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<P><PB REF="00000001.tif" SEQ="00000001" RES="600dpi" FMT="TIFF6.0" FTR="TPG" CNF="893" N="">
320631-1-F
THEORETICAL STUDY OF THE DISTRIBUTION
OF POLES AND ZEROS OF THIN BICONICAL ANTENNA
FINAL REPORT
P. 0. No. T-2150
1 July - 31 November 1976
February 1977
Prepared by:
Professor Chen-To Tai
The University of Michigan
Radiation Laboratory
Ann Arbor, Michigan 48109
Prepared for:
Dikewood Industries
1009 Bradbury Drive, S.E.
Albuquerque, New Mexico 87106
320631-1-F = RL-2528
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<P><PB REF="00000002.tif" SEQ="00000002" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="871" N="1">
THEORETICAL STUDY OF THE DISTRIBUTION
OF POLES AND ZEROS OF THIN BICONICAL ANTENNA
1. Introduction
In recent years extensive investigations have been done in the area of transient
electromagnetic wave phenomena. The new book edited by Leopole B. Felsen [1]
summarizes some of the basic works which have been accomplished so far. The book
also contains a very substantial bibliography of articles and reports written before
1976.
One may recall that transient analysis for discrete networks in the hands of
Cauer [2] and Guillemin [31 developed into one of the major disciplies in electrical
engineering curriculum in the late nineteen-forties. The technique involved in that
discipline is based mainly upon the concept of zeros and poles. This approach, however, has not yet been extended too far to distributed network, including transmission
lines and antennas. Therefore, if one can formulate and solve some canonical problems
involving a distributed network based on this approach, such an endeavor certainly
enhance our knowledge in a much wider area. It is for this reason that we chose the
biconical antenna as the model in this study because methods are now available to
analyze this problem comparable to the ones used for analyzing transients in discrete
networks. This report will summarize the formulation, and the result which we have
obtained; and at the end, some suggestions are made about the research to be done in
the future.
2. Transients on Terminated Line
Before we discuss the biconical antenna problem it is desirable to outline the
methods which are available to study the transients on a terminated line. Our work
on the transmission line was started as a result of a grant from the National Science
Foundation. In the final stage of this work the research was also being supported by
this contract. A technical report [4] has been written on this subject and a copy of
this report is attached. As indicated at the end of that report, the same methods
used for the transmission line analysis are also applicable to the biconical antenna.
1
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<P><PB REF="00000003.tif" SEQ="00000003" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="884" N="2">
We wish to call attention to the fact that the Volterra integral equation method, and
the method of singularity expansion based on the Mittag-Leffer theorem are two new
formulations which we have obtained in analyzing the transients for this class of
problems. The so-called F-series method has also been modified to include the
remainder in the series expansion. This inclusion removes the non-rigorous approach
used by many authors for this problem. The work as a whole, therefore, represents
a thorough treatment of the transient phenomena on an arbitrarily terminated line.
3. Input Transient Current of Thin Biconical Antenna
The theory of thin biconical antenna is well known for harminically oscillating
field [5, 6. To study the input current of such an antenna subject to an input
transient voltage one can transform the harmonic solution to the Laplace-transform
domain and then evaluate the inverse transform. The analysis is most conveniently
done by introducing a normalized Laplace-transform variable s and a normalized
time r which are defined as follows:
S c -
tc
T= half-length of the bicone
c = velocity of light in free space.
The Laplace-transform of the input current to a biconical antenna is then defined by
I(s) = Li(T) = i(T)e dTi, (1)
and the inverse transform is given by
s +jOo
i(T) = L-1I(T) = I(s) e ds. (2)
S -jo00
0
For a unit-step input voltage applied to the input terminal of a biconical antenna, the
expression for the input current in the Laplace-transform domain can be written in
the form
2
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<P><PB REF="00000004.tif" SEQ="00000004" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="842" N="3">
s 2s
Z I(s) 2s (3)
F + r(s)e
where Z = characteristic impedance of a biconical antenna
C
Z / 0 \
= -T n ot )ohms,
Z = free-space wave impedance
o
= 1207T ohms,
0 = half-angle of the biconical antenna,
0
r(s) -- y(s) (4)
1 - - y(s) '
y(s) = normalized terminal admittance
For thin biconical antennas (0 < 5 ), Z is approximately given by
o c
Z
o 2
Z - n (5)
c O
0
and y(s) is represented [5, 6by
Z
y(s) = 4- (2E(2s) + e [In2 +E(2s) - E(4s)]
C
+ 2 - tn2 +E(-2s)] (6)
z
where 1 - e dt
The function is the same as the exponential function E i n(z) denoted by Abramoweitz
and Stegun[7]. It can be shown that the function y(s) vanishes at s = 0 and is unbounded at infinity.
4. Poles of I(s)
In order to find i(T) by means of the singularity expansion method, our first
task is to find the poles of I(s). According to Eq. (3), the poles of I(s) are given by,
in addition to s = 0, the roots of the equation
3
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<P><PB REF="00000005.tif" SEQ="00000005" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="872" N="4">
1+ F(s) -2s =0. (7)
(U)
These roots will be denoted by s. The meaning of the superscript (-) and the subn
script n will be explained later when these roots are displayed graphically. The
numerical values of these roots were first found by a rather tedious searching method,
since then they have been calculated by a contour integration method due to Singaraju,
Giri and Baum[8]. In evaluating the function r(s), we have used the following series
for the experimental function E (z) for values of Jzf not too large
O 1 n+l n
E = (-1) z (8)
E(z) 2 n -- n (8)
n 1 n
For large values of zl the asymptotic expansion of E (z) is used, namely,
E(z) =- nv - e Z (-1) )n '(9)
zn
n=1
where Inv= Euler's constant = 0.577215664...
