4
\$\begingroup\$

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:

  1. assume initially circuit is driven with a drive of frequency \$\omega\$
  2. interpreted all resistors and capacitors as complex frequency dependent impedances
  3. reduced the circuit via standard (series/parallel) rules
  4. 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):

  1. assumed that the input is no longer a drive of some frequency \$\omega\$, but instead my original "short" pulse.

  2. 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)

  3. 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)

  4. 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:

  1. Do you see anything obviously wrong with this approach?
  2. Are items 5-8 unreasonable?
  3. Could I expect the transients of this reduced system to be representative of the full, unsimplified circuit?

thanks!

\$\endgroup\$
3
  • \$\begingroup\$ I got about half way down and realized I didn't have a clue what you are trying to do. Maybe you or I should take a step back and re-evaluate. \$\endgroup\$
    – Andy aka
    Dec 24, 2014 at 23:22
  • \$\begingroup\$ Well, in short, I have a circuit with multiple degrees of freedom, and I want to write it as a single degree of freedom (as a simple harmonic oscillator). I've done this, in way outlined by the list in my post. \$\endgroup\$
    – user2562235
    Dec 25, 2014 at 5:22
  • \$\begingroup\$ Do you have schematics showing where you started from and at least some of the steps you have taken? This would help people visualize the problem and take a better stab at it. \$\endgroup\$
    – FullmetalEngineer
    Jan 16, 2015 at 2:27

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.