FiLE COPY

MBSIR 76-841

AN ANALYTICAL AND EXPERIMENTAL DETERMINATION

OF THE CUTOFF FREQUENCIES

OF HIGHER-ORDER TE MODES IN A TEM CELL

and

Myron L. Crawford

Electromagnetics Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

June 1976

Prepared for:

Electronics Systems Division (AFSC)

Hanscom Air Force Base, Mass. 01731

John C. Tippet

David C. Chang

University of Colorado

NBSIR 76-841

AN ANALYTICAL AND EXPERIMENTAL DETERMINATION

OF THE CUTOFF FREQUENCIES

OF HIGHER-ORDER TE MODES IN A TEM CELL

John C. Tippet

David C. Chang

University of Colorado

and

Myron L. Crawford

Electromagnetics Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

June 1976

Prepared for:

Electronics Systems Division (AFSC)

Hanscom Air Force Base, Mass. 01731

U.S. DEPARTMENT OF COMMERCE, Elliot L. Richardson, Secretary

Dr. Betsy Ancker-Johnson , Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Acting Director

a*

CONTENTS

Page

FOREWORD iv

1. INTRODUCTION 1

2. FORMULATION OF THE INTEGRAL EQUATION 2

3. SMALL GAP APPROXIMATION FOR THE GREEN'S FUNCTIONS 6

4. SOLUTION OF THE APPROXIMATE INTEGRAL EQUATION 9

4.1 Cutoff Frequencies of the Higher-Order Modes 9

4.2 Field Distribution of the Higher Order Modes 12

5. EXPERIMENTAL DETERMINATION OF THE RESONANT FREQUENCIES OF TEM CELLS 13

6. RESULTS AND CONCLUSIONS 14

7. REFERENCES 17

APPENDIX. COMPUTER PROGRAM FOR FINDING THE ROOTS OF AN EQUATION

USING THE BISECTION METHOD 18

LIST OF FIGURES AND TABLES

Figure 1. Cross-section of a TEM cell 19

Figure 2. Nomograph for determining the cutoff wavelength of the first

higher-order perturbed mode in a TEM cell given the ratios g/a

and b/a 20

Figure 3. Functional diagram of an automatic network analyzer (ANA) system- 21

Figure 4. Typical data obtained from an automatic network analyzer 22

Figure 5. Comparison of Mittra and Itoh's exact solution and our approximate solution for the cutoff wavelength of the first higherorder perturbed mode in a TEM cell 23

Table 1. Calculated and measured resonant frequencies of the first higherorder perturbed mode in nine different NBS TEM cells 24

iii
FOREWORD

This report describes the theoretical and experimental determination of

the cutoff frequency of the first higher-order mode for several TEM cells 6f

differing geometry developed at the National Bureau of Standards (NBS) . In

addition, the field distribution of the first higher-order mode is found

explicitly

.

The effort is part of a program sponsored by the Electronics Systems

Division, Hanscom Air Force Base, under contract number Y75-917 with NBS.

The purpose of this effort is to evaluate the use of TEM cells for determining

the radiated emission characteristics Of small electronic devices.

The theoretical analyses were developed by staff from the University of

Colorado under contract with NBS. The experimental work was performed at

NBS. Myron L. Crawford of the Electromagnetics Division was the technical

monitor for NBS and Charles E. Wright of the Electronics Systems Division was

the technical monitor for the Air Force. The period of performance covered by

this report extends from November 1975 to April 1976.

Results of this effort will be used to develop higher^order mode suppression

or rejection techniques to increase the useful frequency range bf a TEM cell.

iv
AN ANALYTICAL AND EXPERIMENTAL DETERMINATION OF THE CUTOFF FREQUENCIES

OF HIGHER-ORDER TE MODES IN A TEM CELL

John C. Tippet, David C. Chang and Myron L. Crawford

ABSTRACT

In addition to the TEM mode, higher order TE and TM modes can propagate in a TEM cell. Since only the TEM mode is desired, the

higher-order modes restrict the useful frequency range of the TEM

cell. In this paper, the cutoff frequency of the first higherorder mode is obtained both analytically and experimentally for

several TEM cells of differing geometry. The difference between

the experimental and theoretical results is shown to be only a few

percent. In addition the field distribution of the first higherorder mode is found explicitly.

Key words: Shielded strip line; TEM cell.

1. INTRODUCTION

In our previous report on the TEM cell, an effort was made to investigate

the properties of the dominant TEM mode [1] . In this paper, the first few

nigher-order TE modes in the TEM-cell will be studied. They are of interest

since they determine the upper frequency limit below which only the TEM mode

can propagate. Thus, knowledge of these modes may enable us to increase the

useful frequency range of a TEM-cell, for example, by placing selectively

absorbing material in the cell at locations where the effect is least on the

propagation of the TEM mode.

As is obvious from the sketch shown in figure 1, the TEM cell can be

viewed as a special case of a shielded strip line for which the strip is

embedded in a homogeneous dielectric. Solutions leading to the cutoff frequencies of the higher-order modes in shielded strip line structures usually

involve a numerical search for the roots of an infinite determinant obtained

from a mode matching at the interface between the partial regions separated

by the center septum in the cell. A problem arises, however, since the

infinite determinant must be truncated in order to obtain numerical results.

As Mittra has shown [2], the partitioning of the set of equations (i.e. the

number of modes used in each of the partial regions) must be chosen in the

correct ratio in order that the solution of the truncated set of equations

converges to the correct value. In fact, an incorrect choice of the partitioning ratio leads to a violation of the edge condition [3].

In order to bypass the difficulties associated with relative convergence,

an alternative approach was sought in this report. An integral-equation formulation for the electric field in the gap was found, and an approximate, but

analytical solution of the integral equation was obtained using the assumption

of a small gap between the center septum and side walls. (All of the TEM cells

currently in use at NBS indeed have small enough gaps so that the analysis
applied.) This allows us to represent the cutoff frequency of the first higher

order mode as a perturbation to the TE^q mode in a rectangular waveguide. The

method developed in this paper is applicable to TEM cells for which the center

septum is displaced vertically from its center position, and may also apply to

TEM cells for which the regions above and below the center septum have different dielectric constants (i.e. a shielded strip line). In addition to the

cutoff frequencies, the electric field distribution of the higher-order modes

can be found explicitly.

