J**'

0F c

?*

a NBS TECHNICAL NOTE 1037

U.S. DEPARTMENT OF COMMERCE/ National Bureau, of Standards

Excitation of a TEM Cell

by a Vertical Electric

Hertzian Dipole
NATIONAL BUREAU OF STANDARDS

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Excitation of a TEM Cell

by a Vertical Electric Hertzian Dipole

Of 9TANDABDS

LIBBAltY

Perry F. Wilson*

David C. Chang'

Mark T. Ma**

'University of Colorado

Boulder, Colorado 80309

**Electromagnetic Fields Division

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303

f^ "" Co,

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued March 1981
NATIONAL BUREAU OF STANDARDS TECHNICAL NOTE 1037

Nat. Bur. Stand. (U.S.), Tech. Note 1037,44 pages (March 1981)

COD'EN: NBTNAE

*U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON; 1981

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402

Price $2.25 (Add 25 percent additional for other than U.S. mailing)
TABLE OF CONTENTS

Page

LIST OF FIGURES AND TABLES iv

FOREWORD v

1

.

I NTRODUCTI ON 1

2. FORMULATION OF THE INTEGRAL EQUATION FOR THE UNKNOWN APERTURE FIELDS 2

2.1 Introduction of the Field Transformations 2 2.2 Integral Equation for the TE Gap Field 4

3. SMALL GAP APPROXIMATION FOR THE GREEN'S FUNCTION 7

4. SOLUTION OF THE APPROXIMATE INTEGRAL EQUATION 9

5. GAP EXCITATION OF THE RCTL MODES 11

5.1 Electric Field Distribution Due to TE and TM Type Modes 11 5.2 TEM Mode Excitation 13 5.3 Characteristic Impedance of the Cell 16

6. NUMERICAL RESULTS 17

7

.

CONCLUDING REMARKS 18

8. REFERENCES 19

APPENDIX A. APPROXIMATE TM GAP FIELD DERIVATION 22 APPENDIX B. EVALUATION OF THE SERIES S 2n 27 APPENDIX C. EVALUATION OF THE UNPERTURBED EXCITATION TERMS AS a+k 30

o APPENDIX D. SUBROUTINE GUIDE FOR COMPUTING THE FIELD DISTRIBUTION 31 APPENDIX E. SUBROUTINE FOR COMPUTING THE CHARACTERISTIC IMPEDANCE 33

-m-
LIST OF FIGURES AND TABLES

Page

Figure 1. Diagram of an NBS TEM cell 34

Figure 2. RCTL cross section 35

Figure 3. Normalized characteristic impedance z /n for g/a=0.1 35

Figure 4. Normalized characteristic impedance zQ /n for g/a=0.2 36

Table l.a. The normalized x-component of the electric field with

bj = b

2

= b, b/a=1.0, and g/a=0.17 (Wilson) 20

Table l.b. The normalized x-component of the electric field with

bj = b

2 = b, b/a=1.0, and g/a=0.17 (Tippet) 20

Table I.e. The percentage difference between the results

given in tables l.a., and l.b 20

Table 2. a. The normalized y-component of the electric field with

bj = b

2

= b, b/a=1.0, and g/a=0.17 (Wilson) 21

Table 2.b. The normalized y-component of the electric field with

bj = b

2

= b, b/a=1.0, and g/a=0.17 (Tippet) 21

Table 2.c. The percentage difference between the results

given in tables 2. a., and 2.b 21

i

IV-
FOREWORD

This report describes theoretical and experimental analyses developed by staff of the

University of Colorado at Boulder in collaboration with the Electromagnetic Fields Division

of the National Bureau of Standards (NBS), under a contract sponsored by NBS. Professor

David C. Chang heads the University team. Dr. Mark T. Ma of NBS serves as the technical

contract monitor. The period covered by this report extends from July 1979 to July 1980.

The work described in this report represents a further aspect of establishing a theoretical basis for the technical analyses of transverse electromagnetic (TEM)

transmission line cells developed at NBS. The general purpose of pursuing theoretical

studies is to evaluate the use of TEM cells for (1) measuring the total rf radiated power by

a device inserted into the cell for test, or (2) performing necessary susceptibility tests

on a small electronic device.

The particular topic addressed herein discusses the manner in which a vertical

electrical Hertzian dipole excites a TEM cell under the assumption that the gap between the

septum and the side wall is small. This premise is consistent with the cell geometries

designed to provide a good impedance match, presently of interest to NBS, and leads to significant simplifications in the mathematical derivation. The formation of the problem

also allows a vertical offset for the septum position so that the size of the test area may

be increased to accommodate larger pieces of test equipment.

Approximate expressions are found for the field distribution inside a TEM cell and for

its characteristic impedance. The latter result consists of a dominant gap dependent

logarithmic term, plus a correction series that accounts for the vertical offset of the

septum. This vertical offset is allowed to be arbitrary, and therefore the results

contained in this report will supplement previous efforts which were restricted to small

offsets. In addition, because of the delta function nature of the source considered here,

the results lend themselves naturally to an analysis of more complex source configurations

such as a practical monopole via Greens function methods. The monopole result may then be

used to model probes inserted into a TEM cell to measure or excite fields.

Previous publications under the same effort include:

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

inside a rectangular coaxial transmission line, NBSIR 75-829 (Jan. 1976).

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

determination of the cut-off frequencies of higher-order TE modes in a TEM cell, NBSIR

76-841 (June 1976).

Tippett, J. C. and Chang, D. C, Higher-order modes in rectangular coaxial line with

infinitely thin inner conductor, NBSIR 78-873 (March 1978).

Sreenivasiah, I. and Chang, D. C, A variational expression for the scattering matrix

of a coaxial line step discontinuity and its application to an over moded coaxial TEM

cell, NBSIR 79-1606 (May 1979).

Tippett, J. C. and Chang, D. C, Dispersion and attenuation characteristics of modes in

a TEM cell with a lossy dielectric slab, NBSIR 79-1615 (Aug. 1979).

Sreenivasiah, I., Chang, D. C, and Ma, M. T., Characterization of electrically small

radiating sources by tests inside a transmission line cell, NBS Tech Note 1017 (Feb.

1980).

EXCITATION OF A TEM CELL BY

A VERTICAL ELECTRIC HERTZIAN DIPOLE

Perry F. Wilson

David C. Chang

and

Mark T. Ma

The excitation of a transverse electromagnetic (TEM) cell

by a vertical electric Hertzian dipole is analyzed where the

gap between the septum and side wall is assumed to be small.

Approximate expressions for the field distribution and

characteristic impedance are derived. These expressions are numerically evaluated for some typical geometries, and good

agreement with previously published results is shown. The

formation also allows a vertical offset for the septum

position, thus offering more flexibility of increasing the

size of the test area to accommodate larger pieces of test equipment.

Key words: Characteristic impedance; field distribution;

Hertzian dipole; integral equation; rectangular coaxial

transmission line; TEM cell.

1. INTRODUCTION

Increased levels of electromagnetic radiation in the environment have made desirable

the ability to predict the effect of low level interference on an electronic device.

Conversely, the device itself may act as a source of low level radiation contributing

additional electromagnetic pollution. Thus the electromagnetic interference (EMI) community

is interested in testing both the susceptibility, and emission properties of electronic

equipment. One approach in use at the National Bureau of Standards (NBS) is the transverse

electromagnetic (TEM) cell which provides an isolated, shielded environment to perform such

testing. Basically, the TEM cell, as shown in figure 1, consists of a section of

rectangular coaxial transmission line (RCTL) coupled at each end to standard 50ft coaxial

line by a tapered section. The results obtained inside a TEM cell may then be related to the free space environment via a result given by Tippet [1, p. 61], and verified

experimentally by Crawford [2].

This report is concerned with analyzing the fields excited in the TEM cell by a vertical electric Hertzian dipole. We view this as the first step toward solving more complex problems, such as the excitation of the TEM cell due to a vertical monopole. The

Hertzian dipole result may be used as the Green's function in formulating a variational

integral for the input impedance of the monopole, an approach employed by Collin [3, pp. 258-261] in treating the excitation of a rectangular waveguide by a coaxial -line probe. The

monopole result may then be used to model probes inserted into the cell to measure or excite

fields.

Our analysis assumes the RCTL to be of infinite extent with perfectly conducting walls,

so in effect we are analyzing the cross section depicted in figure 2.
If we view the cell as two rectangular guides coupled through a pair of apertures, then

in each separate guide we expect the final field to be made of the ordinary rectangular

waveguide modes, plus a perturbation field due to the presence of the gaps. The method of

solution is to formulate integral equations in section 2 for the unknown aperture field.

