Electronics World Cover,TOC,and list of posted Popular Electronics articles QST Radio & TV News Radio-Craft Radio-Electronics Short Wave Craft Wireless World About RF Cafe RF Cafe Homepage RF Cafe in Morse Code Google Search of RF Cafe website Sitemap Electronics Equations Mathematics Equations Equations physics Manufacturers & distributors Engineer Jobs Twitter LinkedIn Crosswords Engineering Humor Kirt's Cogitations Engineering Event Calendar RF Engineering Quizzes AN/MPN-14 Radar 5CCG Notable Quotes App Notes Calculators Education Magazines Software, T-Shirts, Coffee Mugs Articles - submitted by RF Cafe visitors Simulators Technical Writings RF Cafe Archives Test Notes Wireless System Designer RF Stencils for Visio Shapes for Word Search RF Cafe Sitemap Advertising Facebook RF Cafe Forums Thank you for visiting 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.

Here is an online VSWR mismatch calculator.

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-jβlΓ2

This signal travels back and reflects again at V1 :

V1r = V2r • e-jβl Γ1 = Vi • T1 • e-jβlΓ2 • 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-jβlΓ2 • e-jβlΓ1 • e-jβl • T2

Voe = Vi • T1 • T2Γ1Γ2 • 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|

RF Cafe Software

   Wireless System Designer - RF Cafe
Wireless System Designer

RF & EE Symbols Word
RF Stencils for Visio
Calculator Workbook
RF Workbench
Smith Chart™ for Visio
Smith Chart™ for Excel

About RF Cafe

Kirt Blattenberger - RF Cafe WebmasterCopyright
1996 - 2022
Kirt Blattenberger,

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 Internet was still largely an unknown entity at the time and not much was available in the form of WYSIWYG ...

All trademarks, copyrights, patents, and other rights of ownership to images and text used on the RF Cafe website are hereby acknowledged.

My Hobby Website:

Try Using SEARCH
to Find What You Need. 
There are 1,000s of Pages Indexed on RF Cafe !