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In an RLC circuit,what criteria could be used to decide whether the system is overdamped or underdamped? Could we compare the maximum energy stored during one cycle to the energy dissipated during one cycle?

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    \$\begingroup\$ this is really a question for the electrical engineering stack exchange \$\endgroup\$ – user11547 Mar 14 '13 at 10:02
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To establish the damping calculate ζ (damping ratio).

\$\zeta = \dfrac{R_{series}}{2}\sqrt{\dfrac{C}{L}} \$

If ζ is <1 then it is underdamped. If ζ=1 it is critically damped etc..

Q (quality factor) is defined as the peak energy stored in the circuit divided by the average energy dissipated in it per cycle at resonance and, Q is \$\dfrac{1}{2\zeta}\$.

You decide what's the best method for you but for me it's easier to use the ζ formula providing you know the values for R, L and C. If you don't know them then there are other methods that involve looking at the step transient response or the frequency response.

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Here's a link showing the transfer function of a RLC circuitry. If you input your values into this transfer function and look for the poles, you can have clear idea if your system is over-damped or under-damped. If the poles are pure imaginary your system will oscillate if the poles of the system are one value, real and negative, you have critically damped system. I suggest you to read a control theory book to have a good idea on the topic, such as Control Systems engineering from Norman Nise.

Sure you can make a comparison by calculating apparent, active and reactive power consumed by the system. You can do whatever you like as long as your system fits in linear system theory and you know the transfer function.

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Damping ratio is driven from characteristic equation. It's different according to circuit's configuration

$$ζ={R_{series}\over2} \sqrt{C\over L}$$

$$ζ={1\over 2R_{parallel}} \sqrt{L\over C}$$

If value is bigger than 1, we say the system is overdamped. Less than 1, we say underdamped. Equal to 1, we say critically damped.

Here is video on how to derive this equation. It also shows how the graph would like for these 3 cases. https://youtu.be/dGc-ozvwnjE

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