Understanding RLC Damping Coefficient for Non-Standard Circuits

[samthedummy]

Hey,

Quick question for you guys. How do you find alpha (damping coefficient, Neper frequency) for a circuit that's not strictly parallel or in series? For instance, α for a series RLC circuit is R/2L whereas α for a parallel circuit is 1/2RC; but what if it's different?

Homework Statement

Here's an example of what I mean. Suppose you have an inductor and a capacitor in series that's in parallel with a resistor and a current source (see attached). Although R, L, and C, are in series with each other (in their own loop), R and LC are in parallel with the source. How would you know which α to use? Is there a way to get α in terms of impedance?

Homework Equations

As mentioned before:

$\alpha_{series}=\frac{R}{2L}$

and:

$\alpha_{parallel}=\frac{1}{2RC}$

For what it's worth:

$\zeta=\frac{\alpha}{\omega_{0}}$

where ζ is the damping ratio and:

$\omega_{0}=\frac{1}{\sqrt{LC}}$

where ω0 is the natural or resonant frequency of the circuit (the same in series and parallel).

The Attempt at a Solution

Since this is more of a conceptual type question, I've tried looking it up in different sources but they all seem to only look at independent series or parallel cases. More curious for myself than anything. Maybe one of you EEs can point me in the right direction/explain this for me. Much appreciated.

Peace.

 

[gneill]

Here's an example of what I mean. Suppose you have an inductor and a capacitor in series that's in parallel with a resistor and a current source (see attached). Although R, L, and C, are in series with each other (in their own loop), R and LC are in parallel with the source. How would you know which α to use? Is there a way to get α in terms of impedance?

Convert the current source and the parallel resistance into its Thevenin equivalent. This will give you a purely serial RLC circuit driven by a voltage source

There are configurations of R, L, and C that are more tricky. For those you could determine the differential equation governing the circuit and hammer it into standard form (it's a second order differential equation). Then you can pick out the damping ratio, etc., from the parameters.