The distribution of s )for values of 0 = 0.001~ and 0.573~ are shown in
n o
Figures 1 and 2. Since these roots are distributed in two distinct layers or branches,
we have used the superscript. to distinguish them. Figures 1 and 2 only show the
roots situated in the second quadrant, the conjugate ones existed in the third conjugate
are not shown. For the real root s(2) it turns out to be a simple root as ascertained
0 (1)
by the contour integration method. For the case 0 = 0. 573~, the values of s are
o n
very close to the ones found by Tesche [9 for a cylindrical antenna with a radius over
height ratio equal to 1/100. When a biconical antenna is inscribed in such a cylinder
0 = (180/7r ) a/h = 1. 8/7T= 0.573~. Except the equivalence between the roots for the
0 (2) (2)
first layer, we have found no similarity between our s and Tesche's s. In fact,
n n
Tesche's calculation based on an integral equation formulation yields more than two
layers while we have found only two layers. Based on the contour integration method
we are reassured that there are only two layers for a thin biconical antenna. At the
present moment we are unable to ascertain the significance of the different layers of
4
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<P><PB REF="00000006.tif" SEQ="00000006" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="858" N="5">
these poles of I(s) nor could we explain the difference between the distributions of
these poles for a biconical antenna and a cylindrical antenna of comparable dimension.
In Figure 1 and 3 we have also plotted the roots
-2s
1 - F(s)e = 0. (10)
These roots are denoted by s. They correspond to the zeros of I(s). The link
(f) A(~) n
between s( and s will be discussed in the section dealing with the theory of
n ii
receiving antenna.
5. Expansion of Z I(s)
c
Once the poles of I(s) are known one can expand Z I(s) in terms of a residue
series based on Mittag-Leffler theorem [10]. However, the expansion is not unique;
there are at least two alternative expansions one can formulate. The first one is to
write Eq. (3) in the form
Z I(s) = 1- Es) (11.(11)
1 + F(s)e
Now the function
F(s) = -- 2(s) (12)
1 + r(s) e
is finite at s = 0 and unbounded at s -- oo, hence it satisfies the conditions under
which Mittag-Leffler theorem holds. According to this theorem
F(s) = F(o) + P(s) - P(o), (13)
where
P(s) dt
A
(14)
S -
n n
where s denotes the simple poles of F(s), previously denoted by s n and
nn
5
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<P><PB REF="00000007.tif" SEQ="00000007" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="803" N="6">
- 2 (s )
n
A =
n
- 2r(s )
n
2 + e n (Sn)
(15)
d 1 + (s) eS
ds Js = s
n
In Eq. (15), rt(s ) denotes the value of the derivative of f(s) with respect to s evaluated at s = s. Thus,
n
-2 s):- -1+ A+S -. I]
-2s n s - s s
1 + r(s) e n n n nJ
and
- 2F(s) n I An
s -2r(s) -1 s ( - )
s 1 + r(s) e ] n n n
(16)
The last expression results from the identify
1 1 1
s(s - s ) s s s(s - s )
n n n n
The singularity expansion of Z I(s) based on Eq. (11) is therefore given by
C
Z I(S) = - + e
c s
- A
-2s - 1 +: n
s +s (s - s )
[ n n n
-
(17)
- 2s
where A is given by Eq. (15) and s denotes the roots of 1 + F(s) e =. The
n n
time domain solution based on Eq. (17) yields
Z i(T) = L — 1 I() = U(T- 0)+ U(T - 2) -1 + en n
L n n a
(18)
It should be recalled that s occurs in conjugate pairs, hence Eq. (18) may be
writ in the for
written in the form
6
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<P><PB REF="00000008.tif" SEQ="00000008" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="816" N="7">
Z i(T) = U(T - 0)+ U(T - 2) -1 +2Re ' nL e ( 2 (19)
n n
The prime on the summation sign in Eq. (19) means that the sum is taken for these
poles lying in the second quadrant in the s-plane. For convenience we shall identify
Eq. (18) as Solution (A). This solution shows clearly that the casuality condition is
met, namely, for 7 < 2 Z i(T) = U (T - 0), corresponding to the initial response of
c
the antenna before the reflected wave from the terminals of the antenna reaches the
input end.
An alternative expansion of Z I(s) is to split Z I(s), Eq. (3), into two terms
of the form
- 2s
1 P(s)e
s I() + r(s) e2s s l + F(s)e-22s (20)
By expanding the function 1/[1 + F(s)e as before, one obtains
n n -2s -1 _
cIts) 2s +
Z I s(ss) + + e + (21)
— S^, I (21)
C 2 s n s ns + Sn ( 2s s - sn
where 1
n -2s
2+e r'(s )
n
- (s)
n 1
n +-2s r'(s) 2 n'
A being defined by Eq. (15). The corresponding time domain solution is then given
by
Z i(T) =U(T - 0) [+ 2R n e n
c 2 s
n n
'c s (T - 2)
n n (23)
+ U(L - 2) -n2Re (23)
2 s
n n
7
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<P><PB REF="00000009.tif" SEQ="00000009" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="876" N="8">
We shall identify Eq. (23) as Solution (B). This solution does not offer the immediate
impression that the casuality condition is satisfied unless the function within the
bracket attached to the unit-step function U(-r - 0) is numerically equal to unity for
2 >T> 0. However, numerical calculation based on these two alternative solutions
seems to support this identity as will be presented graphically later.
