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RF Cascade Workbook

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1996 -
2022

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

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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, *εθ _{max} = |*Γ

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.

*V _{1 }= V_{i} • T_{1}*

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 * V_{2}*
as:

*V _{2r }= V_{i} • T_{1} • e^{-jβl}
• *Γ

This signal travels back and reflects again at **V**_{1}
:

*V _{1r }= V_{2r }• e^{-jβl} • *
Γ

Finally, this error signal * V_{oe}*
is transmitted and superimposed on expected output signal, causing phase and amplitude error:

*V _{oe }= V_{1r} • e^{-jβl} • T_{2}
= V_{i} • T_{1}• e^{-jβl} • *Γ

*V _{oe }= V_{i} • T_{1} • T_{2}•
*Γ

We can represent these signals in complex plane as:

V| _{o}|
= |V_{i}| • |T_{1}| •
|T_{2}|
Γ|V_{oe}| =
|V_{i}| • |T_{1}| • |T_{2}|
• |Γ_{1}|
• |_{2}| |

It follows that we can write the worst-case phase error *
εθ_{max}* as:

Since *
εθ_{max}* will be a very small
angle, can say that:

*tg(εθ _{max})
≈ εθ_{max}*

Finally, we can write the worst-case phase error (in radians) due to reflections at the source and at the load as:

*εθ _{max} = |*Γ