NAT L INST, OF STAND 1 TECH RT C

AlllOM DEaoMe

NBSIR 78-873

NBS

PUBLICATIONS

HIGHER ORDER MODES IN RECTANGULAR COAXIAL LINE

WITH INFINITELY THIN INNER CONDUCTOR

John C. Tippet

David C. Chang

Department of Electrical Engineering

University of Colorado

Sponsored by:

Electromagnetics Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80303

100

.U56

78-873

1978

c. 2

March 1978

Ot Standajroi LIRRAHT

IMBSIR 78-873

OCT 5 1981

HieilER ORDER MODES IN RECTANGULAR COAXIAL LINE

WITH INFINITELY THIN INNER CONDUCTOR

John C. Tippet

David C. Chang

Department of Electrical Engineering

University of Colorado

Sponsored by:

Electromagnetics Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80303

March 1978

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

Sidney Harman, Under Secretary

Jordan J, Baruch, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler. Director
1
CONTENTS

Page

ABSTRACT 1

I. INTRODUCTION 1 II. FORMULATION 3 III. EXTRACTION OF THE SINGULAR PARTS OF THE KERNEL FUNCTIONS 5 IV. SOLUTION OF THE SINGULAR INTEGRAL EQUATIONS 8 V. DERIVATION OF THE SECULAR EQUATIONS 11

VI. CONCLUSIONS 15 VII. ACKNOWLEDGMENTS 17 VIII. REFERENCES 18

APPENDIX 1, EXPRESSION FOR C.'. IN TERMS OF P . . 21

APPENDIX 2. EVALUATION OF THE CANONICAL INTEGRALS 27 APPENDIX 3. DERIVATION OF THE INFINITE MATRIX EQUATIONS 30

iii

HIGHER ORDER MODES IN RECTANGULAR COAXIAL LINE

WITH INFINITELY THIN INNER CONDUCTOR

JOHN C. TIPPET, Student Member, IEEE, and

DAVID C. CHANG, Senior Member, IEEE

ABSTRACT

The singular integral equation approach is used to derive

the secular equations for both TE and TM waves in a rectangular

coaxial line with zero thickness inner conductor. Approximations

for the secular equations are found that reduce to simple expressions in terms of well-known special functions (elliptic integrals).

When the strip width is exceedingly small or nearly equal to the

width of the outer conductor, cloSed form expressions for the cutoff frequencies can be found by replacing the elliptic integrals

by their asymptotic forms for modulus either near zero or one.

Key words: Higher order modes; rectangular coaxial line; striplines.

I. INTRODUCTION

In many applications involving rectangular coaxial waveguiding structures

an understanding of the propagation characteristics of higher order modes is

just as important as that of the fundamental TEM mode. One such application

is that of the so-called "TEM cell" which is used in some EMI measurement

systems [1]. This device consists of a section of rectangular coaxial line

which is used as a transducer for coupling EM energy from a device under

test into the TEM mode of the transmission line. The useful frequency range

of the TEM cell is limited by the cut-off frequencies of the higher order

modes. Thus, in order to design a cell for use over a specified frequency

range, these cut-off frequencies must be known.

This work was supported by the U.S. Department of Commerce, National Bureau

of Standards, under Contract No. NOAACST-8393.

The authors are with the Electromagnetics Laboratory, Department of Electrical

Engineering, University of Colorado, Boulder, CO 80309.
The device analyzed in this paper consists of a zero thickness metal

plate located inside a rectangular waveguide. The inner conductor may be

offset vertically from its central position but is located symmetrically

about the y-axis. Both the inner and outer conductor are perfectly conducting

and the medium separating the two conductors is a homogeneous dielectric. A

cross-sectional view of the rectangular line is shown in Fig. 1.

Although the transmission line properties of the TEM mode in these

structures have been extensively studied, comparitively little work has

been done in analyzing the higher order mode structure. One can find in the

literature, however, this problem [2-4] as well as related problems, such

as shielded strip lines [5-7], various coupling configurations [8-9], rectangular lines with thick inner conductors [10-13], and ridge waveguides

[14-16] analyzed using a variety of techniques. These include functional

equation techniques, finite difference techniques, mode matching techniques,

integral equation techniques, and methods based on transmission line theory.

