Page 331 - Engineering Electromagnetics, 8th Edition

CHAPTER 10 Transmission Lines 313

In real instantaneous form, this becomes

V(z, t) = Re[2V 0 cos(βz)e jωt ] = 2V 0 cos(βz) cos(ωt)
We recognize this as a standing wave, in which the amplitude varies, as cos(βz), and
oscillates in time, as cos(ωt). Zeros in the amplitude (nulls) occur at fixed locations,
z n = (mπ)/(2β) where m is an odd integer. We extend the concept in Section 10.10,
where we explore the voltage standing wave ratio as a measurement technique.

10.6 TRANSMISSION LINE EQUATIONS AND
THEIR SOLUTIONS IN PHASOR FORM
We now apply our results of the previous section to the transmission line equations,
beginning with the general wave equation, (11). This is rewritten as follows, for the
real instantaneous voltage, V(z, t):

2 2
∂ V = LC ∂ V + (LG + RC) ∂V (38)
∂z 2 ∂t 2 ∂t + RGV
We next substitute V(z, t)asgiven by the far right-hand side of (37b), noting that
the complex conjugate term (c.c.) will form a separate redundant equation. We also
use the fact that the operator ∂/∂t, when applied to the complex form, is equivalent
to multiplying by a factor of jω. After substitution, and after all time derivatives are
taken, the factor e jωt divides out. We are left with the wave equation in terms of the
phasor voltage:

2
d V s 2 (39)
dz 2 =−ω LCV s + jω(LG + RC)V s + RGV s
Rearranging terms leads to the simplified form:

2
d V s 2 (40)
dz 2 = (R + jωL) (G + jωC) V s = γ V s

Z Y
where Z and Y,as indicated, are respectively the net series impedance and the net
shunt admittance in the transmission line—both as per-unit-distance measures. The
propagation constant in the line is defined as
√

γ = (R + jωL)(G + jωC) = ZY = α + jβ (41)
The significance of the term will be explained in Section 10.7. For our immediate
purposes, the solution of (40) will be

− +γ z
+ −γ z
V s (z) = V e + V e (42a)
0
0