I have a "complicated" circuit that I want to approximate as a simple harmonic oscillator. More info below.
I have a parallel LRC circuit, with some extra input and output load resistors and coupling capacitors. The inductor is really nonlinear (made up of a DC-SQUID). The input voltage consists of a cosine drive with a Gaussian envelope (so it's basically a "short" pulse). I can solve for the output ringdown voltage numerically by directly solving the corresponding differential equations, and everything works fine.
Next, I would like simplify this circuit by writing it in terms of a single degree of freedom. In the weak excitation regime (i.e. low input pulse amplitude), the inductance can be approximated as being linear.
What I've done is this:
- assume initially circuit is driven with a drive of frequency \$\omega\$
- interpreted all resistors and capacitors as complex frequency dependent impedances
- reduced the circuit via standard (series/parallel) rules
- arrived at a simple parallel LRC circuit, but now the effective resistance \$R_{\rm eff}\$ and capacitance \$C_{\rm eff}\$ are of course frequency dependent.
Next (and these are the step that may be a bit sketchy):
assumed that the input is no longer a drive of some frequency \$\omega\$, but instead my original "short" pulse.
I looked at the effective resistance \$R_{\rm eff}\$ and capacitance \$C_{\rm eff}\$ as a function of frequency and concluded that these quantities are almost "flat" over the frequency range of the input pulse (which I Fourier transformed to see it in frequency space)
next, set \$R_{\rm eff}\$ and capacitance \$C_{\rm eff}\$ as constants (i.e. took their values at the central frequency of the input pulse)
Arrive at a harmonic oscillator with constant coefficients driven by a complex input pulse (also frequency dependent, so I also approximate \$\omega\$ to a constant).
The final note is that I am hoping that this simple harmonic oscillator form can be useful to study some other properties of the inductance (which i'm not getting into here).
Questions:
- Do you see anything obviously wrong with this approach?
- Are items 5-8 unreasonable?
- Could I expect the transients of this reduced system to be representative of the full, unsimplified circuit?
thanks!