Before we conclude this section, another alternative solution should be mentioned. This solution is based on the F- series method as discussed in the attached
paper on transmission line transients [4]. The only difference is that F(s) is a more
complicated function of s for the antenna problem. According to this method we can
write Z I(s) is the form
c
Z I(s) = - r(s)e 2 - rF(s) e-]
c S n=0
= ~ 1 2 r2(s)e2s + 22(s) e4s+. (24)
For T < 4, the time domain solution is given by
Z i(T) = U(T - 0) - L e-]. (25)
In view of the integral equation method [4] the higher order solution for T > 4 can be
obtained by a successive integration of the low order solutions. It is therefore
sufficient to discuss the solution represented by Eq. (25), which will be designated
as Solution (C). To evaluate the inverse Laplace-transform of the term contained in
Eq. (25) we will expand the function F(s) in the form of a residue series. Since
r(s) = (s) as defined by Eq. (4), the poles of F(s) are given by the roots of
the equation
1 + y(s) = 0. (26)
8
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<P><PB REF="00000010.tif" SEQ="00000010" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="874" N="9">
These roots will be denoted by s. The distribution of these roots is shown in
ny
Figure 4, for 0 = 0.001 and 0.573. There is only one layer. The roots also
0
appear in conjugate pairs. Only those in the second quadrant are shown in the figure.
(i) (2) (1)
Comparing to s(, they are distributed more or less like s, but not like s. The
n n n
singularity expansion of the function - 2 F(s)/s contained in Eq. (25) can be obtained
by applying the Mittag-Leffler theorem to the function F(s). The results yields
- 2F(s) 1 dn ] (27)
s s s (s-s )
n ny ny
where d = y(s ) Thus the time-domain solution as described in Eq. (25) is given
n y s
by ny
d+ 2Re)
Zi(T) = U(r - 2- U(T -2) e s (28)
ny
Comparing with Solutions (A) and (B) as represented by Eqs. (19) and (23) we
see that Solution (C) involves a completely different set of singularities. This immediately raises the question as to which set is more desirable or preferable? From
an analytical point of view it seems the question cannot be answered definitively. In
fact, it is even difficult to attach much physical significance to these sets of poles.
Facing this dilemma we could only accept these alternative solutions as equally valid.
A preference perhaps could be chosen if we could examine more critically the rate
of convergence of these series. This work which is related to Prony's method of
synehesis using an infinite set of exponential functions needs further investigation in
the future.
6. Numerical Calculation
Based on Eqs. (19), (23) and (28), which will be designated as Z iA Z iB
c A' C
and Z ic, we have computed these solutions for a thin biconical antenna with 0 = 0. 573~
The results are shown in Figure 5 for T > 2. In general, the wave forms based on
these different representatives are comparable. It appears that Z i departs considerably from Z i and Z i for near 2 and 4. Since the high order poles in
cA c B
9
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<P><PB REF="00000011.tif" SEQ="00000011" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="873" N="10">
this representation have a real part which is smaller than that of the low order poles,
it is very likely that we need more terms in the series to obtain an accurate solution.
For Z i Eq. (23), we have also, computed its values for the time interval 2 >r> 0.
The result is shown in Figure 6. The values are very close to unity except for T near
0 and 2, where a phenomenon similar to the Gibbs' phenomenon in Fourier series
analysis appears to be existing. This is another area where further research is needed
to ascertain the degree of overshooting.
7. Biconical Antenna as a Receiving Antenna
In this section we give a brief discussion of the formulation using thin biconical
antenna as a receiving antenna.
According to the equivalent circuit of a receiving antenna the terminal current
of a receiving antenna in the Laplace-transform domain can be written in the form
[ Ei(s) h(s)]
L(S) + Z ) (29)
where
E (s) = incident electric field,
h(s) = effective height of the receiving antenna,
Z.(s) = input impedance of the receiving antenna,
Z (s) = load impedance.
L
All these parameters are defined in the Laplace-transform domain. For a thin biconical
antenna we can express Z.(s) in terms of F(s), i.e.,
1
-2s
1 +?N(Ose e
z.(s): +s) z (30)
1T -2s c r
1 - F(s) e
The poles of I L(s), therefore, are found by setting Z.(s) + ZL (s) = 0, or
10
</P>
<P><PB REF="00000012.tif" SEQ="00000012" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="876" N="11">
-2s.+(s)-2s + L(S)=0 (31)
1 - r(s)e
where zL(s) denotes the normalized load impedance. For a purely resistive load we
let z (s) = a to be a real positive constant then Eq. (31) can be written in the form
-2s a + 132)
a- 1
We shall denote the roots of Eq. (32) by s
n
In the special case when a = 0, corresponding to a short circuit terminal,
(S.) (f)
s becomes the same as s defined previously. For a - oo, corresponding to
n n
an open circuit terminal, s are identical to the zeros of the input impedance
() n
function. Like s, there are two layers of these zeros. Figure 7 shows a plot of
s for a thin biconical antenna with 0 =0.001 for different values of a. It is
n o
interesting to observe that these contours do not cross each other in the complex s
domain. For s, corresponding to the curve at the bottom right corner, the curve
n
splits into two branches when a is larger than 1.145. This is analogous to the condition of critical damping for a series R-L-C circuit.