There are two types of mode presentations; namely, the transverse electric (TE or h-type)

and the transverse magnetic (TM or e-type). Detailed derivations for the more important TE type are presented in section 2.2 with the corresponding derivations for the TM type given

in Appendix A. A simplified approximate expression for the TE integral equation is then

given in section 3 by assuming an electrically small gap. The solution of this approximate

integral equation is presented in section 4. More specifically, we will assume that G

2 £n(G)

is much less than unity, where G is some normalized gap size. The TEM cells presently in use at NBS are consistent with this assumption. In formulating the problem we will allow

the center conductor, or septum, to be vertically offset but centered horizontally. A horizontal offset could also be treated at the expense of additional burdensome mathematics,

but offers no practical advantage. Therefore, it is not treated in this report.

Of special concern is the manner in which the vertical Hertzian dipole excites the TEM mode. We give expressions in section 5 for both the electric field distribution of the TEM mode in the cell, and the transmission line characteristics of an RCTL. Sample numerical

results are given in section 6 with the relevant FORTRAN programs listed in Apprndices D and

E. These results may be compared to those of Tippet [1], obtained via different methods.

Using the Schwarz-Christoffel transformation, he solves for the electric field distribution

in an RCTL with a centrally located septum. He also finds expressions for the

characteristic impedance of an RCTL using an integral equation approach. In both cases his

solutions are expressed in terms of Jacobian elliptic functions, whereas our solutions

involve infinite sums of more common functions. Over ranges where both Tippet's expressions

and ours are applicable, the agreement is excellent. Together they provide information

about TEM cell characteristics for a large class of problems of practical interest. In fact, most situations may be treated using simplified special case solutions (e.g., small

gap, small septum width, etc.).

2. FORMULATION OF THE INTEGRAL EQUATION

FOR THE UNKNOWN APERTURE FIELDS

2.1. Introduction of the Field Transformations

The RCTL to be analyzed is depicted in figure 2. The outer walls are of lengths 2a and

2b, and the septum is of width 2w. The septum is located a distance g from the side walls,

and is displaced vertically by an amount b^ from the upper wall, and b£ from the lower

wall. The regions above and below the septum will be designated by the superscripts (1) and

(2) respectively. The guide is filled with a homogeneous dielectric with permittivity e Q

, and permeability m q

. In order to simplify the later analysis, we will remove the z-dependence of field

expressions via the Fourier transform pair defined by
F(x) -^ /.F (x

t )e

iaZ

da (la)

-co

F(x

t

) =

/ F(x)e~ iaz

dz (lb)

where

x

= (x, y, z) (lc)

x

t

= (x, y) (Id)

the subscript t refers to the transverse plane, and a is the propagation constant. In addition, we assume time variation according to exp (-iwt). Thus we may solve for TE and TM type modes by finding the transformed z-components of the magnetic and electric fields. The

TEM mode will then be seen as a special case of each mode type.

In terms of the above transforms we find that H (x. ) and E (x.) satisfy the following

wave equations

(< + c 2 )H

z

(x

t

) - -8 x

3y(x

t

) (2a)

< V

t

2 + ^

K < x't> = ^t 9 y

3y

< x t> < 2b > o

2 2 2

C

= k Q

- a (2C)

2 a2 3 Z 3 where k = u • p e is the free space wave number, 3 = —, 3 = —, v = —=- + —r- , and

o oo _ x 3x y 3y t 3x2 3y2 we have assumed that the current density (J) has only a vertical component.

For the TE case, the remaining field components are related to H (x ) via

iaHJ o

E

t (x.) = —£— v. H (x.) x a , a = unit vector (3a)

?

Ht(xt

) =^Vt H

z

(x

t

) (3b)

and for the TM case the transverse field components are given by

W =F vt W < 3c )

- i we

H

t (xt

) - -p-A v t

H

2 (x t

) x a

z (3d)

Boundary conditions for H (x ) and E (x ) on the guide walls follow from our assumption of

perfect conductors. Thus we require that on the guide walls

3 n H (5L) = 0, 3 n

= normal derivative (TE) (4a)
E

z (xt

) = (TM) (4b)

These then are the basic equations satisfied by the transformed fields.

2.2. Integral Equation for the TE Gap Field

In order to find expressions for H (x ) and E (x ) in the cell, our basic approach will

be to expand these longitudinal fields in terms of the guide modes in the upper and lower

chambers. By guide modes we mean the TE and TM type rectangular waveguide modes which would

exist in each region if there were no gap. If we expressed each cell mode as the doubly

infinite sum of the corresponding guide modes in both the upper and lower chambers, and then

represented the fields excited by the dipole as doubly infinite sums of the cell modes, the

resulting expressions for H (x ) and E (x ) would involve extremely cumbersome infinite

quadruple sums. However, by not attempting to exhibit the cell modes explicitly, we avoid a pair of infinite sums, and instead replace them with an infinite integral. This offers some flexibility. The excitation of individual modes will appear as poles, and therefore may be

found from residue calculations. However, as it will become apparent later, a field

component may be calculated efficiently by numerically evaluating an infinite integral

without looking at its modal composition. The TE formulation (h-type) will be given here in some detail. A similar TM derivation is given in Appendix A. Later it will be shown that

the TM contribution to the dominant TEM mode is small compared to the TE contribution.

We begin by introducing a Green's function G (x , x') which satisfies the following

wave equation and boundary condition

(V

t

2 + c 2

) GJ

h)

(x

t

, i[) = - 6(x

t

- x|) (5a)

j = { „ for the , chamber

2 lower

9 n^j

h)

(V 7

t)

=0 ( 5b >

The boundary condition for the Green's function applies to the gap as well as the septum and

cell walls. In (5), we have used the unprimed coordinates for the field points and the

primed coordinates for the source points. The solution to (5) is well known [1,3], and

includes the following TE guide mode expansion:

Gi

h) (x\, x|) = (J-) I

(j.y

n cos £ (x+a) cos £ (x'+a) cos S& cos 2p\ (6a)

^mn where

3 <v v

= (ib-> i m? cos 1 < x+a ) cos IE < x

' +a ) cos

b cos

J l l ab Tj m,n=0 (KpJ -c

2

) 2a 2a

J J

Km

Jn )= t®2 + e)

2

] V2

(6b)

J

1 mO (6c)
Equation (2a) yields a pair of equations for FP

J '(x. ). If we multiply (2a) by

g| (xt

» x'), multiply (5a) by H^(x.), and then subtract, we find

GJ

h)

(x

t

, x() v;

2 H«)(x

t

) - H

z (x') v| 2 ej

h )(x

t

. x-)

= -6J

h) (\. x£) a x

.3y

(xt) + H<

j) (i{) 6 (xt -x|) (7)

If we now integrate both sides over the respective guide cross sections, and apply Green's

Second Integral identity [4, p. 96], eq (7) yields

/,

^

h)

(x

t

, x() 3 n ,H^(x

t

) - H^(x|) 3 n .GJ

h >(x

t

, xi')] d*'

=Hl

J) (xt

) -/, G]

h )(x

t ,x') 3 x ,5

y

(xj) ds' (8)

where s' is the respective guide cross section, and *' its boundary. Equation (8) may be

simplified by applying the boundary conditions for G. (x., x!) (5b) and

"(i) - j t t H^ (x') (4a). An additional application of (5b) allows the integral on the right-hand side

of (8) to be integrated by parts. Thus we find (8) reduces to

gaps L__

G

;|<V*t> 3 n

H

z <*P d*

= Hp' } (x

t

) + / 5y

(x') a x

. ~GJ

h )(x

t ,x') ds' (9)

The RCTL is to be excited by an elementary vertical dipole located at a point

x = (x ,y ) in the z=0 transverse plane. The source current density is given by

3 (x) = iyId* 6(x

t

- xQ

) 6(z), (10)

where Id* represents the dipole moment.

If we take the transform (lb) of J (x), and then substitute J (xM into (9), we find

y t

/ G<

h)

(x x') 3 n ,H( j) (x;) d*'

gaps

J t i n z t

= H^ J) (x ) + Id* 3 x .G|

h)

(x., xM| (11)

t We now require that the tangential fields be continuous across the gap. From (3a) and (3b)

we see that H (x.), and a H (x. ) must be continuous across the gap. The second condition

may be imposed by dropping the superscript in the left-hand side integral, and the first by

equating hT '(x,o) to H^ '(x,o). Thus noting that n = + a , d*' = dx', and letting the

dipole be in the upper chamber, we find
j G

(h) (x,x') 3 ,H (x'.o) dx' = -Id* 9 ,G{

h) (x,o,x')| , (12a)

gaps x^ = x

Q

where

G

(h) (x,x') = G[

h) (x,o,x',o) + G^

h) (x,o,x\o) (12b)

We may now simplify both the Green's function and the driving term by performing an infinite sum. The Green's function is given by

~ /u \ 2 0° A A G^(x.x') =

I I (^-) —gp

-cos % (x+a) cos-^ (x'+a) (13)

j=l m,n=o j (K^ y - c

2

)

The summation on n is known [5, p. 40] and yields

Jo ^TFhr

b

o cot W2v (14a) mn where

:/2 Km

= [C 2 - (g ) 2

] (14b)

