Modelling a variable (i.e. nonlinear) capacitor in Simulink: how to make it stable?

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I am modelling a variable capacitor, which is described by the following nonlinear equation: . To implement this, I have been taking the standard approach of measuring the current i through the variable capacitor, integrating it to get the charge Q, and then dividing it by the capacitance at that time instant (necessary, since the capacitance varies) to get the voltage V across the capacitor, and then finally feeding that into a Controlled Voltage Source to generate that instantaneous voltage. See the image below for what this looks like in Simulink, and a well-done similar implementation on the FileExchange.
While this approach is relatively straightforward and simple, it is extremely sensitive and will produce voltages that approach infinity at the beginning of simulations. Even if I try saturating the output voltage (or alternatively, saturating the integrator), I still get unrealistic results. The only thing that makes it run "correctly" is making the sample time incredibly small, on the order of , which of course makes the simulation very slow and generates many data points (sometimes causing Simulink to crash). I should note that even at these small sample times, the voltage across the variable capacitor is unrealistically noisy. I have tried a number of other tricks but to no avail:
  • Initializing the integrator and the voltage source (and other elements in the larger circuit that this variable capacitor is a part of)
  • Trying a continuous vs. discrete integrator
  • Using a unit delay (or transfer function as advised by Guy on Simulink in this post)
  • Using ode15s or other nonlinear solvers
  • Making the capacitance a static value for several seconds to let the circuit settle into a steady-state before varying it
Note that is generated by a Sine Wave block and is never equal to zero.
I am guessing this is a broader issue pertinent to other applications, since I am feeding a measured quantity directly into an integrator, and the Current Measurement block has no reference for currents prior to simulation beginning (like the "chicken vs. egg").
Does anyone have any tips on how to either model variable capacitors or more generally, make integrators less sensitive? Thank you!
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Sam Chak
Sam Chak 2022년 7월 20일
If your capacitance is varying with time, then you can include it in the dynamical RC circuit model as an exogenous variable.

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Jon 2022년 7월 19일
It looks like you may be using some circuit block set that I don't have available, however just going back to fundamentals you could model it as shown in the screenshot below from the attached model. In this model I have put the capacitor in series with a voltage source
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