How do you analyse a series RLC circuit and decide if it is inductive or capacitive?

In one line

In a series RLC circuit \( Z = \sqrt{R^2 + (X_L - X_C)^2} \); if \( X_L > X_C \) it behaves inductively (current lags), if \( X_C > X_L \) capacitively (current leads), and if they are equal it is at resonance.

Explained simply

The inductor and capacitor pull the current in opposite directions in phase, like two people in a tug of war. Whichever is stronger decides the outcome. Their voltages are exactly 180° apart, so they partly cancel each other, and the net reactance is the difference \( X_L - X_C \).

Technically, with the same current through all three elements, \( V_R = IR \) in phase, \( V_L = IX_L \) leading by 90°, \( V_C = IX_C \) lagging by 90°. The supply is \( \bar V = \bar V_R + \bar V_L + \bar V_C \).

How it works

1. Find \( X_L = 2\pi fL \) and \( X_C = 1/(2\pi fC) \).
2. Net reactance \( X = X_L - X_C \); \( Z = R + jX \).
3. \( I = V/Z \); \( \phi = \tan^{-1}(X/R) \).
4. Case \( X_L > X_C \): inductive, lagging pf. Case \( X_L < X_C \): capacitive, leading pf. Case \( X_L = X_C \): resistive, unity pf (resonance).
5. Find element voltages \( IR, IX_L, IX_C \) and draw the phasor diagram with \( I \) as reference.

Key formula

\( Z = \sqrt{R^2 + (X_L - X_C)^2}, \quad \phi = \tan^{-1}\dfrac{X_L - X_C}{R}, \quad \cos\phi = \dfrac{R}{Z} \)

\( Z \) = impedance (\( \Omega \)), \( R \) = resistance (\( \Omega \)), \( X_L, X_C \) = reactances (\( \Omega \)), \( \phi \) = phase angle (positive means current lags).

Quick example

\( R = 10\,\Omega \), \( X_L = 50\,\Omega \), \( X_C = 40\,\Omega \), \( V = 100 \) V. \( X = 10\,\Omega \), \( Z = \sqrt{100 + 100} = 14.14\,\Omega \), \( I = 7.07 \) A, \( \phi = 45^\circ \) lagging (inductive). Note \( V_L = 7.07\times50 = 353.6 \) V, much larger than the 100 V supply. This is possible because \( V_L \) and \( V_C \) largely cancel.

Common mistakes

• Adding \( X_L \) and \( X_C \) instead of subtracting.
• Thinking an element voltage can never exceed the supply voltage.
• Deciding lead/lag from \( L \) and \( C \) values rather than from \( X_L \) and \( X_C \) at the given frequency.

How MU asks it

A favourite 7–10 mark numerical: "A series RLC circuit with R, L, C across 230 V, 50 Hz. Find Z, I, pf, power, voltage across each element, and draw the phasor diagram." For full marks: compute reactances first, state whether the circuit is inductive or capacitive, give pf with lag/lead, power \( P = I^2R \), and a labelled phasor diagram.