An a.c. source of voltage V = V_{0} sin ωt is connected to a series combination of L, C, and R. Use the phasor diagram to obtain the expression for an 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?

#### Solution

i)

Voltage of the source is given as

I0→" data-mce-style="position: relative;">V=V_{0}sinωtI0→

Let current of the source be " data-mce-style="position: relative;">I=I_{0}sinωt

The maximum voltage across *R* is `vec(V_R)=vec(V_0)R` represented along OX.

The maximum voltage across L is `vec(V_L)=vec(I_0) X_L`represented along OY and is 90° ahead of `vec(I_0)`

The maximum voltage across *C* is `vec(V_C)=vec(I_0) X_C`represented along OC and is lagging behind `vec(I_0)`by 900

The voltage across L and C has a phase difference of 180° ^{ }

Hence, reactive voltage is`vec(V_L)-vec(V_C)`represented by OB

The vector sum of`vec(V_R), vec(V_L) "and "vec(V_C)`resultant of OA and OB', represented along OK

`OK=V_0=sqrt(OA^2+OB^2)`

`=>V_0=sqrt(V_R2+(V_L-V_C)^2)=sqrt(I_0R^2+(I_0X-V_C)^2)`

`=>V_0=I_0sqrt(R^2+(X_L-X_C)^2)`

The impedance can be calculated as follows:

`Z=V_0/I_0=sqrt(R^2+(X_L-X_C)^2)`

When *X*_{L} = *X*_{C}, the voltage and current are in the same phase. In such a situation, the circuit is known as the non-inductive circuit.