In order to check our solution, we. compared our results to those obtained

by other authors [4,5,6]. In particular, Mittra and Itoh [4] used a singularintegral equation formulation. Their results reduce exactly to ours when a small

gap approximation is used in their equations. However, the results obtained by

Brackelmann, et al. [5] and Bezlyudova [6] were not useful for comparison since

their result for a zero- thickness septum reduces to the cutoff frequency of the

unperturbed TE^q mode in a TEM cell, independent of the width of the center

septum. If we regard the upper and lower regions of the TEM cell as two rec- tangular waveguides coupled through a small aperture, then the use of the so- called coupled mode theory predicts that two system modes (symmetric and antisymmetric) can exist, both of which have approximately the same propagation

constant. The symmetric mode is unaffected by the edge condition at the ends of

the septum, while the anti-symmetric mode is more strongly affected by the

width of the septum [7]. It is clear that the results of Brackelmann et al . [5]

and Bezlyudova [6] were intended for the symmetric mode. However, because the

excitation of the TEM cell is basically of a TEM nature which yields an oppositely-directed electric field in the upper and lower regions of the cell,

it is more relevant to obtain results corresponding to the anti-symmetric case. The resonant frequency of the first higher-order perturbed mode was also

obtained experimentally in this paper. Using the well-known relationship between

the resonant frequency and the cutoff frequency, excellent agreement was obtained

between the theoretically and experimentally measured resonant frequencies.

2. FORMULATION OF THE INTEGRAL EQUATION

A cross-section of a TEM cell with an off-center septum is shown in

figure 1. The strip, of width 2w, is located a distance g from each vertical

sidewall and is assumed to have negligible thickness. In addition it is

embedded in a homogeneous dielectric of permittivity, e Q

and permeability, y Q. The outer wall of the cell, of width 2a and height 2b = b

1 + b2

, and the strip

are assumed to be perfect conductors. The region above the strip is designated

region 1 while that below the strip is designated region 2.

2
We will attempt in this paper to obtain only the cutoff frequencies of the

TE-type modes since the TE^g mode is usually the lowest order mode next to the

dominant TEM-mode in the cell. For the TE modes it is well known that the

fields in region 1 can be found in terms of an auxiliary scalar function,

¥ ^ , which satisfies the following relations.

(V 2 + K

2 )Y (1) =0 in S,, region 1 (1)

and

where

3 =o on the metal walls, i.e., (2) n B-C and A-F-E-D

K

2 = k

2 _ y2; (3)

K = -

2 >Vo = (4)

and we have assumed that the fields propagate as e

1U)t - 1 Y z

> ^ s tjie transverse

Lapiacian, which, for our geometry is given by 3

2 /3x 2 + 82 /9y 2

, and 3 n represents the outward normal derivative. cj is the angular frequency, A is the wavelength, k is the wave number and y 1S the propagation constant which is equal

° (1) to zero at cutoff. In addition to (2), we will also require that 3 ¥ be non-

(1) zero in the gap regions, that is, A-B and C-D. If 9 * were zero there also,

then we could write the solution to (1) immediately as just an unperturbed

rectangular waveguide mode unaffected by the presence of the center septum.

In

follows

terms of ¥ , the field components in region 1 can be written as

H = K

2 Y (1)

i" (5)

z z and

E

t

= iwy

Q

V t(T

(1) )x i"z (6)

where a is a unit vector in the z-direction and V represents the transverse

z — 3 — 3 gradient 3 (a -r— + a -r— ) . x 3x y 3y

In order to determine Y ^ , we will write it as a superposition of a complete set of basis functions \l>^' T as follows

mn

In S, = I hllUil? (X f y) • (7)

1

L mn mn

m, n

A

ma

are the unknown amplitudes, and the basis functions, <p^' , must satisfy

the following

+ «di

>2

»ii'

J = ° (8 > t mn mn

3
and

9 *

(1 > =

n

r mn

0 (9)

on £, , the boundary of S, , i.e. A-B-C-D-E-F. It may appear that (7) is logically

(1) .

' (1) inconsistent since the basis functions, satisfy (9) while their sum, 4* r mn

satisfies 3 f / 0 in the gap regions. As shown by Friedman [8], however,

(1) n

f is discontinuous at y = 0 and thus the limit y = 0 and the sum in (7)

cannot be interchanged. In other words, the requirement of 9 ¥ ^ 0 at the gap

should be understood as applied to the plane y = 0 + where 0 +

= e and e * 0.

The solution of (8) and (9) is obviously

\b (1) (x,y) = r mn 1 2

1 h cos (x+a)

L 2a cos

J (10)

with given by mn ^

J

mn ©

2 + (11)

and the normalization constant appearing in (10) is included for convenience.

