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A need often arises in phased arrays to feed two or more radiators with
equal inphase 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 I_{out}=V_{in}/Z_{0}.
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.
I_{out}=I_{f}'(1Γ') where I_{f}' 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, I_{f}'=I_{f}e^{jπ/2}=I_{f}∠90°,
so I_{out}=I_{f}(1Γ)∠90°.
V_{in}=V_{f}(1+Γ) where V_{f} 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 V_{in}=V_{f}(1Γ').
Lets divide V_{in} by I_{out}.
V_{in}/I_{out}=(V_{f}(1Γ'))/(I_{f}(1Γ)∠90°)=(V_{f}))/(I_{f}∠90°)=Z_{0}∠90°.
Rearranging, I_{out}=(V_{in}/Z_{0)}∠90°.
I_{out} is independent of load impedance both in magnitude and
phase (which lags V_{in} 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 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.
Version  Date  Description 
1.01  07/10/2010  Initial. 
1.02  
1.03  
1.04  
1.05 
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