Electronic – Series RLC circuit dynamics

circuit analysis

I'm just getting my head around RLC dynamics.

After some analysis and math i get,

\$\frac{d^{2}v_{\text{c}}}{dt^{\text{2}}} + \frac{R}{L}\frac{dv_{\text{c}}}{dt} + \frac{1}{LC}v_{c} = \frac{1}{LC}v_{\text{S}}\$

and the characteristic equation:

\$s^{2} + 2\alpha s + \omega_{0}^{2}\$

and

\$ v_c = V_S + A_1e^{s_1t} + A_2e^{s_2t}\$

3 cases:

case 1: \$ \alpha = \omega_{0}\$ — critically damped,
then,

\$v_c = V_S + A_1e^{-\alpha t} + A_2te^{-\alpha t}\$

I understand that no sinusoids here. But I note texts drawing some overshoot before the response continues to the final value, with no clear explanation. I tried to crunch my head with little maths to explain the overshoot, but failed. Any one to explain the overshoot?

EDIT:

i.e. isn't the figure below the case?
enter image description here

or in the case of source-free RLC.

enter image description here

Best Answer

The usual blanket statement of no overshoot for a critically damped system makes an implicit assumption about the initial conditions. A critically damped system will overshoot at most once.

To intuitively see this, imagine a mass spring damper system where the mass is released at t=0 and finds an equilibrium condition. That response will have no overshoot (say a door is release and it closes automatically).

On the other hand, if the mass is moving rapidly at t=0 (say the door is given a mighty push just before t=0) it may well overshoot before reaching a final equilibrium.

For a detailed mathematical treatment, minus hand waving, see, for example, this.

If it's a step response from equilibrium there should be NO overshoot, and any overshoot is an indication of underdamping.

As Dave Tweed mentions, if some overshoot can be tolerated, then a faster rise time can be achieved by slightly underdamping. For example, for a tolerable overshoot of 5%, \$\zeta \$ should be about 0.69. The 10% to 90% rise time is then about 35% faster than the critically damped case. Perhaps they are illustrating such a design decision, which is pretty common.

There seems to be quite a few bad examples of formulas out there, so be careful. Here's one example from here, which is totally incorrect.

enter image description here

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