As far as the time domain solution is concerned, the response, of course,
-i
depends on our knowledge of h(s) and the given functional form of E (s). If we assume
E (s) * h(s) to be equal to 1/s, a unit-step voltage excitation, then the response can be
found in a similar manner as the transmitting case. From the distribution of s,
it seems reasonable to predict that the transient response would be weak when the
load impedance is approximately matched to the characteristic impedance of the antenna. This corresponds to the value of 'a' in the neighborhood of 1.02.
8. Conclusion
In this report we have discussed the transient input current of a thin biconical
antenna based on three distinct methods: (i) the method of F-series, (ii) the Volterra
integral equation method and (iii) the method of singularity expansion (SEM). In the
case of the last method, there are two sets of poles one can use to formulate the
11
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<P><PB REF="00000013.tif" SEQ="00000013" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="884" N="12">
problem. Both formulations yield comparable results. There is no simple criterion
for us to judge which set of poles are physically more meaningful or mathematically
more convenient. Convergence rate of the series involved may be used as a criterion.
This property still needs further investigation.
From the point of view of Prony's method of synthesis, our work shows that the
representation of a function by an infinite set of exponential functions is certainly not
unique. The appearance of a Gibbs' phenomena for a non-periodic function also deserves investigation. We hope that these topics will be considered in our future work
dealing with the transient response of antennas.
9. Acknowledgement
The support of this work by Dikewood Industries is very much appreciated.
The author wishes to thank Dr. Calvin Lee of Dikewood and Dr. Carl Baum of Kirtland Weapons System Laboratory for the encouragement which he received from
them. Dr. David Giri of Kirtland Weapons System Laboratory has contributed significantly to the evaluation and the positive identification of the singularities involved
in this problem. The assistance of Mr. Soon K. Cho is gratefully acknowledged.
10. References
1. Felsen, L. B., Editor, Transient Electromagnetic Fields, SpringerVerlag, Berlin, (1976).
2. Cauer, W., Synthesis of Linear Communication Networks, Vols. I and
II, McGraw-Hill, (1958).
3. Guillemin, E.A., Synthesis of Passive Networks, John Wiley, (1957).
4. Tai, C. T., "Transients on Lossless Terminated Lines", A technical
report issued by the Radiation Laboratory, University of Michigan,
(1976).
5. Schelkunoff, S.A., "Theory of Antennas of Arbitrary Size and Shape",
Proc. I.R.E., 29, p. 493, (1941).
6. Tai, C. T., "Theory of Biconical Antennas", J. of Appl. Phys., 19,
p. 1155, (1948); see also, King, R. W. P., Theory of Linear
Antennas, Harvard University Press, Chap. VIII, (1956).
12
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<P><PB REF="00000014.tif" SEQ="00000014" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="888" N="13">
7. Abramowitz, M. and I.A. Stegun, Handbook of Mathematical Functions,
U. S. Government Printing Office, (1964).
8. Singaraju, B.K., D.V. Giri and C.E. Baum, "Further Developments
in the Application of Contour Integration to the Evaluation of the
Zeros of Analytic Functions and Relevant Computer Programs",
Mathematics Notes 42, Air Force Weapons Laboratory, Kirtland
Air Force Base, New Mexico, (1976).
9. Tesche, F. M., "On the Singularity Expansion Method as Applied to
Electromagnetic Scattering from Thin Wires", IEEE Trans. A-P
21, p. 53, (1973).
10. Whittaker, E.T. and G. N. Watson, Modern Analysis, Cambridge
University Press, p. 143, (1943).
13
</P>
<P><PB REF="00000015.tif" SEQ="00000015" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="541" N="">
j
r
I
I
10
Figv rl
s(^l and
n
o for s(O
n
x for s(Q
n
t(l) for 80 = 0.001 '
n 09
i
80
) I
i
I
I
6 o
5 0
I
I
I
-"- x II
I
I
10I
I
I
I
9
x
I
I
I
X 8
I
I
I
I, 7
J6
I
I, 5
I
I
[4
10
It
I In
c
( I
I I
1 i
ii "
I Il
I I
I 8 I
' I; 7o
I 6 I
'I
I 5Q
5 > 'I
I
1 40
I I
I Io
'. 30
3; *I
I
\ *
\20
n-2 x.
\ I
\6o
\
I s
10 x7T
9
8
7
6
4
3
2
II
30 " 3
/ /
/ /
/ /
/
I
II
I
I.-I- - a * - - --— I ----I
Now
-2.
-1.6
-1.2
-0.8
-0.4
0
Re(s)
</P>
<P><PB REF="00000016.tif" SEQ="00000016" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="344" N="">
I 0 Or77
Figure 2
S ) for
In
80 = 0 5730
n =1
I - -
10 0
1
1
1
1
9 o
I
I
I
1
8 0
1
1
1
r
7 0
1
1
1
1
6 d
I
I
I
J, =I
1(0
1
1
1
1
90
1
1
1
1
1
80
1
1
1
1
70
1
1
1
1
60
1
1
1
1
50
1
1
9
8
7
6
'I
4
0~
5 0
4 0
I
I
/
0
1
4;
I
I
1
3
I
II
1
2 0
1
1
I
I
I
II = 1 44
I
I
I
I
I
I
I I -- I
- 2.
1.6
-1i.2
- 0.8
-0.4
Re(s)
</P>
<P><PB REF="00000017.tif" SEQ="00000017" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="448" N="">
nfor 8FO. 57 730
IO I?X 10
9)(g
8
1~x
4)xi
I
/X
/
/
n:
10 xl7r
9
7
6
5
4
3
2
0
3)'
/
/
2I —
len I - - -- -- — I - -
I
-A.
