Coplanar transmission lines lose energy to surface waves when the propagation constant of the surface-wave mode exceeds that of the transmission line. This happens when the substrate thickness is an appreciable fraction of a wavelength. The losses should become important in integrated circuits at near-millimeter wavelengths because it is hard to make the substrate thickness small compared to a wavelength. In this paper we have developed a theory based on reciprocity for predicting these losses. We also utilized the quasi-static approximation method to derive expressions for propagation constants and line impedances. Experimental measurements were made for the surface-wave losses in the two strip line, the two slot line and the three wire line, and the results obtained were consistent with the theory

Kasilingam, Dayalan P.

Rutledge, David B.

English

Caltech Authors - Main

C-5 
SURFACE-WAVE LOSSES OF COPLANAR TRANSMISSION LINES 
Dayalan P. Kasilingam and David B. Rutledge 
Department of Electrical Engineering 
California Institute of Technology 
Pasadena, California 91125 
Abstract. Coplanar transmission lines lose energy to 
surface waves when the propagation constant of the 
surface-wave mode exceeds that of the transmission 
line. This happens when the substrate thickness is an 
appreciable fraction of a wavelength. The losses 
should become important in integrated circuits at 
near-millimeter wavelengths because it is hard to make 
the substrate thickness small compared to a wavelength. 
In this paper we have developed a theory based on 
reciprocity for predicting these losses. We also 
utilized the quasi-static approximation method to 
derive expressions for propagation constants and line 
impedances. Experimental measurements were made for 
the surface-wave losses in the two strip line, the two 
slot line and the three wire line, and the results 
obtained were consistent with the theory, 
Integrated circuits are now eeing fabricated 
at near-millimeter wavelengths. 1- Coplanar trans-
mission lines (Fig, l) are often the most convenient 
guides at these frequencies, but it can be hard to 
make the substrate thickness small compared to a 
wavelength. This means that the lines will radiate 
into substrate modes, in constrast to the way these 
lines behave at microwave frequencies. Much infor-
mation is available on other engineering characteris-
tics of these lines,5,6 but not on surface-wave losses. 
Understanding these losses is important in designing 
integrated-circuits at these wavelengths, not only to 
minimize losses, but also because this radiation 
could be useful. In this paper we study the losses 
of various transmission lines on different substrates. 
-i.- w __.!.._ w 
---(a) (b) 
.. 
(c) 
Fig. 1 Coplanar Transmission Lines: (a) two-slot 
line (b) two-strip line (c) three wire 
1 ine. 
It is easy to understand how this radiation 
occurs. At an interface, waves propagate along metals 
at a velocity that is intermediate between the velocity 
of light in air and the velocity of light in the 
dielectric. As far as the dielectric is concerned, 
0149-645X/83/0000-0187 $01.00 © 1983 IEEE 113 
the wave is fast, and radiation will occur in the 
dielectric in the manner of a leaky wave antenna. 
The radiation peak is at an angle ~. given by 
(l) 
where k is the guide propagation constant along the 
line ana ~ is the propagation constant of the 
substrate mode, This is illustrated in Fig. 2a. In 
a thick substrate radiation is in a semi-cone of 
angle~ (Fig, 2b), This formula indicates that the 
a 
F,i g. 2 
radiation 
b 
Radiation by coplanar transmission 1 ines. 
(a) into surface-wave modes 
(b) into a thick substrate in a semi 
cone of angle 1/J. 
line will begin to suffer losses to a particular 
mode when the mode propagation constant 8 exceeds the 
guide propagation constant kz. Each mode has a 
frequency at which ~ first exceeds kz and this turn-on 
frequency is somewhat larger than the usual cut-off 
frequency, This is shown for a two strip line in 
Fig. 3. The TE1 and ™1 modes' cut-off thickness is 
indicated by the broken line. Also shown on the 
figure are the experimentally measured values of k . 
The radiation patterns of these surface waves can te 
observed by putting the transmission line on a 
quadrant of a circular disk (Fig. 4). The edge is 
tapered to prevent reflections, A vertically 
polarized receiving horn picks up the TM waves and 
a horizontally polarized horn picks up the TE waves. 
Typical surface-wave radiation patterns for a 
coplanar two strius transmission line and a coplanar 
slot line are sh~n in Fig, (5). Surface wave 7 
radiation patterns for elementary dipoles and slots 
have also been derived from reciprocity. 
We have recently developed a theory, based on 
reciprocity and the quasi-static approximation method 
which gives the surface-wave attenuation constant of 
coplanar transmission lines. In all cases we find 
simple formulas with losses propo.rtional to the 
inverse of the effective guide thickness he• The 
effective guide thickness is a term from the ray 
picture of waveguide propagation in integrated optics, 
1983 IEEE MTT-S DIGEST 
0.2 0.4 
4k0 
kd 
en_ 
.: 
c: 
0 
-(/) kz kz c: 0 
u 
c: 2k0 
.Q 
-0 Cl 
0 
0. 
0 
.... ko 0.. 
0.2 0.4 
Substrate Thickness, h/~0 
Fig. 3 Dispersion curves for surface-waves as 
a function of thickness with Er = 12. 
kd and kz are tl:le propagation constants 
in the dielectric and the line 
respectively. 