Using the orthogonality properties of the basis functions, the amplitudes,

A* , can be written as follows

mn

A

(1) = A A mn m n /^

(x- #y , )*

I^

) (x' ,y«) ds' (12)

where

A. =

l

i = 0

otherwise

(13)

In order to find an alternative expression for the surface integral appearing

in (12), we multiply (1) by and (8) by Subtracting the resulting

equations and integrating over S^, we obtain the following

(K (1)

2 mn

- K

2

) / ds = / N

fa l

b

l

(D V

2 ¥(1) . ,(1)^(1)] ds mn t t

r mn

(14)

Using the two-dimensional form of the divergence theorem and Green's second

identity [9], the right-hand-side of (14) can be written as

R.H.S. = r mn n n

r mn

(15)

The second term in (15) is zero by virtue of (9). Thus the unknown amplitudes

given in (12) reduce to

(1) _

A A m n mn 7 r mn n

(16)
Because of (2), however, the line integral in (16) only contributes in the

gap. Thus,

4 *

(1)

3 * r mn n dl = - -w a

/ + / -a w

\b (1) (x,0) 3 *

(1)

y (x,0) dx mn y

(17)

since

3 yd) = _ |_ yd) at y = 0 n 3y 1 (18)

In order to combine the two integrations appearing in (17), we will consider

two cases: (1) the odd modes for which ¥

(1) (x,y) = -4f (1

)

(-x ,y) ; and (2) the

(1) (1) even modes for which 4* (x,y) = yVJ"' (-x,y) . The following analysis will apply

for the odd modes for which (17) can be written as

a ^d) 3 yd) dl _ _r 3 ^u/ ( 0) [4/-^ ( x o) - rii (-x,0) ] dx. * r mn n y

1T mn ' rmn ' £. w

2

(1) (1) (1) (19)

The term in the square brackets in (19) is zero if m is even. Thus (16) gives

"VrP / 3Y (1) (x,0)^ } (x,0) dx (m=l,3,5, . . .) A (20)

(1 > = mn

(K 2 - K

(1)

) w mn

0 (m=0,2,4,...)

Equation (20) can now be inserted into (7) which then determines the scalar

function, ¥ in region 1 in terms of the unknown cutoff wave number, K, and

an integration of the unknown electric field in the gap (3 ¥ (x,0)« Ex

(x,0))

Thus

(1) (x,y) = 2 I ~

2 '(1) 2, *2pk,n<*'y> ^ V(1)(X '' 0) *2pll,n (x, ' 0) dx ' p,n (K^ - „) * ' w

2p+l ,

n

where p = 0, 1, 2, (21)

A similar relation can be written in region 2 by replacing all subscripts and

superscripts 2 by 1 and by replacing y by -y. In the gap, we require con- tinuity of H

z and E^ which is equivalent to requiring the following

4- (1) (x,0) = * (2) (x,0) (w < x < a) (22)

and

3 V (1) (x,0) = 3 * (2) (x,0)

y y

(w < x < a) (23)

When (22) and (23) are imposed on (21) and its equivalent expression in region

2, we obtain the following integral equation.

/ 3 4* (x 1 , 0)G

q

(x,x' )dx' =0 (w < x < a)

5

(24) w
wnere

G (x,x') = I I n j=l p,n (K' K (j) 2. *2 P+1 (X ' 0) *2 P+ 1 (X, ' 0)

2p+l,n ; (25)

and we have dropped the superscript on 3 ¥(x*,0) since it is continuous across

tne gap as well as the subscript "n" on the basis functions since they are evaluated at y = 0 . The Green's function given in (25) is singular at x = x'.

Thus, as shown in [10, pp. 197-198] the integral equation (24) is to be

interpreted in the principal value sense. A similar expression can be obtained for the even modes. The resulting

integral equation is identical to (24) if we replace G

Q by Gg

which is given as

2 A A G (x,x') = I I E-Jlpj *

9 _(x f 0)ip 7 (x',0). (26) e j=l p n (K 2 - K^ )2

) 2p 2p

J tfl 2p,n

If the septum is located in the center of the TEM cell (i.e. b^ = - b)

then the summation on "j" in (25) and (26) can be dropped and replaced by the

factor 2.

3. SMALL GAP APPROXIMATION FOR THE GREEN'S FUNCTIONS

Tne integral equation given in (24) could be solved numerically at this

point using, for example, the moment method. In this method, the unknown funcUsing the

orthogonality properties of the basis functions, an M x M matrix equation

ensues, and the requirement that the determinant vanish then determines the

cutoff frequencies of the higher-order modes. However, since a strictly

numerical technique does not furnish much insight as to how the cutoff frequencies change as the gap varies and may also lead to relative convergence

difficulties [11], we will solve (24) by approximating the Green's function

using a small gap assumption, i.e. Kg << 1. The solution to the approximate

integral equation can then be obtained analytically.

In order to find an approximation for G

q

, we note that the summation on

tion, 3 f(x',0), can be approximated by an M-term Fourier series

in (25) can be written as follows

n=0 [K 2 - K^* 2

] 2p+l,n

b .1 2 _1

T\ j 2p+l

- I n=l (n- _ a (j>2

a ; 2p+r

(27)

wnere

l 2p+l

b .

TT - (2p+l) 2 (^) (28:

6
The summation on the right-hand-side of (27) is given as [12]

a 2p+l ; ^ a 2p+l Z 2p+l

so that (27) reduces to

An

b

*

J fM t— = ^-r-t COt(TT a^-3 ,.]

n=0 [K 2 - k<3> 2

] 2 it a^) 2 P+1

2p+l,n 2p+l

(30)

Thus, G

Q given in (25) can be written as

2 b

2 cotdrai^,)

G

o

(x,x') = I ^ I (jf^- ^2p+ l (x ' 0) *2p+ l (x, ' 0) ' (31 > : =1 2 tt p a

2

J +1

Tne summation on "p" in (31) cannot be done in closed-form; however, we notice

tnat for large "p"

cot (ira^l- ) r _ i , -2£±L_ * - 2a _i (32 ) aipll V (2P+1)

so tnat if we add and subtract the right-hand-side of (32) to G

q we obtain the

following

G

o

(x,x') =G^1) (x,x') +G^2) (x,x') (33)

wnere m 2 b

2

. r .x g», (x,0) ip„ +1 (x' ,0)

j = l 2tt b. J p (2p+l)

and

, 0

. 2 b

2 G^2) (x,x«) - J -1 J j=l 2 TT p

cot (Tra^i, ) 2p+l'

+ 2a

a

2 Pll (2 P+1) J ^ 2p+1 (x,0)^ p+1 (x',0). (35)

-2 (2) The term in the square brackets in (35) now decays rapidly as p . Hence, G^

approacnes a constant whenever | x-x 1

| * 0. On the other hand, the terms involved in decay only as p and thus will have a logarithmic singularity

as | x-x '| 0. Therefore under the assumption of a small gap the dominant

contribution to G will come from G^. o o Substituting for the basis functions from (10) , and making the following

change of variables

t = a - x (36)

and

f = a - x' (37)