I I A I
-1.8
-1.6
-1.4
-1.2
- I.
-0.8
-0.6
Re(s)
</P>
<P><PB REF="00000018.tif" SEQ="00000018" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="425" N="">
16 ir
0
0
0
G
0
0
0
0
1
1
1
1
1
0
1
1
1
0
1
1
0
1
1
0
1
1
0
1
Figure 4
I + Y(S;&86) = O.10
o —o —o
14 7r
12 ir
lOir3
C'
8 7r
67r
47r
2 ir
I I
o I
I *
i I
0 I
/
j
0
/
/
/
/*
/0 1
/
0.. -I~
I I
I
I
a
I
I
_u
0
Q m JL
-1.6
- 1.4
L 12
- I.
0.8
-0.6
- 0.4
-,0.2
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</P>
<P><PB REF="00000019.tif" SEQ="00000019" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="452" N="">
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</P>
<P><PB REF="00000020.tif" SEQ="00000020" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="485" N="">
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</P>
<P><PB REF="00000021.tif" SEQ="00000021" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="292" N="">
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<P><PB REF="00000022.tif" SEQ="00000022" RES="600dpi" FMT="TIFF6.0" FTR="TPG" CNF="892" N="1">
Transients on Lossless Terminated Transmission Lines*
Chen-To Tai
Radiation Laboratory
The University of Michigan
Ann Arbor, Michigan
Abstract
This work contains a general investigation of the methods
in analyzing the transients on lossless terminated transmission
lines. After reviewing the conventional method we present two
alternative methods one in the form of a Volterra integral equation
and another corresponding to the so-called singularity expansion
method. Based on these methods several specific problems are
treated in detail. For the case of a short-circuited termination
it is shotwn that all the solutions are equivalent although they
appear different in analytical form. For a resistively terminated
line we have observed a Gibbs' phenomenon associated with the
series solution obtained by the singularity expansion method although
the series is by no means an ordinary Fourier series. Application
of these methods to study transients on biconical antennas is
briefly outlined.
Introduction
The excitation and the propagation of transients on transmission lines have been studied by several authors. In particular
we like to mention the work by Levinson [1], Bewley [2], Weber [3],
Kuzmetsov and Stratonovich [4]. While the formulation for lossy
line terminated by an arbitrary load is known, a general solution
is not available because of the difficulty in evaluating some of
the inverse Laplace transforms. When the line is lossless the
situation in quite different. However, no detailed treatment seems
to be available for arbitrary terminations except for the case of
a resistive load. It is therefore desirable to present a general
treatment by which one can solve the problem for an arbitrary
termination in a systematic way. The research conducted here is
*The work reported here is supported by National Science
Foundation under Grant ENG 75-17967 and a grant from the Dikewood
Corporation to the Radiation Laboratory of The University of
Michigan.
</P>
<P><PB REF="00000023.tif" SEQ="00000023" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="866" N="2">
-2 -
partly motivated by our desire to investigate the transient
phenomena on biconical antennas which can be interpreted as a
pair of biconical transmission lines terminated by a distributed
load [5,6]. Before we present the general methods let us review
first the conventional treatment for a pair of lossless lines
terminated by a resistive load.
Conventional Method of Treatinq a Lossless Line Terminated by
a Resistive Load
We consider apair of lossless lines terminated by an impedance
load Z shown in Figure 1. The lines are assumed to be excited by
an unit step voltage at the input end.
For convenience we introduce several normalized variables
defined as follows:
= x/ = normalized distance
T = tc/- = normalized time
where Z = length of the line
c = velocity of propagation on the lossless line being
equal to 1/(L'C') /2
L',C' = inductive and capacitive line constants of the line
s = aZ/c = normalized Laplace tranform variable
a = the ordinary or conventional Laplace transform variable
being equal to jw where I denotes the complex angular
frequency
In terms of these normalized variables we deonte
v(,T) = instantaneous line voltage
i(i,T) = instantaneous line current
V(Es) = Laplace transform of v(U,T)
=<?[v(~,IT)] = fv(S,,)e dr
0
I(U,s) = Laplace transform of i(t,T)
(( =-ST
= [i(,T) ]= i(,T)e dT
0
</P>
<P><PB REF="00000024.tif" SEQ="00000024" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="831" N="3">
-3 -
For a unit step voltage applied at the input end we have
V(O,T) = U(T - 0)
hence
V(0,s) = fU(T - 0)-st =
In terms of these normalized variables and V(0,s) the line voltage
and the line current in the Laplace transform domain can be written
in the form
V(e,s) = e + I(s)e (2 (1)
s[1 + r (s)e ]
_ e-, S - (s)e-(2 - E)s
Z I(' s) = e — (s)e-( (2)
c s[l + F(s)e 2s]
where Zc denotes the characteristic impedance of the line, being
equal to (L'/C') /, and F(s) the voltage reflection coefficient
defined in the s-domain at the output end.
The conventional method of determining v(E,T) or i(E,T) is
to express (1) or (2) in a series using the expansion
1 -2s n
1-2s= E [-F(s)e ] (3)
1 + r(s)e n=0
Substituting (3) into (1) we have
1 Es (2- - -2s n
V(,s) = l[e I + r(s)e (2 ~)s] Z [-((4)
n=0
For a resistive load F(s) is a real constant which will be denoted
by r and its value is given by
r r- 1
r + 1
</P>
<P><PB REF="00000025.tif" SEQ="00000025" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="849" N="4">
-4 -
where
r = R/Z
R = load resistance
Z = characteristic impedance of the line
The inverse Laplace transform of (4) with F(s) = r yields
00 +
n n+l
v(,T) = Z [(-F) U(T - 2n + ) - (-I')n (T - 2n - 2 + i)] (5)
n=0
where U(T-Tn) denotes an unit step function commencing at T = Tn.
