# How to calculate the resonance frequency of a parallel LC circuit with L having a DC resistance?

Here is the parallel LC circuit with inductor having also r:

How can I calculate the resonance frequency?

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Hint (I'm too tired to attempt a complete answer): write the differential equation for this circuit, solve it, and compare with the equations/results for a parallel LC circuit without the resistance. – Renan Dec 7 '12 at 2:35
Check, e.g. en.wikipedia.org/wiki/RLC_circuit#Other_configurations (looks like the first case is what you want) – Renan Dec 7 '12 at 2:43
it is really much more complicated than in series circuit – user16307 Dec 7 '12 at 2:53
@user16307, Renan is correct. I've added an answer with more detail. – Alfred Centauri Dec 9 '12 at 2:13

How can I calculate the resonance frequency?

It's quite straightforward if you'll take the time to think about it.

By definition, the resonance frequency is, in this context, the frequency at which the impedance of the equivalent impedance is real.

The branch with the R and L has an impedance of: $R + j\omega L$

The branch with the C has an impedance of: $\dfrac{1}{j \omega C}$

Now, the equivalent impedance is just:

$(R + j\omega L)||\dfrac{1}{j \omega C}$

The resonance frequency is the frequency for which the above is purely real, i.e., set the imaginary part of the above equal to zero and solve for the frequency.

When you do this correctly, you should get

or equivalently, as the Wikipedia link Renan gives:

$\omega_0 = \sqrt {\frac{1}{LC} - \left ( \frac{R}{L} \right )^2}$

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