Computer aided analysis of characteristic impedance of rectangular coaxial lines

Computers & Elect. Engng Vol. 16, No. 2, pp. 95-98, 1990 Printed in Great Britain. All rights reserved

COMPUTER IMPEDANCE

AIDED OF

0045-7906/90 $3.00+ 0.00 Copyright © 1990 Pergamon Press plc

ANALYSIS

OF

RECTANGULAR

CHARACTERISTIC COAXIAL

LINES

HASSAN A. KALHOR Electrical Engineering Department, State University of New York, New Paltz, NY 12561, U.S.A.

(Receh~ed 10 January 1990; in revised 20 March 1990; accepted I 1 April 1990) Abstract--The characteristic impedance of a coaxial line with rectangular conductors and a homogeneous dielectric is computed by a numerical technique. Results are compared against those of a special line for which analytical solution is possible, Good agreement establishes the adequacy and the accuracy of the proposed computer model and numerical technique.

1. I N T R O D U C T I O N The characteristic impedance of coaxial lines with polygonal and circular conductors and homogeneous dielectrics is of much interest in transmission of communication signals. Very few coaxial lines which have simple geometrical shapes can be solved analytically. Many investigators have devised analytical methods based on transformations and numerical techniques to calculate parameters of coaxial lines of more complex geometrical forms [1-5]. In this paper a simple numerical analysis technique is described that can solve coaxial lines of arbitrary geometry. The problem is formulated into an integral equation for the charge distribution on the conductors. The integral equation is solved numerically by the method of moments [6] to obtain the charge distribution and the line capacitance, from which the characteristic impedance is calculated. This general method is then applied to coaxial lines with rectangular conductors. A special configuration of this line namely one with square conductors of 2:1 side length ratio lends itself to exact analytical solution [1]. Our numerical results are compared against analytical results for this special case to test the convergence and accuracy of our numerical method. Numerical results are presented for a variety of coaxial lines and microstrip lines of interest.

2. M A T H E M A T I C A L

FORMULATION

AND

SOLUTION

Consider a coaxial line with rectangular conductors and filled with a homogeneous lossless dielectric, as shown in Fig. 1. To calculate the characteristic impedance of the coaxial line, we employ a two-dimensional Green's function which is the electric potential due to uniform line charge. Consider a uniform line charge of 1 C/m located at ~' in a homogeneous lossless medium with parameters E and #0- The potential at any point ~, G(p), is (1) vhere/5 0 is a constant vector depending on the zero potential reference chosen. For our purpose 'e choose I~01 to be unity for simplification. To calculate the potential at every point ~ within our ~axial line, we use this Green's function in the following form v(~)=~

,(lnl~-~'l)psdl

(2)

~.c is the periphery of the inner and the outer conductors and Ps is the as yet unknown charge y on the conductor surfaces. 95

96

HASSAN A. KALHOR

[b ~r

40

O

Fig. 1. T h e c o a x i a l line u n d e r c o n s i d e r a t i o n .

To solve the integral equation (2) numerically, we divide the peripheries of the two conductors into a total of N equal segments, each having a length of Ac;. Then 1

N

~ lnl17 -~;Ips.;Aci

v(fi) = ~

i=l

ps.;Ac; is the total charge per unit length of segment i, and we call it q; 1

v(~) = ~

N

;~t ln1¢5 - ~5~1q,.

(3)

This equation has N unknowns q;. These can be calculated by forcing the equation to hold at N different points pj v (fly) = 2--~n E

l n l f i j - fi;Iq;.

(4)

i=1

This is a matrix equation of the form [V] = [A ][q]

(5)

in which V is the potential at points on the boundaries; A is a matrix of known coefficients and q can, therefore, be calculated by inversion of an N x N matrix [q] = [A ]-'[ V].

(6)

The total charge on the inner conductor Q can then be calculated. If the inner conductor contains segments 1 to M, we get M

Q = ~ q~.

(7)

i=l

To calculate the characteristic impedance, we use the two equations for characteristic impedance and the phase velocity in terms of the line inductance and capacitance

(8) (9) Multiplying (8) by (9) we get 1 Zov = or

1 Z o --

Cv

Z o - 3 x 108C

(lo)

Characteristic impedance of rectangular coaxial lines

I

4O

u~

39

--O~O~O

u~ 49 co 48

\

38

37 ~36

°"""~ °~ • ~

T 0

I 20

I 40 NO.

•

I 60 of

o--

•

h 80

I 100

97

./

~" 46 ~ 45 "~4/

I 120

I 600

e J

.J

"°

I 800

segments

I 1OOO

I 1200

I 1400

A-/~m

Fig. 2. Variation of the characteristic impedance with number of segments for a coaxial line with square conductors of side ratios 2 : 1 .

