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AlllQH 514205

NBSIR 79-1606

A VARIATIONAL EXPRESSION FOR THE SCATTERING MATRIX

OF A COAXIAL LINE STEP DISCONTINUITY AND ITS

APPLICATION TO AN OVER MOOED COAXIAL TEM CELL

1. Sreenivasiah

David C. Chang

University of Colorado

Sponsored by:

Electromagnetic Fields Division

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303

May 1979

NBSIR 79-1606

A VARIATIONAL EXPRESSION FOR THE SCATTERING MATRIX

OF A COAXIAL LINE STEP DISCONTINUITY AND ITS

APPLICATION TO AN OVER MOOED COAXIAL TEM CELL

I. Sreenivasiah

David C. Chang

University of Colorado

Sponsored by;

Electromagnetic Fields Division

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303

May 1979

U.S. DEPARTMENT OF COMMERCE, Juanita M. Kreps, Secretary

Sidney Harman, Under Secretary

Jordan J. Baruch, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director
FOREWORD

This report describes a theoretical analysis developed by staff of

the University of Colorado at Boulder under a contract sponsored by the

National Bureau of Standards (NBS). Professor David C. Chang heads the

University team. Mark T. Ma of the NBS Electromagnetic Fields Division

serves as the current technical contract monitor. The period covered by

this report extends from August 1978 to January 1979.

The work described in this report represents a further aspect of

theoretical analyses of Transverse Electromagnetic (TEM) Transmission line

cells developed at NBS. The purpose of this effort is to evaluate the use of TEM cells for (i) measuring the total RF radiated power by a device

inserted into the cell for test, or (ii) performing necessary

susceptibility tests on a small electronic device.

Previous results indicate that the useful frequency range of such

TEM cells is limited by the higher-order modes that may be excited at

the junction and propagate in the cell. A knowledge of the relative

strengths of these higher-order modes might enable designers to extend

the usable frequency range and/or to minimize the effect of these

modes on the transmission characteristics of the cell.

In this report, the transition region of the cell is considered to

consist of coaxial transmission lines with different cross-sectional areas.

The coupling between the different modes is analyzed by a scattering-matrix

formulation. An equivalent circuit is developed for the general case where

the number of propagating modes in each transmission line is finite but

arbitrary. A network representation for the entire cell including the

transition region is also given.

ii
TABLE OF CONTENTS

Page

I. INTRODUCTION 1

II. SCATTERING MATRIX OF A STEP DISCONTINUITY IN A

COAXIAL TRANSMISSION LINE 5

III. EQUIVALENT CIRCUIT FOR THE GENERAL GASE 12

IV. ANALYSIS OF AN OVER-MODED TEM CELL 16

V. NUMERICAL RESULTS 18

VI. CONCLUSIONS 24

VII. ACKNOWLEDGMENTS 25

REFERENCES 25

APPENDIX I 27

APPENDIX II 29

APPENDIX III 33

APPENDIX IV 35

iii
LIST OF ILLUSTRATIONS

Page

Figure 1. Cross-sectional views of a typical TEM cell 2

Figure 2. Cross-sectional view of a coaxial TEM cell 3

Figure 3. Equivalent network of a coaxial TEM cell with a small source and loading material in place 4

Figure 4. (a) Coaxial transmission line with a step discontinuity

at z = 0 6

(b) Representation of the discontinuity when only one higher order mode is propagating 6

Figure 5. Equivalent circuit of the discontinuity when only one higher order mode is propagating 13

Figure 6. Equivalent circuit of the discontinuity when arbitrary

number of higher order modes are propagating 15

Figure 7. (a) C^ ^vs frequency characteristics of a step discontinuity

in a coaxial line 19

(b) n^^ vs frequency characteristics of a step discontinuity

in a coaxial line 20

Figure 8. Transmission coefficient vs frequency characteristics of an empty coaxial TEM cell 21

Figure 9. Transmission coefficient vs frequency characteristics of

a coaxial TEM cell with internal resistive loading 22

Figure 10. Transmission characteristics of a coaxial TEM cell

with an Internal source 23

IV
Previous publications under the same effort include:

J. C. Tippet and D. C. Chang, Radiation characteristics of dipole sources

located inside a rectangular coaxial transmission line, NBSIR 75-829,

January 1976.

J. C. Tippett, D. C. Chang and M. L. Crawford, An analytical and

experimental determination of the cut-off frequencies of

higher-order TE modes in a TEM cell, NBSIR 76-841, June 1976.

J. C. Tippet and D. C. Chang, Higher order modes in rectangular coaxial

line with infinitely thin inner conductor, NBSIR 78-873, March 1978.

V
A VARIATIONAL EXPRESSION FOR THE SCATTERING MATRIX

OF A COAXIAL LINE STEP DISCONTINUITY AND ITS

APPLICATION TO AN OVER MODED COAXIAL TEM CELL

I. Sreenivasiah

and

David C. Chang

A step discontinuity in a coaxial transmission line, where one of the

sections is large enough to support both TEM and TM^^ modes, may be modeled as a

3-port junction. A variational expression for the (3x3) scattering matrix of

such a junction is obtained in simple closed form. The scattering matrix, so

obtained, is used to analyze the transmission characteristics of a coaxial TEM

cell beyond the cutoff point of the TM^^mode. Finally, an equivalent circuit,

along with the expressions for the circuit parameters, is given for the general

case where the number of propagating modes in each section is finite but

arbitrary

.

Key words: Coaxial transmission line; cut-off frequency; equivalent circuit;

modal analysis; scattering matrix; step discontinuity; TEM-cell; variational

formulation.

