Hello, I would like to go over the transient states of a LRC circuit, starting f

Question

Hello,

I would like to go over the transient states of a LRC circuit, starting from under damped, to critical damped, and finally over damped. I am familiar with the characteristics themselves. What I am seeking is the mathematical model as the LRC goes into these different states (when is changed for each impulse).

When the circuit becomes critically damped where 0= , then = 0. The circuit no longer oscillates, and the above equation doesn't hold. Peak amplitude is achieved, with the response reaching stable state the quickes. How would I model this mathematically?

When the circuit becomes over damped where > 0, then becomes imaginary. I remember an explanation using the above current equation, that the sin becomes hyperbolic. Could someone elaborate on this, show me how this is express mathmatically?

Thanks!

Explanation / Answer

The equations in my text book slightly differ from yours, however, it looks as if they are essentially the same thing but instead of , is used:

Over damped:
When the characteristic equation for the circuit has real and distinct roots, i.e.

s2-s-12=(s+3)(s-4)

Equation for this state is:

x(t) = K1Aes1t+K2es2t

x(0+)=K1 + K2

x'(0+) = s1K1 + s2K2

Under damped:

where wo > where =R/(2L) when in series and =1/(2RC) when in parallel

x(t) = (A*cos(wdt) + B*sin(wdt))e-t

x(0+) = A

x'(0+) = -A + wdB

Critically damped:

where wo= and roots are equal; s1=s2

x(t) = (K1 + K2t)es1t

x(0+) = K1

x'(0+) = s1K1 + K2

wd = (wo2-2)

wo = 1/(LC)

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