~ (h) ' and thus we may write G^ ; (x,x') as follows

G

(h) (x,x') = -1 1 I K cos S (x+a) C0S 3a (x

' +a) (15)

j=l m=o m

The driving term on the right-hand side of (12a) may similarly be simplified. We notice

that

- 3 , G '(x.o.x ) = W i (1)2 (^)

1 m,n=o (K^ 1 - c 2

)

rrm , , x

. rmr , , > nny' ,. c \ x cos pT (x+a) sin -*- (x+a) cos —

e

2- (16)

Again the sum on n is known [5, p. 40]

A_ COS

nny' - "n "^ bj -b

2 cos ^ (bj - y')

1 77TTP 77

= ~2~ t K sin K b. J (17) n=o (K^ 1 - c 2

) m ml

and thus, after some manipulation, the right-hand side of (12a) may be written

-Id* 3 x ,G|

h) (x,o,x|)|_ _ = -Id* g (h)

(x) (18a)

x+ = x^

t

,, > , oo cos K (b, - y ) „v n ;/..\ 1 \

p ml o /irnrx rmr / , % rmr , , x ho,,

9 < x > =

" I i. K sin K b. W C0S

2a" < x+a ) Sln 2? < x

o +a

) (18b)

m=l m ml
form

Defining f^ (x

1 ) = 3 ,H (x',o), our integral equation (12) takes the more compact

/ f

(h)

(x') G

(h) (x,x') dx' = -Id* g (h)

(x) gaps

(19)

We next allow the dipole to be centrally located, i.e., x = (o,y ), then g( '(x) given

in (18b) reduces to a summation over odd indices m. Therefore g' '(x) is an odd function in

x which implies that G^ '(x,x') is also an odd function in both x and x'. Thus we need only

consider the odd component of f( h '(x') for the centrally located dipole. Thus, eq (19)

reduces to

:(h), :(h). ,(n: / f^(x') G inj (x,x') dx' = -1/2 Id* g

l ; (x] w

./,, ,2 cot K b

.

G

(h) (x,x') = -\

j=l

I Imo —^m

mux . mux

sin "2? sin IT

(20a)

(20b)

/un i i cos K (b,-y ) __

9 (h)

(x) =-1 li (|L) - .,",V ° sin " mo m

sin Kb,

m 1 2a

(20c)

where the summation over mo denotes the odd indices (m = 1 ,3,5* * * ) . In order to solve our

integral equation (20a), our next step will be to assume the gap is small and then to find

an approximate analytic expression for f("'(x').

3. SMALL GAP APPROXIMATION FOR THE GREEN'S FUNCTION

The integral equation (20a) could be solved in terms of Chebyshev polynominal

expansions as was done by Tippet [1]. However, this leads to an infinite system of

equations in terms of less familiar special functions. The problem may be simplified

considerably by assuming a small gap, and then approximating the Green's function. This

will lead to a fairly simple solution in terms of common functions.

We begin by changing variables, letting t = a-x, and t' = a-x'. In terms of these gap

parameters, (20) becomes

9

( / Q

( o (t,f) f (h)

(t') dt' = Id* g (h)

(t), (21a)

there

(t,f) "I L

cot Kb.

1

j = l mo

m j mirt rmrt

si— COS —77- cos

2a 2a (21b)

f

(h) (t') = f(h)

(x') when x' = a-t' (21c)
/u\ i cos K (b,-y )

.

9 (t) = -2F^

^ K sin K b. Sln T cos "ST

(21d) mo m ml

and the upper integral limit in (21a) represents the gap, g = a-w (see figure 2).

Notice that as m becomes very large we find the following asymptotic behavior

cot K b . „ F

2-^—-

(22) m

Therefore, q' '(t,f) is singular as t»t' since it approaches the harmonic series. We may

extract the singular part of Q( h '(t,t') by adding and subtracting the asymptotic limit (22)

yielding

Q

(h) (t,t') = Q^h) (t,t') - ojh) (t,t') (23a)

Qi

h) (t,f)=^ l I cos H| cos fgl (23b) mo

o<

h) (t,f) -\\ i [ ^i- + &, cos -t cos -t:

(23C ) j = 1 mo m

where the subscripts s and f denote the singular and finite terms respectively. The first

term inside the brackets in (23c) decays exponentially for large m, and therefore is summable as t>t'. However, Q^ '(t,t') has a logarithmic singularity as t+t', and represents

the dominant contribution.

Having split the Green's function we may now take advantage of a small gap assumption. The series (23b) may be summed [6, pp. 96-97] giving

Q<

h) (t,t') =±in [cot (^ |t-t'|) cot (^ |t+f |)] (24)

Note that both |t+t'| and |t-t'| are bounded by 2g. If we assume g7r/2a <1, the cotangents

may be replaced by the reciprocal of their arguments giving

Qi

h) (t,t') - -±in [b£)

2

k |t*-t'2|j (25) 4a

( h\ Next consider Qx '(t,f) for low frequencies. The term in brackets decays rapidly and only

a few initial terms need to be considered for which the cosine terms remain near unity under

the same assumption of a small gap. Thus Qa '(t,t') is approximately independent of t

1

and t

<4

h) (t,t')*i 1 i [-^r^'+f] -= <4

h)

< 26 > j=l mo m

If we let

A

(h) =

j f(

h )(t , )dt' (27;
then our approximate Green's function yields the following integral equation for (21a)

- 7 /

9 f

(h) (t')£n [(^)

2 \%W*l } dt' - A<

h)

Q<

h

> + Ul g (h)

(t) (28)

We may extend the definition of f( n )(t') according to f( h )(t') = f( h )(-t') which

corresponds to introducing an image field. Noting that our solution is only valid for o<s t

<; g, we may split the logarithmic term into the sum of two terms, let t'+ -t' in the |t+t'|

term, and rewrite (28) as

.1 j

9 f(

n

)( t ') ^ [Jg- |t~t'|] dt' * A

(h) Q<h)

* Id* g (h)

(t) (29)

This is the approximate integral equation to be solved. For notational purposes we define

the following quantities

9 {h)

(t) , I ^

h,

(y )cos5g <30a) mo

,,« , m cos K (b,-yj „_ 9m ' W = la ( 1a } Km

sin K^ 51n ~7 (30b)

Then our integral equation (29) may be written as

g

f-g mo J

/

9

f (h) (f) *n (^ M'|] = A<

h > Q<h > + Id* { gm^(y ) cos *£ (3!)

Having formulated an approximate integral equation, we next proceed to solve it via a Fourier series approach.

4. SOLUTION OF THE APPROXIMATE INTEGRAL EQUATION

Our particular problem may be solved directly by taking advantage of a Fourier

expansion for the logarithmic kernel . By expanding the other relevant functions in a similar manner, and applying orthogonality, the coefficients may then be evaluated. »We begin with the following result given in [7, p. 171]

JUt|cose-cos<|i| = An 1/2' 2 J j- cosne cosn<t (32)

n=l

n

ith the substitution of t s gcose, and t' = gcos<j>, (31) and (32) combine to give

I I f(

h) (gcos4>)sin$ [up (||j -2 f 1 cosne cosn^j d*

o n=l

I [A<

h)

Q<

h > + Idit I gmh)

(y ) cos {» cose)] (33) mo
If we now expand f^ ' (gcos<t>)sin<)>, and cos (—5* cose) in cosine series as follows

f(

h Mgcos<|))sin<i) =

I a^cosp* (34a)

p=o P

cos (^|| cose) =

I cj^cosne (34b) n=o

then orthogonality gives

(h)„ /ng\ 1 r/\(h) n ( n ) tj„ v Wr ^ ( n )i /, c \ 3 mo

a (h) I- -I id£ I g (h)

(y ) c (h)

(n>l) (35b)

n n g

L am

XJ o' nm * ; v '

It remains to evaluate c given by

nm

3 J

c l"; = _D. / cos (HM cose) cosne de (36' nm tt J v 2a

This integral is well known [5, p. 402], and we find

which gives

«£}- \ »* ? J

n (^ < 3 ')

g in (g|) mo

a

<»> = . & cos

n.

Id, l g

(h)

(yo) ^

(M), (38b) mo

where J is the Besel function of the first kind of order n. Because of the quick exponential decay of g^ ' (y ) at low frequencies, and the small

argument of the Bessel functions for the first few terms, a reasonable zero''" order solution

may be achieved by retaining only a^ . This may be seen more easily by asymptotically

/ L \ estimating a^ '

. We have, near the septum where convergence is slowest, that

,,, -mY

(h ) , v o • rmr / on \ 9

rn (yo^ ~ e S1n 2~~ ( '

where Y Q

= Tfy Q/2a. We need to look at

cos if 1 gl

h)

(y ) Jn (^)~(-Dn S2n

(40.) mo

S 2n

-

I e

" mY °sin|lj 2n (iSg) (40b) mo

The sum Sgn is evaluated approximately in Appendix B making use of the small gap

10
assumption. If we now compare a^ ' to a;, ' we find

gy -2»4n(ff) (V2 kf) (1 - £) (41a)

a o

-2Y

k

2

. ^ ,°

Y <1 («b)

(l*e V

This bound shows that neglecting a^ ' is consistent with our earlier simplification of

/ 1 \ ii the finite kernel component Qx . Thus our zero*" order approximate solution is given by

f

(h)

(gcos<t>) sin* = a^

h)

(42)

where ^ ' is given by (38a). We may now solve for A'

11 '. Substituting t

1 = gcos* into (27)

gives

tt/2 A

(h)

= g / f (h)

(gcos<j,) sin* d* = g | a<

h)

(43)

o

and thus we find that a^ ' is given by

.W-f ["©-iO?"]"

