# How do you calculate phase shift induced by a shunt capacitor?

Electrical Engineering Asked by KJ7LNW on September 7, 2020

Given a simple loaded-line shunt-capacitor phase shifter, how can I calculate the phase shift and insertion loss at S21 caused by the shunt capacitor (varicap in my case, but for any cap) assuming I know the capacitance value C1? I can model this in Sonnet (or perhaps some circuit theory program) but I want to put the calculations in a spreadsheet and see if my simulated model (nearly) matches the mathematical model.

1. According to Rick Sturdivant the "normalized shunt susceptence" of the capacitor ($$B$$) must be known—but I can’t find such a value on varicap/varactor datasheets. If I can get $$B$$ then Rick says insertion loss is $$10log_{10}({1+{B^2 over 4}})$$ and phase shift is $$-tan^{-1}({B over 2})$$.
• Can $$B$$ be calculated from the value of C(v)? I tried $$B = {1 over X_c} = 2pi f C$$ but my values don’t line up to his unless his $$f$$ is about 100GHz, which seems unlikely. (I’m shooting for a center frequency of 146MHz.)
1. At microwaves101.com they hint that "the reflection coefficient from a shunt capacitor" might be used to calculate phase shift, in which case I don’t need susceptence: $${displaystyle Gamma ={1-j2pi fCZ_0 over 1+j2pi fCZ_0} }$$ . But once I have $$Gamma$$, how do I calculate shift?

2. and Wikipedia:Reflection Coefficient says that "if that load, $$Z_L$$ were measured not directly but through a transmission line, then […] its phase will have shifted according to
$${displaystyle Gamma ‘=Gamma e^{-i,2phi }}$$ where ϕ [is phase]"—but I think thats the S11 reflection phase not the S21 phase. Even if it is the S21 phase, how do I calculate $$phi$$?

If you can help me understand what is right, what is irrelevant, and what other information I need to answer the question and how to calculate phase shift and insertion loss then I would greatly appreciate it!