Transmission Line with Complex Load: Compute the Voltage Transmission Coefficient A lossless line has Z0 = 300 Ω and is terminated with ZL = 300 − j300 Ω. Find the voltage transmission coefficient T at the load (magnitude and angle).
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A1.265 ∠ -18.43°
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B1.01 ∠ -10°
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C1.14 ∠ 66.68°
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D1.09 ∠ 66.68°
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E0.447 ∠ -63.43°
Answer
Correct Answer: 1.265 ∠ -18.43°
Explanation
Introduction / Context:On a transmission line with characteristic impedance Z0 and load ZL, the reflection coefficient Γ and transmission coefficient T describe how waves reflect from and transmit into the load. Here we compute T for a complex load.
Given Data / Assumptions:
- Z0 = 300 Ω
- ZL = 300 − j300 Ω
- Lossless line; standard voltage-wave definitions.
Concept / Approach:
For voltage waves at the load: Γ = (ZL − Z0) / (ZL + Z0). The transmission coefficient for voltage at the load is T = 1 + Γ.
Step-by-Step Solution:
1) Compute Γ: ZL − Z0 = (300 − j300) − 300 = − j300.2) ZL + Z0 = (300 − j300) + 300 = 600 − j300.3) Γ = (− j300) / (600 − j300) = (0.2 − j0.4). Magnitude |Γ| ≈ 0.447, angle ≈ −63.43°.4) Transmission coefficient T = 1 + Γ = (1.2 − j0.4). |T| = sqrt(1.2^2 + (−0.4)^2) ≈ 1.265, angle ≈ arctan(−0.4/1.2) ≈ −18.43°.Verification / Alternative check:
Compute Γ on a Smith chart at ZL / Z0 = 1 − j1 and then add 1 vectorially to obtain T; the numerical result matches.
Why Other Options Are Wrong:
Option B and D: Magnitude/angle do not match T. Option C: Wrong quadrant/angle. Option E: That is |Γ|∠∠ for the reflection coefficient, not the transmission coefficient.
Common Pitfalls:
Confusing Γ with T; forgetting that T = 1 + Γ for voltage at the load; mixing degrees and radians.
Final Answer:
1.265 ∠ -18.43°.