VI
I . INTRODUCTION

Recently there has been considerable interest in developing methods for

measuring the electromagnetic emissions by and/or penetration into electronic

equipment [I]. Among others is a TEM cell consisting of a section of rectangular

coaxial transmission line connected at both ends to 50 coaxial lines through

tapers as shown in Fig. I. The details of the design and evaluation of such cells

are given elsewhere [2] - [5]. -The useful frequency range of such a cell is

limited by the higher order modes that may be excited at the junctions and

propagate inside the cell. A knowledge of the strength and the effect of these

modes on the transmission characteristics of the cell might enable one to find

ways to minimize such effects; for example, by placing absorbing material at

selective places, or to calibrate the cell with due consideration to the

propagating higher order modes. Since a taper of the type shown in Fig. I is

difficult to analyze it was felt that a qualitative understanding may be gained

by analyzing a simpler coaxial TEM cell of the type shown in Fig. 2. When the

frequency is high enough that the first higher order mode TM^^^ may propagate inside

the cell, one can analyze the transmission characteristics of the cell by the

equivalent transmission line network shown in Fig. 3 where the TEM cell is replaced

by two transmission lines corresponding to the two propagating modes. These lines

are coupled to the output ports through 3-port junctions which correspond to the

step discontinuities in Fig. 2. It is assumed that the dimensions of the output

lines are such that only the TEM mode can propagate. In Fig. 3 two voltage sources

are indicated to account for the possibility of exciting the cell with a small

source within the cell, which could be the radiating test object itself. The load

impedances Z and Z could account for selective placement of a sheet of absorbing

ij ^ Lj o material to modify or suppress the resonances within the cell.

In order to analyze such a circuit, one obviously needs to develop first the

scattering matrix representing the coupling between different modes in the transition
Figure 1. Cross-sectional views of a typical TEM cell

2
Figure 2. Cross-sectional view of a coaxial TEM cell

3
A|-

Figure 3

3-PORT

JUNCTION

Equivalent network of a coaxial TEM cell with a small source and loading material in place

4
region. In this report, a variational formulation of the scattering matrix elements

is derived and later generalized to a finite but arbitrary number of propagating

modes in each coaxial section. Numerical results for the transmission of TEM mode

to and from a coaxial TEM cell capable of propagating TEM as well as TMq^ mode are

then given.

II. SCATTERING MATRIX OF A STEP

DISCONTINUITY IN A COAXIAL TRANSMISSION LINE

Consider an infinitely long coaxial transmission line with a step discontinuity

in the plane z = 0 as shown in Fig. 4 (a) . The radii of the inner and the outer

conductors of the line are given by and for z < 0 and and for z > 0.

The dimensions of the line and the frequency are assumed to be such that only the

TEM mode can propagate in the left-hand side section A (z - 0) and two modes, namely

the TEM and the can propagate in the right-hand side B (z - 0) . In such a

case the discontinuity may be represented by a 3-port junction, with a scattering

matrix S, as shown in Fig. 4 (b)

.

A. Expression for the scattered magnetic field

Let the incident and the reflected magnetic fields and . , i = 1,2,3,

(pi (pi as shown in Fig. 4 (a), be given by

^(Jl^

Z^(r^ ,r^,p)e*-'^^^^o,A^

'o" 1’ 3^ (2. la)

^(|)2 +h^ H2^?q) Z^(r

2 ,r^;p)e o,j (2.1b)

Zpr2,r^;p)e("5iq,B^ (2.1c)

where = /y/e is the characteristic impedance of the free space; Zp(a,b;p) is

the orthonormal function for the magnetic field of TM mode, p = 0 being the TEM

op

5
(b)

02

b2

b3

Figure 4. (a) Coaxial transmission line with a step discontinuity at z = 0

(b) Representation of the discontinuity when only one higher order

mode is propagating

6
mode in a coaxial line, with inner and outer conductors of radii a and b respectively

^ and3p g are the propagation constants of the mode in regions A and B

respectively, and a time dependence of exp(imt) is assumed and suppressed. We note

that the incident waves at ports 2 and 3 are incident from the right to yield a

propagation factor of exp( + i3^ gZ) and exp( + i3^ that of port 1.

The modal functions Z are given by

Z_(a,b;p) = ^

1/ (p An(b/a)l

ApZp(a,b;p)

Z (a,b;p) = J (A p) + B N (A p)

p

V » > K ^ 1 ^ pM ^ p 1 ^ p*^

A = /2 {b^Z^(A b)-a^Z^(A a)}"^^^

p P P P P

P

= 0

P

^ 1

B = -J (A a)/N (A a)

p o p o p

(2.2a)

(2.2b)

(2.2c)

(2. 2d)

where J and N represent n^^ order Bessel functions of the first and the second

n n

^

kind and A^ are the roots of the equation

J (A a) N (A b) - J (A b) N (A a) = 0 (2.2e)

o p o p o p o p

The propagation constants 3p and

3^ g are given by

p ,A = - B

3 = m/c

/o2 ,2, ,,1/2

(B -XpCrj, r, ))

2 4

•/.2, , ^2 .1/2 •i{Ap(rj, r, )-6 }

2 4

; p

= 0

; p

> 1 ; 3 > A,

; p : 1 ; Ap >

(2.3)

"k k We have chosen a. and b. such that a. a. and b.b. give the incident and the reflected

1 1 11 11

powers respectively, in the i^^ port. Then the scattering matrix S relates the

coefficients a^ and b^ through

b. = E S. a.