1 I 9i

h '(y ) J t^J) (44)

a mo

In terms of our gap parameter t' we find

f

(hV)-—^

Un @ - f q<V I g< h)

(y ) o

o <^f) (45)

/ g

z -t * mo

Notice that f'"'(t') is singular according to (g-t 1 ) '2 at the septum edge, t'=g, which is

as expected since f' '(f) is proportional to Ex in the gap and therefore must satisfy the

appropriate edge condition. We will now use this approximate gap field expression to examine how the Hertzian dipole excites the cell, and specifically the TEM mode.

5. GAP EXCITATION OF THE RCTL MODES

5.1 Electric Field Distribution Due to TE and TM Type Modes

We begin with the expression given in (11) for H (x.), which generates the TE modes.

For convenience of presentation, it is rewritten as follows:

H* j, (x\) = / G^

h)

(x x') 3 , ^(xMdf

gaps

J

- & u

Ida 3 x

.i[

h)

(xt

, x£) |_

( _ , (46a)

x

t

= x o

11
where 6^,- is the Kronecker delta.

The corresponding expression for E (x. ), which generates the TM modes, can also be

obtained by using steps similar to those leading to (11),

^

j) (x

t

) = / 3 n

.G^. e)

(x , x') Ep)(xM d*'

z L gaps

" J l u z t

+ 6 ij-^r ldi V

G

i

(V

"x

i

)l -. _-

(46b)

x: = x„

t o

In each expression the first term represents the fields excited by the gap, and the

second term represents the usual rectangular guide modes excited in each chamber by the

vertical Hertzian dipole. If there were no gap, then the RCTL would not support a TEM mode,

therefore we expect the TEM mode information to be contained in the gap pertubation term.

In Appendix C we show explicitly that as a-»-k , E and E x

, due to the superposition of the

unperturbed TE and TM components, tend toward zero. Thus in order to analyze the TEM mode

we need only consider the perturbed terms.

Beginning with the y-component of the electric field due to TE-type modes, we have

6e£j) (x ) --2*-/ 3 x G| h)(x- xM a ,H (x'.ojd*' (47;

y z c gaps

x J t i y

z

where in the gap

3 x^) (x

-

t

, gy . I j ,M, "S

syVy) cos B% sin

^

(48) mo m m j

If we let x' = a-gcosi)) and perform the integration [5, p. 402], the TE-type contribution to the y-component of the perturbed electric field takes the following form

«<%t

) =^9^h)

f

e<

h >(x

t

) (49a)

e (h)(x ) = (-) V (—) m J e ix sin — cos BA j tBl) ( 49b ) t

; ^ a; i k z& ) k sip

„

b sin2 cos2fl o Q

( 2a ) (wd>

m m j £n *2a ; K sin Kb,mo m m j

A similar analysis of the x-component of the perturbed electric field due to TE-type

contributions yields

^

J) (*- t )=^ga(Mief>(x- t

) (50a)

mo m j

In order to evaluate the perturbed electric field due to TM contributions, we again

begin with the y-component given by

12
6^j) (x

t

) -f /

aps

3y3y'GJ

e)

(x

t ,xi) i^txiJdl' (51)

where in the gap

Vy,6J (X

t ,Xt' a I

sing.

K cos K (b .-y) — C0S ^a- C0S ^r (52)

If we combine the previous two equations and integrate by parts, 6E^ J '(x, ) may be given as y *• follows

6Eij) (xJ

2ia

mo

K cos K (b.-y)

y

x "t y acj ^

v rrm' sin Kb.

2a \ m "m x ~

j

"* ' rmrx COS —jr j f

v (X ) COS :(ei w

hvitx 2a dx'

Letting x' = a-gcos<(> and integrating [5, p. 403], we find

5eJ

j, (3L) =4^-g a{ e US<e, (xJ

y

v t' C z ^e a

1 2 y

v t'

(53)

:54a)

;(e) 1 ,2a x m " "m v "j " . rrnr rrmx . ,rnirg> sin -^ cos -^ Jj (-gf).

(54b)

K cos K b,-y

"g k t

; Sg ; L

3 3 m W sin Kb.

mo m j

( e) where ai ' is defined by (a-17) in Appendix A. ~(i) - A similar calculation gives the following result for 6E Vd '(x.) due to TM-type modes

:ti),Z ia2 (e) 7T -(e)

6E

x

( V

_ F^r" 9 a

l 2

e

x

( V (55a)

;(e) , Sin Km

< b ry)

p( e )(x ) = 1-2.) V (h m J e sin — sin — J

x

l V Sig' ^[ m> sin Kb. sin

2 sin2a

J

l

Clearly the total field distribution is the superposition of TE and TM type contributions

We next examine the manner in which the above expressions contribute to the TEM mode.

5.2 TEM Mode Excitation

In order to characterize the TEM mode, we need to take the transform (la), and then

analyze the poles at a = +k

Q

which represent the forward and backward travelling waves. A

TEM mode component arises both from TE and TM type contributions. The TEM mode has an associated gap voltage which may be determined by integrating the x-component of the

electric field across the gap. Both f("'(x') and f( e ^(x') are proportional to E (x',o),

thus the result of the gap integration is already known for both cases. The total voltage

is the sum of the two results. Before analyzing the electric field distribution we will

examine further these gap voltages.

We begin with the TE case by referring to the x-component of the electric field given

in (3a). The voltage due to the TE type contribution is

-(h)

iwu a -7-2- j 3 .H (x',0)dx'

w y z (56)

13
Recalling that f^ '(x 1 ) = 3 ,H (x',o), making the substitution x

1 = a-gcos4>, and using

(42), we obtain

V (h)

= -77^ 9 / f

(h) (gcos«) sin* o> = —^ g f

a<

h

> (57)

o

Thus, after evaluating the appropriate residue, we find that the forward and backward

voltages are given by

V

(hTL= J 9 nrt aih

>

I . „ (58) TEM" 4

a "o °o 'a = k o

where n is the characteristic impedance of the free space TEM wave. A similar analysis of

the TM contribution presented by (a-25) in Appendix A gives

v( e ) = JL n n (—- a( e

V MI (S9\ TEM 4

9 n O( 2a

a

l ;i a = k Q

[ ^>

The gap voltage excited by the TEM mode is thus given by

where we have suppressed the z-dependence.

The electric field distribution of the TEM mode may be readily found from the previous

expressions, (49a), (50a), (54a) and (55a). If we normalize each expression by the

appropriate voltage expression we find

u i \ u cosh K (b .-y) W Ey

(x

t

} = 2(

a

} L ~i i : sin - cos i? J 1-& (61a)

V TEMm0 Sinh Kmb

j

to Ey

< x

t>

= <Tg> i I — g

-

J Sln r cos la J

l <~2a> (61b)

V TEMmo sinh K mb

j

/ b x r(J)^ \ o/bi v m ' j *' .„ rmr „. rmrx 7 W /miry* fei„\ Ei < x

t> "

-2

<a' > — ' ,/

Sln ~7 Sln —25" J o <-2J> < 61c >

(-A) E<%t

) . -(||) J I ^ ""'Vy)

sin S| sin Bg 0, <=§) (61d,

»teA

s -" »«"Vj

where

Km

- (f|) (61.)

The x- and y-components arising from the TE and TM derivations look quite similar. We may show that, up to the order of our previous approximations, both are equivalent. We make

14
use of the following Bessel function recurrance relationship

2 J

: (z) = z (J Q (z) + J

2 (z)) (62)

which allows us to write the TM contribution to the y-component of the electric field as follows

, , . * . cosh K (b .-y)

(-Tiy) E)J '(x ) = 2(-) I

J sin -^ cos — (J (-^) + J (-^)) (63)

V TEM

m0 Slnh K mb

j

id in justifying

asymptotically and find

As we did in justifying a zero order solution, we may compare the two series

(-&) Ej^it") -

2(f)

S {1 + ^} (64)

V TEM

°

where S?

n has been defined previously in eq (40b). As was shown in Appendix B, the ratio

S2/S

Q

is on the order of (irg/2a) 2

, and therefore may be neglected. Thus (63) reduces to

, i \ l cosh K (b .-y)

(-T^y) E"'(x ) = Z(-) i

J sin -^ cos — J

Q (-^) (65)

V TEM

mo Sinh K mb

j which is the same as we found for the TE excitation. An analogous treatment of the x- components shows they also are equivalent up to the order of our approximations. Thus, we find that the electric field distribution of the TEM mode is given by

E (j) (xJ = (villi + vi!i) e

(j) (xj (66a)

y

v t' v TEM TEM' yv t' v '

(j) (x\) = (vffi + vifh e(j) (xj (66b)

x

v t' v TEM TEM' xv t'

E v x

where

mo sinh K b . m J and

(j)/- ^ t?-\ ° 0S m j • niTT mux , /miTg> ,„j\

y l a mo sinh Kb. Z 2a 2a m j

We need to know how strongly viTM and vlpA are excited. If we evaluate v|

F A given by

(58), and vifZ given by (59) at a = k Q

, and normalize the result, we find

.. cosh K (b.-y ) aSH * m sinh Kb,

S1n T J

o

( 2a> TEM ml (67a)

Id£n Q

1 2 mirb.

