Page 346 - Engineering Electromagnetics, 8th Edition

328 ENGINEERING ELECTROMAGNETICS

g g

in in L in in

z = −l z = 0

Figure 10.7 Finite-length transmission line configuration and its equivalent circuit.

Figure 10.7 shows the basic problem. The line, assumed to be lossless, has
characteristic impedance Z 0 and is of length l. The sinusoidal voltage source at
frequency ω provides phasor voltage V s . Associated with the souce is a complex
internal impedance, Z g ,as shown. The load impedance, Z L ,is also assumed to be
complex and is located at z = 0. The line thus exists along the negative z axis.
The easiest method of approaching the problem is not to attempt to analyze every
reflection individually, but rather to recognize that in steady state, there will exist one
net forward wave and one net backward wave, representing the superposition of all
waves that are incident on the load and all waves that are reflected from it. We may
thus write the total voltage in the line as
+ − jβz
− jβz
V sT (z) = V e + V e (93)
0
0
in which V 0 + and V 0 − are complex amplitudes, composed respectively of the sum of
all individual forward and backward wave amplitudes and phases. In a similar way,
we may write the total current in the line:
+ − jβz
− jβz
I sT (z) = I e + I e (94)
0
0
We now define the wave impedance, Z w (z), as the ratio of the total phasor voltage to
the total phasor current. Using (93) and (94), this becomes:
− jβz
V sT (z) V e + V e
+ − jβz
Z w (z) ≡ = 0 0 (95)
+ − jβz
− jβz
I sT (z) I e + I e
0
0
We next use the relations V 0 − = V , I 0 + = V /Z 0 , and I 0 − =−V /Z 0 . Eq. (95)
+
+
−
0
0
0
simplifies to
e + e
− jβz jβz
Z w (z) = Z 0 (96)
e − jβz − e jβz
Now, using the Euler identity, (32), and substituting = (Z L − Z 0 )/(Z L + Z 0 ),
Eq. (96) becomes
Z L cos(βz) − jZ 0 sin(βz)

Z w (z) = Z 0 (97)
Z 0 cos(βz) − jZ L sin(βz)
The wave impedance at the line input is now found by evaluating (97) at z =−l,
obtaining