" j=l ^ i = 1,2,3 (2.4)

7
We let H and H,„ be the total magnetic fields in regions A and B respectively

(J) A (p D

S S and define the scattered fields H . and as

(})A cp D

, z < 0 (2.5a)

(2.5b)

s s Defining the transverse electric fields E . , E , E , , and E ^ in a similar

pA pB pA pB

s s s s manner we can obtain (Appendix I) H (p,z) and H „(p,z) in terms of E . and E as

(j)A (j)B "pA p B V. B

= * 3,4

1,2 P U g

expC+iBp,A O

B

( 2 . 6 )

B. Scattering matrix of a 3-port junction

In view of equations (2.1a-c) we identify the coefficients b^,b

2

, and b^

from (2.6) to be

1 1,2 1/2

z,

' - 1,2 r 1,2

(2.7a)

fT,

S^1,B

(2.7b)

where

^1

"

^2 " ('?Q/2TT)£n(r^/r2)

(2.8a)

(2.8b)

Further, by noting that

^pa'-P’^^^^pa'^P’^^ ^ Ep^(p,z) ; z < 0

and

(2.9a)

(2.9b)

8
we may write

rr 3,4rr.

1/2

( 2 . 10 ) 1,2 2

which, upon the use of the boundary condition

E

y^(p ,0) = E g(p ,0) = 0 for r^ < p

< and r^ < P < r^ (2.11)

yields

{Si2ai+(1+S22)a2+S23a^ } n^ [ (1+S^^)a^+S^2^2'"^13^3^

where

"o

= ‘"f "2’

1/2

( 2 . 12 )

(2.13a)

(2.13b)

for arbitrary values of a^ , a^, and a^.

In obtaining (2.13) use has been made of (2.4) and the symmetry property of

the scattering matrix of a reciprocal junction

S = S (2.14)

where S is the transpose of S. Now, setting a^ = a^ =0; a^ = a^ = 0; a^ = a

2 = 0

consecutively enables us to obtain the following three relations among the coefficients

of S:

(2.15a)

(2.15b)

(2.15c)

Further, for a lossless junction the scattering matrix is unitary,

S“^ = S* (2.16)

Combining (2.16) with (2.14) we can write

SS* = I (2.17)

where I is an identity matrix. The above identity provides the following relations

1 + s,. = S, „/n

11 12 .

1 + = n S, „ 22 0 12

= n S, _ 23 0 13

9
that are independent of the equations (2.15a-c).

* * s , . + S .,3 ^ + 35

11 11 12 12 13 13

* * * s + 5 .,5., + 5 .,5

11 13 12 23 13 33

(2.18a)

(2.18b)

To determine S uniquely we therefore only need two additional relations as derived

in Appendix II;

CO

^ {2n_^-(lV)Sjp/Sj2 = Bln(r3/rp{

j ,0) Zy ,r3;p )pdp ) hSp

2 rT

+ P=1

^ E (p,0)Z (r r ;p)pdp)^/3^ .}/(

P p ^ 4 p, b

rr Ep(p,0)dp)^ (2.19a)

{(1-S33)(1- Sir%Si2)-(l-%)Si3S^3)/((l.S^^)(l-S33 ^^^

13

^ 13

^

= 3i!.n(r3/rp{ ^|^ ( |

^ E^ (p ,0) (r^ , r3;p )pdp ) ^73^ ^

p=2' E^ (p , 0 )Z_(r„,r .;p)pdp) /3^ A / (. J E^(p, 0 )dp) p" 2’ 4 p,B

(2.19b)

where E (p.O) and E (p,0) are the total transverse electric fields in the plane

p P

z = 0 corresponding to the cases i) a^ = a^ = 1 , a

3

= 0 and ii) a^ = ( 1

- 333 ) 75

^ 3

,

a^ = 0, a

3

= 1 respectively. The above relations (2.19a,b) are exact , although

they involve the unknown fields £^(0,0) and Ep(p,0). However, as shown in Appendix

III, these relations are variational with respect to the electric field distribution,

which means that the right-hand side expressions are stationary for small variations

in the electric field distribution from its correct value. It therefore follows

that a first-order approximation to the electric field distribution will yield a

second-order approximation to the desired quantities on the left-hand sides of these

relations. Further, the expressions on the right-hand sides are homogeneous in

the sense that they do not depend on the amplitude of the electric field distribution

but only on its functional form. In view of this, we can use the following first10
order approximations

E (p,0) = C,/p (2.20a)

P

1

i (p,0) = C./p (2.20b)

P 2 where the constants and need not be known. For small discontinuities, we

note that in case i) the incident fields are completely that of the TEM mode and

in case ii) the amplitude (1-S^2)/S^2 oE Ehe TEM mode In the Incident fields

relative to the incident mode is quite large since is close to -1 and

is quite small compared to unity. Hence the above approximation, where the

electric field distribution is assumed to have a functional form corresponding to

the TEM mode, is indeed quite reasonable. Upon substituting (2.20a and b) in

(2.19a and b) respectively, we obtain

= "l

* i/b.!

where

n^ =

^ ^ (r^ , r^ ;p ) dp

hi r^

+ E^(

p

= 2^ J Zp(r2,r4;p)dp)b(iep^g)}

(2.21a)

(2.21b)

(2.22a)

(2.22b)

To determine all the elements of the scattering matrix we first substitute

the value of obtained from (2.21a) into (2.15a and b) , and obtain and ^22*

The magnitude of then be established through (2.18a). Substituting the

values of S and S in (2.21b) we can write an expression for S in terms of

1 J. X Z which, when substituted in (2.18b), along with (2.15c), determines uniquely.

Thus, all the elements of the scattering matrix can be determined and the final

result is

11
s = —A

where

i2n X

o 0,1

A = 1-i (1+n + n, ) X,

l-i(l-n2.n2)Xg^j

i2n n X

o 1 0,

1

i2n X„ , o 1 0 ,

1

(2.23a)

(2.23b) r 0,1

and the subscripts on X indicate the number of higher order modes that can

propagate in the left- and the right-hand sections of the transmission line.

