Page 329 - Engineering Electromagnetics, 8th Edition

CHAPTER 10 Transmission Lines 311

We next consider a point (such as a wave crest) on the cosine function of Eq. (27a),
the occurrence of which requires the argument of the cosine to be an integer multiple
of 2π. Considering the mth crest of the wave, the condition at t = 0 becomes
βz = 2mπ
To keep track of this point on the wave, we require that the entire cosine argument be
the same multiple of 2π for all time. From (27a) the condition becomes
ωt − βz = ω(t − z/ν p ) = 2mπ (31)
Again, with increasing time, the position z must also increase in order to satisfy (31).
Consequently the wave crest (and the entire wave) travels in the positive z direction
at velocity ν p . Eq. (27b), having cosine argument (ωt + βz), describes a wave that
travels in the negative z direction, since as time increases, z must now decrease
to keep the argument constant. Similar behavior is found for the wave current, but
complications arise from line-dependent phase differences that occur between current
and voltage. These issues are best addressed once we are familiar with complex
analysis of sinusoidal signals.

10.5 COMPLEX ANALYSIS
OF SINUSOIDAL WAVES
Expressing sinusoidal waves as complex functions is useful (and essentially indis-
pensable) because it greatly eases the evaluation and visualization of phase that will
be found to accumulate by way of many mechanisms. In addition, we will find many
cases in which two or more sinusoidal waves must be combined to form a resultant
wave—a task made much easier if complex analysis is used.
Expressing sinusoidal functions in complex form is based on the Euler identity:
e ± jx = cos(x) ± j sin(x) (32)
from which we may write the cosine and sine, respectively, as the real and imaginary
parts of the complex exponent:
1 1
cos(x) = Re[e ± jx ] = (e jx + e − jx ) = e jx + c.c. (33a)
2 2
1 1
sin(x) =±Im[e ± jx ] = (e jx − e − jx ) = e jx + c.c. (33b)
2 j 2 j
√
where j ≡ −1, and where c.c. denotes the complex conjugate of the preceding
term. The conjugate is formed by changing the sign of j wherever it appears in the
complex expression.
We may next apply (33a)to our voltage wave function, Eq. (26):
1
jφ
e
V(z, t) =|V 0 | cos[ωt ± βz + φ] = (|V 0 |e ) e ± jβz jωt + c.c. (34)
2
V 0
Note that we have arranged the phases in (34) such that we identify the complex
amplitude of the wave as V 0 = (|V 0 |e ). In future usage, a single symbol (V 0 in the
jφ