The purpose of this paper is to obtain a relatively simple, closed form

expression for the characteristic equation for both TE and TM modes which is

valid for arbitrary strip widths. Thus, very little computer programming is

necessary to calculate the cut-off frequencies. In addition, approximate

solutions for either small or large strip widths can be obtained without

resorting to any numerical analysis. This is achieved by using the

singular integral equation technique similar to that used in waveguide

diaphragm problems [17]. This method has the advantage of handling the

edge condition exactly and eliminates the problems encountered in any

numerical solution associated with the discontinuities of the fields near

the sharp edges of the inner conductor. More specifically, the problem

is formulated using an integral equation—Green's function type of approach.
The singular part of the kernel of the resulting integral equation is

extracted (as is done, e.g., in [18-20]) and the nonsingular part of the

kernel that remains is expanding in terms of Chebyshev polynomials as

suggested by [19]

II. FORMULATION

The cut-off frequencies of the higher-order modes of a rectangular

coaxial line are just the eigenvalues K of the reduced wave equation

(V2 + k2)

z

z

= 0

(1)

where

V2 = 92/8x2 + 32/ay2

and the unknowns satisfy the following boundary conditions on the metal vralls

and

an =0

n z

E = 0.

z

In the first case, represents the z-component of the magnetic field of a

TE mode; and in the second case, E^ represents the z-component of the electric

field of a TM mode.

In order to solve this eigenvalue problem, we will convert (1) into an

integral equation. This is accomplished by expanding „ into a complete set

^ 7.' of basis functions appropriate to each of the subregions above and below the

strip (j=l,2, respectively), as

(j = 1.2)

„(J)- mn ^mn

= I m,n 0 B^j>

z mn ^mn

3
where the A ^ ' s and B ^-^^'s are as yet undeterminded coefficients, mn mn

For our geometry, 'I'^j^^^''^ ^^<^ Vn^"'^ just the TE and TM basis functions

of a rectangular waveguide of height b^ , i.e.,

^^^(^)(x.y) =

and

lab.. hcos 21 ^^-^^^ cos mryb.

• r ' s 2 ab.

I J.

sin 2l ^^"-^^ sin mry

b.

These basis functions also satisfy (1) but with known eigenvalues

h

K

(j) _ mn +

They do not, except in the rather trivial case of an unperturbed rectangular

waveguide mode, satisfy the same continuity conditions as the unknowns

in the gap regions. We can still expand

H

in terms of these basis functions.

however, since "^^^ or need not be uniformly convergent at y = 0. In this

case the limit y -> 0 cannot be taken inside the appropriate expansion. See

[21] for a discussion of this phenomenon.

Using the orthogonality properties of the basis functions to solve for

the unknown expansion coefficients, applying Green's theorem, and matching

boundary conditions in the gap regions, results in the following integral

equations for the TE and TM modes, respectively.

- . . .... E^(x' ,0)G'^^(x,x')dx' = 0 (2)

and

TM

3^,E^(x',0)G (x,x')dx' = 0 (3)

4
where P denotes that the integral is to be interpreted in the principal value

TE TM

sense, E^(x,y) <^ d^R^(x,y) , and the kernels G and G which contain the

unknown eigenvalues K^^ and are given by

j=l m.n

mn mn

(4)

and

G (x,x ) = ) —

j=l ^ I m,n K

K ^ ^j^(x,y).|; ^^^(x',0) on mn •' ^mn

(j) mo

(5) y = 0

where

A. =

1

i = 0

i > 0

The synmetry about the y-axis allows the integrations appearing in (2)

and (3) to be written only over the one gap from x = w to a and permits

a decomposition of the TE and TM polarizations into even and odd modes.

This is manifest in the summations on "m" in (4) and (5). For TE modes

with odd symmetry, i.e., H^(x,y) = -H^(-x,y), (4) is summed over odd

values, while for TM modes with odd symmetry, i.e., E^(x,y) = -E^(-x,y),

(5) is summed over even values. The reverse is true for modes with even

symmetry.

III. EXTRACTION OF THE SINGULAR PARTS OF THE KERNEL FUNCTIONS

The summations on "n" in (4) and (5) can both be summed exactly using

the residue series technique [22]. The derivative in (5) can then be

taken and when evaluated at y = 0 one finds

5
-> (i

)

TF 2 h.- Am cot TTY -" f.. . G^^x,x')= I ^,^J)u,o)*J^>(x',0) (6)

and

GTM(x,x') =

2 r^j^cot ^r^j)

j = l m K_mo

where

m fr TE mo (8)

and

^m " ^ f'^TM \o J (9)

Although the summations on "m" in (6) and (7) cannot be done in

closed form, we can extract the singular part in each sum by replacing

the coefficients in front of the basis functions by their asymptotic

form for large "m", i.e.,

cot TTY (j)

2a

3'

and

r^j^cot TTF^j^ ^ m m

rb.- J.

2a

The kernels will then consist of two parts, a singular part G which

can be summed in closed form (see, e.g., [22]) and a correction series

G = G-G. Recognizing that the summations on "m" are either over even or

odd indices, four cases need to be considered. These are not independent,

however, since it is easily verified that

6
GTM(x,x') = 3^g'^^(x,x') (10)

Thus, only the kernels for the TE polarization are needed, i.e..

