# 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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This is basic classroom or web material solution. f= sqrt (1/LC) / 2pi

Here is another way. http://www.testecvw.com/carl/images/ImpedanceNomograph.pdf Find the intersection of the same impedance for L and C.

Then look at the L/r ratio at resonance. This will be the Q or gain at resonance.

If you had a parallel R the R/C impedance ratio will another limiting factor on Q.

This is the exact answer and also listed in Wiki as same. The definition of resonance is when the impedance of both reactive elements are equal in scalar and inverse polarity in vector reactance. The phase shift is also zero at resonance. $f_0 = { \omega_0 \over 2 \pi } = {1 \over {2 \pi \sqrt{LC}}}.$

R does not affect this result, only the Q factor.

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 it is more complicated then it seems. f= sqrt (1/LC) / 2pi is an approximation but i need the exact solution for all values. what should be the approach to calculate resonance frq. here? In series XL=XC so we find resonance frequency from this, but what is the idea when it is parallel? – user16307 Dec 7 '12 at 2:32 dude i need how it is "derived not" the table. thats why i asked here. if the reactance of the inductor is less than 10r the formula you gave doesn't work. it is an approximation. i just couldnt find how the exact formula(not the one you wrote) is derived mathematically.. – user16307 Dec 7 '12 at 2:48 and here is the formula im talking about:img268.imageshack.us/img268/4858/resonancelc.jpg I just dont know the derivation.. – user16307 Dec 7 '12 at 2:50 I'll let you do your homework electronics.stackexchange.com/questions/50674/… good luck.. if this is a neglected factor it is due to Q>>1 – Tony Stewart Dec 7 '12 at 3:04 it is not homework dude. im not even studying anymore. im just learning myself. if you dont want me to learn you dont have to answer such way. – user16307 Dec 7 '12 at 14:22
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