Page 330 - Engineering Electromagnetics, 8th Edition

312 ENGINEERING ELECTROMAGNETICS

present example) will usually be used for the voltage or current amplitudes, with the
understanding that these will generally be complex (having magnitude and phase).
Two additional definitions follow from Eq. (34). First, we define the complex
instantaneous voltage as:

e
V c (z, t) = V 0 e ± jβz jωt (35)
The phasor voltage is then formed by dropping the e jωt factor from the complex
instantaneous form:

V s (z) = V 0 e ± jβz (36)

The phasor voltage can be defined provided we have sinusoidal steady-state
conditions—meaning that V 0 is independent of time. This has in fact been our assump-
tion all along, because a time-varying amplitude would imply the existence of other
frequency components in our signal. Again, we are treating only a single-frequency
wave. The significance of the phasor voltage is that we are effectively letting time
stand still and observing the stationary wave in space at t = 0. The processes of
evaluating relative phases between various line positions and of combining multiple
wavesis made much simpler in phasor form. Again, this works only if all waves under
consideration have the same frequency. With the definitions in (35) and (36), the real
instantaneous voltage can be constructed using (34):
1
V(z, t) =|V 0 | cos[ωt ± βz + φ] = Re[V c (z, t)] = V c + c.c. (37a)
2
Or, in terms of the phasor voltage:

1
V(z, t) =|V 0 | cos[ωt ± βz + φ] = Re[V s (z)e jωt ] = V s (z)e jωt + c.c. (37b)
2

In words, we may obtain our real sinusoidal voltage wave by multiplying the phasor
voltage by e jωt (reincorporating the time dependence) and then taking the real part of
the resulting expression. It is imperative that one becomes familiar with these relations
and their meaning before proceeding further.

EXAMPLE 10.1
Twovoltage waves having equal frequencies and amplitudes propagate in opposite
directions on a lossless transmission line. Determine the total voltage as a function
of time and position.
Solution. Because the waves have the same frequency, we can write their combina-
tion using their phasor forms. Assuming phase constant, β, and real amplitude, V 0 ,
the two wave voltages combine in this way:

V sT (z) = V 0 e − jβz + V 0 e + jβz = 2V 0 cos(βz)