It can be easily verified that the scattering matrix as given by (2.23a and

b) is exactly the same as that one obtains from the equivalent circuit shown in

Fig. 5, where all the 3-ports are assumed to have the same characteristic impedance

of z^. To verify this we note that the impedance matrix Z of the equivalent

circuit shown in Fig. 5 is given by

Z =

^^0,1 in X_ , 0 0,1

in^X

0 0,1

^^1^0,1 in n X

0 1 0,

1

(2.24)

and the scattering matrix is related to Z through

S = (Z - 1) (Z + I)'^ (2.25)

111. EQUIVALENT CIRCUIT FOR THE GENERAL CASE

In Section 11 we have shown that the equivalent circuit of a step discontinuity,

in a coaxial transmission line, with one mode propagating on one side and two modes

propagating on the other side, is as shown in Fig. 5 where the parameters n^, n^,

Xq

^ are given by (2.13b), (2.22a), and (2.22b) respectively. The reactance X^

^

12
Figure 5. Equivalent circuit of the discontinuity

when only one higher order mode is

propagating

13
and the turns ratios n^ and n^ of the transformers in the equivalent circuit

determine the amount of power coupled into different modes. From the expression

(2.22b) for

^ we notice that the term containing Z^(r

2 ,r^;p) is missing from

the infinite sum. As more numbers of higher order modes start propagating in either

section of the transmission line, the corresponding terms will not appear in X.

This allows us to represent the junction when arbitrary number of modes are

propagating by the general equivalent circuit shown in Fig. 6, where the circuit

papameters are given by

M,N

^3 ^

(5-n ) 2 p=ivl+l

Zp"(^1r

.r„;p)dp)

3

/(i3^p,AJ

+ ( q=N+l^

fr

^Zq(r2,r4;p)dp)2/(i3p^B^ (3.1)

r 1/2 r 1/2

n = (£n — ) / (£n — ) o r^ ^ r^

1/2 r 1/2 r n = (3/3 n) (5-n — ) (£n — ) q

^ q,B^ r^ " r

2

-1

(3.2a)

Zq(r

2 ’r

4 ’P)dP ; q=l,2,...N (3.2b)

1/2 ^3 ^3 ^ mp = (6/Bp_p ' (.n^) (ta ^

) Zp(r^,r^;p)dp; p=l,2,...M (3.2c)

It is straightforward to write down the impedance matrix of the equivalent

circuit shown in Fig. 6, and the corresponding { (M+N+2) X(M+N+2) } scattering matrix

may be obtained through the relation (2.25). The size of the matrix gets bigger

as the frequency is increased, allowing more and more numbers of higher order modes

to propagate. As we decrease the frequency, the transmission lines corresponding

to the modes under cutoff will be replaced by equivalent shunt reactances which

manifest themselves as additional terms in the series (3.1) for X,, and when the

M,N

frequency is low enough that only the TEM mode can propagate, the (2X2) scattering

matrix will be entirely determined by X. _ and n . Further, we note that our

U , (J o expression for the discontinuity capacitance, as given by = 1 /(wXq

^ reduces

14
Figure 6 Equivalent circuit of the discontinuity when

arbitrary number of higher modes are propagating

15
to the special result derived by the others [6,7] for the case when r4*

IV. ANALYSIS OF AN OVER-MODED TEM CELL

As we noted in the introduction to this paper, the characteristics of the

coaxial TEM cell may be analyzed, with the knowledge of the scattering matrix of

the junction, through the equivalent transmission line network shown in Fig. 3,

where all the transmission lines are assumed to have a characteristic impedance

of z^. In Fig. 3,V and V represent the external sources with internal impedances

and Zj^' respectively. In actual practice of test measurements, one or more of

these sources are replaced by detectors or terminations, in which case z and z

'

Li L represent the internal impedances or load impedances, depending on the situation.

We assume that

^L*’ ^L2’ ^L3 normalized impedances with

respect to z^.

Denoting A^ ,

' , and (i = 1,2,3) to be the voltage amplitudes

corresponding to the incident and the scattered power coefficients; viz. A^ = a^/z^,

etc., we derive the following relations in Appendix IV:

-1

^^ 11^22 ^ 12

^

21 ^ [’’22'' ’’12'^'^

'

['’n'’22 - hAnV [DA' - Vfl

where

A = [AJA2A3]

A' = [Aj'Aj'Aj'f

= I - RS

P12 = = -TS

P22 = I - ''s

R =

-i26 L

(4.1a)

(4.1b)

(4.2a)

(4.2b)

(4.3a)

(4.3b)

(4.3c)

(4.4a)

16
R' =

R

’ c

R _e

c 2

-i23 L

' o r

R „e

c 3

-i23, L

' 1 r

T = -i3 L o

1

T T© c3

-i3

V =

V =

v/(l + Z^)

Vj {e + R^2® ’

-i3 L -i3 (2L - L )

V'/(L + Zj_')

''2 ^C2*

-i3 L '

. o s

V, T _e

3 c3

iBjLs'

= (z, - l)/fz + 1) C Li Li

R

' = (z ' - l)/rz ' + 1)

"c2

- ^ 2 z^2 )

^c2

2 ^l2^*'^

^ 2 ^l2 ^

'1 "^1,A

1 = 3 = w/c

with L , L', L , L' and L shown in Fig. 3. r r s s

(4.4b)

(4.4c)

(4.4d)

(4.4e)

(4.5a)

(4.5b)

(4.5c)

(4.5d)

(4.5e)

(4.5£)

17
Once A and A' are known, B and B' may be readily found through the relations

B = [B^ B^ B^]^ = S A (4.6a)

B'= [B^' B^' B^']^ - S A' (4.6b)

and the transmission characteristics of the cell may be completely determined.