I Jin (M) + y 1 y

( coth _^I _ X

L ) Sg y ^ m . A , v 2a ' mo j = l 15
cosh Km (b,-y v „. „_„

n ), m - L sin -z J. (-**)

Vici 8 mo sinh Kb, *

x *a TEM _ ___ ml /c7k\

IdlTr " — 2 rrmb . ~~ ( 67b >

3 mo j-1

If the septum is not offset too radically, both the series in the demoninators Will be much

less than the gap dependent term. Thus, in forming the ratio of the voltages they may be

neglected. As before in seeking bounds, we look at the asymptotic terms of the numerator

series and find

v (e)T

V TEM°

where G = ng/2a, Tj is defined by (a-19b) in Appendix A, and S

Q

is defined by (40b). What

we find is MI J? I'M* **"(£> (69)

VTEM

(e) The ratio tends toward zero as G+0, but more importantly, we see that neglecting Vjru

in comparison to VJF m is consistent with our previous approximation (small gap). The bound

is quite good for reasonably small gaps. Even if g/a = .25, no longer y/ery small, we find

that the bound is still less than 18%.

5.3 Characteristic Impedance of the Cell

The characteristic impedance of the cell may now be found using our gap voltage

expression. The power in the forward wave is given by

P - |f (W)

An alternate expression given by Tippet [1, p. 59] is

zn

Id*E n

2 P=4 h^] (71)

where VQ

is the gap voltage that excites a field EQ

at the source point. Neglecting

V

TEM ' we ma^ solve for zo» insert ( 6 ? a ) *or v » substitute (61a) at x fdr E /V

Q

, and We find

z o ,2 rnirb

.

t* * I [m {% -

I i I (1-coth -5-J-)] (72)

n o

8 * g mo m

j=l

2a

For normal geometries the series term will be mudh smaller than the logarithmic term. Thus,

we have a >/ery simple formula which demonstrates that th6 dominant effect on the

characteristic impedance comes from the gap size, and not the vertical offset. Some

numerical results on the field distribution and the characteristic impedance for various

cell dimensions are presented in the next section.

16
6. NUMERICAL RESULTS

Tippet [1] has done much work in characterizing an RCTL. In order to check the

validity of our approximate analysis, we may compare our results with his for some typical

geometries. For the special case of a symmetrically located septum, Tippet [1, p. 52] found

an expression for the normalized magnitude of the electric field given by

,b>

F

I m'b

i

| dn2 (m'z)

1 V2 n,% ¥ Eo

" ^K(TTj Un* (m'w) - sn* (m'z)J (73)

where dn and sn are JaCobian elliptic functions. The x- and y-components of the electric

field are found from the real and imaginary parts of E , and the remaining notation is

defined in his derivation. Two basic differences appear between the above expression and our result (61). First, our expression is in terms of more common functions, and secondly

(61) allows for a vertical offset. A FORTRAN program to evaluate (61) was written, and is listed in Appendix D. In Tables 1 and 2 our result is compared to numbers given by Tippet

[1, p. 54] for a typical geometry given by bj = b

2

= b, b/a = 1.0 and g/a = 0.17. As we

see, there is a large area of excellent agreement, which represents a large working area for

analyzing equipment. The slow convergence of the series near the septum is treated by

adding and subtracting the asymptotic term. In the case of the y-component of the electric

field (61a), we find

cosh K (b.-y) -n \ J ~ 1 t 2e2mB

j (74)

sinh K b

.

m J for y-o, where B. = Tib. /2a. By adding and subtracting this term and evaluating the

J J resulting asymptotic series, we can get faster convergence near the septum. However, we

still do not get good results at the gap, as evidenced by the results in Table 1. Our expression for the characteristic impedance of the cell (72) may also be compared

to a similar result found by Tippet [1, p. 35] and given by

Z Q

( K [1 + Al (2-32)] - 2A lE f

1

7£ 4( K' (1 - 3* A

x

) + 2A

X E'

J (75)

where again his expression is in terms of Jacobian elliptic functions of different kinds K and E. The modulus and notation are that of his thesis. His solution assumes that

e

" mb

3 /d

m0 , J - 1 and 2 (76)

for m>l, and is therefore not applicable for a septum with a large vertical offset. The

expression given in this report (72) is not subject to this restriction, although the series

will converge more slowly as the offset grows. A large offset might be useful to increase

the size of the test area to accommodate larger pieces of equipment. Again, a program was written to evaluate (72), and is given in Appendix E. In figures 3 and 4 we compare our

result to Tippet's for gaps of g/a = .1 and g/a = .2. As expected, we find good agreement

17
for a nearly centered septum. However, the curves separate more as the offset grows. It should be noted that if small argument expressions are substituted for the elliptic

functions in (75), it reduces to

ir

= |L*n (ff)

- j (l-coth^L)] (77)

A comparison of the above expression to (72) shows that they are similar, except that only

the first term of our correction series appears in Tippet's result. The correction series

becomes more important as the offset increases. Tippet also provides an expression for the

normalized characteristic impedance when the septum width 2w is small (large gap),

z I [in (% + 2AJ, (78a)

%

2

irft^+bp) n(b,-bJ

cosh 7) cosrh 5 A i= —nn*r\ —-l (78b)

1 Tr(b,+b 9 ) sinh

2a

where again the condition (76) fs assumed. Together (78) and (72) provide good approximate

solutions for a wide range of geometries.

7. CONCLUDING REMARKS

This report examines the excitation of a TEM cell with a small gap by a vertical

Hertzian dipole. Numerical results on the detailed field distribution inside the cell and

the characteristic impedance of the cell have been obtained. The results are applicable to the TEM cells now in use at NBS. The analysis allows a large vertical offset of the center

conductor which might be useful to increase the size of the test area to accommodate larger

pieces of equipment. The problem of a narrow septum could be solved in an analagous fashion

by formulating an integral equation for the current on the septum. It is believed that the

results given here will prove useful in solving for the monopole excitation of a TEM cell.

Once the method of solution has been established, other similar problems, such as a probe in

a shielded strip line, could perhaps be analagously treated.

18
8. REFERENCES

[1] Tippet, J.C. Modal Characteristics of Rectangular Coaxial Strip Line, Ph.D. Thesis,

University of Colorado, Boulder, CO, 1978.

[2] Crawford, M.L. Experimental evaluation of the radiation characteristics of dipole

sources enclosed in a TEM cell, IEEE Conference on Precision Electromagnetic

Measurements, Boulder, CO, June 28-July 1, 1976; pp. 57-59.

[3] Collin, R.E. Field Theory of Guided Waves, McGraw-Hill, New York, NY, 1960.

[4] Johnk, C.T.A. Engineering Electromagnetic Fields and Waves, Wiley and Sons, New York,

NY, 1975.

[5] Gradshteyn, I.S. and Kyzhik, I.M. Table of Integrals, Series and Products, Academic

Press, New York, NY, 1965.

[6] Jolley, L.B.W. Summation of Series, Dover Publ . Inc., New York, NY, 1961.

[7] Lewin, L. Theory of Waveguides, Wiley and Sons, New York, NY, 1975.

[8] Standard Mathematical Tables, S.M. Selby, ed., Chemical Rubber Co., Cleveland, OH, 18th

edition, 1970.

19
Table 1. Sample Results of Field Distribution Inside an RCTL (x-component)

y/b

1.0

.6

.4

.2

.000 .000 .000 .000 .000 .000

.000 .060 .129 .208 .279 .308

.000 .108 .244 .422 .601 .682

.000 .126 .310 .618 1.030 1.244

.000 .090 .246 .640 1.678 2.316

.000 .000 .000 .000 .000 .000

.2 .4 .6 .8 1.0 x/a

l.a. The normalized x-component of the electric field with

b

x = b2

= b, b/a = 1.0, and g/a = 0.17 (Wilson).

.000 - .000 .000 .000 .000 .000

.000 .060 .129 .208 .278 .307

.000 .108 .245 .422 .600 .680

.000 .127 .311 .620 1.029 1.237

.000 .090 .248 .647 1.684 2.285

.000 .000 .000 .000 .000 3.603

y/b

1.0

.8

.6

.4

.2

.2 .4 .6 .8 1.0 x/a

l.b. The normalized x-component of the electric field with

b

1 = b2

= b, b/a = 1.0, and g/a = 0.17 (Tippet)

y/b

1.0

.8 1%

.6

.4

i .2

'

| 2% • ?