Fig. 4 Disk for measuring surface-wave losses 
for a two strip line. 
and is given by the actual thickness plus the apparent 
penetration depth of the ray (Fig. 6). At high 
frequencies the effective thickness tends towards the 
physical thic~ness of the substrate. The quasi-static 
approximation enables us to derive analytic express-
ions for the characteristic impedances of the coplanar 
transmission lines. It also gives estimates for the 
propagation constants. The_ attenuation constant due 
to TM and TE modes in a two slot line are as following 
149.6 
11+ 1/E:r 
Sin 3~J Sined 
h KK' 
e 
Sin4 Cos 2~ Sined Cos 2ed W 2 
-----ch--;-;;KK,..,,----''---...;! (:~.--) dB /m 
e d 
(2) 
(3) 
where ~ and e are those shown in Fig. 2a and Fig. 6. 
W is the widtR of the slot and K, K' are elliptic 
integrals defined by Wen,9 Fig. 7 show the losses as 
114 
1.0 X\ 
X 
~TE ... Q) 0.8 ): 0 0.. 
X , 0.6 
\ 
Q) 
N 
0 
E 
... 
0 
z 
Fig. 5 
X 
Angle, degrees 
Measured radiation pattern for a two 
strip line (strip width 2mm, strip 
spaci:1g 1 mm, 5 GHz, actual thickness 
is 12. 5 mm, tr = 12) . 
a function of width for both two slot line and two 
strip line on different dielectrics. Fig. Sa and 
Fig. 8b show the frequency dependences of the losses 
for a two slot line and a two strip line, respectively. 
Air 
Fig. 6 Effective thickness: ray picture, 
h = h + 2z . 
... 1.0 
.< 
' m 
'0 0.8 
~ 
rn 
~ 0.6 
c: 
0 
-
0.4 
0 
'0 
0 
0::: 0.2 
0 
e o 
0 o.os 0.10 0.15 0.20 0,25 0.30 
Transmission-line width, w/~d 
Fig. 7 Theoretical losses for lines with a 
strip or slot spacing equal to l/3 the 
total width. The dielectric constants 
are indicated. The substrate thickness 
is assume·d to be much greater than :\d. 
20 
E 
.......... 
Ill 
"0 
,; 40 
0 
0 
::J 
c: 
Q) 
.,_ 60 
.... 
<t 
eo 
2 
0 
E -5.0 
' Ill 
'0 
-10.0 
..: 
c: 
.!!! 
-~ -15.0 
-
-
., 
8 -20.0 
c: 
0 
; -25,0 
"' c: .,
~ -30.0 
2,0 
3 
~ 
lmm 
, 2.5cm~--
' 
' 
' 
' ', Theory 
' 
' ' 
4 
Frequency, GHz 
' 
' 
' 
' \ 
5 
' 
' 
' 
' 
' 
' 
' 
6 
----------------~ 
3,0 
Total Surfac'l. 
Wave Losses 
4,0 
Frequency, GHz 
5,0 6,0 
Fig. 8 (a) Measured losses, compared to theory 
for a two slot line. 
(b) Measured losses (x) and theoretical 
losses for two strip line. Broken 
line indicates conduction losses. 
In all the cases the attenuation constant a was 
calculated utilizing the equation 
a = Power loss/unit length 
2(Power transmitted) (4) 
The power loss per unit length was calculated using 
reciprocity. 10 In effect we relate the surface wave 
power per unit length to the line voltage (or current), 
The transmitted power may also be represented in terms 
of voltage (or current) and the characteristic imped-
ance. The characteristic impedances is found by the 
quasistatic approximation method, and are given by 
ztwo strip 
K(k) 2 
K'(k) £ +l no 
r 
ztwo "~ot K
1 (k) 2 
K(!<.;) e; +~ no 
r 
where n is the free space impedance, £ the dielectric 
constan~, and from Fig. 1. r 
k s S+2W 115 
As shown in Fig, 3, the measured values of the 
propagation constant were found to be smaller then 
their quasi-static approximation, by 5-10 percent. 
This means the wave in the transmission line propa-
gates at a velocity raster then the phase velocity 
predicted by the quasi-static approximation. A 
plausible explanation for this discrepancy is yet to 
be found. Daviesll et. al, show that the propagation 
constant and impedance of the coplanar transmission 
line are not equal to their quasi-static values when 
the substrate is of finite thickness, but the values 
we measured still do not agree with the modified 
values they obtained. Kitazawal2 et al have predicted 
that the propagation constant decreases with increases 
in metal thickness, Even this premise does not ex-
plain the magnitude of our discrepancy. 
The conduction losses for the two strip line are 
also shown in Fig, Sb, These losses are comparatively 
small in the S band. They remain small even at near-
millimeter frequencies although the surface resistance 
does increase with frequency, At 90 GHz, the conduc-
tion loss attenuation constant, ac is approximately 
45 dB/m for the line we experimented with. 
We have in this paper tried to quantifY the 
effects of surface waves in coplanar transmission 
lines, The most significant conclusion is that at 
frequencies above the "turn on11 frequencies the 
surface wave losses dominate all other loss mechamisms, 
up to submillimeter frequencies, The theory derived 
for surface wave losses also apply to lines at near-
millimeter wavelengths and the losses are therefore 
indicative of the losses one would experience at the 
higher frequencies, We have also critically examined 
the quasi-static approximation method used frequently 
in micro strip line and coplanar transmission line 
analysis, 
Acknowledgements 
This research was supported ty Army Research 
Office, 
[J,] 
[2] 
[3] 
[4] 
[5] 
[6] 
[8] 
[9] 
[10] 
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116 


Surface-Wave Losses of Coplanar Transmission Lines

11Coplanar Waveguide: A Surface Strip Transmission Line Suitable for Non Reciprocal Qyromagnetic Device Application, 11 I:Ei:Ei:Ei Tre.n&quot; Microwave Theory Tech., Vol,

A Coplanar Waveguide with thick metal casting&quot;

Infrared and Millimeter Waves

Microstrip Lines and Slotlines, Aertech House,

http://authors.library.caltech.edu/32102/1/KASims87.pdf

Surface-Wave Losses of Coplanar Transmission Lines

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