Electronics World articles Popular Electronics articles QST articles Radio & TV News articles Radio-Craft articles Radio-Electronics articles Short Wave Craft articles Wireless World articles Google Search of RF Cafe website Sitemap Electronics Equations Mathematics Equations Equations physics Manufacturers & distributors LinkedIn Crosswords Engineering Humor Kirt's Cogitations RF Engineering Quizzes Notable Quotes Calculators Education Engineering Magazine Articles Engineering software RF Cafe Archives Magazine Sponsor RF Cafe Sponsor Links Saturday Evening Post NEETS EW Radar Handbook Microwave Museum About RF Cafe Aegis Power Systems Alliance Test Equipment Centric RF Empower RF ISOTEC Reactel RF Connector Technology San Francisco Circuits Anritsu Amplifier Solutions Anatech Electronics Axiom Test Equipment Conduct RF Copper Mountain Technologies Exodus Advanced Communications Innovative Power Products KR Filters LadyBug Technologies Rigol TotalTemp Technologies Werbel Microwave Windfreak Technologies Wireless Telecom Group Withwave RF Cafe Software Resources Vintage Magazines RF Cafe Software WhoIs entry for RF Cafe.com Thank you for visiting RF Cafe!
Temwell Filters

Windfreak Technologies Frequency Synthesizers - RF Cafe

Innovative Power Products Passive RF Products - 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

Innovative Power Products Couplers

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)

Innovative Power Products Couplers
everythingRF RF & Microwave Parts Database (h1)

Exodus Advanced Communications Best in Class RF Amplifier SSPAs

Windfreak Technologies Frequency Synthesizers - RF Cafe