In one line
In a series RC circuit the impedance is \( Z = \sqrt{R^2 + X_C^2} \) and the current leads the supply voltage by \( \phi = \tan^{-1}(X_C/R) \), giving a leading power factor \( R/Z \).
Explained simply
A capacitor first takes in charge and only then shows voltage, so in a capacitor the current is ahead of the voltage by 90°. When a resistor is added in series, the total circuit is partly "resistor-like" and partly "capacitor-like", so the current leads the supply voltage by some angle between 0° and 90°.
Technically, the same current flows through R and C. \( V_R = IR \) is in phase with \( I \), \( V_C = IX_C \) lags \( I \) by 90°. The supply voltage is their phasor sum, which lags \( I \); equivalently, \( I \) leads \( V \).
How it works
- Take \( I \) as reference. Draw \( V_R \) along it and \( V_C \) vertically downward.
- \( V = \sqrt{V_R^2 + V_C^2} \); impedance triangle has \( R \) horizontal and \( X_C \) downward.
- Complex form: \( Z = R - jX_C \).
- \( \phi = \tan^{-1}(X_C/R) \); the power factor \( \cos\phi \) is leading.
- Real power is used only in R: \( P = I^2R \).
Key formula
\( Z = \sqrt{R^2 + X_C^2}, \quad X_C = \dfrac{1}{2\pi fC}, \quad \cos\phi = \dfrac{R}{Z}\text{ (leading)} \)
\( Z \) = impedance (\( \Omega \)), \( R \) = resistance (\( \Omega \)), \( X_C \) = capacitive reactance (\( \Omega \)), \( f \) = frequency (Hz), \( C \) = capacitance (F), \( \phi \) = angle by which current leads voltage.
Quick example
\( R = 40\,\Omega \), \( X_C = 30\,\Omega \), 100 V supply. \( Z = 50\,\Omega \), \( I = 2 \) A, \( \cos\phi = 0.8 \) leading (\( \phi = 36.87^\circ \)). \( V_R = 80 \) V, \( V_C = 60 \) V, and \( \sqrt{80^2+60^2} = 100 \) V. \( P = 100\times2\times0.8 = 160 \) W \( = 2^2\times40 \). RC circuits are used in timers, filters and coupling networks.
Common mistakes
- Writing \( Z = R + jX_C \); the capacitive term carries \( -j \).
- Calling the power factor lagging in an RC circuit.
- Forgetting to convert \( \mu F \) to farads when finding \( X_C \).
How MU asks it
Asked as a 5–8 mark numerical: "A resistor and capacitor in series across 230 V, 50 Hz; find impedance, current, power factor, voltage across each element and draw the phasor diagram." For full marks: show \( X_C \) with unit conversion, give the leading power factor, draw the phasor and impedance diagrams, and check that the phasor sum of drops equals the supply.