Analysing the transient response of an LCR circuit

[Froskoy]

Hi,

I'm trying to determine when the following cases occur for the attached circuit:

1) underdamped
2) critically damped
3) overdamped

There's loads of sources that quote results for a series lcr circuit (for example the Wikipedia article covers it thoroughly, but I've got no idea how to do it for this circuit).

So, I guess what I'm asking is, for this circuit, how do you calculate α, and therefore the damping factor, of this circuit. (I assume the cases for damping factor >1, damping factor<1 and damping factor=1 still apply to overdamped, underdamped and critically damped respectively).

I'm applying a low frequency square wave to the circuit to examine the oscillations that occur on the falling edge, but want to know how to theoretically derive the results.

With very many thanks,

Froskoy.

 

[MrAl]

Hi there Froskoy,


First thing is you can analyze this circuit in the frequency domain and come out with a transfer function H(s).
Once you have that, it will have a denominator in this form:
a*s^2+b*s+c

What you can do next is get the s^2 term alone without any coefficient by dividing by 'a', this gives us:
s^2+s*b/a+c/a

which with simple change of variables can be written:
s^2+B*s+C

But this can also be written in this new form:
s^2+2*d*wn*s+wn^2

and to get this far all we had to do was equate B and C as:
B=2*d*wn
C=wn^2

This form is especially useful because 'd' is the damping factor, so all we have to do now is solve for the damping factor 'd' knowing B and C. Since B is made up of factors of wn and 2 besides 'd', if we divide by the square root of C we'll get 2*d, from which all we have to do is divide by 2 and we'll have the damping factor 'd' and can use the greater than or less than 1 criterion to determine what form of response we'll have.
So the damping factor comes out to:
d=B/(2*sqrt(C))

So the idea is to analyze the circuit for it's transfer function, then simplify the denominator and solve for the damping factor. This works for any second order transfer function regardless of the connection of the components.