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. AC Circuit

And here's the resonant frequency which supposed to be the answer. The Resonant Frequency of AC circuit

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

Find the input admittance enter image description here

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

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

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