Question

Establish the phase relation between voltage and current in an a.c. circuit containing capacitance C and resistance R and determine the impedance of C-R circuit.

Answer 1

Let a capacitor of capacitance C and a resistance R be connected in series to a source of an alternating e.m.f. E as shown in Fig. (a).

The alternating current I at all points in the circuit has the same amplitude and phase, but the voltage across each element will be of different amplitude and phase.

(i) If V_R is the voltage across R, then

V_R = IR

Voltage across the the resistor is in phase with current.

(ii) If V_L is the voltage across C, then

V_C = IX_C

Voltage across the capacitor lags behind the current by a phase π/2.

Let us construct the phasor diagram for this circuit. A single phasor I is used to represent the current in each element and V_R and V_C represent voltage phasor. If V_R is represented along OX by OL (current is also along OX) then, V_C will be represented along OY' by OM because V_C lags behind I by a phase π/2. This is graphically represented in Fig. (b). The resultant of V_R and V_C is represented by E and makes an angle Φ with current I

From the phasor diagram, we find that

where Z_C = \(\sqrt{R^2+1/\omega^2 C^2}\)  is the effective opposition offered by C and R and is called impedance of CR circuit.

Since Φ is the angle made by the resultant of V_C and R with x-axis so from phasor diagram, we have

Eq. (2) gives the phase angle Φ by which alternating e.m.f. (voltage) lags behind the current in a CR circuit.