# Phase Relationships of Currents in a Parallel RC Circuit A parallel RC circuit consists of a resistor (R) and a capacitor (C) connected in parallel across an AC voltage source. ## 1. Circuit Description - **Applied voltage:** \( v(t) = V_m \sin(\omega t) \) - **Resistor current:** \( i_R \) - **Capacitor current:** \( i_C \) - **Total current:** \( i_{total} \) ## 2. Currents Through Each Branch ### Resistor Branch - Voltage across resistor: \( v_R(t) = v(t) \) - Ohm’s Law: \[ i_R(t) = \frac{v(t)}{R} = \frac{V_m}{R} \sin(\omega t) \] - **Phase relationship:** The resistor current is **in phase** with the voltage. ### Capacitor Branch - Voltage across capacitor: \( v_C(t) = v(t) \) - Capacitor current: \[ i_C(t) = C \frac{dv(t)}{dt} = C \frac{d}{dt}[V_m \sin(\omega t)] = V_m \omega C \cos(\omega t) \] or \[ i_C(t) = V_m \omega C \sin\left(\omega t + 90^\circ\right) \] - **Phase relationship:** The current through the capacitor **leads** the voltage by \( 90^\circ \). ## 3. Total Current - The total current is the sum of the resistor and capacitor currents: \[ i_{total}(t) = i_R(t) + i_C(t) \] - This total current **leads** the voltage by an angle \( \theta \), where \[ \tan\theta = \frac{I_C}{I_R} = \omega RC \] where \( I_C = V_m \omega C \) and \( I_R = \frac{V_m}{R} \). ## 4. Phasor Diagram  *Alt: Phasor diagram showing voltage along the horizontal axis, resistor current in phase, and capacitor current leading by 90 degrees.* - \( i_R \): Along the voltage phasor (in phase) - \( i_C \): 90° ahead of voltage (vertical, upward) - \( i_{total} \): Vector sum, leads voltage by angle \( \theta \) (between ° and 90°) ## 5. Summary Table | Quantity | Phase Relation to Voltage | |------------------|---------------------------------------| | \( i_R \) | In phase (\( ^\circ \)) | | \( i_C \) | Leads by \( 90^\circ \) | | \( i_{total} \) | Leads by \( \theta = \tan^{-1}(\omega RC) \) | ## 6. **Final Answer** - **Resistor current (\( i_R \))** is in phase with the voltage. - **Capacitor current (\( i_C \))** leads the voltage by \( 90^\circ \). - **Total current (\( i_{total} \))** leads the voltage by an angle \( \theta = \tan^{-1}(\omega RC) \), which is between ° and 90° depending on the values of \( R \), \( C \), and frequency (\( \omega \)).