V. NUMERICAL RESULTS

Using the result obtained in Section II, we have plotted the discontinuity

capacitance = 1/(50 as a function of frequency in Fig. 7, with the

ratio (r^ - r

2 )/(r^ - r^) as the parameter, choosing the ratios

^4^^2 equal and such that the characteristic impedance is 50fl on both sides

of the coaxial line. The results on the analysis of a coaxial TEM cell with

dimensions r. = 1.0 m, r„ = 0.80 m. r,/r_ = x^/x, = 2.3 are shown in Figures 8

4 3 s- 2 * through 10. Fig. 8 shows the voltage transmission coefficient, |b^,/A^| with z' = l, XX Xi of an empty cell as a function of frequency with the length L as a parameter. The

null at 263 MHz corresponds to the cutoff point of the TM^^^ mode. We observe

deep nulls in the neighborhood of points where the parameter is an integral

multiple of it. For example, the null at about 296.5 MHz for the case where

L = 2.0 m corresponds to B^L = 2ir

,

and B^^L = tt corresponds to a fr-equency of 273

MHz, where we notice a shallow null. The effect of placing loading resistors is

shown in Fig. 9, where the length of the cell is taken to be 2.0 m and resistors

of normalized values ~ 10 z^^ = 1.0 are assumed to be connected at a

distance of 1.5 m from the source end. As anticipated, the overall insertion

loss is increased with a filling of the deep nulls. However, the shallow null

at the first resonance frequency has not been affected.

In Fig. 10 we have shown the results corresponding to a small dipole source

midway between the cell. In this figure the magnitude of the output voltage, I

U

normalized with respect to an arbitrary constant, is plotted against frequency.

18
200

220

240

260

280

300

320

340

360

380

FREQUENCY

MHz

^0. N— ro

f\) cn O O O

Figure 7a. ^ vs frequency characteristics of a step

discontinuity in a coaxial line

19

100
Figure 7b. vs frequency characteristics of a step discontinuity

in a coaxia'' line

20
with

FREQUENCY MHz

Figure 8. Transmission coefficient vs frequency characteristics of

an empty coaxial TEM cell

21
3l/A,

with

L = 2.0 m

NO LOADING

2[_2 = I0.0, Z|_ 2

=I.O; Lr=l.5m

Figure 9. Transmission coefficient vs frequency characteristics of a coaxial TEM cell with internal resistive loading

22
FREQUENCY MHz

Figure 10. Transmission characteristics of a coaxial TEM cell with an internal source

23
One end of the cell is assumed to be terminated with a matched load (z “ Zt) ^rid

1j i. the other end with a matched detector (zj ' = z^)

•

The dotted curve shows the results

for the cell with just the source and the solid curves show the effect of internal

loading. Two resistors with the normalized values Zj ^ - 10.0 and Zr , = 1.0 are

assumed to have been connected at a distance from one end. As expected, the

overall insertion loss and the shape of the frequency response is seen to depend

on the location of the loading point. However, a drastic improvement in the

uniformity of the frequency response seems not possible with such a simple scheme.

We should note that in the actual design of the TEM cell with rectangular crosssections, the transition region is usually tapered smoothly from a standard coaxial

line. Consequently, it should yield smaller intercoupling between modes than the

step discontinuity we studied here. Even so, the deep nulls, particularly the one

when TM is near cutoff, appears to be inevitable as confirmed by the experimental

01

measurement [8]

.

VI. CONCLUSIONS

In this report we have obtained the scattering matrix of a step discontinuity

in a coaxial transmission line where one section of the line is large enough to

support TMq^ mode. Because of their variational nature, the expressions for the

scattering coefficients are expected to be accurate over a wide range of discontinuities. When this is not true, more accurate results can be obtained by substituting an appropriate series for the field distribution in the exact equations

(2.19a and b) . We have extended the concept in a straightforward manner to the

more general case where arbitrary numbers of modes are assumed to be propagating

in either section of the line, and presented simple expressions for the parameters

of the equivalent circuit which can be used to obtain the scattering matrix in any

frequency range.

24
Since the scattering matrix is characteristic of the junction itself, i. e.

,

independent of the external connections to the ports, one can use the same

scattering matrix to analyze TEM cells of different lengths through the method

presented in Section IV without having to solve the entire problem every time

the length is changed. The method can be used to analyze coaxial structures

with a number of step discontinuities spaced at irregular intervals, in any

frequency range of interest, with an increasing amount of complexity as the

frequency is Increased. However, the method by itself is simple and straightforward.

VII. ACKNOWLEDGMENTS

The authors are very grateful to Drs. M. T. Ma and M. Kanda for their

technical discussions and helpful suggestions during the course of this work.

The cooperation and assistance received from M. Crawford is also acknowledged.

The work reported herein would not have been possible without the constant

support and encouragement from C. K. S. Miller and F. X. Rles.

REFERENCES

[1] J.W. Adams, "Electromagnetic Interference measurement program at the

National Bureau of Standards, USNC/URSI, Commissions I-VIII, annual

meeting, p. 33, Oct. 20-23, 1975.

[2] M.L. Crawford, "Generation of standard EM field using TEM transmission

cells," IEEE Trans, on Electromagn. Compat ., Vol. EMC-16, No, 4,

pp. 189-195, Nov. 1974.