;te ^ 1 "odd TT Jln[tan J^|t+t'ltan ^|t-t'|] 4a I (11)

and

G^^ = i an [4 sin ^|t+t' lain ^|t-t' I ] even ir 2a '

' 2a '

' (12)

where the following change of variables was introduced

t = a-x (13)

and

t' = a-x' (14)

The singular kernels for the TM polarization can then be found using (10).

Although the correction series G cannot be summed in closed form,

they are rapidly convergent and are given as

and

G'^^(t,t') = y A cos m

m

mirt 2a cosmut'

2a

(15)

GTM(t,t') = 78 sin m

m

mirt 2a cosmTTt' 2a

(16)

where

A = I ^

Am cot iry^^^ 2a

.(j) lb.

(l-« ) (17)

« = 4- m . 1 mb

3=1 J

r (j),„^,^(j) cotfrr _ f_i

ra m (18)

7
and £ is the Kroneker delta defined as mo

6mo

'l m = 0

[o m =^ 0

IV. SOLUTION OF THE SINGULAR INTEGRAL EQUATIONS

Having extracted the singular parts of both kernels we can move the

nonsingular correction series to the right-hand sides of both integral

equations and treat these terms as forcing terras. Since the singular

parts of the kernels of (2) and (3) are related according to (10), it is

convenient to differentiate (2) with respect to x. Both equations will

then be of the same canonical form. In terms of the new variables defined

in (13) and (14), (2) and (3) can be rewritten as follows

P U(t')GTM(t,t')dt' = U(t')8 G'^^(t,t')dt' (19)

and

where

and

P I V(t')GTM(t,t')dt' = -f^ V(t')GTM(t,t')dt' (20)

U(t') = E^(x',0)

V(t') = 3^,E^(x',0)

Equations (19) and (20) are both singular integral equations. The

standard form of the singular integral equation, however, has integration

limits -1 and +1 respectively. These limits can be achieved through the

use of the Schwinger transformation [17]

8
fTTt'' = a+gu (21)

and

cos I —I = a+3v (22)

where

and

a = i [cos(^) + 1]

3 = i [cos(^) - 1]

Defining

and transforming to the new variables u and v defined in (21) and (22) we

find that (19) and (20) can be rewritten in standard form as

T [F..(u)] = H, .(v) (i,j = 1,2)

ij ij

(23)

where F^^(u) = U(t')cos

2a

/sin

7^2^u) = U(t')/sin IT t

F2-,^(u) = V(t')cos ^^*^'Vsin

2a

TTt'a

TTt'a

F22(u) = V(t')/sfn

11 H,,(v) = |%(t')9j.G^^(t,t')dt'/2sin|g

H (v) =

I U(t')3^G'^^(t,t')dtVsin

H2]^(v) = - 'v(t')GTM(t,t')dt'/2sin

TTt Ua

and H22(v) V(t')GTM(t,t')dt'/sin

TTt

9
In (23) the subscript "i" refers to the type of mode, TE or TM respectively,

and the subscript "j" refers to the type of symmetry, odd or even respectively

As shown in [17] (23) can be inverted exactly for the unknown F^j (v) as

F. (v) = —^

{C. - T^[/I^ H (u)]} (24)

where the C 's are constants to be determined.

In order to make use of (24) we need to express H. . in terms of u.

The most convenient expansion to use is one in terms of Chebyshev polynomials

of the second kind U . In terms of these polynomials H. . can be written as

n 13

H..(u) =

I P.. U (u) (25)

n=0

where the P.. 's are expansion coefficients to be determined.

To express as a function of v we can use the following identity

[23]

T^[/1=^I^ U^(u)] = -Vi(v) (26)

where T^ is the Chebyshev polynomial of the first kind.

Thus, F.. is given as

iJ „ . -,•

F..(v) = —

(C. + J P.. T^, (v)}. (27)

10
V. DERIVATION OF THE SECULAR EQUATIONS

In order to complete the solution given in (27) we need to evaluate

the unknown constants C. . and P., . We begin by first expressing C. in

terms of the P.. 's. For the TM polarization this is accomplished by

requiring

[ 3^,E^(x',0)dx' = E(a,0) - E(w,0) = 0. (28)

For the TE polarization, we substitute the solution given in (27) back

into the undifferentiated form of the integral equation, i.e., eq. (2). As

shown in Appendix 1, one then finds for C^^ the following

oo C. . = - :r-^^ y R. . P. . (29)

\j,0 m=0 ^^'"'-^^ ^J'"'

where

77 ^11, m

"

'"11, m

" 4A~ ^2n+l ^mn m n=m

00

^12, m

"