Fig. 3. Variation of the characteristic impedance of a rectangular coaxial line having a = 3 0 0 / a m , b = 1/am, B = 5 0 9 / a m , Er = 3.75, w i t h A.

where C is the line capacitance which is easily found from

C =--a

(11)

v0

where Q is the total charge per meter length o f the inner conductor given by (7) and V0 is the potential o f the inner conductor, above the outer conductor. 3. N U M E R I C A L

RESULTS

AND

DISCUSSION

To test the accuracy of the proposed techniques, results were compared against those of Terakado [1] which are exact for a square coaxial line with ratio o f its sides equal to 2. For this line with air dielectric, the exact characteristic impedance is 36.8368 Q. We have plotted our calculated characteristic impedance versus the number o f segments used, in Fig. 2. As is evident, the impedance converges to the exact value very fast. This establishes the accuracy and the adequacy o f the method. Next, a line having a = 300 # m , b = l # m , B = 509 # m , Er = 3.75 was analyzed for different values o f A. This models a strip metallization between two flat ground planes on each side o f the substrate, as A gets large. The variation o f the characteristic impedance with A is shown in Fig. 3. As is also evident, the characteristic impedance of this line approaches a limit slightly below 50 f~. Any further increase in A, beyond A = 1200 # m , does not affect the capacitance or the inductance and consequently does not affect the impedance. For this particular substrate, to design lines with higher characteristic impedance, we have to reduce the metallization thickness b or the strip width a. Results for a line having a = 300 p m, b = 0.5 # m , B = 509 # m , /~r 3.75 are shown in Fig. 4, for different values of A. The reduction of the metallization thickness has led to slightly higher impedances. To see the effect o f strip width on the impedance, a line with a = 250/~m, b = 1/~m, B = 509 ~tm, Er = 3.75 was analyzed for different values of A. Larger values of impedance are then obtained. =

54 .t:: 52 o I

I

=u 5o c 49 o

f b-

E48 . /

o

r

50 c o 4s 4a

O~O

oj ° ~

47 46

I 600

51

./

,n,J

° ~

47

46 I 800

I

I

I

I

I

I

I

I

1000

1200

14OO

600

800

1000

1200

1400

A-/~m

A-/~m

Fig. 4. Variation of the characteristic impedance of a rectangular coaxial line having a = 3 0 0 # m , b = 0 . 5 / a m ,

Fig. 5. Variation of the characteristic impedance of a rectangular coaxial line having a = 2 5 0 # m , b = l/am,

B = 5 0 9 / a m , Er = 3.75, w i t h A.

B = 5 0 9 / a m , Er = 3.75, w i t h A.

98

HASSANA. KALHOR

In conclusion, we have presented a simple n u m e r i c a l m e t h o d which is c a p a b l e o f calculating the characteristic i m p e d a n c e o f r e c t a n g u l a r coaxial lines as well as that o f strip lines with a h o m o g e n e o u s dielectric. F r o m the c o m p a r i s o n o f Figs 3 a n d 4 it is c o n c l u d e d that for small thicknesses the m e t a l l i z a t i o n thickness does n o t play an i m p o r t a n t role in d e t e r m i n i n g the characteristic impedance. F o r simplifying the analysis o f the strip lines filled with two different dielectrics, therefore, we assume the strips have negligible thickness. T h e G r e e n ' s function given by (1) is for a h o m o g e n e o u s m e d i u m a n d is not suitable for these p r o b l e m s . T h e a p p r o p r i a t e G r e e n ' s function can be f o u n d by representing the p o t e n t i a l in each region by a c o m p l e t e infinite series and enforcing the b o u n d a r y c o n d i t i o n s at the interface. The m e t h o d described in this p a p e r can then be a p p l i e d to the i n h o m o g e n e o u s strip line as well. REFERENCES 1. Terakado Ryuiti, Exact wave impedance of coaxial regular polygonal conductors. IEEE Trans. Microwave Theory Tech., Vol. MTT-33, Feb. (1985). 2. Weigan Lin, Polygonal coaxial line with round center conductor. IEEE Trans. Microwave Theory Tech., Vol. MTT-33, June (1985). 3. H. E. Green, Higher-order mode cutoff in polygonal transmission lines, analytical solution. IEEE Trans. Microwave Theory Tech., Vol. MTT-33, Jan. (1985). 4. Sheng-Gen Pan, Characteristic impedance formulation for coaxial systems consisting of irregular outer conductor with circular inner conductors. IEEE Trans. Microwave Theory Tech., Vol. MTT-35, Jan. (1987). 5. Sheng-Gen Pan, Characteristic impedance of a coaxial system consisting of circular and non-circular conductors. 1EEE Trans. Microwave Theory Tech., Vol. MTT-36, May (1988). 6. R. F. Harrington, Field Computation by Moment Method. Krieger, New York (1985). AUTHOR'S BIOGRAPHY Hassan A. Kalhor--Hassan A. Kalhor obtained his Ph.D. degree in electrical engineering from the University of California

in Berkeley in 1970. He taught at Shiraz University in Iran from 1970 to 1986. Since 1986 he has been a professor of electrical engineering at the State University of New York in New Paltz. His areas of interest are electromagnetics and energy conversion. Dr Kalhor is a member of Sigma-Xi and a senior member of the IEEE.