Page 334 - Engineering Electromagnetics, 8th Edition

316 ENGINEERING ELECTROMAGNETICS

is often true in practice. Before we apply these conditions, Eq. (41) can be written in
the form:

γ = α + jβ = [(R + jωL)(G + jωC)] 1/2

1/2 1/2
√ R G
= jω LC 1 + 1 + (50)
jωL jωC
The low-loss approximation then allows us to use the first three terms in the binomial
series:
√ . x x 2
1 + x = 1 + − (x 1) (51)
2 8
We use (51) to expand the terms in large parentheses in (50), obtaining:
R R 2 G G 2
. √
γ = jω LC 1 + + 1 + + (52)
2
2
j2ωL 8ω L 2 j2ωC 8ω C 2
2
2
All products in (52) are then carried out, neglecting the terms involving RG , R G,
2
2
and R G ,as these will be negligible compared to all others. The result is
1 R G 1 R 2 2RG G 2
. √
γ = α + jβ = jω LC 1 + + + 2 2 − + 2
j2ω L C 8ω L LC C
(53)
Now, separating real and imaginary parts of (53) yields α and β:
. 1 C
L
α = R + G (54a)
2 L C
and

2
. √ 1 G R
β = ω LC 1 + − (54b)
8 ωC ωL
We note that α scales in direct proportion to R and G,aswould be expected. We
also note that the terms in (54b) that involve R and G lead to a phase velocity,
ν p = ω/β, that is frequency-dependent. Moreover, the group velocity, ν g = dω/dβ,
will also depend on frequency, and will lead to signal distortion, as we will explore in
Chapter 12. Note that with nonzero R and G, phase and group velocities that are
constant with frequency can be obtained when R/L = G/C, known as Heaviside’s
√
.
condition.In this case, (54b) becomes β = ω LC, and the line is said to be distor-
tionless. Further complications occur when accounting for possible frequency depen-
dencies within R, G, L, andC. Consequently, conditions of low-loss or distortion-free
propagation will usually occur over limited frequency ranges. As a rule, loss increases
with increasing frequency, mostly because of the increase in R with frequency. The
nature of this latter effect, known as skin effect loss, requires field theory to understand