Page 476 - Engineering Electromagnetics, 8th Edition

458 ENGINEERING ELECTROMAGNETICS

2
2
2
center conductor is πa and that of the outer is π(c −b ). Adding the two resistance
values, we have
1 1 1
R = 2 + 2 2 (16)
σ c π a c − b
Only one of the four parameter values remains to be found, the inductance per unit
length. The external inductance that we calculated at high frequencies is the greatest
part of the total inductance. To it, however, must be added smaller terms representing
the internal inductances of the inner and outer conductors.
At very low frequencies where the current distribution is uniform, the internal
inductance of the center conductor is the subject of Problem 43 in Chapter 8; the
relationship is also given as Eq. (62) in Section 8.10:
µ
L a,int = (17)
8π
The determination of the internal inductance of the outer shell is a more difficult
problem, and most of the work was requested in Problem 36 in Chapter 8. There,
we found that the energy stored per unit length in an outer cylindrical shell of inner
radius b and outer radius c with uniform current distribution is
µI 2 4c 2 c
2
2
W H = b − 3c + ln
2
16π(c − b ) c − b 2 b
2
2
Thus the internal inductance of the outer conductor at very low frequencies is
2
µ 2 2 4c c
L bc,int = b − 3c + ln (18)
2
2
8π(c − b ) c − b 2 b
2
At low frequencies the total inductance is obtained by adding (11), (17), and (18):
µ b 1 1 2 2 4c 2 c
L = ln + + b − 3c + ln (19)
2
2
2
2π a 4 4(c − b ) c − b 2 b
13.1.3 Coaxial (Intermediate Frequencies)
There still remains the frequency interval where the skin depth is neither very much
larger than nor very much smaller than the radius. In this case, the current distribution
is governed by Bessel functions, and both the resistance and internal inductance are
complicated expressions. Values are tabulated in the handbooks, and it is necessary to
use them for very small conductor sizes at high frequencies and for larger conductor
sizes used in power transmission at low frequencies. 1

1 Bessel functions are discussed within the context of optical fiber in Section 13.7. The current
distribution, internal inductance, and internal resistance of round wires is discussed (with numerical
examples) in Weeks, pp. 35–44. See the References at the end of this chapter.