Question

Draw vector diagram (phasor diagram) for a series RLC circuit which is connected with an alternating voltage source and determine the expression for impedance of the circuit. 

Answer 1

Series LCR - circuit : -

Suppose a resistance R, an inductance L and capacitance C are connected in series to a source of alternating emf given by

1.  Voltage V_R = RI Racross the resistance R will be in phase with current I  . So phasors V_R  and I  are is same direction. The amplitude of V_R  is

V^R₀  = I₀R

2.  Voltage  V_L =  X _LI across the inductance L is ahead of current I in phase by π/2 rad. So phasor V_L  lies π/2 rad anticlockwise w.r.t. the phasor I . Its amplitude is 

V₀^L  = I₀X_L

3.  Voltage  V_c  = X_cI  across the capacitance C lags behind the current I  in phase by  π/2 rad. So phasor V_C  lies π/2 clockwise w.r.t. the phasor I . Its amplitude is 0 0 

V₀^c  = I₀ X_C 

As V_L  and V_C  are in opposite directions, their resultant is (V_L - V_C). By parallelogram law, the resultant of V_R  and  (V_L - V_C) must be equal to the applied emf ε , given by the diagonal of the parallelogram.

 

Clearly, √(R²+(X_L-X_C)²) is the effective resistance of the series LCR - circuit which opposes or impedes the flow of current through it and is called its impedance. It is denoted by Z and its SI unit is ohm (Ω). Thus