Question

Reactance Frequency Dependence: Sketch a graph of the frequency dependence of a resistor, capacitor, and inductor. RLC Circuit Resonance Frequency: Identify the relation between the capacitive reactance (Xc) and inductive reactance (XL) that will minimize the total impedance (Z) of an RLC circuit. Using this condition, derive the resonance frequency () of an RLC circuit.

Answer 1

Answer:

$f=\frac{1}{2\pi \sqrt{LC}}$

Explanation:

We know that impedance of a RLC circuit is given by $Z=R+J(X_L-X_C)$

So $Z=\sqrt{R^2+(X_L-X_C)^2}$ here R is resistance $X_L$ is inductive reactance and $X_C$ is capacitive reactance

To minimize the impedance $X_L-X_C$ should be zero we know that $X_L=\omega L\ and \ X_C=\frac{1}{\omega C}$

So $\omega L-\frac{1}{\omega C}=0$

$\omega ^2=\frac{1}{LC}$

$\omega =\sqrt{\frac{1}{LC}}$

We know that $\omega =2\pi f$

So $\omega =2\pi f=\frac{1}{\sqrt{LC}}$

$f=\frac{1}{2\pi \sqrt{LC}}$

Where f is resonance frequency