[3] J.C. Tippet and D.C, Chang, "Radiation characteristics of dipole sources

located Inside a rectangular, coaxial transmission line," National Bureau

of Standards Internal Report, NBSIR 75-829, January 1976.

25
[4] J.C. Tippet, D.C. Chang and M.L. Crawford, "An analytical and experimental

determination of the cutoff frequencies of higher order TE modes in a TEM

cell,” National Bureau of Standards Internal Report, NBSIR 76-841, June 1976.

[5] J.C. Tippet and D.C. Chang, "Higher order modes in rectangular coaxial line

with infinitely thin inner conductor," National Bureau of Standards Internal

Report, NBSIR 78-873, March 1978.

[6] R.N. Chose, Microwave Circuit Theory and Analysis , New York; McGraw-Hill

Book Company, Inc., 1963.

[7] N. Marcuvitz, Waveguide Handbook, Vol. 10 of M.I.T. Rad. Lab. Series,

New York; McGraw-Hill Book Company, Inc., 1951.

[8] M.L. Crawford, J.L. Workman, and C.L. Thomas, "Expanding the bandwidth

of TEM cells for EMC measurements," IEEE Trans, on Electromagn. Compat .

,

Vol. EMC-20, No. 3, pp. 368-375, Aug. 1978.

26
APPENDIX I

Expression for the scattered magnetic field

The total magnetic field in Region A (z < 0), H satisfies the differential “

(p A

equation

2 { . i 3

. 2 p 6^^^} H^,(P,Z) = 0

op P o Z (Al.la)

and the boundary condition

^

“ for p = rj and Tj (Al. lb)

The scattered magnetic field in the same region, H^^(p,z) is given by

(A1.2)

Since the incident field H satisfies (Al.la and b) , the scattered field should

?!

also satisfy the same equations and the radiation condition

' al " 0 as z ^ CA1.3)

Keeping the above requirements in view, we /can write the scattered fields as

s - H^.(p,z) = C Z (r,,r 'p)e

(j)A^ p=0 p p 1 3

(A1.4)

where the orthonormal functions Z and the constants 6 . have been defined in

P P,A

Section II. In view of the orthonormal property of Z^, we can easily verify that

C = (i3 J

p P,A^

-1 3 dti. (p' ,z=0)

3 z Zp(ri,r3;p ' )p ' dp' (A1.5)

Further noting that

i al »*A(o.n (A1.6)

we can immediately write

27
H(J)a(p.z)=-W£

Q 00 — 1 Ep^Cp'

.

0 )p'dp' E Z (r^,r

2 ;p)Z (r^,r^;p') (3 exp(+i3 (A1.7)

p_0 r P P> Pj

1

s s We can obtain an expression for H in a similar manner by noting that H should

(j)D (pB

satisfy the following radiation condition

as z ^ + °o (A1.8)

28
APPENDIX II

Variational relations for the scattering parameters

In this appendix we derive the relations (2.19a and b) . We use the expansions

obtained in Appendix I for the scattered magnetic fields, and, noting that the total

fields are given by the sum of the scattered and the incident fields, we write

H^^(p , z) = (p , z) -COE (p ' ,0)p 'dp ' E Z (r ,r ;p) _ps p 1 3

1

ZpCr^ ,r

3

;p ’ ) (A2. la)

rr

,

Eppp',0)p'dp' pEgZpCr^.r^lP)

Zp(r2,r^;p’)(6p_P‘hxp(-i(ip^gZ) (AZ.lb)

" Epj(p,z) (A2.2a)

(A2.2b)

where the incident magnetic fields for i=l,2,3 are given by (2.1a-c) and

the incident transverse electric fields are related to through relations

of the form (A1.6). In view of the equations (2.1a-c) we identify the coefficients

bj^,b

2 ,b^ of the propagating reflected fields, in the expressions for and H^g

to be - 1/2 r r b = hi^i " h2*2 " = ‘ N ’ ^ Ep^CP',0)dP’ (A2.3a)

1/2

^2 ^ 12^1 ^ 22^2 ^ ^ 23^3 ^^

2

^ E%(p',0)dp'

pB

(A2.3b)

2-ni ^3 ^13^1 ^23^2 ^33^3 ^ C o,B

1/2 rl - 4

o'^l,B

E^g(p',0)Z^ (r

2 ,r^;p')p'dp' (A2.3c)

29
where

Zl = U^/2T\)ln(T^/T^ ) 2 4 2

(A2.3d)

Using (A2.2a and b) , we can write

rr. rr.

j Ep^(p-, 0 )dp- = J E (p',0)dp' -a^yTz.

p 1 (A2.4a)

rr

,

Epg(p', 0 )dp> E (p',0)dp' -a/’z^ -a3( 5^)o , B

1/2

Z^(r

2 ,r^;p')p'dp’

where

(A2.4b)

Ep^(p, 0 ) Epg(p,0) = Ep(p,0) (A2.4c)

and we used the boundary condition

E (p> 0 ) = 0 for r, < p <r„ and r„ < p

< r.

p

i. ^ 4

(A2.5)

To obtain a variational relation involving 3^25 3^^*^ ^22 we consider the case

when a^ = a

2 = 1 and a^ = 0 and denoting the field quantities corresponding to

this case with on top, we write

= (2„y-W2zjrj,r3;p)e‘ * Sjp(2.?^)-W2

7 f

a ^^^o,A^ f^3 ;s Z^(r^,r^;p)e -03£ Ep^(p-,0)p'dp' pZ^

(A2.6a)

e ’ +coe 4~

E=g(p’,0)p'dp' ! :,p(r2,r^;p) Zp(r2.r^;p')

(Bp_g)’ exp(-iBp_gZ) (A2.6b)

30
Using the orthogonal property of and (A2.5) we can establish the following

relations

:

-1

E^^(p'0)p'dp' ZpCrj,rj;p) Zp(rj,r3;p')

2 B B 2 4 2 4

r^3

- 1 . E (p',0)p'dp' E (3^,) Z (r ,r ;p)Z (r ,r ;p ’) (A2.7)

P P»A p

1 3 p 1 3 B 2 4 2 4

where t (p,0) is the total transverse electric field at the junction for the case

P when a^ = a

2

= 1 and a^ = 0.

