Page 480 - Engineering Electromagnetics, 8th Edition

462 ENGINEERING ELECTROMAGNETICS

because it provides a way of unifying the effects of the dielectric and the conduc-
tor geometry. To see this, consider the two extreme cases involving large and small
width-to-height ratios, w/d.If w/d is very large, then the line resembles the parallel-
plate line, in which nearly all of the electric field exists within the dielectric. In this
.
case r,eff = r .On the other hand, for a very narrow top strip, or small w/d, the
dielectric and air regions contain roughly equal amounts of electric flux. In that case,
the effective dielectric constant approaches its minimum, given by the average of the
two dielectric constants. We therefore obtain the range of allowed values of r,eff :
1
( r + 1) < r,eff < r (29)
2
The physical interpretation of r,eff is that it is a weighted average of the dielectric
constants of the substrate and air regions, with the weighting determined by the extent
to which the electric field fills either region. We may thus write the effective dielectric
constant in terms of a field filling factor, q, for the substrate:
r,eff = 1 + q( r − 1) (30)

where 0.5 < q < 1. With large w/d, q → 1; with small w/d, q → 0.5.
Now,thecharacteristicimpedancesoftheair-filledlineandthelinewithdielectric
substrate are, respectively, Z 0 air = √ L ext /C 0 and Z 0 = √ L ext C. Then, using (28), we
find
Z air
Z 0 = √ 0 (31)
r,eff
A procedure for obtaining the characteristic impedance would be to first evaluate the
air-filled impedance for a given w/d. Then, knowing the effective dielectric constant,
determine the actual impedance using (31). Another problem would be to determine
the required ratio w/d for a given substrate material in order to achieve a desired
characteristic impedance.
Detailed analyses have led to numerous approximation formulas for the evalua-
air
tion of r,eff , Z , and Z 0 within different regimes (again, see Reference 2 and the ref-
0
erences therein). For example, with dimensions restricted such that 1.3 < w/d < 3.3,
applicable formulas include:
 

d d 2 w
Z 0 air .  w + 16 w + 2  d < 3.3 (32)
= 60 ln 4
and
. r + 1 r − 1 d −0.555 w
r,eff = + 1 + 10 > 1.3 (33)
2 2 w d
Or, if a line is to be fabricated having a desired value of Z 0 , the effective dielectric
constant (from which the required w/d can be obtained) is found through:
. w
r,eff = r [0.96 + r (0.109 − 0.004 r ) (log (10 + Z 0 ) − 1)] −1 > 1.3 (34)
10
d