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By inserting a matched (nominal system impedance) attenuator in front of a mismatched load impedance, the mismatch "seen" at the input of the attenuator is improved by an amount equal to twice the value of attenuator. The explanation is simple.
Return loss is determined by the portion of the input signal that is reflected at the load (due to impedance mismatch) and returned to the source. A perfect load impedance (complex conjugate of the source impedance) would absorb 100% of the incident signal and therefore reflect 0% of it back to the source (return loss of ∞ dB).
For the sake of illustration, assume that the load is an open (or short) circuit, where 0% of the incident signal is absorbed by the load and 100% is reflected back to the source. The reflected signal would therefore have a return loss of 0 dB. Insert a 3 dB attenuator in front of the load. Now the incident signal is referenced to the input of the attenuator.
As signal at the input of the attenuator will experience a 3 dB reduction in power by the time it reaches the load. That 3 dB less power will be 100% reflected by the load and experience another 3 dB reduction in power by the time is returns back to the input, for a total loss of 6 dB. The same principle applies for a load anywhere(§) between zero and infinite load impedance (short and open circuits, respectively).
Calculate the improved VSWR as follows. Note that by my convention the loss value is returned as a positive number,
since the word "loss" implies the negative. If it were to be termed "return gain," then the result would be reported
as a negative number. Equally qualified experts will disagree on whether return loss should take on a negative value
or a positive value; the important thing is to keep the sign correct in your calculations; i.e., if you use a positive
value, then subtract it, and vice versa.
RL_{Improved} = 2*Attenuator + RL_{Load} [dB] (where RL=Return Loss)
Of course, the method can be reversed to predict the attenuator required to improve a load VSWR by a predetermined amount. To do so, calculate the desired return loss and subtract the known load return loss. Divide the answer by two to get the attenuator value needed.
Here is a JavaScript calculator for VSWR / Return Loss / Reflection Coefficient / Mismatch Error / Improvement
See the VSWR Calculator page.
§ Actually, the attenuator is only rated for its specified attenuation level when it is connected between two nominal impedances. Therefore, the attenuator will either have to be designed to closely match the two impedances at its input and output (source and load, respectively), or an adjustment will need to be made in the specified attenuation value to compensate for the mismatched load impedance.
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 VSWR Mismatch Errors
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