(i) An a.c. source of voltage V = V_o sin ωt is connected to a series combination of L, C and R. Use the phasor diagram to obtain an expression for impedance of a circuit and the phase angle between voltage and current. Find the condition when current will be in phase with the voltage. What is the circuit in this condition called?

ii) In a series LR circuit, X_L = R and the power factor of the circuit is P₁. When capacitor with capacitance C, such that X_L = X_C is put in series, the power factor becomes P₂. Calculate P₁ / P₂.

Voltage of the source is given by, 

V = V_o sin ωt 

                                           

Let current of the source be I = I_o sin ωt

Maximum voltage across R is V_R = V_o R, represented along OX

Maximum voltage across L = V_L  = I_O X_L, represented along OY and is 90^o ahead of I_o.

Maximum voltage across C = V_C = I_o X_C, represented along OC and is lagging behind I_o by 90^o

Hence, reactive voltage is V_L - V_C, represented by OB'



the vector sum of V_R, V_L and V_C is resultant of OA and OB', represented along OK.

OK = V_o = 
i.e., V_o = 



Impedance, Z = 

When, X_L = X_C ,  the voltage and current are in the same phase. 

In such a situation, the circuit is known as non-inductive circuit. 

ii) 

Given,

Power factor, P₁ = R/Z


Thus,