The solution can be displayed most conveniently by a characteristic
diagram as shown in Figure 2.
Although (5) is known to be a valid solution by physical
reasoning its derivation is considered to be unsatisfactory from
the mathematical point of view because expansion (3) holds true
only if I '(s)e 1<1, and in executina the inverse Laplace transform
the contour of integration lies in the left-half plane where
-2s
( I(s) -2s could exceed unity. This conventional method of finding
v(~,T), however, has been adopted by many authors including
Levinson [1] and Weber [2].
One way of removing this weak step is to expand the same
function in terms of a finite series with a remainder, i.e.,
N -2s N+1
1 = N [ ^ (s e 2s]n - [-(s)e ] (6)
Z [_-r(s)e ]-s) n -(s)e (6)
1 + O 1 + r(s) (s)e
when substituting (6) into (1) the remainder would yield a term of
the form
N+2
[-r(s) ] -s[2(N + 2) - ](
-2s e
s[l + (s)e ]
Because of the shifting theorem the inverse Laplace transform of
(7) vanishes when T<2(N+2) - i. In other words if one evaluates
</P>
<P><PB REF="00000026.tif" SEQ="00000026" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="848" N="5">
the series (5) up to T<2(N-2) - ) the remaining terms do not
enter the picture. From this point of view (5) is an exact
solution since N can be fixed any value. From now on we will
designate the solution based on (4) as the F - series solution.
In addition to the F - series method there are two alternative methods
to formulate the problem and to find the solution. One is an
integral equation method and another is the so-called singularity
expansion method or the method of residues.
Integral Equation Method
We consider the general case where F(s) is a function of s.
-2s
If (1) is multiplied by 1 + r(s)e the following equation results
V(,s) [ + (s)eV(s) [e- + (s)(2-)s] (8)
By taking the inverse Laplace transform of (8) we obtain
V(ST) =- 1l[-r(s)e V(E,s)] + v (F,T) (9)
0
where
V, (,T) = -1 [e-s + ](s)e-(2 - s]
= vof(S,T) + Vob(,T) (10)
with
Vof(C,T) =J [-e ] = U(T - C) (11)
Vob(,T) = -l(s)-( -.)s (12)
vof(~,T) represents the initial forward wave propagating on the
line and vob(S,r) represents the first reflected wave or backward
wave from the termination. For a given F(s) we assume (12) can be
evaluated. Thus v o(,T) is a known function. On account of the
0
</P>
<P><PB REF="00000027.tif" SEQ="00000027" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="807" N="6">
-6 -
convolution theorem in the theory of Laplace transform (9) can
be written in the form
V(,T) = J k (T - T')V(,T)dT' + V (S,T)
13)
where
k(r) = c-1 [- (s )e- 2S]
(14)
Equation (13) with v(S,T) as the unknown function corresponds to
the Volterra integral equation of the second kind. Its solution
is given by Picards' series [7], namely,
v(,T) =
00
Z vn (IT)
n= n
(15)
where
v (gIT) = |T k(T - T')vnl(C,T')dT'
0
(16)
n = 1,2,...
In the case r(s) is a real constant, previously denoted by r, we
obtain, from (12),
Vob(,T) = TU(T - 2 + 5)
ob
hence
Vo(,T) = U(T - 5) + FU(T - 2 + E)
(17)
and from (14) we have
k(T) = -r (T - 2)
where 6 (T - 2) denotes the delta function defined at T = 2.
Substituting (17) and (18) into (16) we obtain
2
Vl(S,T) = -rU(T - 2 - ) - rU(T - 4 + 5)
2 3
v2(E,T) = r U(T - 4 + 5) + ' U(T - 6 + 5)
(18)
</P>
<P><PB REF="00000028.tif" SEQ="00000028" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="830" N="7">
-7 -
The solution represented by (15) is obviously the same as (5).
To illustrate the application of the integral equation method
to more complicated terminations let us consider the case where
the terminal load consists of a series R-L lumped circuit then
- z(s) - 1
r(s) = z(s) + 1
where
z(s) = Z[R + jwL) = -[R + s( )L]
Zc c
= r + as
CL L
r = R/Z, - -
R/Zc ' = Z c L'Z
c
L' = inductive line constant
The coefficient a is a measure of the load inductance in terms of
the total line inductance. The reflection coefficient F(s) can
now be written in the form
s - s
r(s) = (19)
where
s _ (r- 1) - (r + 1)
thus
F(s) = 1( - + 1 - p (20)
s s s- s1 s s- s1
where
o r - 1
p s r + 1
</P>
<P><PB REF="00000029.tif" SEQ="00000029" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="822" N="8">
-8 -
Using (12) and (14) one finds
s1(T - 2 + i)
Vob(,T) = U(T - 2 + e)[p + (1 - p)e ] (21)
s (T - 2)] 2)
k(T) = - (T - 2) - U(T - 2)[(1 - )sle1 (22)
Knowing Vob(~,T) and k(T) we can find v1(~,T), using (16). The
result is given below:
s (T - 2 -)
v1(,T) = - U(T - 2 - ) [p + (1 - p)e ]
2 2 2 S(T - 4 + )
+ U(T - 4 + )ip{ + [1 - P + (1 - P) Sl(T - 4 + )e ]} (23)
The successive terms of vn(~,T) for n > 2 can be found accordingly.