'-12,m 2A~ f ^2nVi

m n=m

and

^21, m ''21,

R„„ =0 (m > 0) ,m

''11' ''12' ^21 three canonical integrals, the recursion formulae

for which, are derived in Appendix II. For modes with odd symmetry, the

integrals I^^ and 1^2 '^^^ be expressed in terms of complete elliptic integrals

of modulus /a for the TE polarization and /-3 for the TM polarization,

while for modes with even symmetry, 1^2 contains no special functions.
p^^ and are the coefficients of the following expansions

2a

(-Tt'i ' V, 'ran ^m" cos (2^) m=0

cos(2n+l)(||-) n

and

m=0

fmrt' cos

We are now in a position to substitute our solution for F^^ (expressed

solely in terms of the unknown P.. 's), back into the definition of the

H..'s defined in (23). Upon matching coefficients of U (v) we then obtain

an infinite set of equations for the unknown P.. 's. Setting the

13,

determinant of the coefficients to zero we obtain the secular equation

from which K can be determined. The details of this procedure are outlined

in Appendix III. Four infinite sets of equations are derived of which the

following for the odd TM polarization is representative.

aB r V ^22,m 2~ ^ , ^2n+2 ^mn "^k+l.n+l ^22, k (30) n=m k=0

where s^ is the coefficient in the following expansion

sin[(n+l)—] n —^ = I

s U (v) . sin — m=0

Equation (30) is the simplest of the four equations given in Appendix III

since ~ 0 ^'^^ this case only. Fortunately, however, these equations can be

greatly simplified. In most cases, only a few of the A^'s or B^'s, given respectively in (15) and (16), are significant, and thus the infinite matrix equations

given in Appendix III reduce to equations of very small order. The number

12
of Amm's or B 's that we must keep depends upon the number of higher order

modes that we allow to propagate. We can see from (8) or (9) that once

"m" becomes large enough so that K^^ > K^^, or respectively, then Y^^'^ ^

or r^-' ^ becomes imaginary. The cotangents then in (15) or (16) can be

replaced by hyperbolic cotangents and thus the succeeding A^'s or B^'s

become negligibly small. Thus, we can truncate our infinite matrix

equation to a finite one at the sacrifice of being able to calculate all

of the higher order modes. We will only find those modes for which

or K^, < K„ where we have truncated the matrix to one of the finite

TE TM Ho

order M.

In many applications (e.g., [24]) one is only interested in the first

few higher order modes. This method is especially applicable for finding

these modes since we can usually truncate the matrices to matrices of order

one. In this case, (30) reduces to

1 = ^ B2 (31)

since s^^ = 1 and q^^ = g. Substituting for B^ from (16) we then find the

following secular equation for the odd TM modes

2 ar

I cotirr^J^ = 1-1? (32)

j=l j

where r^''^ was defined in (8). Similarly, for the even TM modes we find

J=l 2

13

VI, CONCLUSIONS

In addition to the higher-order modes whose cut-off wavelengths are

plotted in Figs. 2-4, there are also some higher-order modes which are unperturbed by the presence of the inner conductor. If b^ = b^ = b, these modes

are iust the TE „ and TM . modes of the full guide of cross-section m,2n m,2n

2a X 2b. Alternatively, we can view this system as two rectangular subwaveguides of cross-section 2a x b which are coupled through the gap. The

sub-waveguide modes can then be combined to form a system mode of the entire

structure which is either symmetric with respect to the inner conductor

(i.e., E^(x,y) = E^(x,-y) or antisymmetric. As found also in [27] only the

symmetric combination is unperturbed by the presence of the inner conductor.

If the inner conductor is offset then there exists fewer unperturbed modes.

Since the perturbed modes reduce to rectangular waveguide modes in the

limit of either zero strip width (3 -1) or zero gap (3 0) we can obtain

approximate solutions for the cut-off wavelengths by replacing the cotangents

and elliptic integrals in (32) to (34) by their asymptotic forms for beta

either near 0 or -1. One can then find that the cut-off frequencies are

perturbed as follows. For small strip widths

'TM11

l/£n 8a

TM,21

w

and

TE

11

while for small gaps one finds

15
1^

2

4

CC TM,21

and

TE,11

where

K - K"

The significance of the logarithmic perturbation of the cut-off wavelength

of the TE^^ mode for small gap and the T^-^l small strip width is

apparent in Figs. 3 and 4. For any finite gap width or strip width, respectively, the perturbation is quite significant. Thus, in contrast to the other

cases for which the perturbation is algebraic, the cut-off frequencies are

not accurately predicted by their unperturbed values. This same type of

logarithmic or algebraic behavior was found by [8] in an analysis of a related

problem of two rectangular waveguides coupled through a longitudinal slot.