.2 .4 .6 .8 1.0 x/a

I.e. The percentage difference between the results given in Tables l.a. and l.b.

20
Table 2. Sample Results of Field Distribution Inside an RCTL (y-component)

y/b

1.0 .824 .794 .699 .531 .289 .000

.8 .853 .826 .737 .569 .316 .000

.6 .935 .918 .853 .701 .412 .000

.4 1.049 1.052 1.051 .979 .657 .000

.2 1.153 1.185 1.296 1.499 1.364 .000

1.180 1.223 1.382 1.808 3.292 .000

.2 .4 .6 .8 1.0 x/a

2. a. The normalized y-component of the electric field with

'1 b, b/a = 1.0, and g/a = 0.17 (Wilson).

y/b

1.0 .824 .793 .698 .530 .289 .000

.8 .853 .825 .736 .568 .315 .000

.6 .935 .917 .852 .699 .410 .000

.4 1.049 1.052 1.051 .977 .652 .000

.2 1.153 1.186 1.298 1.499 1.343 .000

1.196 1.245 1.431 1.986 6.640 .000

.2 .4 .6 .8 1.0 x/a

2.b. The normalized y-component of the electric field with

bj = b

2

= b, b/a = 1.0, and g/a = 0.17 (Tippet)

y/b

1.0

.8

.6

.4

.2

1%

2% 10%

2%

.2 .4 .6 .8 1.0 x/a

2.c. The percentage difference between the results given in Tables 2. a. and 2.b.

21
APPENDIX A

APPROXIMATE TM GAP FIELD DERIVATION

The major steps in the derivation of the approximate gap field expression for the TM

(or e-type) case are similar to those taken in the TE case. Thus, most of the intermediate

derivations will be omitted. We again begin by introducing an associated Green's

~(e) - - function G- '(x, x') which satisfies the following wave equation and boundary conditions

J t (gap as well as walls)

and is given by

(v2 + ^) ~GJ e)

(x

t

, x() = -6 (x - x() (a-la)

GJ

e)

(x

t

, x() = 0, (a-lb)

^v^ -^ l

J —m-—

J i»,n-l (K" J - c

2

) mn

sin tt- (x+a) sin *- (x +a) sin —rj- sin -^~~ (a-2)

J J

We next multiply the wave equations for E^ (x!) given in (2b) and G- (x., x!), by

-(e) - - ~(i) - z J G- (x. , xM and E Vd/ (xj), respectively, subtract the two results, and integrate over the

respective chamber cross sections S

1 . An application of Green's Second Integral identity

and the boundary conditions satisfied by G- and E^ J ' will considerably simplify the

integral equation. Inserting the current density distribution transform of (10) will yield

-

J V^e)

(x

t

, x') E< d) (x')d£'

gaps" J t u z ^

jy («t)-=-"Wy«ie, («V«t)l«;--*

<a " 3)

J to

We next wish to impose the continuity of the tangential fields across the gap, which

requires that E (x. ) and 3 E (x. ) be continuous at the gap. However, a slight difficulty

z t y z t ~^ e

\ _ arises. The y-derivative may not be taken inside the integral because 3 , G

^- (x

t

, x!)

involves asymptotic behavior according to

mn_ ~ g e'Y 2a (a-4)

which will diverge as y+o. This problem may be avoided by first integrating the left-hand

side of eq (a-3) by parts. We then find the following result

22
-J 3 n

,G (e)(x

t ,x() 3 x .EJx) djt' gaps

= -£- Id* 3 v

3 v ,GJ

e)

(x

t

, o, xMU, _

- we y y i t t 'x. - x (a-5)

where

G^ = GJ

e ^(x 5 o,x',o) + G

2 (x,o,x',o).

Analagous to the TE case, the Hertzian dipole is centrally located, the summations over n are performed to simplify the Green's functions, and the integral equation (a-5) takes the

following form

»/vM r.(e), ,(e)

/ f^

e; (x') G^e;

(x, x') dx' = - 1/2 Id£g^

e;

(x) w(a-6a)

where

f

(e)

(x') = 3 x ,iP(x', o)

G^(x,x') = - ?

v

' 7i >,\ iL ^ \m

C ot Kb,

m j cos JSg

2a sin

^2a j=l mo

(a-6b)

(a-6c)

.(e),

3 v ' me a

L o mo

K cos K (b.-y ) m m

v 1 •'o' sin K b. m 1 cos

miTX 2a

:(e)

(a-6d)

Next, the singular component of G

v '(x, x

1 ) is extracted and a small gap is assumed.

In terms of the gap variables t and t

1 we find after some manipulation

9

r

-g

where

I j 9 f

(e)

(t ,, _d^ . A

(e)

Q

(e)

(t) + Id£ _|a

g (e)

(t) (a _ 7)

f (e)(f) = f( e )(_ t ')

A

(e) =

j f (e)(

t .) dt .

(a-8a)

(a-8b)

mo

(a-8c)

(e) __ 2

2 *m w, ^TM " . L - L m mi ?a J 'fm tt >,

J = l m j 2a (a-8d)

(ei/.v v (e), x. rrmt 9 (t) =

* 9m (y J Sin2T

mo

(a-8e)

23
Q (e),

, =

1 m m 1 J o' . rrm

f

. 9 m

( V 4a sin i^ET

Sln ^ ( a " 8f )

Finally, if we let

-"i

e) W * A(e) Q^} + w* — gie)

(y„) U-9) m o' fm toe 3m w o

x ' o

our integral equation may be written

g

-g mo i / f

(e) (t')^,- I h(e »(y )sin^ (a-10,

Again a Fourier series approach will be used to solve (a-10). We make the change of

variables t = gcos0, and t

1 = gcos<|>, and (a-10) becomes

I J

f (e) (gcos»)

* in *!iJL I hi

e) (yj sin (SJ cose) (a-11)

tt J V3 T/ cose - cos* £ m w o' v 2a ' * r ' o mo

If we now differentiate (32) with respect to <j> we find

sint)) „. = 2 ) cos ne sin n$ (a-12)

cose - cos* L . T v » Y n=l

which gives a Fourier expansion to be used analagously to (32). We next expand f' e '(gcos4>)

in a sine series, and sin (—~f cose) in a cosine series

La

f

(e)

(gcos<|>) =

I a(

e) sin p* (a-13a)

p=l p

then orthogonality gives

sin (^v cose) = V 6*J cos ne (a-13b)

2a '

j; , nm

a (e)

= y c

( e Ve)

(y ) (a-14)

n i mn m w o mo

(e)

It remains to evaluate C

x ' given by

nm J J

ZTM = ~ I sin (^cose) cos ne de (a-15)

'nm tt J v 2a

o

which is a known integral given in [5, p. 402]

,~(e) r, . m\ , /mrrg, , , c i CL = 2 sin -x J (-**) (a-16) nm 2 n ^ 2a'

Thus, we find

a'*'- 2 sin f £

„ (.,

( , -j, „_„,

mo

(e) Again, as with the TE case, we wish to base our solution on the initial term a: ', and

24
(e)

neglect the higher coefficients. Referring to h: (yo

)» we find

hi

e)

(y )-e"

mYo

sin^| (a-18)

where Y Q

= vy

Q /Za. Thus, we need to look at

mo

-mY

T 2n+1

-

I e ° m sin ^ J 2n+1

(mG) mo

(a-19a)

(a-19b)

where G = irg/2a. We may make use of the following two Bessel function relationships [8, p. 519]

J„_i(z) + J„*i(^) =^J„U) n-r n+1 z n'

J n-1<*> " Wz

> = 2 J

n<

z >

(a-20a)

(a-20b)

to generate the following pair of relationships between T2 n+ ^ and S£

n [which is given in

(40b)],

T + T = — S ' 2n-l ' 2n+l G5 2n

T - T = 2 S' '2n-l 2n+l * ^2n

(a-21a)

(a-21b)

and thus we find

T = — S - S

1

1 2n+l G5 2n ^n (a-22)

where derivatives are with respect to G. Substituting this in the result (b-19) found in Appendix B gives

Y

L 2n ,.- ,. „2n+l

T

2n+1 " TT=pT

«- 1 >" (^ k ) 2"

( k2 "!) G (a-23a)

VTT^T (k2 - 1/2)G

(e) (e) Therefore, comparing a~ '. to a] ' we find the following

a (e)

2n+l (1/ 2n (k^-l) 2n

a

i

(a-23b)

(a-24)

25
so neglecting higher order terms in the TM solution is consistent with the small gap approximation made in the TE derivation. Thus, our approximate gap field is given by

f( e

)(gcos<|>) = a[

e

^sin<j) (a-25)

