Page 478 - Engineering Electromagnetics, 8th Edition

460 ENGINEERING ELECTROMAGNETICS

Finally, using the capacitance and the external inductance expressions, we obtain a
value for the characteristic impedance,

L ext 1 µ −1
Z 0 = = cosh (d/2a) (24)
C π
13.1.5 Two-Wire (Low Frequencies)
At low frequencies where a uniform current distribution may be assumed, we again
must modify the L and R expressions, but not those for C and G. The latter two are
again expressed by (20) and (22):
π
C =
−1
cosh (d/2a)
πσ
G =
−1
cosh (d/2a)
The inductance per unit length must be increased by twice the internal inductance of
a straight round wire,
µ 1

−1
L = + cosh (d/2a) (25)
π 4
The resistance becomes twice the dc resistance of a wire of radius a, conductivity σ c ,
and unit length:
2
R = 2 (26)
πa σ c
D13.3. The conductors of a two-wire transmission line each have a radius of
7
0.8 mm and a conductivity of 3 × 10 S/m. They are separated by a center-to-

center distance of 0.8 cm in a medium for which = 2.5,µ r = 1, and σ =
r
4×10 −9 S/m. If the line operates at 60 Hz, find: (a) δ;(b) C;(c) G;(d) L;(e) R.
Ans. 1.2 cm; 30 pF/m; 5.5 nS/m; 1.02 µH/m; 0.033 /m

13.1.6 Microstrip Line (Low Frequencies)
Microstrip line is one example of a class of configurations involving planar conduc-
tors of finite widths on or within dielectric substrates; they are usually employed
as device interconnects for microelectronic circuitry. The microstrip configuration,
shown in Figure 13.5, consists of a dielectric (assumed lossless) of thickness d and of
permittivity = r 0 , sandwiched between a conducting ground plane and a narrow

conducting strip of width w. The region above the top strip is air (assumed here) or
a dielectric of lower permittivity.
The structure approaches the case of the parallel-plate line if w d. In a
microstrip, such an assumption is generally not valid, and so significant charge den-
sities exist on both surfaces of the upper conductor. The resulting electric field, origi-
nating at the top conductor and terminating on the bottom conductor, will exist within