7
can be written as

o

(1) (t,f) = I p=0 (2p+l)

which can be summed as [13]

cos

(2p+l)TT (t+f ) 2a

+ cos

G

(1) (t,t') = ^jln tan — It+t'l

n l 4a ' J + In tan

(2p+l)TT (t-t '

)

2a

^- |t-f

(38)

4a

(39)

The largest value of | t+t 1

| is 2g so that if we assume that g/a << 1 we can replace the tangents in (39) by their arguments and obtain

(t,f j In 4a

•2_4- > 2 (I « 1] (40)

(2) In order to obtain an approximate form for G^ we notice that if we restrict attention to only the first few higher-order modes for which K is not

too large, then the sum in (35) converges very rapidly, so rapidly in fact that

the arguments of the cosines appearing in the basis functions in (35) remain

(2

)

close to zero. Thus, G^ is essentially independent of x and x' for gK > 0 and we can write

(2)

2 . b . »

I ?

-1

I j=l a p=0

cot(irot 2pll )

2p+l

2a

(2p+l)

(41)

(2gK << 1) (42)

Since we have assumed a small gap, it seems reasonable to represent the modes

in the TEM cell as perturbations of rectangular waveguide modes; that is, we

let

K

2 = K

(S) + 6 (S)

qr

(q=l,3,5. . .) (r=0,l,2 ) (s=l,2)

with

5 (s)

qr

<< K

(s)

(43)

(44!

and K

(s)

qr

cannot be so large that (42) is violated. Then the main contribution

to the summation in (41) occurs when ai^l, = n or when K - Ki^'- . There may 2p+l 2p+l,n 2 be more than one term for which this occurs since it is possible to have two or more rectangular waveguide modes with the same cutoff frequency. We will take,

for example^ q = 1 and r = 0 which corresponds to a perturbation of the TE^q

mode in a rectangular waveguide which is the first higher-order mode that we

(s) would expect to appear,

and only one term in (41) is dominant.

,(2) ,

K_l~' does not depend on "s For this case,

qr

in this case since r = 0,

(2)

(G,

TE10

2

I

j = l

cot (bj6 1Q ) + -2 a6

10

*

2- I

TT p=l (2p+ l)

can be written as

TTb . 1 - coth (2 P+i;

2a

(45]

8
Tne summation on "p" in (45) is very small compared to the dominant p = 0 term,

so we will neglect it, subject to a consistency check when is determined.

In any case the summation is rapidly convergent and can be evaluated numerically

if needed. Thus,

i + cot Vio + cot VlO

(46) TE.10 ""10

In the general case for non-degenerate odd modes, we obtain in a similar way

n . 2 fcot(b -C )

qr J

[

1 ^qr

+ - L jl - coth — (2p+l)

11 p=0 (2p+l) 2a

(q-li

(47)

wnere

(s) = qr

(48)

The preceding analysis has dealt with the approximation of G . A similar

analysis also applies for G , and we therefore only quote the results.

g]

1) (t,f) - J ln[(J) |t 2 -t' 2 |] 6 a. (49)

and in the general case for non-degenerate even modes we obtain

(Ge

> TE (s,

qr

2 'cot(b.E

y J L'<* r

r (S)

^qr

2

irq + 1 - 1 - coth —J—

it p=l p [ a j p^q/2

qtrb 1

1 + —

IT

, COt i—= 1

i 2 2 a

-

(50)

It is interesting to note that all of the dependence on position (through the

variables t and t

1 ) is contained in G^^ while all of the dependence on the as

( ) ' f 2

)

yet 2 undetermined K (through 5 ) is contained in G . ^ qr o,e

4. SOLUTION OF THE APPROXIMATE INTEGRAL EQUATION

4 . 1 Cutoff Frequencies of the Higher-Order Modes

Inserting the approximate form for G

q

into (24) we obtain

/ f(f) ia[(fe) o t

2 -t' 2

|] + G^2)

J df = 0 (51)

where

f (f ) = 3 yY(a-t' ,0) . (52)

9
If we make the following substitutions

TTt 4a

'"

TTt' 4a

'"

(53)

(54)

U

o

= & (55)

.9 A

Q

= / g(t') dt; (56) o h(u') = f'(^^-); (57)

then (51) reduces to

J~° h(u') In |u 2 - u

,2

| du' = - G

o^ )ho' (58)

TT 2 „(2)

o But the solution of (58) is known to be [14]

h(u') = ^ ( 59)

[u 2 - u' o

2 P

where C is an arbitrary constant which is related to A

Q

defined in (56)

.

Inserting (59) into (58) we obtain

f U

° ^ r- In |u 2 -u' 2

| du' = — G

(2) A (60) o [u 2 - u' 2 P 4a °

°

which can be written alternatively as follows

f U° - r In |u-u' I du ' = — G

(2) A • (61) r2 12!^ a O O -u \m - u

1

2

] 2 4a

o o

If we make the following substitutions

and

then (61) reduces to

u = u cos 8 (62) o

u' = u cos d> (63) o

C / ln[u Icos 6 - cos d>|]d<f> = G

( 1

' A . (64) o o' Y|T 4aoo

However, since [10, p. 171]

00 In I cos 0 - cos <j> | = -ln2 - 2 \ — cos mG cos m<J> (65)

m=l

10
(64) reduces to

TrCln °l = -JL- G

(2) A . (66) 4a o o

A

q in (66) can be found in terms of C by changing variables and substituting

(59) into (56) with the result that

A = 2aC. (67) o Thus, (66) and (67) give

G

(2 u

o 1 , ~ + — In

2 TT 5~ = 0. (68)

^2

(2

)

Substituting for G

Q

and u

q

from (46) and (55) respectively we obtain the following characteristic equation for the perturbed TE,q mode.

|[cot b

1

6 1Q+ cot b

2

5 1Q ] = [ln(^|) - 2]. (69)

Provided that the remaining sum in (45) is indeed small under the small gap

assumption, this is then the equation for determining &-^q- In particular, when

the gap ratio (g/a) is much smaller than one, the perturbation solution can be

obtained simply from the small argument expansion of the cotangent to yield

r 2 Tib/a

6 10

= - ~

\ ,8a, (70)

b

l b2 ln( ig>

and from (43) the cutoff wavenumber is then given by

K

TT Trb/a

2a b,b 0 ln(—

)

1 2 "rrg

(71)

It is of interest to point out that the form of K as given by (71) is essentially the same as the one obtained in [15] which treats a similar problem in

which coupling occurs through only one aperture.

