A complex load (in Ω) is represented as ΓL = 0.5∠30° on the Smith chart. A co-axial cable with a characteristic...

 

Q. A complex load (in Ω) is represented as Γ_L = 0.5∠30° on the Smith chart. A co-axial cable with a characteristic impedance of 50 Ω is connected to the load.

The new input impedance of the load now moves to a diametrically opposite point on the same Γ circle on the Smith chart.

Which option is the nearest input impedance of the cable-connected load?

Options

(A) 20.7 − j5.1 Ω

(B) 17.7 − j11.8 Ω ✓

(C) 97.5 − j65.0 Ω

(D) 97.5 + j65.0 Ω

Step-by-Step Solution

Step 1: Given Data

Load reflection coefficient:

Γ_L = 0.5∠30°

Characteristic impedance:

Z₀ = 50 Ω

Step 2: Find the New Reflection Coefficient

A diametrically opposite point on the same VSWR circle represents a phase change of 180°.

Γ_in = −Γ_L

Γ_in = 0.5∠(30° + 180°)

Γ_in = 0.5∠210°

Γ_in = −0.433 − j0.25

Step 3: Calculate Input Impedance

Using the reflection coefficient and impedance relation:

Z_in = Z₀ × (1 + Γ_in) / (1 − Γ_in)

Z_in = 50 × (1 − 0.433 − j0.25) / (1 + 0.433 + j0.25)

Z_in = 50 × (0.567 − j0.25) / (1.433 + j0.25)

Z_in = 50 × (0.354 − j0.236)

Z_in = 17.7 − j11.8 Ω

✓

Final Answer: Option (B)

17.7 − j11.8 Ω

Summary

A diametrically opposite point on the same VSWR circle corresponds to a 180° phase change in the reflection coefficient. This is equivalent to a quarter-wavelength shift along a lossless transmission line.