The same result, of course, can also be found by the r-series
method. If we follow this route then it is necessary to expand
the function [F(s)]n/ in partial fractions;after that the inverse
Laplace transform of (4) can be evaluated.
One unique feature of the integral equation method should be
pointed out. It concerns the relationship between Vob(~,T) and
k(T), the kernel of the integral equation. Since
'Vb = (1 r(s)e-(2 - 5)s
Vob(~ ') s (s)e
and
K(s) = [k(T)] = -(s)e2s
so
K(s) = -sVb(Os)
thus
Vob (O,T)
k(T) = -.. (24)
where we interpret the derivative in the generalized sense that for
a discontinuous function
- U(T - 0) = 6(T - 0) (25)
Equation (24) suggests that once the characteristics of V (
Equation (24) suggests that once the characteristics of Vob(tT),
the first reflected wave, is known one can determine the kernel of
</P>
<P><PB REF="00000030.tif" SEQ="00000030" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="865" N="9">
-9 -
the integral equation, subsequently the complete solution. As
an example, from (21) one finds
Vo b(0,T) s1(T - 2)
- -6(T - 2) - U(T - 2) [(1 - p)sle ]
D1
which is the same as k(T) given by (22). This completes our
discussion of the integral equation method. Our next topic deals
with the singularity expansion method or the residue method.
The Singularity Expansion Method Supplied to a Short-Circuit Line
The terminology of this method was suggested by Baum [8]
in his work dealing with the scattering of electromagnetic waves by
objects. This method when applied to transmission lines was treated
previously by Weber [9] for a short-circuit terminiation. For our
purpose we shall present our treatment in a different manner
emphasizing the casuality condition which is inherently
in our solution. We consider just (1) with r(s) = -1, corresponding
to a short-circuit termination, then
-is - (2 - E)s
V(5,s) = e -e- (26)
-2s
s(l - e )
Because of the retarded factors e- s and e(2 )s we will take
advantage of the shifting theorem and treat V(~,s) as consisting
of two terms with
e-s
V (,s) -2s (27)
1 s( - e )
and
V-e-(2 - E)s
V2 (s) = (28)
s(l - e )
Observing that both (27) and (28) have a double pole at s = 0 and
</P>
<P><PB REF="00000031.tif" SEQ="00000031" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="776" N="10">
-10 -
simple poles at s = jnn, n = +1, 2,... By means of MittagLeffler theorem [10] we can derive the following expansion
1 1
( — 1 -- - J_)
-2s 2s
1 - e
= [1i +
2
+o00
1 0
( - + 1 ) ]
s - s s
n=-+l n n
where
hence
S = jnTT,
n
1
s(l - e )
1 1
2 2
s
+ 00
1 n- 1
s s (s - s )
n=-+l n n
As a result of the shifting theorem we find
vl(E,T) =S~l[Vl(,s)]
00
1 + 1
= U(T - ) [1 + 1(T - ) + Z - sin n7r(T - ) ]
2 n=l n17
(29)
similarly,
00
v2(5,T) = -U(T - 2 + S) [ + 1(T - 2 + 2 ) + Z 1 sin n7T(T - 2 + ) ] (3(
n=l
Except for the negative sign v2(,T) is merely a delayed reproduction of
vl (,T). The sum of (29) and (30) gives v(U,T). Now the Fourier
series representation of the periodic function
f(T) = 1 - T, 2 > T > 0
is given by
00
2
f(T) = - sin nr T,
n=l
(31)
</P>
<P><PB REF="00000032.tif" SEQ="00000032" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="865" N="11">
-11 -
the two functions v1(~,T) and v2(r,T) can therefore be plotted
out easily as shown in Figure 3. The sum of the two functions
produces the repeated square wave shown in Figure 4. The last
figure checks with the characteristic diagram shown in Figure 2
if we let r = -1.
As we mentioned before the same problem was previously treated
by Weber who obtained the solution by evaluating the inverse Laplace
transform for the function V(i,s) without separating into V1 and
V2. His result is given by
V((T) - U(T -0) [ E 2 sin no- + sin nr(T —)
n=l n1 n=l n
00
+ ~ sin n7 (T+) ] (32)
n=l
In appearance his formula is quite different from ours, but
if we make use of (31) it can be shown that the result is
identical to ours in spite of the fact that his formula involves
a unit step function starting at T=0 while ours involves
U(T-L) and U(r-2+E). These two different approaches have important bearing to more complicated cases. In our treatment
the casuality condition is automatically met, namely, U(i,T)=
0 for T<. If Weber's approach is followed enough terms in the
series must be included in the numerical calculation to ensure
the vanishing of v(E,Tr) for T<E. We have experienced this behavior for the case of a resistive loading with r different
from -1.