As a trial solution for any numerical or graphical search of the root of

(69) , we note that since the value of the cotangent can vary from 00 to 0 when

t>1^10 increases from 0 to tt/2 (assuming b^ > b^) , the expected solution therefore lies in the range where 6^^ varies from 0 to Tr/2b, which is the midpoint

between the unperturbed TE^q mode and the unperturbed TE2Q mode. For g/a

ratios from .01 to .3 and a/b ratios from .5 to 1, a nomographic solution of

(69) was obtained which allows one to easily determine the cut-off wavelength

graphically. This is included in figure 2.

The integral equation with G

e instead of G

q

is identical if we replace "a"

by a/4. Thus, for example, for the perturbed TE

2

q mode we obtain

i[cot bl 6

2() + b2

6 2Q ] = L|°[ln(§f) - 2] (72)

from which the perturbed cutoff wavenumber can be obtained as above.

11
4 . 2 Field Distribution of the Higher Order Modes

The field distribution of the higher-order modes can be obtained by substituting the aperture field found in (59) into (21) , but first we can evaluate

advantageously the unknown constant, C, in terms of the "voltage" drop across the

gap. Substituting for f(t') from (52) into (56) and realizing from (6) that

E (x',0)

d *(x' ,0) = Y iwu (73)

we obtain for A . o

o

= -J——— J

3

ltOU J EX (x',0)dx' = —cou (74) o a-g o where V is the potential difference across the gap. Thus, C in (67) is given as iV

C =

2acoy (75)

and the aperture field from (59) can then be written as iV r .. 2 ..,21-^ h(u') = [u - u' 2

] 2atou o

(76)

Transforming the integration in (21) from x' to u' via (37) and (54) and

inserting h(u') from (76) we obtain

(1)

; 2p+l

2 h

{-} 7T ^ab^

given as [16]

\

j o cos [ (2p+l) 2u' ]du'

(?7) o /u z - u' 2 o

j

U

o cQ8[( 2p+l)2u' ]du' = !j [2 (2p+l)u ]

(78;

where JQis a Bessel function of order zero. Thus by analogy with (21) the

fields in regions 1 and 2 can be written as

A (-l)P j r (2p+l)g] sin [ (2p+l)^x

-] costj^)

(3 (x,y) = \-*-\\**-\ I ab . ; y

ix>\i ' p,n (K 2 - K< j > 2

) 2p+l,n

The summation on "n" in (79) can be written as follows

1

A cos

n

y n=0 (K 2 - K^i, 2

) 2p+l,n

b

2

TT - I

n?ry cos | g—*-

2a (j)2 n=l (n 2 - a (j)2a ) 2p+l n x

^ r a 2p+l ;

(j = 1,2;

(79)

(80)

12
wnere ai was defined in (21 2p+l

is given as [12]

The summation on the right-hand-side of (80)

cos cos

2p+l' 2p+l 2p+l

J 2p+l

sin(7ra ( ^+1 ) (81)

so that (79) reduces to

'2b jW; „ i - (-l) PJ^[(2p+l)J|]sint(2p+l)^] cos

Y (D) (x,y)

Tra iY_|

i mu ' p=0

2a J

From (28) and (43) we have that a

(j)

2p+l

VlO

k b . - 2p+l

sin(7ra 2pil )

(82:

for r = 0 so that restricting

attention to the perturbed TE^g mode we have

2 (x,y)

,-

| cos[6 10 (y+b.)]

a6

lo' ^y o sm

sin 6 , „b . 10 3 2 a

2b .

TT iV

I 9=1

(-D PJ0

[ (2p+l) JflsinI (2p+l)g]

ITM2p+l sinUa2p!l } (83)

Since the potential difference, V, is still arbitrary, we can normalize V in such

a way that the amplitude of sin (•=—) is unity with the result that

where

(j)

b .6. sin 6 , n b . Q

I 10 j TTX , j 10 (x,y) = sin | — + —' 2a j

tt cos [& 1Q (y+bj ) ]

(84)

. (-1) [ (2 P+l)If] sin

( ( 2P+1) g cos

[ (^l) . a g+1

q =

I

. L p=l 2p+l sin(TTa

2 P!l )

(j = 1,2: :85)

In this form the first term in (84) can be recognized as the dominant TE^q mode

of a rectangular waveguide and the second term as a small perturbation. As

g/a -> 0, 10 0, and the perturbation term in (84) vanishes

5 . EXPERIMENTAL DETERMINATION OF THE RESONANT FREQUENCIES OF TEM CELLS

Measurements were made using an automatic network analyzer to determine

the network scattering (S) , parameters for a number of TEM cells as a function

of frequency. These S-parameters comprise a total characterization of the cell

as a two terminal network from which the cell's complex impedance, insertion

loss, and resonant frequencies can be determined.

