Derivation of Phase Angle Error Due to VSWR Mismatch
by Haris Tabakovic
Try finding the equation for phase angle error due to VSWR mismatch, and you will likely fail. Extensive keyword searches for related terms will turn up websites that present the formula for amplitude error due to VSWR mismatch, but not for phase angle error due to VSWR mismatch. If you are fortunate enough to find the equation, you almost certainly will not be given the derivation.|
The actual equation, εθmax = |Γ1 | • |Γ2|, is so simple that it seems unbelievable, but here its validity is demonstrated.
Well, the search is over thanks to Haris Tabakovic, who was kind enough to provide this excellent derivation for the benefit of RF Cafe visitors.
Here is an online VSWR mismatch calculator.
V1 = Vi • T1
V2 = Vi • T1 • e-jβl
Vo = Vi • T1 • T2• e-jβl
Vo is expected output signal.
At the same time, the reflected signal is being bounced around on the connecting transmission line. First order reflections are going to be dominant, and higher order reflections are not taken into account. Note that the transmission line is assumed to be lossless.
Then we can express reflected signal at V2 as:
V2r = Vi • T1 • e-jβl • Γ2
This signal travels back and reflects again at V1 :
V1r = V2r • e-jβl • Γ1 = Vi • T1 • e-jβl • Γ2 • e-jβl • Γ1
Finally, this error signal Voe is transmitted and superimposed on expected output signal, causing phase and amplitude error:
Voe = V1r • e-jβl • T2 = Vi • T1• e-jβl • Γ2 • e-jβl • Γ1 • e-jβl • T2
Voe = Vi • T1 • T2• Γ1 • Γ2 • e-j3βl
We can represent these signals in complex plane as:
||Vo| = |Vi| • |T1| • |T2||
|Voe| = |Vi| • |T1| • |T2| • |Γ1| • |Γ2|
It follows that we can write the worst-case phase error εθmax as:
Since εθmax will be a very small angle, can say that:
tg(εθmax) ≈ εθmax
Finally, we can write the worst-case phase error (in radians) due to reflections at the source and at the load as:
εθmax = |Γ1| • |Γ2|