Enforcing the continuity of the total tangential magnetic field across the

boundary, that is,

(A2.8)

and making use of (A2.7), we obtain

{(1 ^12 ^ 22 ^^ ^0 ^^

1 ’^

3

’^-

1/-

(2ttC^) (oac);

rr.

00 E (p',0)p'dp' E Z (r ,r ;p)Z (r ,r ;p ') (3 J P

L.p ^ P

-1

+ E Z (r2,r^;p)Z (r2,r^;p')(3„ d)

P

=1 ^ E

-r

p>® Jj

(A2.9)

Multiplying (A2.9) by E^(p,0)pdp and integrating between the limits r

2 and

we obtain

{(1-Sji-Si2) * n^(l-Si2-S22)} E Cp,0)dp

P

rr.

) ^ Z (r ,r ;p)E (p,0)pdp)

p J. o p

rr.

+ ^ ( 3^ J ( q=l

p,B^ Zp(r2,r4;p)Ep (p ,0)pdp) (A2.10)

31
Further, from (A2.3a) and (A2.4a), we can establish the relation

(l*Sji.Sj2) Ep (p,0)dp (A2.11)

Combining the last two equations, and using (2,15a and b) , we can immediately

obtain (2 . 19a)

.

To obtain (2.19b) we repeat a similar procedure with a^=(l-S^^)/S^

2

, 3.^= 0 ,

and a^=l with the modification that we isolate the term containing Z^(r

2 ,r^;p) in

the expression for the total magnetic field in region B (z > 0).

32
APPENDIX III

Proof that the expressions (2.19a and b) are variational.

To verify that the expiession (2.19a) for { 2n^- (I+n^)S^

2

^ variational

in E we need to prove that a small variation in E about its true value causes

P P

2 no change in { 2n -(I+n )S-,„}/St„.

o o 12 12

Let

= { 2n^- (A3. I)

From (2.19a)

r^- E (p ,0)dp ) = E ( P p

= I Ep(p,0)2p(r^,r3;p)pdp)

ft.

+ E ( p

= I

E (p ,0)Z (r r ;p )pdp) /3

P n ^ 4 p,B

(A3. 2)

Small variations in y and E give

1 P

Ayp E (p,0)dp)^ + 2y

P 1 E (p,0)dp

P

. To J

rr. AE (p ,0)dp

P

= 2 E ( p

= I

E (p ,0)Z (r ,r ;p)pdp ) ( P p

I 3

AE (p ,0)Z (r ,r ;p)pdp)

P p

1 o

+ 2 E ( p

= I

(p ,0)Z (r r ;p)pdp) (! AE (p , 0) Z (r ,r ;p )p dp ) p pz4 p pz4

Tz 2

(A3. 3)

From (A2.II) and (2.15a) we can verify that

- 1/2

12

= ( I+n

)(zp Ep (p ,0)dp (A3. 4)

Multipliying (A2.9) by AE (p,0)pdp and integrating between the limits r^ and r^

we obtain, after using (2.15a and b)

,

33
{2n -(1+n )S „}

o o 12

rr __ nr 1/2

AE (p,0)dp = ( °

) (2 tt 5

5,n — ) (ue;) p 1+n r,

^ ^ o 1

rr.

S ( P=1 J

rr.

+ E ( p

= l

E (p',0)Z (r r ;p')p'dp')(

P p 1 o

E (p',0)Z (r r ;p')p’dp')(

P p

2 4

rr.

AE (p,0)Z (r ,r ;p)pdp)/3

P p i j p

AEp(p;0)Zp(r2,r^;p)pdpySp

g

V (A3. 5)

Using (A3 . 1) , (A3 .4) , and (A3. 5), we can immediately verify that the term

2y

1 E (p,0)dp

P AE (p,0)dp

P

2 2

is identical to the expression on the right-hand side of (A3. 3), which implies that

rr. Ayp Ep(p,0)dp) = 0 (A3. 6)

which establishes the variational nature of the expression given in (2.19a).

Proceeding in a similar manner, we can prove that (2.19b) is also variational in

34
APPENDIX IV

Relations characterizing the transmission properties of the TEM cell.

Denoting the incident and the reflected voltage amplitudes at different

points along the cell as indicated in Fig. 3, we recognize the following relations

-i3 (L -L ) -i3 L

~A o^rs „ '^or

A

2 - V

2 e

^ ^ ® 1 (A4.1a)

B2 = -V2 e

^ . A2 e

^ (A4.1b)

where

’o ‘"o,A ^o,B

‘1 = h,A

= (jo/c (A4.2a)

(A4.2b)

— - “ —

82’ 1 - 1/C2Zj^2 ) -1/(2Zl2 ) b

3 3 3 3

A ' ^2 1/C2z,2 ) 1 - 1/(2z^2 ) h

3 3 3 3

*

-i3 L'

0 r

" 1 r "1

A

’ 2 0 ^2’

3

^2'

=

0

+ i3 L'

0 r

" 1

3

A

' ^2

3 3 \

- l)/rz^ + 1)

A^' = V^V(1 + H- B^'fz^’ - D/Cz^' + 1)

(A4.3)

(A4.4)

(A4.5a)

(A4.5b)

Combining (A4.1) - (A4.4) we obtain the following relations

-i3 L i3 (L -L ')

A2' = B2 12(1 , l)e ° + A2T2(l,2)e ° ^ ^

3 3 3 3

-i3 L

’ -iB (2L '-L ') + V2 {T2(1 , l)e

° ^ - T2(l,2)e ?