13
The operational details of the system used to make these measurements has

been published [17] and are not repeated here. However, the test system's

functional diagram is shown in figure 3. The system has three main sections,

namely, source, measurement, and computer. The signal generator provides

the RF power over the frequency range of interest (0.1 GHz and above) required

to pre-calibrate the system and test the unknown device or the TEM cells. The

frequency and level of the RF source is stabilized, level controlled, and preselected by command to the computer. The RF is applied to the device under

test via the S-parameter test set. Then depending upon how and are set,

the parameters S-^, S 22' S 12' or S 21 are measured * The "complex ratio detector" measures the amplitude ratio and phase difference between the reference and test channels. This information is digitized

and routed to the computer via the instrument interface.

The computer then takes the S-parameter data and stores it either as calibration data, if measuring standards, or raw, corrected data if measuring the TEM

cell. This data is then processed for correction and routed to the displays

and to other computer peripherals

.

Summarizing, the measurement sequence essentially is broken into two

steps. The first is the preparation of the system through pre-calibration to

make the desired measurement. The second is to measure, digitize, and store

the S-parameter data in the computer and then convert, correct, and output the

data in the desired form. Examples of data obtained for a typical TEM cell are given in figure 4. Resonant frequencies are indicated by the high VSWR.

Note how narrow the bandwidth is for each resonance. This is indicative of the

small RF losses within the cell's conductive walls and center plate and is an indication of the cell Q factor.

In this section we will discuss the result obtained for the resonant frequency of the first higher-order, perturbed mode in a TEM cell with a symmetrically placed center septum via three different techniques; 1) the experimental measurement as described in the last section, 2) theoretical calculations

based upon (69), and 3) a numerical result based upon an exact integral equation

formulation as obtained from equation (58) in the work of Mittra and Itoh [4].

In both of the last two methods, the cutoff frequency, f , is actually calculated, and this must then be converted to the resonant frequency, f res via

the following well known result

6 . RESULTS AND CONCLUSIONS

(86)

where

c = l//yK e o o

(87)

and d is the resonant length.

14
As shown in the figure included with table 1, the two ends of the TEM cell

are tapered, so that the resonant length is not very well defined. As a first

approximation, however, we will take the average overall length as the resonant

length.

In order to find the zero of (69) , a root-finder based upon the bisection

method was used. This is contained in the Appendix. This same root-finder

was also used to find the zeroes of Mittra and I ton's equation (58), which for

a symmetrically-placed center septum and a homogeneous dielectric filling

reduces to the following for the TE modes

Y

1

(0)

where

1 —£ K.J =0

k

l

K

1 = a2 = |(1 + cos TM) ; (89)

and

Y,(0) S'(0) + 2d,(0)E — 1

* = —x . (90)

k

1 k1 [S(0) - 21 ]

The symbols in the above equations are defined in Mittra and Itoh's paper and

in general must be evaluated numerically. However, for a small gap, it can be

shown that the solution of (88) is given approximately by

S (0) - 21 = 0 (91)

or just the denominator of (90) equal to zero. Note that * 0 in (88) as w -> a. Furthermore (91) can be shown to reduce to our result given in (69)

for g/a + 0

.

Shown in figure 5 are the calculated cutoff wavelengths computed via

Mittra and Itoh's eq. (58) and our result, eq . (69) for two different TEM cell

geometries. As expected, the two curves become indistinguishable for very

small gaps. The w/a ratios of the TEM cells currently being used at NBS are

all greater than 0.72. Thus figure 5 shows that only a few percent error

occurs even for those cells with the largest gaps. It is also interesting to

note that Mittra and Itoh's result predicts a cutoff wavelength given by that

of the TE^^ mode in the full cell for septum widths up to nearly half of the

width of the cell.

15
Shown in table 1 are the calculated and experimentally measured resonant

frequencies of the first higher-order, perturbed mode in nine different TEM

cells used at NBS. The column labeled "Form Factor Group" refers to those

cells which are scaled versions of one another. As can be seen, the largest

percentage error between our approximate solution and the measured value is

6.4% for w/a = .72 or g/a = 0.28. For smaller gaps (w/a = .83) the percentage

difference is typically 1%.

Thus we are able to predict the useful frequency range of a TEM cell within

a few percent. Using the same technique, additional work needs to be done to

extend our solution to TEM cells which have different dielectrics above and

below the center septum, i.e., a shielded strip line. In addition, if we re- formulate the problem in terms of an integral equation for the current on the

septum, then we should be able to obtain an equivalent form for small center

septums, i.e., w/a << 1. The formulation of the problem for TM modes has yet

to be accomplished. When this is completed, we should be able to look at the

TEM mode solution (a degenerate TM mode) and obtain a formula for the characteristic impedance of a TEM cell with an off-center septum. Thus, the solution

of the conventional strip-line problem can be obtained by the same technique

described in this report.

16
7 . REFERENCES

[I] Tippet, J.C. and D.C. Chang, "Radiation characteristics of dipole sources located inside a rectangular , coaxial transmission line," NBSIR 75-829

(Jan. 1976)

.

[2] Mittra, R, "Relative convergence of the solution of a doubly infinite set

of equations," J. Res. Nat. Bur. Stand. (U.S.), 67D, (Radio Propagation)

No. 2, 245-254 (March-April 1963).

[3] Jones, D.S., The Theory of Electromagnetism, pp. 566-569 (Pergamon Press,

New York, N.Y. , 1964)

.

[4] Mittra, R. and T. Itoh, "A new technique for the analysis of the dispersion characteristics of microstrip lines," IEEE Trans, on MTT, Vol. MTT-19,

No. 1, 47-56 (Jan. 1971)

.

[5] Brackelmann , W. , D. Landmann, and W. Schlosser, "Die grenzf requenzen von hoheren eigenwellen in streifenleitungen, " Arch. Elek. Ubertragungstech .

,

Vol. 21, No. 3, 112-120 (March 1967).

[6] Bezlyudova, M.M. , "Cutoff wavelength of waveguide-type oscillations in

transmission lines having internal and external conductors of rectangular

cross section," Radio Engineering and Electronic Physics, Vol. 8, No. 11

1727-1733 (Nov. 1963).

[7] Kopp, E.H., "Coupling between dissimilar waveguides," IEEE Trans, on MTT,

Vol. MTT-16, No. 1, 6-11 (Jan. 1968).

