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A simple explanation of the voltage forcing property of a half wave transmission line

An interesting property is that where a load is connected via an electrical half wavelength of lossless transmission line (of any characteristic impedance):

In fact Vout=-Vin.

Taking the general case of the load, there is a mismatch at the load end of the line, and the complex reflection coefficient Γ' is the ratio of the reflected voltage wave to the forward voltage wave at the load. For more information on the meaning of Γ see Telegrapher's Equation.

Vin=Vf(1+Γ) where Vf is the forward wave voltage at the input to the line and Γ is the reflection coefficient at the input to the line.

Since the line is a lossless line an electrical half wave long, Vf'=Vfe-jπ=Vf∠-180° and  Γ'= Γ ,where Vf' is the forward wave voltage at the load end of the line and Γ' is the reflection coefficient at the load end of the line. So Vout=Vf'(1+Γ')=Vf∠-180°(1+Γ).

Vout/Vin is (Vf∠-180°(1+Γ))/(Vf(1+Γ))=1∠-180°=-1, or Vout=-Vin

Lossless lines do not exist in the real world, but short sections of low loss lines will deliver almost ideal results. For example, using TLLC to find Vo/Vi for a half wave of LMR400 with a load of 100+j0Ω at 144MHz we get Vo/Vi=9.975e-1∠-180.0°, less than 3% departure from ideal.

Changes

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1.01 12/08/2012 Initial.
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