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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.

Phase error due to VSWR mismatch

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|

Phase error due to VSWR mismatch

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|

Here is an online VSWR mismatch calculator.

 

 

Posted June 6, 2023
(updated from original post on 2/15/2009)

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Webmaster:

    Kirt Blattenberger,

    BSEE - KB3UON

RF Cafe began life in 1996 as "RF Tools" in an AOL screen name web space totaling 2 MB. Its primary purpose was to provide me with ready access to commonly needed formulas and reference material while performing my work as an RF system and circuit design engineer. The World Wide Web (Internet) was largely an unknown entity at the time and bandwidth was a scarce commodity. Dial-up modems blazed along at 14.4 kbps while tying up your telephone line, and a nice lady's voice announced "You've Got Mail" when a new message arrived...

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