In reference [11] are plotted similar curves for the case of a thick

center conductor. The curves show the same qualitative behavior as a function of strip width. In particular, the cut-off wavelength of the TE^^^^

mode for a small gap tends to be that of a "coaxial TE^q mode" in each

sub-waveguide.

Although we have only presented the numerical results for a one-term

approximation to the secular equations, higher-order solutions can be

easily found, since the canonical integrals needed in the computation of

the matrix elements are given recursively. For example, if we keep two

of the B ' s in the expression for the secular equation for the odd TM m

mode, (30) reduces to

16
The first term in (36) gives the same roots as we found earlier in our

one-term approximation as given in (31). In addition, however, we have an

extra term which gives rise to additional higher-order modes. One can see

how continuing this process would ultimately recover all of the higher-order

modes.

VII. ACKNOWLEDGMENTS

The authors would like to thank Dr. Leonard Lewin and Dr. Paul Wacker

for their critical comments and helpful suggestions and Mr. Chuck Miller

and Mr. Myron Crawford for their constant support and encouragement.

17
VIII. REFERENCES

Crawford, M. L. , Generation of standard EM fields using TEM transmission

cells, IEEE Trans. Electromagnetic Compatibility, vol. EMC-16, pp. 189-

195 (Nov. 1974).

Belousova, L. I., and Sal'nikova, L. P., Computation of the cutoff

frequencies of higher order modes in shielded sjrmmetric stripline,

Radiotekhnika (Khar'kov), No. 17, pp. 72-76 (1971).

Shestopalov, V. P., The Method of Riemann—Hilbert Problems in the

Theory of Diffraction and Propagation of Electromagnetic Waves ,

Izdat. Khar'kov. University (1971).

Belousova, L. I., and Sal'nikova, L. P., Higher wave modes in a shielded

nonsymmetrical strip line. Radio Eng. and Elect. Physics, Vol. 16,

No. 7, pp. 1199-1202 (July 1971).

Lerer, A. M. , and Mikhalevskii, V. S., Using paired integral equations in

certain problems involving the diffraction of waveguide modes by bodies

of finite size, Radiophys. Quantum Electron., vol. 18, pp. 859-865

(1975).

Mittra, R. , and Itoh, T. , A new technique for the analysis of the dispersion characteristics of microstrip lines, IEEE Trans. Microwave Theory

Tech., vol. MTT-19, No. 1, pp. 47-56 (Jan. 1971).

Davies, J. B. , and Corr, D. G., Computer analysis of the fundamental and

higher-order modes in single and coupled microstrip. Electronics Letters,

Vol. 6, No. 25, pp. 806-808 (Dec. 1970).

Garb, Kh. L. , and Fridberg, I. Sh. , Dispersional properties of a Rectangulat

waveguide, loaded in half-space by a narrow slot, in a wall of finite

thickness. Radio Eng. and Elect. Physics. Vol. 19, No. 1 (1974).

18
[9] Voytovich, N. N. , Katsenelenbaum, B. Z. , and Sivov, A. N. , The excitation

of a two-dimensional metal cavity with a small opening (slotted

cylinder), Radio Eng. and Elect. Physics, Vol. 19, No. 12, pp. 8-17

(Dec. 1974).

[10] Bezlyudova, M. M. , Cut-off wavelength of waveguide-type oscillations in

transmission lines having internal and external conductors of rectangular

cross section. Radio Eng. and ELect. Physics, Vol. 8, No. 11, pp. 1727-

1733 (Nov. 1963).

[11] W. Brackelmann, D. Landmann and W. Schlosser, Die Grenzfrequenzen von

HSheren Eigenwellen in Streif enleitungen. Arch Elek. Ubertragungstech.

,

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

[12] Gruner, L. , Higher order modes in rectangular coaxial waveguides, IEEE

Trans. MTT, Vol. MTT-15, No. 8, pp. 483-485 (Aug. 1967).

[13] Brackelmann, W. , Wellentypen auf der Streif enleitung mit rechteckigem

Schirm, Arch. Elek. Ubertragungstech., Vol. 21, No. 12, pp. 641-647

(1967).

[14] Pyle, J. R. , The cutoff wavelength of the TE^q mode in ridged rectangular

waveguide of any aspect ratio, IRE Trans. Microwave Theory Tech., Vol.

MTT-14, No. 4, pp. 175-183 (April 1966).

[15] Collins, J. H., and Daly, P., Orthogonal mode theory of single ridge

waveguide, J. Electronics and Control, vol. 17, pp. 121-129 (Aug. 1964).

[16] Cohn, S. B., Properties of ridge wave guide, Proc. IRE, pp. 783-788

(Aug. 1947).

