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,
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
Here is an online
VSWR mismatch calculator.
V1 = Vi
V2 = Vi • T1 • e-jβl
Vo = Vi
• T1 • T2• e-jβl
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
V2r = Vi
• T1 • e-jβl • Γ2
This signal travels back and reflects
V1r = V2r
• e-jβl • Γ1 = Vi • T1 • e-jβl • Γ2
• e-jβl • Γ1
Finally, this error signal
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
Voe = Vi • T1 • T2• Γ1
• Γ2 • e-j3βl
We can represent these signals in complex plane as:
= |Vi| • |T1|
= |Vi| • |T1|
• |T2| • |Γ1|
It follows that we can write the worst-case phase error
will be a very small angle, can say that:
Finally, we can write the worst-case phase error (in radians) due to reflections at the source and at the load as: