Problem 2.26 A 50-Ω lossless transmission line is connected to a load composed
of a 75-Ω resistor in series with a capacitor of unknown capacitance (Fig. P2.26). If at
10 MHz the voltage standing wave ratio on the line was measured to be 3, determine
the capacitance C.
RL = 75 Ω
Z0 = 50 Ω
C=?
Figure P2.26: Circuit for Problem 2.26.
Solution:
|Γ| =
S−1 3−1 2
=
= = 0.5
S+1 3+1 4
ZL = RL − jXC ,
where XC =
ZL − Z 0
ZL + Z0
·µ
¶µ ∗
¶¸
ZL − Z 0
ZL − Z 0
2
|Γ| =
ZL + Z0
ZL∗ + Z0
1
.
ωC
Γ=
|Γ|2 =
ZL ZL∗ + Z02 − Z0 (ZL + ZL∗ )
ZL ZL∗ + Z02 + Z0 (ZL + ZL∗ )
Noting that:
ZL ZL∗ = (RL − jXC )(RL + jXC ) = R2L + XC2 ,
Z0 (ZL + ZL∗ ) = Z0 (RL − jXC + RL + jXC ) = 2Z0 RL ,
|Γ|2 =
R2L + XC2 + Z02 − 2Z0 RL
.
R2L + XC2 + Z02 + 2Z0 RL
Upon substituting |ΓL | = 0.5, RL = 75 Ω, and Z0 = 50 Ω, and then solving for XC ,
we have
XC = 66.1 Ω.
Hence
C=
1
1
=
= 2.41 × 10−10 = 241 pF.
ω XC 2π × 107 × 66.1