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

[18] Chang, D. C. , Equivalent-circuit representation and characteristics of

a radiating cylinder driven through a circumferential slot, IEEE Trans.

Ant. Prop. Vol. AP-21, No. 6, pp. 792-796 (Nov. 1973).

19
[19] Gladwell, G. M. L. , and Coen, S., A Chebyshev approximation method for

microstrip problems, IEEE Trans. MTT, vol. MTT-23, No. 11, pp. 865-870,

(Nov. 1975).

[20] Chang, W. , and Otto, D. V., Current distributions on electrically thick

tubular antennas of limited length. Int. J. Electronics, Vol. 39,

No. 4, pp. 393-413 (Apr. 1975).

[21] Friedman, B. , Principles and Techniques of Applied Mathematics , New York:

Wiley, p. 271 (1956).

[22] Collin, R. E. , Field Theory of Guided Waves , New York: McGraw-Hill (1960).

[23] Tricomi, F.G., Integral Equations , New York: Interscience (1957).

[24] Tippet, J. C, Chang, D. C, and Crawford, M. L. , An analytical and experimental determination of the cutoff frequencies of higher-order TE modes

in a TEM cell. National Bureau of Standards Internal Report NBSIR 76-841

(June 1976). .

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

New York: Academic Press (1965).

[26] Byrd, P. F., and Friedman, M. D. , Handbook of Elliptic Integrals for

Engineers and Physicists , Berlin: Springer-Verlag (1954).

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

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

20
APPENDIX 1. EXPRESSION FOR C. IN TERMS OF P . . ij ij,n

We begin with the odd TM Polarization and transform (28) into an

equation in terms of u as follows

3 ,E (x',0)dx' = X z V(t')dt' =

f 1

-1

F22(u)du = 0 (1.1)

If we now substitute for Y^^iu) from (27), we obtain

^22 I , ^ ^22,

-1 /1-u n=o -1 / l-u'^ (1.2)

The right-hand side of (1.2) vanishes because of the orthogonality property

of the Chebyshev polynomials, i.e., [25]

1 T^(u) T^(u)du m n TT -1 /l-t?

2A mn m

(1.3)

where

A = m J

m=0

1 m>0

Thus,

S2 = ° (1.4)

For the even TM Polarization, we transform (1.1) into an equation in terms of

a new variable x defined as

X = cos (111) (1.5)

to obtain

21
9 ,E (x' ,0)dx' =2 F„^ (u)sin

l„ X z Jq 21 2a

dt' = ^IT 211^ 6 dX=0.

If we now substitute for F2-j^ from (27) we obtain

1 C^^dx

- I P

/(l-X^)(x''-ot) n=0 ^^'"^ /(l-x^)(x^-a)

since

= - I /(l-X^)(x^-a)

Defining as

21,

21,

'2x^-l-a

/a

dx

/(l-X^)(x^-a)

we find for C^-^ the following

"21 y p I ^21,0 nio 21,n 21,n^-l (1.

Now for the odd TE polarization we transform (2) into an equation in

terms of t and t', and without loss of generality, evaluate the resulting

equation at t = 0 to obtain

U(t')G^^(0,t')dt' = - U(t')G'^^(0,t' )dt' . (1. 0

The left-hand side (L.H.S.) of (1.7) can be transformed into an equation in

terms of the variable x defined in (1.5) as follows

1 . - . 2 TTt' 1 „ Jl- G(0,t') = - £n tan^ = - Jln^^

IT 4a TT |1+X

22
Thus,

L.H.S. = -

, F,^(u)£n

TT Iq 11

1-X

1+X

sin TTt' 2aat = —

11

2x^-l-a

U+xJ

dx

/a

If we now substitute for F^^^ from (27) we obtain

L.H.S. =

2a ( jl-X

dx

Jr^ /(l-X-^) (x'^-a)

11 ^v, 11, n n+ll n=0 '

f 2x^-l-cx

' (1.8)

Defining I, , as

11,

1 11, n TT J n

^ ^2x^-l-a in

^i+x'i li-x. dx

^(l-X^)(x^-a)

we see that (1.8) reduces to

L.H.S. = 2ae

TT f^ll ^11,0 ^ll,n^ll,n+l^ n=0

(1.9)

Now the right-hand side (R.H.S.) of (1.7) can be written in terms of u as

follows

rg sin[ J «>, R.H.S. = - F^^(u) — I A^^^^ cos[(2n+l)|^]df =

11 cos —J n=0

a3

TT I A

2a

r 1

n=0

2n+l

cos[(2n+l)|^]

If we now make use of the following expansion in terms of Chebyshev polynomials

of the first kind T

23
cos[(2n+l)^] n

=

I P T (u)

