
5116 Marburn Avenue
Los Angeles 45, California
Director
Air University Library
ATTN: AUL-8489
Maxwell AFB, Alabama
Stanford Research Institute
Documents Center
Menlo Park, California
ATTN: Mary Lou Fields, Acquisitions