Page 343 - Engineering Electromagnetics, 8th Edition

CHAPTER 10 Transmission Lines 325

The important characteristics of this result are most easily seen by converting it to
real instantaneous form:

V(z, t) = Re[V sT (z)e jωt ] = V 0 (1 −| |) cos(ωt − βz)

traveling wave
+ 2| |V 0 cos(βz + φ/2) cos(ωt + φ/2) (84)

standing wave
Equation (84) is recognized as the sum of a traveling wave of amplitude (1 −| |)V 0
and a standing wave having amplitude 2| |V 0 .We can visualize events as follows:
The portion of the incident wave that reflects and back-propagates in the slotted line
interferes with an equivalent portion of the incident wave to form a standing wave.
The rest of the incident wave (which does not interfere) is the traveling wave part of
(84). The maximum amplitude observed in the line is found where the amplitudes
of the two terms in (84) add directly to give (1 +| |)V 0 . The minimum amplitude
is found where the standing wave achieves a null, leaving only the traveling wave
amplitude of (1 −| |)V 0 . The fact that the two terms in (84) combine in this way
with the proper phasing is not immediately apparent, but the following arguments
will show that this does occur.
To obtain the minimum and maximum voltage amplitudes, we may revisit the
first part of Eq. (81):
V sT (z) = V 0 e − jβz +| |e j(βz+φ) " (85)
!
First, the minimum voltage amplitude is obtained when the two terms in (85) subtract
directly (having a phase difference of π). This occurs at locations
1
z min =− (φ + (2m + 1)π)(m = 0, 1, 2,...) (86)
2β
Note again that all positions within the slotted line occur at negative values of z.
Substituting (86) into (85) leads to the minimum amplitude:
V sT (z min ) = V 0 (1 −| |) (87)

The same result is obtained by substituting (86) into the real voltage, (84). This
produces a null in the standing wave part, and we obtain
V(z min , t) =±V 0 (1 −| |) sin(ωt + φ/2) (88)

The voltage oscillates (through zero) in time, with amplitude V 0 (1 −| |). The plus
and minus signs in (88) apply to even and odd values of m in (86), respectively.
Next, the maximum voltage amplitude is obtained when the two terms in (85)
add in-phase. This will occur at locations given by

1
z max =− (φ + 2mπ)(m = 0, 1, 2,...) (89)
2β