'-^ mn m m=0

(1.10)

and substitute for Fj^-j^Ci^) from (27) we find the following

R.H.S. = — I I A2^_^^

n=0 m=0

1 T (u)du CO

k=0

(1.11)

The integrations in (1.11) can be evaluated by using the orthogonality

relation given in (1.3) and thus (1.11) reduces to

V no r* R.H.S. = ag ) A„ [P C,, + 5 ) , , , p, ] 2n+l on 11 2 , ^. ll,k-l '^kn n=0 k=l

(1.12)

where 6 is the Kroneker delta. Finally (1.9) and (1.12) when equated and

no solved for C^^ gives

, CO CO n

- y P^, In j_i - T I A„ ,,(1-6 ) y P^, , , p, 11, n ll,n+l 4 ^„ 2n+l no , ^ ll,k-l '^kn n=0 n=0 k=l

'11 ^11,0 2 ^„ ^2n+l Pon

' n=0

(1.13)

For the even TE polarization we again start with (1.7), but we transform

the left-hand side of that equation into an equation in terms of u as follows

G^^(0,t') = i !in[(-2B)(l+u)]

Thus,

3g r 1 L.H.S. = -

J ^ F^2(^)^^[(-2^)(l+'j)]du

24
If we now substitute for ¥^^(u) from (27) we obtain

-1 n=0

Defining

^

r 1 T (u)«,n(l+u) =- "

du + 6 to(-2B)

we see that (1.14) reduces to

L.H.S, =

TT^ ^^12 '12,0 ^„ ^12, n -""U.n+l^

' n=0

(1.15)

Now the right-hand side of (1.7) can be written in terms of u as follows

R.H.S. = - F^^Cu) sin Y cos n=0 2n ^^Idt'

= ^ I A, F,,(u) cosP^ du.

If we now make use of the following expansion

fmrt'

= y q T (u) ^„ ^mn m

m=0

and substitute for F^^(u) from (27) we find the following

R.H.S. = ^ y J A

n=0 m=0

2n %in

1 T (u)du

-1 k=0 '

25
Again, the integrations in (1.16) can be evaluated using the orthogonality

relation given in (1.3) to obtain

R.H.S. = aB y A„ [q C, „ +

^ 2n ^on 12 no n=0 ^2.k-l (1.17)

Finally (1.15) and (1.17) when equated and solved for C^^ gives

CO n ^12,n^l2.n+l +1 ^2n(l-^no>

n=0 n=0 J-k=l ^2,k-l ' V

^12 (1.18)

lio n ^ I q 12,0 . 2n n=0

Equations (1.4), (1.6), (1.13) and (1.18) can all be expressed in one

equation as

where

:. . = - ^— y R. . P. . ij , 0 m=0

7T r ^11, m

" '11, m

" 4A~ ^ ^2n+l ^mn m n=m

^12, m

"

'"12, m 2A~ ^ ^2n'^inn m n=m

R =1

21, m 21,

and

R„T =0 (m > 0)

26
APPENDIX 2. EVALUATION OF THE CANONICAL INTEGRALS

The canonical integrals appearing in (29) are defined by

11, n 77 J n[ B /a

In 1+X

1-X /(l-X^)(x^-a)

^1 T (u))ln(l+u)

[,„ = -

"

du + 6 £n(-2B)

and

21 ,n J n ^ e

^ /a

dx

Ai-x^)(x^-a)

The evaluation of 1^^ '^^'^ found in [19] and is given by

12,

to(-B)

(-1)

n+1

n = 0

n > 0

The remaining two integrals can both be evaluated by recognizing that [25]

T (u)-T (u) = 2(l-u )U (u) n n+z. n

where

/I^ = 4 v/(i-x^)(x^-a)

and

2x2-l-a

27
Thus, if we take I.^ - I.-, . o we can convert the integral into OM for

il,n il,n+2

which the square root is now in the numerator. This integral can then be

integrated by parts to obtain the following recursion relation

2(^] (2n+2) I.. . - (2n+l) I.^ +J.,

T _ ^l-g^ il.n+l xl,n il.n /, i ox r? il,n+2 " (2n+3) " ^'"^^ ^

'

where

J =il

11, n TTg^

C 2 X -g

X cJx

and

/a

21,

is zero because the integrand of 12-^ does not contain a logarithmic

function as does

^-^i' evaluation of J^^ is easily accomplished by

substituting ^ ' "

= cos^G + asin^e

to give

4 (-1)"-^^

11, n B

The starting values for the recurrence relations given in (2.1) are

given as [26]

^11,0 = ^

I,, = i ;2[l+K(^)-E(^)] - (1+a) K(^)

11,1 3

28
and

^21,1 " i

2E(/^) - (1+a) K

where K and E are complete elliptic integrals of the first and second kind,

respectively.

