. . .

Sensor and Simulation Notes

Note go.

July 1969

Impedance and Fields of Two Parallel Plates of Unequal Breadths

by

G. W. Carlisle

Northrop Corporate LaboratoriesPasadena, California

Abstract

The method of conformal transformation

and fields of two parallel plates of unequal

type make up the transmission”lines that are

‘#ggis used to determine the impedancey%J#%

breadths. FParallel plates of this ““--~.::&-.,.=,z,-w..used to transport electromagnetic

energy along certain portions of some nuclear EMP simulators. Pertinent data

are displayed graphically.

.. ..._ ..=>..,,~.... ,,.

● ✎

Acknowledgment

Thanks are due Mr. R. W. Sassman for his expert programming, Drs. R. W.

Latham and K. S. H. Lee for their helpful advice, and Mrs. G. Peralta for

preparing the manuscript.

. ,6’

●

(--1. Introduction

In simulating the phenomenon of a nuclear electromagnetic pulse (EM2)

by artificial means, a parallel-plate transmission-linehas proved to be a

practical way of transporting the electromagnetic energy along certain

portions of the simulator. In a previous notel the properties of two

parallel plates of equal breadths were considered. This model also represents

the electromagneticallyequivalent configuration of a single plate parallel

to an infinite ground plate --- a configuration that approximates an actual

physical set-up of a plate parallel to a large, but finite, ground plate.

In this note the actual geometry of two parallel plates of unequal

breadths, as shown in figure 1, is considered, and the impedance of such a

structure is computed. Curves are presented to show how the impedance depends

on the size of the ground plate, which varies from a breadth equal to that

of the upper plate (a1= a) to a breadth of infinite extent (al=@) .

In addition curves are presented to show the values of a/h and

(-al/h corresponding to standard impedance values of 50, 75, 100 and 150 ohms

ewhere a and a

1are the half-breadths of the plates and h is the total

spacing between the plates. The latter set of curves is particularly useful

because it can be used to design a set of parallel plates of different breadths

--- one plate perhaps much larger than the other --- whose impedance is exactly

a prescribed value, without resorting to the infinite plate approximation.

Finally the magnitude of the electric field beneath the finite ground

plate is compared with the magnitude of the electric field between the parallel

plates. This consideration is important if electronic equipment is placed

beneath the ground plate.

1

. ._, . ..__., ,.... — ——

. ●

*

11.

A. E. H. Love gives~

Exact Conformal Transformation

the conformal transformationfor

of unequal breadths.~ For the geometry depicted in figure

is

[

cn(wlml)dn(wlml)iz K(m) Z(wlml) +.—= —

2K(mY(ml)+

h~ v + sn(w ml)

Associated with this transformationare the equations

K(m) - v2E(m)C1C2 =

mlv2K(m) - E(m)

two parallel plates

1 the transformation

T

1‘Zzm “

(v’ - 1)(mlvL - l)K(m)

(v + cl)(v + C*) = .

mlv2K(m) - E(m)

(1)

(2)

(3)

In these equations Z(wlml) is Jacobi’s Zeta-function; K(m) is the complete

elliptic integral; sn(wlml) s Cn(wlml) , and dn(wlrnl) are Jacobian

elliptic functions. The notation used here is consistent with that used in

reference 3. The parameter v relates to the difference in the breadths of

the two places. As V+m, (1) reduces to the transformation for the case

of two plates of equal breadth; as v + I/q , the breadth of the larger

plate becomes infinite while the smaller plate remains finite. The range of

v , therefore, is v > l/~ and throughout this range ~azab

Evaluating the real part of (1) at (-a,h) and (-al,O) gives

(4)

-.

.*“

.

/--{

al K(m) I ‘- #M3Fm “)E(m)— = — ‘2 K(m)h ‘lT

E(v21m) -

where v1

and v~ satisfy

and

dn(v2\m) + ~ = O 9 0~v2~K(m) .C2

r’

9

The constantsc1

and‘2

are obtained from (2) and (3) such that

(6)

(7)

(8)

where ~c~ is the positive value of (8) and C2 is the negative value. The

quantities a and 8 are given by

V2E(m) - K(m)‘a=

E(m) - mlv2K(m)

$V(l - m/2)K(m) - E(m)= ‘.E(m) - mlv2K(m)

(9)

(lo)

. ●

●

The impedance of a parallel-plate transmission-lineimmersed in free-1

space is

Cz00

‘L=-T--

where in this particular case C , the capacitance per unit length, is

given by

coK(m)c= K(m,) “

It is convenient to define a normalized impedance

z.

which, if expressed explicitly, is

K(ml)z;=-—

h K(m).

