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}