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A simple explanation of the so called current forcing property of a quarter wave transmission line

A need often arises in phased arrays to feed two or more radiators with equal in-phase currents, even if the load impedance presented at each feed point is not necessarily identical.

An interesting property identified by Roy Lewallyn (W7EL) is that where a load is connected via an electrical quarter wavelength of lossless transmission line (of any characteristic impedance):

In fact Iout=Vin/Z0.

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.

Iout=If'(1-Γ') where If' is the forward wave current at the load end of the line and Γ' is the reflection coefficient at the load end of the line.

Since the line is a lossless line and an electrical quarter wave long, If'=Ife-jπ/2=If∠-90°, so Iout=If(1-Γ)∠-90°.

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 and an electrical quarter wave long, Γ=-Γ', so Vin=Vf(1-Γ').

Lets divide Vin by Iout.

Vin/Iout=(Vf(1-Γ'))/(If(1-Γ)∠-90°)=(Vf))/(If∠-90°)=Z0∠90°.

Rearranging, Iout=(Vin/Z0)∠-90°.

Iout is independent of load impedance both in magnitude and phase (which lags Vin by 90°), so if two elements of an array are fed from a common point using electrical quarter waves of lossless line, the current at each feed point is identical in magnitude and phase, even if the impedances are different.

The above explanation relies on the line section being an electrical quarter wave and having no loss. A line section that has very low loss will behave approximately the same.

Fig 1:


Fig 1 shows the sensitivity of load current and phase with a low loss practical coax (LMR400) quarter wave at 3.8MHz. The flatter lines are the amplitude response, and the others, the phase response. Lossier line has a greater variation, particularly in phase. Note also that changing frequency (or conversely, error in cutting the quarter wave) will degrade performance.

Changes

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1.01 07/10/2010 Initial.
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