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.

loaded line phase shifter

  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 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!

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