29
APPENDIX 3. DERIVATION OF THE INFINITE MATRIX EQUATIONS

The procedure for deriving the infinite matrix equations for the four

cases, even and odd TE and TM, is basically the same. Therefore, we will

derive only the odd TE case and quote the results for the remaining cases.

We begin with the definition at H^^ defined in (23), i.e.,

H^-^(v) =

j U(t')3j.G^^(t,t') dt' /2sin(|J) (3.1)

^TE

From the definition of G given in (15) we can derive the following result

TE

2sin (-;;—) ^2a'^ 4a

oo

I (2n+l)A2^^, cos (g^-)

n=0

sin[(2n+l)||-] cos [(2n+l) ^]

'2a -

\- (3.2)

If we use the following expansion in (3.2),

sin [(2n+l)^]

la.

TTti ;in f^l

I r U (v)

''rt mn m

m=0

insert (3.2) into (3.1), and use the expression for H^^ given in (25) we

obtain the following

00 n

I, \l,n \^^^ = I (2n+l)A2^^^

-mn L "^^'^^^^if

)

n=0 n=0 m=0 0

cos[(2n+l) ^]

cos

^t' 2a

• dt'U (v) m

We can now equate coefficients of U (v) to obtain

n

I (2n+l)A,^^, r_ |~ V(t') cos (^)

cos[(2n+l)||^]

cos(2^J

dt'

(3.3)

(3.4)

30
If we now transform the integration in (3. A) from t' to u via (21) and use the

expansion given in (1.10), we obtain the following

\. = -r y y (2n+l)A„ r p, 11, m 4 ^ 2n+l mn "^kn

' n=m k=0

F^^(u) Tj^(u) du . (3.5)

Next, we insert the expression for

^-^-i^^") given in (27) and note that the

integration in (3.5) can be performed using the orthogonality relation

given in (1.3). Equation (3.5) then reduces to the following

. «= n (2n+l) A„ r p, , , , _

7r_g y

V 2n+l mn '^kn ll,k-l

11, m

"

8 ^ A

' n=m k=0 k

where we have defined P^^ _^ = C^^ .

The final step involves substituting for C^^ from (29) with the result that

P„ = ^ y (2n+l) A. r 11, m 8 ^ 2n+l mn n=m

n-1 2P,

^1-^no^ Jo "'^+1'- '^^'^ hl,0 k=0

^ ^ll,k+l ^11,

(3.6)

Equation (3.6) is thus the infinite matrix equation desired. The following

equations for the other three cases can be derived in an analogous manner.

12, m 2 ^ 2n+2 mn n=m

o,n+l

n 2q

tk=0

'"^^I'^-^l 'l2,k - ^^2,0 k=0

^ ^12,k+l ^12,

ag

I

n-1

(l-6_) I p,., P. 2P,

21, m 4 ^ 2n+l mn T no' , ^„ '^k+l,n 21, k R„, _ , ^„ 21,k+l 21,

n=m k=0 21,0 k=0

31
and

'22, m ^ , ^2n+2 \n \+l,n+l ^22,

' n=m k=0

where we have made use of the following expansion

. f(n+l)TTt sxn J n

= I

s U (v)

f-nt^ n nin m

sin(

IL 32
© 0

s to

CM

J2 ~

J3

CM

33

Fig. 3. Cutoff wavelength vs. strip width of the TM mode.

33

NBS-114A (REV. 7-73)

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

1. m'lM.K A riON OR RI-l^ORT NO.

NBSIR 78-873

2. Gov't Accession

No.

3. Recipient's Accession No.

4. TiTLf-: AND suBTH LiHigher Order Modes in Rectangular Coaxial Line with

Infinitely Thin Inner Conductor

5. Publication Date

•W_ -^r^U 1 Q 7 o riarcn ly / o

6. Performing Organization (..ode 276.00

7. AUTHOR(S)

John C. Tippet and David C. Chang

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAMF AND ADDRKSS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project /Task/Work Unit No.

11. Contract/Grant No. NOAACST-839 3

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

Same as #9.

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.)

The singular integral equation approach is used to derive the secular

equations for both TE and TM waves in a rectangular coaxial line with zero thickness inner conductor. Approximations for the secular equations are

found that reduce to simple expressions in terms of well-known special

functions (elliptic integrals) . When the strip width is exceedingly small

or nearly equal to the width of the outer conductor, closed form expressions

for the cut-off frequencies can be found by replacing the elliptic integrals

by their asymptotic forms for modulus either near zero or one.

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

)

Higher order modes; rectangular coaxial line; striplines.

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