(11)

(12)

(13)

(14)

In figures 2 through 4 the normalized impedance Z: is plotted versus

a/h with d/h = (al - a)/h as a parameter. The data in figures 2 and 3

were obtained by choosing a set of values for ml and v and by using these

values to compute the correspondingvalues of a/h , al/h and Z; . To

compute these three quantities, (9) and (10) are used to obtain a and $

which are substituted into (8) to compute c1 and C2 . The values of c1

4

-“

●

‘1

/--

[

and‘2

are substituted into (6) and (7) and a root-finding technique is

used to determine‘1

and v2“

With all of these constants known, (4)

and (5) are evaluated to find a/h and al/h . The impedance Z~ is

computed from (14).

A different method was used to obtain the data in figure 4. The

method, as well as the reason for using it, is discussed in the next section.

The data presented in figures 5 through 7 were obtained in the same

way as the data in figures 2 and 3 were obtained exeept that a root-finding

technique was used first to determine from (14) what values of ml correspond

to specific values of‘L “

In these figures A = a/afi is plotted versus

‘1=a/a~ ~ with Z = 50, 75, 100 and 150 ohms as a parameter. The quantity

L

aQ is measured in meters and is assigned the subscript $2 to emphasize

that it takes on a different value for each ohmic value of ZL . Below is

listed a table of the values of 2ao/h that correspond to each of the four

values of ZL“

/---

Table 1. Values ofan

and‘L

2afiZL (ohms)

T

50 5.998

75 3.618

100 2.457

150 1.337

...

Physicallyan

is the half-breadth of two parallel plates of equal breadth

that are one meter apart and have an impedance ‘L “Multiplying the values

of A and Al that correspond to a particular value of‘L

by the appropriate

value of 2aQ/h listed in Table 1 gives 2a/h and 2al/h , respectively,

which can be used to design an actual parallel plate structure,

5

._—--- ..-— .-— ._.

.

III. Approximate Transformation for Plates Close Together

For the case of two parallel plates that are very close together

(a/h >> 1) , the parameter ml is very small. For example when 2a/hss 10 ,

ml w 1~14 ; when 2a/h s 20, ml s 1~24 . Numerical computation becomes

difficult for such small values of ml; hence, it is convenient to use an

approximate conformal transformationfor 2a/h> 10 . The approximate

transformationis4

TT(Z - a)2

h=*+ (1-T)w-Tlllw++ -’c

where the parameter T satisfies

The real and imaginary parts of (15) are

27rT(x- a)

h=>cos28+(l-T)r cos~-~lnr++-~

2y => sin 2f3+(l-T)r sin 6--r6

Setting x =Oande=n in (17) gives

?rTas;-(1-T)r

1

T- T hi r +—- -r

2

(15)

(16)

(17)

(18)

(19)

-0’

J

,-(

/’--

For a set of values of a/h and T , two values of r satisfy (19); these

radii are denoted‘A

and‘B

such that‘A

< 1 and rB> 1 . Values of

T are obtained from (16), which depends on d/h .

In terms of .4‘A

and‘B

the transmission line impedance‘L ‘s

‘nZ.

‘L = Y ln(rB/rA) “(20)

The curves in figure 4 were obtained by using this approximate transformation.

In figure 4 the normalized impedance Z: defined by (13) is plotted versus

a/h with d/h as a parameter. Note that the values of d/h in figure 4

are different from those in figures 2 and 3.

. . ..——

●

Iv. Electric Field below the Lower Plate

In an actual simulator it may be that electronic equipment is located

beneath the lower plate, i.e. the plate of greater breadth. In placing

such equipment in this position it might be assumed that the field of a

simulated EMP is contained wholly within the transmission line and that

none of it leaks around the ends of the lower plate into the region where

it could interfere with the electronic equipment. In the case of a finite

lower plate a portion of the field does of course penetrate into the region

beneath the lower plate. In this section of the note the magnitude of the

electric field just beneath the center of the lower plate (ZA = O + iO-)

is compared with that just above the center of the lower plate (ZB = O + iO+) .

