Download SSN 90, G. W. Carlisle, Impedance and Fields of Two Parallel Plates

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. .
Sensor and Simulation Notes
Note go.
July 1969
Impedance and Fields of Two Parallel Plates of Unequal Breadths
by
G. W. Carlisle
Northrop Corporate Laboratories
Pasadena, California
Abstract
The method of conformal transformation is used to determine the impedance
and fields of two parallel plates of unequal breadths. Parallel plates of this
‘#gg
y%J#%
~.:
:&““-z,
F.,.=,
-w..
type make up the transmission”lines that are 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-line has 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 electromagnetically equivalent 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
(a = a) to a breadth of infinite extent (al=@)
1
In addition curves are presented to show the values of a/h and
of the upper plate
al/h
(e
where
.
corresponding to standard impedance values of 50, 75, 100 and 150 ohms
a
and
a
are the half-breadths of the plates and h is the total
1
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.
Exact Conformal Transformation
A. E. H. Love gives the conformal transformation for two parallel plates
~
of unequal breadths.~ For the geometry depicted in figure 1 the transformation
is
iz
.—=
h~
K(m) Z(wlml) +
—
cn(wlml)dn(wlml)
2K(mY(ml)
+
v + sn(w ml)
T
‘Zzm
[
1
“
(1)
Associated with this transformation are the equations
K(m) - v2E(m)
C1C2 =
(2)
mlv2K(m) - E(m)
(v’ - 1)(mlvL - l)K(m)
.
(3)
In these equations Z(wlml) is Jacobi’s Zeta-function; K(m)
is the complete
(v + cl)(v + C*) =
mlv2K(m) - E(m)
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
the two places. As
V+m,
(1)
relates to the difference in the breadths of
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
and throughout this range azab
~
v , therefore, is v > l/~
Evaluating the real part of (1) at
(-a,h)
and
(-al,O)
gives
(4)
-.
.
*“
.
{/--
I ‘-
al
E (m)
K(m)
—h = — ‘lT ‘2 K(m)
where v
1
and
v~
E(v21m) -
#M3Fm
“)
satisfy
(6)
and
dn(v2\m) + ~ = O
C2
r’
The constants
c1
and
‘2
9
0~v2~K(m)
.
(7)
are obtained from (2) and (3) such that
9
(8)
where ~ c~
is the positive value of (8) and
quantities a
and
8
C2
is the negative value.
The
are given by
‘a=
V2E(m) - K(m)
(9)
E (m) - mlv2K(m)
=
$V(l
- m/2)K(m) - E(m)
E (m) - mlv2K(m)
‘.
(lo)
.
●
●
The impedance of a parallel-plate transmission-lineimmersed in free1
space is
Cz
00
(11)
‘L=-T-where in this particular case C , the capacitance per unit length, is
given by
coK(m)
c
= K(m,)
(12)
“
It is convenient to define a normalized impedance
z.
(13)
which, if expressed explicitly, is
K(ml)
z;=-—
(14)
.
h K(m)
In figures 2 through 4 the normalized impedance Z:
a/h with
d/h = (al - a)/h
is plotted versus
as a parameter. The data in figures 2 and 3
were obtained by choosing a set of values for ml
values to compute the corresponding values of
and
v
a/h , al/h
and by using these
and
Z; . To
compute these three quantities, (9) and (10) are used to obtain a
which are substituted into (8) to compute c1
and
and
C2 . The values of
$
c1
-“
4
●
‘1
/--
[
are substituted into (6) and (7) and a root-finding technique is
‘2
used to determine
and v
With all of these constants known, (4)
‘1
2“
and (5) are evaluated to find a/h and al/h . The impedance Z~ is
and
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
In these figures A = a/afi is plotted versus
‘L “
= 50, 75, 100 and 150 ohms as a parameter. The quantity
to specific values of
‘1
aQ
=a/a
~
~
with
Z
L
is measured in meters and is assigned the subscript $2 to emphasize
that it takes on a different value for each ohmic value of
listed a table of the values of
values of
2ao/h
ZL . Below is
that correspond to each of the four
Z
L“
/---
Table 1. Values of
ZL (ohms)
and
an
2afi
‘L
T
50
5.998
75
3.618
100
2.457
150
1.337
...
