Problem 2.23 A load with impedance ZL = (25 − j50) Ω is to be connected to a
lossless transmission line with characteristic impedance Z0 , with Z0 chosen such that
the standing-wave ratio is the smallest possible. What should Z0 be?
Solution: Since S is monotonic with |Γ| (i.e., a plot of S vs. |Γ| is always increasing),
the value of Z0 which gives the minimum possible S also gives the minimum possible
|Γ|, and, for that matter, the minimum possible |Γ|2 . A necessary condition for a
minimum is that its derivative be equal to zero:
∂
∂ |RL + jXL − Z0 |2
2
|Γ| =
0=
∂ Z0
∂ Z0 |RL + jXL + Z0 |2
=
∂ (RL − Z0 )2 + XL2 4RL (Z02 − (R2L + XL2 ))
=
.
2
∂ Z0 (RL + Z0 )2 + XL2
((RL + Z0 )2 + XL2 )
Therefore, Z02 = R2L + XL2 or
q
Z0 = |ZL | = (252 + (−50)2 ) = 55.9 Ω.
A mathematically precise solution will also demonstrate that this point is a
minimum (by calculating the second derivative, for example). Since the endpoints
of the range may be local minima or maxima without the derivative being zero there,
the endpoints (namely Z0 = 0 Ω and Z0 = ∞ Ω) should be checked also.