
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,
εθ_{max}
= Γ_{1}

•
Γ_{2},
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.
V_{1 }= V_{i}
• T_{1}
V_{2 }= V_{i} • T_{1} • e^{jβl}
V_{o }= V_{i}
• T_{1} • T_{2}• e^{jβl}
V_{o}
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
V_{2}
as: V_{2r }= V_{i}
• T_{1} • e^{jβl} • Γ_{2} This signal travels back and reflects
again at
V_{1}
: V_{1r }= V_{2r
}• e^{jβl} • Γ_{1 = }V_{i} • T_{1} • e^{jβl} • Γ_{2}
• e^{jβl} • Γ_{1}
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_{2} • e^{jβl} • Γ_{1} • e^{jβl}
• T_{2}
V_{oe }= V_{i} • T_{1} • T_{2}• Γ_{1}
• Γ_{2} • e^{j3βl} We can represent these signals in complex plane as:
V_{o}
= V_{i} • T_{1}
• T_{2}
V_{oe}
= V_{i} • T_{1}
• T_{2} • Γ_{1}
• Γ_{2} 

It follows that we can write the worstcase 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 worstcase phase error (in radians) due to reflections at the source and at the load as:
εθ_{max}
= Γ_{1}
•
Γ_{2}



