In one line
Series resonance occurs at the frequency where \( X_L = X_C \), so the impedance becomes purely resistive and minimum, the current is maximum and in phase with the voltage: \( f_0 = \dfrac{1}{2\pi\sqrt{LC}} \).
Explained simply
Pushing a swing at exactly its natural rhythm makes it go higher and higher with small pushes. An RLC circuit also has a natural frequency. Fed at that frequency, the inductor and capacitor cancel each other completely, and only the resistor limits the current, so the current becomes very large.
Technically, \( X_L \) rises with frequency and \( X_C \) falls. At \( f_0 \) they are equal, so \( Z = R \), \( I = V/R \) (maximum), pf = 1. Below \( f_0 \) the circuit is capacitive; above \( f_0 \) it is inductive.
How it works
- Set \( 2\pi f_0L = \dfrac{1}{2\pi f_0C} \), so \( f_0 = \dfrac{1}{2\pi\sqrt{LC}} \).
- At \( f_0 \): \( Z_{min} = R \), \( I_{max} = V/R \), pf = 1.
- Voltages across L and C are equal and opposite, each \( = QV \), which can be much larger than the supply. This is voltage magnification, measured by the quality factor \( Q \).
- A plot of \( I \) against \( f \) is a peaked curve. The frequencies where current falls to \( I_{max}/\sqrt2 \) (half-power points) define the bandwidth.
- Series resonant circuits are called acceptor circuits: they accept the resonant frequency. Used in radio tuning.
Key formula
\( f_0 = \dfrac{1}{2\pi\sqrt{LC}}, \quad Q = \dfrac{\omega_0L}{R} = \dfrac{1}{R}\sqrt{\dfrac{L}{C}}, \quad BW = f_2 - f_1 = \dfrac{f_0}{Q} = \dfrac{R}{2\pi L} \)
\( f_0 \) = resonant frequency (Hz), \( L \) = inductance (H), \( C \) = capacitance (F), \( R \) = resistance (\( \Omega \)), \( Q \) = quality factor (no unit), \( BW \) = bandwidth (Hz), \( f_1, f_2 \) = half-power frequencies (Hz).
Quick example
\( L = 0.1 \) H, \( C = 10\,\mu \)F, \( R = 10\,\Omega \): \( f_0 = 1/(2\pi\sqrt{10^{-6}}) = 159.2 \) Hz. \( Q = \frac{1}{10}\sqrt{0.1/10^{-5}} = 10 \). With a 10 V supply at \( f_0 \), \( I = 1 \) A and the voltage across L (and C) is about 100 V.
Common mistakes
- Saying impedance is maximum at series resonance (it is minimum; maximum is for parallel resonance).
- Forgetting the \( 2\pi \) in \( f_0 \).
- Thinking the circuit is inductive below resonance; below \( f_0 \), \( X_C > X_L \), so it is capacitive.
How MU asks it
MU syllabus says theory only, so expect "Explain series resonance; derive the resonant frequency; draw the variation of \( X_L, X_C, Z \) and \( I \) with frequency; define Q factor and bandwidth" (5–8 marks). For full marks give the derivation, a neat sketch of the resonance curve with half-power points, the conditions at resonance in bullet form, and the term 'acceptor circuit'.