The magnitude of the electric field is given by

which, in this case, is

IEl = A

where the constant A relates

plates and does not depend on

IE[ = $ ‘1

(snw - cl)(sn w - c2)

(snw+ ;)2

to the specific geometry

w . Under the conformal

the points‘A

and‘B

correspond respectively to ‘A

(22)

of the parallel

transformation (1)

= - K(ml) and

‘B= - K(ml) -t-iK(m) . Upon substituting these values of w into (22)

the ratio between the magnitude of the electric field at ‘Aand z~ is

]EA\ (l+c1)(l+c2)Gp 1)2

~=

(l+qcl) (l+~c2) (v-1)2●

(21)

(23)

o

8

0

9-

Plots of \EA[/lEB\ x 100% versus a/h with d/h as a parameter are

presented in figure 8.

9

..—.

Y

+,-xd

1

Figure 1. Two Parallel Plates of Unequal Breadths.

a-J

e

.25

.20

z’ .“1:L

.1(

.o!

(

d/h = O

hz

‘; =zL/(~)

h

I I I I [1 I 1 I I I 1 I 1 1 t t

.01 .02 .03 .04 .05 .06 .08 .1 .2 .3 .4 .5 .6 .8 1.0

.

a/h

Figure 2. Z; versus a/h with d/h as a parameter (.Ol<a/hs 1.0).

.95

.85

.75

.65

z;

.55

.45

.35

.25

.

.

.

hZz; = ZLI(=)2a

k-2a+’l

h

4/ dfh = O

.

. 1 .2 .3 .4 .5 .6 .7 .8 .91.0 2 3 456 78910

a/h

Figure 3. ‘iversus a/h with d/h as a parameter (.1< a/h~ 10).

. I

1.00

.98

.96

.94

z’L

.92K-u

.90

.88

Cl/h= o

5.0

2.0

1.0

0.25

hzz; = zLf(~)

t+

.8610 20 30 40 50 60 80 100 200 300 400 500 600 800 1000

a/h

Figure 4. Z; versus a/h with d/h as a parameter (10 < a/h< 1000).

1.00

.9e

.96

.94

.92

A

*90

.88

.86

.84

.82

.80

.

I I I I 1 i I I t I 1 I i 1

.

.

.

.

.

.

.

150Q

I

1.0 1.1 1.2 1*3 1.4 1.5 1.6 1.7

‘1

. @

.

s

*

-’

(1.OsA@ 1.7).Figure 5. A versus‘1 with

‘L as a parameter

.

-3.86

,.

(-0 .84

.82

.80

.78

A .76

.74

.72

.70

.68

.66

1.

I I

ZL = 50$2

150sl

3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

‘1

(1.3 <Al ~ 2.0).Figure 6. A versus‘1

with ZL as a parameter

15

.85

.80

.75

.70

A .65

‘ .60

.55

\

i’xl

\

100$2

11500

.45 1 I 1 I 1 I I I I I t 1 I I 1 I I I I 1

1 1.5 2 3 456 78910 15 . 20 30 40 50 60 70 80 93100

%●

Figure 7. A versus‘1 with ZL as a parameter (l<AIS 100).

.

&

80

70

60

50

Figure 8. ‘Ai/lEB~xa/h

100% versus a/h with ZL as a parameter.

*

References

● ✍✍✍✍✍ b

1. Carl E. Baurn,Sensor and Simulation Note 21, “Impedances and Field

Distributions for Parallel-Plate Transmission-LineSimulators,”

June 1966.

2. A. E. H. Love, “Some ElectrostaticDistributions in Two Dimensions,’r

Proc. London Math. Sot. ~, pp. 337-369, 1923.

3. Milton Abromowitz and Irene A. Stegun, Editors, Handbook of Mathematical

Functions, National Bureau of Standards, AIM-55, June 1964.

4. F, Assadourian and E. Rimai, “Simplified Theory of Microstrip Transmission

Systerns,rlProc. of IRE, December 1952, pp. 1651-1657.

a

18