Next we may solve for the undetermined constant A^. Substituting t

1 = gcoscj) into (a-8b)

gives

„(e) (a) n , „,-•> k = g aj - (a-26)

and thus we find that

a

l " g cue I \ ( V J

l

( 2a ) L g

" 2i Qfm

J

l

( 2a j J (a " 27)

3 o mo 3 mo

In terms of our gap parameter t

1 we find

3 o mo 3 mo

and this expression, which is proportional to E , will be used in analyzing the TEM mode

characteristics.
APPENDIX B

EVALUATION OF THE SERIES S 2n

We wish to evaluate the series S2n defined by (40b). If we let G = ng/2a, and

p = -e °, our series may be written in the following form

-Y co S 2n

= e

°

I p m

J 2n ((2m+l) G) m=o(b-1)

If we replace the Bessel function by an integral representation given in [8, p. 360],

1 *

J 2n

(z) -fj cos (zsine - 2ne) de (b-2)

o

S 2n takes the following form

IT °° e r j„ v m

2n tt / de I p cos [(2m+l) G sine - 2ne] (b-3)

o m=o

We next expand the cosine function which generates two summable series

TT °° °° S

9

= - / de cos (z-2ne) I pm cos2mz - sin (z-2ne) I pm

sin2mz (b-4) 2n tt m=o m=o

where we have let z = Gsine. These two series are given in [5, p. 40] and S2n becomes

-Y

S

= £ } de cos (z-2ne) - pcos (z+2ne)

(fa

, ^2n tt I aa

1 - 2p cos 2z + V {

'

Because z = Gsine <<1 according to our small gap assumption, we may approximate the

numerator and denominator as follows

cos (z-2ne) - pcos (z + 2ne) =

72

- (1-p) (1 - 4 ) cos2ne + (l+p) z sin2ne

1 - 2p cos2z + p2 - (1 - p)2 (l - k

2 z2

)

(b-6)

(b-7a)

where

k

2 =

" 4p

/ (1-p) 2 ;b-7b)

This reduces S 2n to

'2n e"

V °

5 (i-p;

, (1) (l+£) ( L1 2n JT-p)

l Z

2) 2n (b-8a)

27
(1)

*

(1 -V2

z

2

) cos2ne de

l Zn ~ j TT-k2i2 (b-8b)

o

(2) _ ]

z sin2ne de _ l 2n - * 1 - k

2 z2

( b " 8c > o

It remains to evaluate these two integrals.

We begin with ii ' which in terms of 8 is given by

(1) ! (1 -V2 G

2 sin2 e) cos2ne de 2n ~

J 1 - k

2 G

2 sin2 e

( y;

o

The integral may alternately be written as

(1)

.

J

/2 (Hfeg .i^e) c.,a,e de

2n J 1 - k

2 G2 sin2 e

v ' o

We now expand the numerator in terms of cosines as follows

(1 -V2G2 sin2 e) cos2ne (b-11)

=

-^ G

2 cos 2(n-l)e + (1 -V4G2

) cos2ne + -g G

2 cos 2(n+l)6

Thus our integral may be expressed as follows

l 2n

= i

G2 K n-1

+ (I-V4G2

) Kn+ |

G

2 K n+1 (b-12a)

where

~ r cos 2n6 de ,, ,_. . K = 2 / .,., . 9q b-12b

n

J 1 - k

2 G2 sin2 e

This integral is given in Gradshteyn and Ryzhik [5, p. 368] and we find

2n

K = (- 1 )"-

( 1 - /- 1 - k2tf) Cb-13)

" / 1 - k2 G

2 b

Recalling the definition of k in (b-7b), the small gap assumption gives kG «1, thus the

square root may be approximated by:

and K

p reduces to

/ 1 - k2 G

2 - 1 - V2 k

2 G2 (b-14)

2n (-1)", (V2 kG) . , K

n (1 -V2k2 G

2

) (b " 15)

Thus we find ii ' approximately given by

28
I ln'- tl

,-4"^) '^8)2n( '-^- V4tf) n > o

(b-16a)

(b-16b)

where approximations of the order of (b-14) have been made.

(?) We next need to evaluate Ia ' given by

(2)

V

r sin2n6 sine de 2n J 1 - k

2 G2 sin2 e o (b-n;

If we split the integration path and let e

1 = ir-e over (tt/2, tt), as we did in generating (b10), we find that the integrand is the negative of (b-17). Thus we find

l(2) x . 2n

u

This finally gives us a small gap approximation for $2 n -Y

o

(b-18)

(b-19a) VlTTpT (i -V2W) {1 _1/4G2)

-Y

S2n * (hT (1 -£*») ( '/2kG)2n

< 1 -2^- 1/4

G2 ' n>1 < b " 19b »

29
APPENDIX C

EVALUATION OF THE UNPERTURBED EXCITATION TERMS AS a+k

Q

The y-component of the electric field from both types of modes is given by

-iwu

Ey(x"t } = -T^ 3 xMX"t> + F

9 yE z( x\) ( C -!)

which upon substitution of the unperturbed expressions for H (x.) given in (11) and E (x.)

in (a-3) gives

We next examine the Green's functions and find

oo 2 vx

.G (

i

h)

(x t'x

()

=

i ® —rrp

g TM

) (x t'

x

t> < c - 3a)x x j t t m=1da

K u;_ 2 mn r r

v mn '

a ya yl Gf)(xt ,x|) . J^^M)^ _^l__ fl (J) (x

-t§Si) (e.»)

^v3^

=

(dH sin S {x+a) sin S (x

' +a) cos T cos T1

( c - 3c ) j j j

Thus, we find that EjJ '(x.) may be expressed as follows:

y z c m=l,n=0 K u

i

- s 2 ^

a o j mn z °

v mn '

For a = k Q (or t, = 0) and with KJi' defined in (6b), our expression for E„ (x t

) as a->k may

be simplified to

y t"«=k a^k

Q

^

lj m= ^ n=0

ymn t o'

Notice that if we expand the delta function 6(x.-xM in a double Fourier series over a chamber cross section, we find

m=l ,n=0

Thus we see that the unperturbed terms do not contribute to the y-component of the electric

field of the TEM mode.

However, the unperturbed terms will contribute a reactance at the source point. A similar analysis of the x-component of the electric field yields the same result, except

that there is no reactive contribution at the source.

30
APPENDIX D: SUBROUTINE GUIDE FOR COMPUTING THE FIELD DISTRIBUTION

1

.

2. OOOOOOBOOOOOOB

3.

4.

5.

6.

7.B.

9. 10.

11 . 12. 13.

14.

15.

16. 17.

18. 19. 20.

21 . 22.

23.

24.

25.

26.

27.

28.

29.

30.

31 . 32.

33.

34,

35,

OOOOOOB

OOOOOOB

OOOOOOB

000127B

000127B

000130B

000132B

000134B

000137B

000141B

000142B

0001E.OB 000153B

000154B

000157B

000161B

000165B

000165B

000174B

000174B

000175B

000213B

000213B

000215B

000221B

000221B

000230B

000230B

000231B

000247B

000247B

000251B

000253B

SUBROUTINE GUIDE(A.B.G)

C THIS SUBROUTINE CALCULATES THE X- . AND Y-COMPONENTS OF THE

C ELECTRIC FIELD IN A CORNER OF THE GUIDE. THE INPUTS ARE.

C A THE HALF WIDTH OF THE GUIDE.

C B THE CHAMBER HEIGHT NORMALIZED BY A. C G THE GAP WIDTH NORMALIZED BY A. C THE SUBROUTINE RETURNS THE FIELD VALUES AT THE LATTICE

C POINTS OF A SIX BY SIX GRID. REAL EXX(6.6).EYY(6.6).XX(6). YY(6) PRINT 9 9 F0RMAT(1X.*ITERATI0NS*.8X.*FINAL STEP VALUES* . 1

1

X.*X* , 9X. *Y*/)

DO 11 1 = 1 .6 DO 10 d=1 ,6 X=( J-1.0)/5.0

Y=(6.0-I)/5.0

XX( J)=X*A

YY( I)=Y*A CALL NORP/!FLD(X.Y.G.B.EX.EY)

EXX(I.d)=EX

EYY(I.J)=EY

10 CONTINUE

11 CONTINUE

PRINT 12 12 F0RMAT(5(/1 ,1X.*THE X-COMPONENT OF THE ELECTRIC .FIELD*/) PRINT 13.(XX(JJ).dd=1 .6) 13 F0RWAT(1X.6F10.7. 12X.*X/Y*/)

DO 15 11=1.6

PRINT 14. (EXX(II.K).K=1 .6).YY(II)

14 FORMAT( 1X.6F10.7.5X.F10.7)

15 CONTINUE

PRINT 16 16 F0RMAT(5(/) .1X.*THE Y-COMPONENT OF THE ELECTRIC FIELD*/)

PRINT 17.(XX(Jd) . Jd=1 .6) 17 F0RMAT(1X.6F10.7.12X.*X/Y*/)

DO 19 11=1.6

PRINT 18. (EYYdl ,K) ,K= 1 .6 ) . YY( 1 1

)

18 FORMATMX.6F10.7.5X.F10.7)

19 CONTINUE

RETURN

END

OOOOOOB

OOOOOOB

OOOOOOB

OOOOOOB

OOOOOOB

SUBROUT INE NDRMFLD ( X . Y . G . B . EX . EY

)

C THIS SUBROUTINE COMPUTES THE ELECTRIC FIELD COMPONENTS AT

C A SPECIFIED GRID POINT. THE INPUTS ARE.