[8] Friedman, B. , Principles and Techniques of Applied Mathematics, p. 271

(Wiley, New York, N.Y., 1956)*.

[9] Jackson, J.D., Classical Electrodynamics, p. 15 (Wiley, New York, N.Y.,

1962) . [10] Lewin, L. , Theory of Waveguides (Newnes-Butterworths , London, 1975).

[II] Mitta, R. , T. Itoh, and T.-S. Li, "Analytical and numerical studies of

the relative convergence phenomenon arising in the solution of an integral

equation by the moment method," IEEE Trans, on MTT, Vol. MTT-20, No. 2,

96-104 (Feb. 1972) . [12] Collin, R.E., Field Theory of Guided Waves, p. 581 (McGraw-Hill, New York,

N.Y., 1960).

[13] Jolley, L.B.W., Summation of Series, pp. 96, 97 (Dover Publ. Inc., New

York, N. Y. , 1961)

.

[14] Chang, D.C, "Equivalent-circuit representation and characteristics of a radiating cylinder driven through a circumferential slot," IEEE Trans, on Antennas and Propagation, Vol. AP-21, No. 6, 792-796 (Nov. 1973).

[15] Voytovich, N.N., B.Z. Katsenelenbaum, and A.N. Sivov, "The excitation of

a two-dimensional metal cavity with a small opening (slotted cylinder),"

Radio Engineering and Electronic Physics, Vol. 19, No. 12, 8-17 (Dec. 1974).

[16] Gradshteyn, I.S. and I.M. Ryzhik, Table of Integrals Series and Products,

p. 419 (Academic Press, New York, N.Y., 1965).

[17] Rytting, D.K. and S.N. Sanders, "A system for automatic network analysis,"

Hewlett-Packard Journal, Vol. 21, No. 6, 2-10 (Feb. 1970).

17
APPENDIX

SUBROUTINE HOOT U.B.FtX.JMAX.E.El)

C THIS SUHHUUTINE USFS THE BISECTION METHOD TO SOLVE FOR ONE ODD

C BOOT OF FIX) = 0 ON THE INTERVAL (At 6) . THE FUNCTION PASSED

C THROUGH F MUST £?F DECLARED EXTERNAL In ALL CALLING PROGRAMS. E IS

C IN TEHvaL OF UNCEpTAImTY DESIRED FOR ThE ROOT. AND MUST HE SMALLER

C THAN THE STARTING INTFRVALt * s b-A. THE NUMBER OF BISECTIONS IS

C DETERMINED NM/\x = LN ( W/E ) /LN ( 2 ) • AFTER BISECTING. THE FUNCTION

C VALUE IS COMPAREn TO El. IF ARCtF(XO)) > El THEN THE SUBROUTINE

c prints: discontinuity' at x = xo. a Random, search occuring jmax

C TIMES TS USFD TO LOOtf FOR A CHANGE OF SIGN IF SIGN(FU)) = C SIGN(F(B>).

RE«L LN? OIHENSION f(3)

C QUESTION; OOES F(A) r 0.

YI=F (A) IFIYI.nE.O.) GOTO 10

X = A GOTO 80

C QUESTION! DOES F (S) c 0.

in Y?=F(B)

IFIY2.NE.0.) GOTO 20

X = B GOTO 80

C QUESTION! ARE THE SIGNS OF F ( A ) AND F(8) DIFFERENT.

?0 I1=SIGN(1. .Yl)

I2=Sl6Mtl. »Y2) WrB-A IFlri.NE.12) GOTO 60

C SEARCH FOR A CHANGE IN SIGN.

DO 30 J=1.JMAX

X=A*flANF (0. ) »W

I3 = SIGN ( 1 . »F I A) ) IF IIJ.NF.I1) GOTO 50

30 CONTINUE

PR 1 NT 40

40 F0RMAT(1X«N0 CHANGE OF SIGN FOUND"/)

RE I URN

50 B = X

C DETERMINE NUMRER OF gl SECT I ONS

60 LN2=0. 693147181

NMAX=AL0G ( W/E) /LN2*1

.

Y ( 2 1 1 ) = A Y(2-U)=H

C BEGIN BISECTION

DO 70 N=1»NMA/

X= ( Y (1 ) *Y (3) ) /2. Y3=F (X) IF (Y3.EQ.0. ) GOTo 80 I3=^IGN(1..YJ)

T 0 Y(2*I3)=X

80 IF IABS(F (A) ) -LE.F1 ) GOTO 85

C CONVERGENCE TO A DISCONTINUITY

PRINT 82.X

8? K0RMAT(1X»DISC0NTINUtTY AT X = CE12.4/1 RETURN

C CONVERGENCE TO A ROOT

PRIN.T qn.X

go FORMAT ( lX»OnE 000 ROoT AT X = *E12.*/) RETURN

END

18
Y

1 Fi E

i

2

' t

> I

(D

-

b*

2

A B «< 2w * ©

«« 2a

Figure 1. Cross-section of a TEM cell.

19
o

vffl LO

kO o LO

r- ooo LO

oo o LOc*. oo

LOo

o LO o

CM

LO

CM o

00

LOCO o

«*

LO

"d- o

LO o

1 o o O1 o o o o —1— 1 l_ L_ L__4_—t-_l__l_—J

LO oo CM o 00 r-» <o LO «3" 00 CM O

OO

=, oo oo oo —1—

oo —CCM CM _1_

CM

1 CM CM -J_

CM CM _l_CM CM CM _l_-Z.

