# How to derive the resonant frequency from the RLC circuit?

Just like the title, i have an AC circuit (RLC) RL in series and they are in parallel with C. How do I derive the resonant frequency from that circuit? It says to derive from first principles. I got the resonant frequency for XL=XC but it is not same as the resonant frequency below... Any help on deriving that resonant frequency?

Here's the circuit.

And here's the resonant frequency which supposed to be the answer.

• I think you are asking for the damped resonant frequency. – Andy aka Sep 23 '13 at 14:09

## 1 Answer

Find the input admittance

$$Y = j\omega C + \frac{1}{R+j\omega L} = j\omega C + \frac{R-j\omega L}{R^2+\omega^2 L^2}$$ $$Y = \frac{R}{R^2+\omega^2 L^2} + j(\omega C + \frac{-\omega L}{R^2+\omega^2 L^2})$$

Then the Resonant Fequency is when the Imaginary component of the input admittance is zero$$Im(Y)=0$$

So $$\omega C + \frac{-\omega L}{R^2+\omega^2 L^2} = 0$$

$$\omega C = \frac{\omega L}{R^2+\omega^2 L^2}$$

$$\frac{C(R^2+\omega^2 L^2)}{L} =1$$

$$\frac{R^2C+\omega^2 CL^2}{L} =1$$

$$\frac{R^2C}{L}+\omega^2 LC =1$$

$$\omega^2 LC = 1 - \frac{R^2C}{L}$$

I bet you can take it form here