C X,Y THE X AND Y COORDS OF THE GRID POINT NORMALIZED BY A. C Q.B THE GAP. AND CHAMBER HEIGHT NORMALIZED BY A. C THE OUTPUTS ARE EX. AND EY. THE X-. AND Y-COYPONENTS OF THE

C ELECTRIC FIELD AT THE SPECIFIED POINT. REAL Pl.MPI

PI-4.0*ATAN(1.0)

31
6.

7.8.

9. 10.

11 . 12.

13. 14.

15.

16.

17. 1G.

19. 20.

21 . 22. 23.

24.

25.

26. 27.

28.

29.

30.

31 . 32.

33.

34. 35.

36.

37.

38.

39. 40.

3.

4.5.

6.

7.8.

9. 10.

11 .

12.

13.

14.

IS.

0000048

000005B

000006B

000010B

0000148

000015B

000016B

000020B

000024B

000025B

000031B

000032B

000044B

000057B

000062B

OO0OG5B

000072B

000102B

0001 1 1

B

000114B

000115B

000115B

000122B

000133B

000133B

000135B

000137B

000141B

000145B

0001528

000160B

000170B

000172B

000174B

000176B

000000B

OOOOOOB

OOOOOOB

OOOOOOB 000001B

000002B

000003B

000004B

000015B

000020B

000023B

000034B

000047B

000053B

0000568

1020

21

30

10

EX=0.0

EY=0.0

DO 10 M=1 .30 MPI=((2.0*M)-1.0)«PI/2.0

X1*MPI*X

X2=MPI*G

X3=-MPI*Y

X4 = -I«PI*( 2.0*B-Y)

X5=-MPI»2.0*B

CALL BJ0(X2.BJ)

ST1 *FLOAT(M)

ST2=(EXP(X3)-EXP(X4))/(1 .0-EXP(X5))

ST3=(EXP( X3)+EXP(X4))/(1.0-EXPfX5))

IF (Y.GT.0.1 ) GO TO 5 ST2=ST2-EXP(X3)

ST3=ST3-EXP(X3)-2.0*EXP(X5)

ST4=S1N(MP1)*S1N(X1 )*BJ*ST2

ST5=SIN(MPI)*C0S(X1 )*Bd*ST3

EX=EX+ST4

EY=EY+ST5

IF (ST3.LT. 0.0001) GO TO 20 CONTINUE

PRINT- 21 ,ST1 .ST2.ST3.X.Y

FORMAT(1X.F10.7.5X.2F10.7.5X.2F10.7)

IF (Y.GT.O. 1 ) GO TO 30

IF (X.GT.0.95) GO TO 30 XPI=PI*X/2.0

ST6=EXP(-PI«B)

ST7«2.0*COS(XPI)*ST6*(1 .0+ST6*ST6)

ST8 = 1 .0+2.0*COS(2.0*XPI )*ST6*ST6

EYrEY+(0.5/COS(XPI))+(ST7/ST8)

EX=2.0*B*EX

EY*2. 0»B*EY RETURN

END

SUBROUTINE BjO(X.BJ)

THIS SUBROUTINE CALCULATES THE BESSEL FUNCTION JO(X) AND RETURNS THE VALUE THROUGH BJ

.

T«X/3

Z*3./

Y*T*T

IF (X BJ*1. 1Y*(.0

RETUR W=SQR

AF=.7

1-2*(. THETA

1-Z*(.

BJ = AF RETUR

END

GE.3. -Y*(2.

039444

NT(X)

978845

000728

=X-.78

000541 *COS(T

N

) GO TO 10 2499997-Y»(1 .2656208-Y»( .3163866-Y*f

-Y*. 0002100) ))))

0444479-

6-Z*( .000000 7 7+Z*f . 00552740+Z* ( . 0000 951 2-Z* ( .001 37237

05-Z*. 00014476))))

)

539B16-Z*( .04166397+Z*(.00003954-Z*f .00262573

25+Z*( .00029333-Z*. 00013558))) )) HETA)/W

32
APPENDIX E: SUBROUTINE FOR COMPUTING THE CHARACTERISTIC IMPEDANCE

1

.

OOOOOOB

2. OOOOOOB

C

c

cc

c

c

c

3. OOOOOOB

4. OOOOOOB

5. OOOOOOB

e. 000004B

7. 000005B

8. 000010B

9. 000015B

10. 000021B

11 . 000027B

12. 000034B

13. 000035B

14. 000037B

15. 000041B

16. 000047B

17. 000047B 20

18. 000054B 21 19. 000055B

20. 000060B

21 . 000063B

22. 000066B

23. 000073B 30

24. 000073B

25. 000104B 22

26. 000104B

27. 000106B

SUBROUTINE Z0( B1 , B2.G. X

)

THIS SUBROUTINE CALCULATES THE CHARACTERISTIC IMPEDANCE OF AN RCTL NORMALIZED BY NO. THE FREE SPACE IMPEDANCE OF THE TEM

MODE. THE INPUTS ARE B1.B2 THE DISTANCES FROM THE SEPTUM TO THE UPPER. AND LOWER CHAMBER WALLS NORMALIZED BY A. G THE GAP LENGTH NORMALIZED BY A. THE NORMALIZED CHARACTERISTIC IMPEDANCE IS OUTPUT VIA X. REAL PI. 00 PI»4.Q*ATAN(1.0)

ST1-0.0

DO 20 M=1 .20 X1«-PI*B1*((2.0»M)-1 .0) X2=-PI*B2*((2.0*V)-1.0)

R1=2.0*EXP(X1)/(1.0-EXP(X1 )) R2=2.0*EXP(X2)/(1 .0-EXPU2))

CHECK1=ABS(R1)

CHECK2=ABS(R2)

CHECK=AMAX1(CHECK1 .CHECK2)

ST1 =ST1+(R1+R2)/(2.0*M-1 .0) IF (CHECK. LT. 0.0001) GO TO 21 CONTINUE

Q0=2.0*ST1

X3=8.0/G/PI

X4=2.0*ALOG(X3)

X=PI/4.0/(X4+Q0)

PRINT 30 FORMAT (//. 4X,«B1».8X.*B2». BX. G«.9X,*ZO*.BX.*00*. 8X.» FINAL STEP*/) PRINT 22.B1 ,B2.G,X. 00. CHECK FORMAT(1X.6F10.7)

RETURN

END

33
—' \

DC

N dc a.

H- ZD

CJ CO

UJ

00 rr- uj 2

uj CD

CJ o B^

<

Q.

UJ

CO

00

CO

<T3 s34
M

1

1

II

b

b

1

J

b

i

2

1

1

11 GAPS OR

APERTURES" ~~\

©

\x "septum"

w -a

Figure 2. RCTL cross section

WILSON

TIPPET

b/o = 1.0

b/a = 0.5

.6 .8 (b-b,)/b

Figure 3. Normalized characteristic impedance z /n for g/a = 0.1

35
WILSON

TIPPET

b/o= 1.0

b/o = 0.5

.2 .4 .6 .8 (b-b,)/b

Figure 4. Normalized characteristic impedance zQ /n for g/a = 0.2

36
NPS-114A (REV. 9-78)

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBS TN-1037i.Gov'L Accession No, 3. Recipient's Accession No,

4. TITLE AND SUBTITLE

Excitation of a TEM Cell by a Vertical Electric

Hertzian Dipole

5. Publication Date March 1981

6. Performing Organization Code

7. AUTHOR(S)

Perry F. Wilson, David C. Chang, Mark T. Ma

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, DC 20234

18. Pmject/Task/Work Unit No.

11. Contract/Grant No.

12. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street, City, state, ZIP) 13. Type of Report & Period Covered

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

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

16. ABSTRACT (A 200-word or less factual summary ol most significant information. If document includes a significant bibliography or

literature survey, mention it here.)

'he excitation of a transverse electromagnetic (TEM) cell by a vertical electric

lertzian dipole is analyzed where the gap between the septum and side wall is assumed

;o be small. Approximate expressions for the field distribution and characteristic

mpedance are derived. These expressions are numerically evaluated for some typical

leometries, and good agreement with previously published results is shown. The

'ormation also allows a vertical offset for the septum position, thus offering more

lexibility of increasing the size of the test area to accommodate larger pieces of

est equipment.

•'• KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unleas a proper name;

separated by semicolons)

haracteristic impedance; field distribution; Hertzian dipole; integral equation;

'ectangular coaxial transmission line; TEM cell.

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