Physically
is the half-breadth of two parallel plates of equal breadth
an
Multiplying the values
that are one meter apart and have an impedance
‘L “
by the appropriate
of A and Al that correspond to a particular value of
‘L
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
ml w 1~14
; when
is very small. For example when
2a/h s 20, ml s 1~24
2a/hss 10 ,
. Numerical computation becomes
difficult for such small values of ml; hence, it is convenient to use an
approximate conformal transformation for
4
transformation is
TT(Z
- a)
h
where the parameter
T
2
=*+
2a/h>
10 . The approximate
(1-T)w-Tlllw++
-’c
(15)
satisfies
(16)
The real and imaginary parts of (15) are
2
7rT(x- a)
=>cos28+(l-T)r
h
cos~-~lnr++-~
(17)
2
y
Setting x =Oande=n
=>
sin 2f3+(l-T)r
sin 6--r6
(18)
in (17) gives
?rTa
s;-(1-T)r
T
1
- T hi r +—-
2
-r
(19)
-0’
J
,-
(
For a set of values of
a/h
and
T ,
two values of
r
< 1 and
and
such that
‘A
‘B
‘A
are obtained from (16), which depends on d/h .
radii are denoted
T
In terms of
‘A
and
‘B
satisfy (19); these
rB>
1 . Values of
the transmission line impedance
‘n
‘L
.4
‘s
Z.
‘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:
a/h with
d/h
defined by (13) is plotted versus
as a parameter. Note that the values of
d/h
are different from those in figures 2 and 3.
/’--
..
..——
in figure 4
●
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
IE[ = $
(21)
‘1
o
which, in this case, is
(snw - cl)(sn w - c2)
IEl =
(22)
A
(snw+
where the constant A
;)2
relates to the specific geometry of the parallel
plates and does not depend on w . Under the conformal transformation (1)
correspond respectively to ‘A = - K(ml) and
and
the points
‘B
‘A
= - K(ml) -tiK(m) . Upon substituting these values of w into (22)
‘B
the ratio between the magnitude of the electric field at ‘A and z~ is
]EA\
(l+c1)(l+c2)Gp
1)2
●
~=
(l+qcl)
(l+~c2)
(23)
(v-1)2
0
8
9-
Plots of
\EA[/ lEB\ x 100% versus
a/h with
d/h
as a parameter are
presented in figure 8.
9
..—.
Y
d
1
+,-x
Figure 1. Two Parallel Plates of Unequal Breadths.
a
-J
e
.
.25
d/h = O
.20
z’
L
.“1:
hz
‘; = zL/(~)
.1(
h
.o!
1
(
.01
.02
I
.03
1
.04
I
I
.05 .06
I
1
.08
I
1
1
.2
.1
t
.3
t
.4
a/h
Figure 2.
Z; versus
a/h with
d/h
as a parameter
(.Ol<a/hs
1.0).
I
.5
I
.6
I
I
.8
[
1.0
.95
hZ
z; = ZLI(=)
2a
.85
k-2a+’l
.75
h
4/
dfh = O
.65 .
z;
.55 .
.45 .
.35 .
.25
.1
.2
.3
.4
.5
.6
.7 .8 .9 1.0
2
3
456
78910
a/h
Figure 3.
‘i
versus
a/h
with
d/h
as a parameter
(.1< a/h~
10).
.
I
1.00
Cl/h= o
.98
5.0
.96
2.0
1.0
.94
0.25
z’
hz
z; = zLf(~)
L
Ku
.92
.90
t+
.88
.86
10
20
30
40
50
60
80
200
100
300
400 500
a/h
Figure 4.
Z;
versus
a/h with
d/h
as a parameter
(10 < a/h<
1000).
600
800
1000
.
1.00
I
I
I
I
1
i
I
I
t
I
1
I
i
.
@
1
.
s
.9e .
*
.
.96 .
.
.94 .
.
.92 .
A
*90
.88
.86
.84
.82
150Q
.80
1.0
I
1.1
Figure 5. A
1*3
1.2
versus
‘1
with
1.4
1.5
‘1
as a parameter
‘L
1.7
1.6
-’
(1.OsA@
1.7).
.
-3
.86
I
I
,.
(-
0
.84
ZL = 50$2
.82
.80
.78
A
.76
.74
.72
.70
150sl
.68
.66
1.3
1.4
1.7
1.6
1.5
1.8
1.9
‘1
Figure 6. A
versus
‘1
with
ZL
as a parameter
15
(1.3 <Al
~ 2.0).
2.0
.85
.80
.75
i’xl
.70
A
\
.65
100$2
‘ .60
\
.55
1
1500
.45
1
1
I
1
I
1.5
2
3
456
1
I
I
I
I
I
t
15 .
78910
1
20
I
30
I
40
%
Figure 7.
A
versus
‘1
with
ZL
1
50
I
I
I
I 1
60 70 80 93100
●
as a parameter
(l<AIS
100).
.
&
80
70
60
50
a/h
Figure 8.
‘Ai/lEB~x
100% versus
a/h
with
ZL
as a parameter.
*
● ✍✍✍✍✍
b
References
1.
Carl E. Baurn,Sensor and Simulation Note 21, “Impedances and Field
Distributions for Parallel-Plate Transmission-Line Simulators,”
June 1966.
2.
A. E. H. Love, “Some Electrostatic Distributions in Two Dimensions,’r
Proc. London Math. Sot. ~,
3.
pp. 337-369, 1923.
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,rl
Proc. of IRE, December 1952, pp. 1651-1657.
a
18