What is the necessary condition for resonance in a series circuit? Derive an expression for resonance frequency.

Condition: an R-L-C series circuit is at resonance when the inductive reactance equals the capacitive reactance, X_L = X_C. Then the net reactance is zero, impedance is minimum (Z = R), current is maximum (I = V/R) and in phase with the applied voltage (unity power factor).

Derivation: for a series R-L-C circuit, \( Z = R + j(X_L - X_C) = R + j\left(\omega L - \dfrac{1}{\omega C}\right) \).

At resonance the imaginary part is zero: \( \omega_rL = \dfrac{1}{\omega_rC} \Rightarrow \omega_r^2 = \dfrac{1}{LC} \Rightarrow \omega_r = \dfrac{1}{\sqrt{LC}} \).

Since ω_r = 2πf_r: \[ \mathbf{f_r = \frac{1}{2\pi\sqrt{LC}}\ \text{Hz}} \]

  I  ^         _
     |        / \      I_max = V/R at f_r
     |       /   \
     |      /     \
     |  ___/       \___
     +---------|----------> f
              f_r
  below f_r: capacitive (X_C > X_L);  above f_r: inductive (X_L > X_C)

At resonance V_L = V_C = QV (voltage magnification), with \( Q = \dfrac{\omega_rL}{R} = \dfrac{1}{R}\sqrt{\dfrac{L}{C}} \); bandwidth = R/(2πL) = f_r/Q.