RF Cafe Homepage
Anritsu Test Equipment - RF Cafe

Anatech Electronics RF Microwave Filters - RF Cafe

withwave microwave devices - RF Cafe

Please Support RF Cafe by purchasing my  ridiculously low-priced products, all of which I created.

RF Cascade Workbook for Excel

RF & Electronics Symbols for Visio

RF & Electronics Symbols for Office

RF & Electronics Stencils for Visio

RF Workbench

T-Shirts, Mugs, Cups, Ball Caps, Mouse Pads

These Are Available for Free

Espresso Engineering Workbook™

Smith Chart™ for Excel

Rigol DHO1000 Oscilloscope - RF Cafe

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)

Rigol DHO1000 Oscilloscope - RF Cafe
dB Control dB-9006 Magnum Opus Synthesizer - RF Cafe

Windfreak Technologies Frequency Synthesizers - RF Cafe

TotalTemp Technologies (Thermal Platforms) - RF Cafe