Page 342 - Engineering Electromagnetics, 8th Edition

324 ENGINEERING ELECTROMAGNETICS

if the load is a short circuit, the requirement of zero voltage at the short leads to
a null occurring there, and so the voltage in the line will vary as |sin(βz)| (where
φ =±π/2).
A more complicated situation arises when the reflected voltage is neither 0 nor
100 percent of the incident voltage. Some energy is absorbed by the load and some
is reflected. The slotted line, therefore, supports a voltage that is composed of both
a traveling wave and a standing wave. It is customary to describe this voltage as a
standing wave, even though a traveling wave is also present. We will see that the
voltage does not have zero amplitude at any point for all time, and the degree to
which the voltage is divided between a traveling wave and a true standing wave is
expressed by the ratio of the maximum amplitude found by the probe to the minimum
amplitude (VSWR). This information, along with the positions of the voltage minima
or maxima with respect to that of the load, enable one to determine the load impedance.
The VSWR also provides a measure of the quality of the termination. Specifically, a
perfectly matched load yields a VSWR of exactly 1. A totally reflecting load produces
an infinite VSWR.
To derive the specific form of the total voltage, we begin with the forward and
backward-propagating waves that occur within the slotted line. The load is positioned
at z = 0, and so all positions within the slotted line occur at negative values of z.
Taking the input wave amplitude as V 0 , the total phasor voltage is

V sT (z) = V 0 e − jβz + V 0 e jβz (79)

The line, being lossless, has real characteristic impedance, Z 0 . The load impedance,
Z L ,isin general complex, which leads to a complex reflection coefficient:

Z L − Z 0 jφ
= =| |e (80)
Z L + Z 0

If the load is a short circuit (Z L = 0),φ is equal to π;if Z L is real and less than Z 0 ,φ
is also equal to π; and if Z L is real and greater than Z 0 ,φ is zero. Using (80), we may
rewrite (79) in the form:

V sT (z) = V 0 e − jβz +| |e j(βz+φ) " = V 0 e jφ/2 ! e − jβz − jφ/2 +| |e jβz jφ/2 " (81)
e
e
!
To express (81) in a more useful form, we can apply the algebraic trick of adding and
subtracting the term V 0 (1 −| |)e − jβz :

e
e
V sT (z) = V 0 (1 −| |)e − jβz + V 0 | |e jφ/2 ! e − jβz − jφ/2 + e jβz jφ/2 " (82)
The last term in parentheses in (82) becomes a cosine, and we write

V sT (z) = V 0 (1 −| |)e − jβz + 2V 0 | |e jφ/2 cos(βz + φ/2) (83)