ID

-Q

O

1=

enoCOo o

_1_

«3O

_l_

O

o

LO

LO

=3- O o

oo

IX) o

CM

LO O LO i— .— o

o

o

LO O LO

o*t oi oo

LOLO

Jx—H—x—]I—T 7 T T

° o o o

_1 L. l_ o o_l_

00 o

CM

n t o

I 1 I

T3

<U .aU

0

•uM

<up.u

T3M

O

1u

oo H

St

>-< fi

a) 4J

4-1O

Si

4J00

(3

<u a 1—

1

§ % (13 00

M-l

>4-( 03 o 0

4J •H 3 4J

CJ RJM

a) si <U u 43

4-1 00c C

•rl OJ C > H rl 0 t>0 H

tu —

1

4J i—

1

CJ OJ -a o

u

o

>u H

Si cfl

n) C M H

00o CJ s XI o o

Z3 9

01

00

•r-l

20
Source Sacbaa Maamramam Secton

DEVICE UNOER

TEST

SIGNAL GENERATOR

.1—18

GHz

Frequency

Control

PORT 2

S-parameter Test Set

COMPLEX

RATIO DETECTOR

S-oorameter Meet

INSTRUMENT

INTERFACE

"Real Time"

0;ta ^ DISPLAY

Data

Output

Computer Display Data

Computer Section COMPUTER

PERIPHERALS

Figure 3. Functional diagram of an automatic network analyzer

(ANA) system.

21
22
3 QO

I— Q

Q_ LU

LU CO go cc=3

GO I—q:

LU X Q. t—

C3 cc Z LU

LU O

_l CC

LU O

3 LUx

U_ i—

i

o x o oo

4.0

3.0

2.0

/

TE

jl M0D E 0 p THE FULL GUIDE

1 .0

1 1 1 1 — MITTRA & ITOH'S SOLUTION

APPROXIMATE SOLUTION FOR SMALL GAP

(g/a «1)

4.0

3.0

2.0

1 . 0

a = b = 1 H9

2w

2a

2b

UNPERTURBED

TE 10 MODE

1 .0

•re 5. Comparison of Mittra and Itoh's exact solution and our approximate solution for

the cutoff wavelength of the first higher-order perturbed mode in a TEM cell.

23
cuca

•H

cu T3O0

"J

CU uu

cuexn

01 13Ho1L4

CU 43

ao

•HX.u

tou

fl

aj &u

of

co

CD *Huc

cu3

cr

cu fr

4-1 d

cd30

COcuu • -a CO

CU I—1 M rH 3 CU

co

u

CO

CUE H

-aC c/^ CO mz

T3

cu 4-1 4J c

CO 0) H (J 3 OJ CJ <-Lj i—1

«-l

CO •H o T3

43

CO H

-e- cc

-D

II

CM CD I CM CO I C71 cc ool t= c

11

— CM .—i CC

IU

—

L. *— OO MO 0 on on

O MO LA

Ul

N 4-J cu 4-1 MO

— -3- MO CM r-» CM r-~ on

L- Si

— pa LA -a- -3- p-- -3- OO on MO

I4_ -—

OO MO LA CM 00 CM PA

Ul

<L) U

<4- 1 CM 1—

—

MO LA

<1

— 1

V -—

L

Ul

N cf

<u

x O- r~«

i_ 2;

x

<+-

—

LU 1 ' O 0 0 O LA

O 1.

Lf\ -3- -3- P«- 1 OO

O on

OO mo LA CM CM -3" MO

*—

V)

cu CM

— -3- PA -3- vD -3"

u 1 M— 1 — MO MO MO CN

—V

X

1/1 0

CD

N l_ -3- MO rA CM PA 00

— PA -a1- X a. M- Cl vO O •— P-. OO 00 CM CM r— < pa MO CO OO r-» on O P~

00 MO LA CM 00 CM -3"

co

O J- OO on MO CM on

u

ON 4-> CM LA CTl CO PA on on P-v

4-1 PA LTV 01 0 LA LA p^ LA

31 P-~ LA LA CM p-- CM PA '

an

LA

u CM PA LA MO LA PA CM PA PA

y-< _

r MO MO MO -a- -3-

X0

1— -3" pa

O -3- -3- MO 0 ISI Cl

UX Q- CM LA on 0 PA

O CM 0 >*- 21 < PA vO PA -a* p-. 00 MO

p^ MO LA CM OO CM PA

cc CM

II

O II -eII l/l CU 0 L. u on

CO Q. U. 3O E 1- u CJ O

T0| CD

3l fO

CM

U

O

PA

O

PA

O

PA

LAI

-3"

OO

vD

O

LA

O

on o

-aCM CM CM MO MO PA CM P^ P^ P^ P^ P~ 00 00

24
NRS-114A (REV. 7-73)

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBSIR 76-841

2. Gov't Accession

No.

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

An Analytical and Experimental Determination of the

Cutoff Frequencies of Higher-Order TE Modes in a TEM Cell

5. Publication Date July 1976

6. Performing Organization Code

276,06

7. AUTHOR(S)

John C. Tippet, David C. Chang, and Myron L. Crawford

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No. 2766281

11. Contract /Grant No.

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP)

Electronics Systems Division (AFSC1

Hanscom Air Force Base f Mass* 01731

13. Type of Report & Period Covered

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

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

bibliography or literature survey, mention it here.) In addition to the TEM mode, higher order TE and TM modes can

propagate in a TEM cell. Since only the TEM mode is desired, the

higher-order modes restrict the useful frequency range of the TEM

cell. In this paper, the cutoff frequency of the first higherorder mode is obtained both analytically and experimentally for

several TEM cells of differing geometry. The difference between

the experimental and theoretical results is shown to be only a few

percent. In addition the field distribution of the first higherorder mode is found explicitly.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons

Shielded strip line; TEM cell.

18. AVAILABILITY [j£ Unlimited 19. SECURITY CLASS

(THIS REPORT)

21. NO. OF PAGES

For Official Distribution. Do Not Release to NTIS

UNCL ASSIFIED 28

_

Order From Sup. of Doc, U.S. Government Printing Office Washineton. D.C. 20402. SD Cat. No. C13

20. SECURITY CLASS

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22. Price

IX X Order From National Technical Information Service (NTIS)

Springfield, Virginia 22151 UNCLASSIFIED $4.00

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