Usually cavity detuning is found by cavity pick up signal and feedforward signal from waveguide directional coupler. Cavity RLC mathematical model leads to 2nd order differential equations and after few approximation first order differential equation for cavity voltage is derived which depends on cavity half bandwidth, cavity detuning and driving current. How to deduce the mathematical relation used for cavitydetuning calculation, depending on forward and cavity voltage amplitude and phase??
For doing so, you need to calibrate the forward signal to the same reference plane as the cavity probe signal. Then you can invert the cavity differential equation to get the detuning info. If the forward signal is zero, you can simply fit the phase slope of the cavity probe signal for estimating the detuning.
There should be lots of papers describing this algorithm, like Alexander Brandt's Ph.D. thesis. In our book on LLRF, the basic algorithms for this problem are also discussed (Book Low-Level Radio Frequency Systems
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