Singularity Expansion Method Applied to a Resistive Load
For a resistive termination with r (s) = F, a constant,
e-ES+re- (2-0 s
V(E,s) = e..+e2- (33)
s(l+Fe )
For definiteness we assume F, being equal to (r-l)/(r+l), to
be a negative constant corresponding to r<l. The poles of (33)
</P>
<P><PB REF="00000033.tif" SEQ="00000033" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="613" N="12">
-12 —
then are given by s =0 and s=S = s -LnjF1 + j nfn, with n=
~1, ~2,'p Again, by means of Mittag-Lef fler theorem one f inds
110 _ _ _ _ _ l3 -sl+e- 2s ( + F)s n=_ sn (s-s) (34
where
- -LnFJr + jn7T
By considering
V(~,fs) =V (~,rs) + IV(2 s (35)
with
Vi(E,s) = - 2s (36)
s(l+J'e
and -(-~
=V-2s (37)
2 ~s(1+Fe 2
the inverse Laplace transform of (36) and (37) yields
v(ET) = (-) 1+ 1 5n (Tv
+' Z2-00 n
+U(T-2+E) [ ___r 5n (-+
1+F +-00 25
-2s..(8
Since Fe n —1, (38) can be simplified to
V (T,) =U(T-~) 1 + 1 en(~
1+F ~2s
co 5 (T+E)
+ t.J( T-2+~)[ F + E e n -o*.(39)
1+F -00
</P>
<P><PB REF="00000034.tif" SEQ="00000034" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="884" N="13">
-13 -
For the case F= 0, (39) reduces to v(S,T) = U(T-~) which is certainly true. Eq. (38) or (39) does not apply to F = -1, a
special case which was treated previously. The solution represented by (39) has been computed and is shown in Figs. 5 and 6
1 1
for r = -2 and E =. In Fig. 5, twenty-five terms are used
to compute the curve. In Fig. 6, fifty terms are used. The result as a whole agrees with the one obtained by the F-series
method or the integral equation method except at the points of
discontinuity where the curves exhibit the Gibbs' phenomenon
commonly encountered in Fourier-series analysis. To the knowledge of this author Gibbs' phenomenon associated with nonperiodic discontinous functions have not been examined in the
past. The subject matter is related to Prony's method of representing a discontinuous function by exponential functions
with complex damping constants. The problem is currently under
investigation.
Application to Transient Analysis of Biconical Antennas
The methods which we have discussed apply equal well to
the transient analysis of small-angle biconical antennas. The
only difference is that the reflection coefficient F(s) is a
transcendental function of s. In fact it is convenient to
write F(s) in the form of [1-y(s) /[l+y(s)] where y(s) is
expressible in terms of exponential integral functions of s.
If the integral equation method is used to investigate the
input current then we have to find the poles of F(s) or the
zeros of l+y(s) = 0 so that i o(0,T) and k(T) can be determined. The Picard series can then be evaluated either analytically or simply by numerical integration. If the singular
-2s
expansion method is used,then the roots of l+r(s)e =0 have
to be determined first in order to build the residue series
for I(0,s). These methods apply equally well to biconical
antennas operating in the receiving mode. This work will be
reported elsewhere in a separate article.
</P>
<P><PB REF="00000035.tif" SEQ="00000035" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="893" N="14">
-14 -
Conclusion
In this paper we have investigated various methods of
analyzing transients on lossless transmission lines with
arbitrary terminations. It is shown that there are three
distinct methods to formulate and to solve the general problems, namely the r-series method, the Volterra integral
equation method and the singularity expansion method, the integral equation method is potentially more appealing because
from the information of the first reflected wave it is possible to construct the kernel of the integral equation and
thereupon to find the complete solution based on quadrature.
In principle all these methods are applicable to study the
transient on thin-angle biconical autennas.
The author wishes to thank Albert Heins and C. Bruce
Sharpe for many valuable discussions. The assistance of
Soon K. Cho in carrying out the numerical calculations and
in checking most of the formulas is appreciated.
</P>
<P><PB REF="00000036.tif" SEQ="00000036" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="853" N="">
REFE RENCE S
1) Norman Levinson, "The Fourier Transorm Solution of
Ordinary and Partial Differental Equations," Jour of
Mathematics and Physics. Vol. 14, pp. 195-227, 1935.
2) L. V. Bewley, "Travelinq Waves on Transmission Systems,"
second edition, John Wiley & Sons, Inc., New York 1951.
3) ErnestWeber, "Linear Transient Analysis," Vol. II, John
Wiley & Sons, Inc., New York 1956.
4) P. I. Kuznetsov and R. L. Stratonovich, "The Propagation
of Electromagnetic Waves in Multiconductor Transmission
Lines," The MacMillan Company, New York, 1964.
5) S. A. Schelkunoff, "Theory of Antennas of Arbitrary Size
and Shape," Proc. I.R.E., 29, 493, 1941.
6) C. T. Tai, "On The Theory of Biconical Antennas," J. Appl.
Phys., 19, 1155, 1948.
7) F. G. Tricomi, "Integral Equations," Interscience Publishers, Inc., New York, 1955.
8) Carl E. Baum, "The Singular Expansion Method," Chap. 3 of
the book, "Transient Electromagnetic Fields," edited by
L. B. Felsen, Springer-Verlag, Berlin, 1976.
9) Ernest Weber, loc. cit., Sec 6.9.
10) E. T. Whittaker and G. N. Watson, "Modern Analysis,"
Cambridge University Press, p. 134, 1943.
</P>
<P><PB REF="00000037.tif" SEQ="00000037" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="824" N="">
-It
L', C'
Z
x =0 x =
Figure 1: A terminated line excited at
the input end by a unit. step
voltage.
1 1
0 2 4 6 8
Figure 2: Characteristic diagram displaying
the solutioun or v (4c, 7) for a resistively terminated line..
</P>
<P><PB REF="00000038.tif" SEQ="00000038" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="545" N="">
vI(t, 7)
-T
v (F~ T)
Fig~ure 3: v1 (~, 7) and ~ (Q, 7) for a
short-ci-rculit terminLation..1111 -0 ~ -
- I - I I I
+ 4- 4+ 6- 6+
Figure 4: Sum of v (~, T-) andyv (S.? T).
</P>
</DIV1>
</BODY>
</TEXT>