"" }

3 3 3

I (A4.6)

35
-i3 (L -L ') i 3 L

B

2

' = B

2 T

2

( 2 ,l)e ° T

2

( 2 , 2 )e

°

3 3 3 3 3

where

i3 (2L • - L ') i3 L

' + {T^ ( 2 ,l)e

° ^

- T

2

( 2 , 2 )e

° ^

}

3 3 3

T2 (1,1) = 1 - 1/(2zj^2 ^

3 3

(1,2) = -T^ (2,1) = -1/(2Zj^2 ^

3 3 3

(2,2) - 1 + 1/(2Zl2 )

3 3

(A4.7)

(A4.8a)

(A4.8b)

(A4.8c)

Equations (A4.5a) and (A4.7) may be cast into the following matrix equation:

[A^ A^ A^] = A = R B + T B' + V

B = [Bj Bjf

B’ = [Bj- B^’ B3']

R = Rc 2

-i23 L o r

R _e c3

0

0

-i2Bjq

T = T „e

c 2

-i 3 L

o

T

c3

-i3^L

V = [V^ V

3

]

= V / (I + Z^)

r -13 L -i3 ( 2 L - L ) M ) o s _ o r s ^

2 =^2 ]^ 1 " \2 " 1

(A4.9)

(A4.10a)

(A4.10b)

(A4.11)

(A4.12)

(A. 13a)

(A4. 13b)

(A4.13C)

36
R = (z^ -1)/U. + 1) CA4.14a)

c L L

^2 * "^L2 ^

^c2 ^^L2

^

^^L2 ^

(A4.14b)

(A4.14c)

Substituting for A^ from (A4.7) we can write (A4.6) as

3

-i2e L

' -i3 L ^ A„' = B ’ R _ e ° + B- T _ e

° + V

' 2 2c2 1 -2c2 1 2 3 3 3 3 3 3

(A4.15)

where -i3 L

' v^' = T

^2

e p (A4.16)

3 3 3

Equations (A4.5b) and (A4.15) may now be put into the following matrix equation

,T

[A ' k' A,'] = A’ = R' B' + T B + V 1 2 3

(A4.17)

where

R' =

R

' c

R ^e

c2

-i23 L

' o r

R ,e c3

-i23,L ' 1 r

(A4.18a)

R

’ = (z/ - l)/fz ' + 1) (A4.18b)

C Li Li

V’ = [V^' V^’ V^']^ (A4.19a)

V^' = V V(1 + (A4.19b)

Noting that

B = S A (A4.20a)

B' = S A' (A4.20b)

we may write (A4.9) and (A4.17) as

[1 - R S]A - T S A' = V (A4.21a)

[I - R' S]A' - T S A = V' (A4.21b)

37
whose solution is given by

where

A =

- h2

'’ 2 ll'' [P22" - b2 ^'l

-1 - A’ = rp p - p p 1 [p V - p VI '11 22 21 12-' ^11 21

(A4.22a)

(A4.22b)

11

12

22

1

21

I -

R S

= -f S

R' S

(A4.23a)

(A4.23b)

(A4.23c)

38
NBS-114A (REV. 9-78)

u.s. DEPT. OF COMM. 1. PUBLICATION OR REPORT NO. BIBLIOGRAPHIC DATA

SHEET

4. TITLE AND SUBTITLE

NBSIR 79-1606

A Variational Expression for the Scattering Matrix

of a Coaxial Line Step Discontinuity ancd Its Application

to an Over Moded Coaxial TEM Cell

7. AUTHOR(S)

No. 3. Accfessioo No,

5. Publication Date May 1979

8. Performing Organ. Report No. I. Sreenivasiah and David C. Chang

9. PERFORMING ORGANIZATION NAME AND ADDRESS

Department of Electrical Engineering

University of Colorado

Boulder, Colorado

•t „n I , I , Ml 1 1 1 ..Iffi'l-.i l 11. Contract/Grant No.

12. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS CSfreer, City, state, ZIP) 13 . Type of Report & Period Covered Electromagnetic Fields Division

National Engineering Laboratory

National Bureau of Standards

Boulder. Colorado 80303

15. SUPPLEMENTARY NOTES

I I Document describes a computer program; SF-185, FIPS Software Summary, is attached.

16, ABSTRACT (A 200"‘word or less factual summary of most significant information. If document includes a signi ficant bibliography or

literature survey, mention it here.) A step discontinuity in a coaxial transmission line, where one of the

sections is large enough to support both TEM and TM^^^ modes, may be modeled

as a 3-port junction. A variational expression for the (3x3) scattering

matrix of such a junction is obtained in simple closed form. The scattering

matrix, so obtained, is used to analyze the transmission characteristics of a coaxial TEM cell beyond the cutoff point of the TM^^^ mode. Finally, an equivalent circuit, along with the expressions for the circuit parameters,

is given for the general case where the number of propagating modes in each

section is finite but arbitrary.

17* KEY WORDS Cs/x to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name;

separated by semicolons)

Coaxial transmission line; cut-off frequency; equivalent circuit; modal

analysis; scattering matrix; step discontinuity; TEM-cell; variational formulation.

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