## What is called an LCR circuit?

*A circuit containing an inductor of pure inductance ( L ) , a capacitor of pure capacitance ( C ) and a resistor of pure resistance ( R ) joined in series across an AC supply is called a LCR circuit.*

### Phasor diagram of LCR circuit

Consider about an LCR circuit as shown in figure.

Let, ( V ) is the *RMS value* of the applied alternating *emf* and ( I ) is the *RMS value* of *current* to the circuit.

Now total effective opposition offered by the circuit is obtained with the help of *phasor diagram* as shown in figure.

The potential difference across the inductor –

V_L = I X_L ……… (1)

This *potential difference* or voltage leads the current ( I ) by an angle \left ( \frac {\pi}{2} \right ) . It is represented by OB perpendicular to the direction of ( I ) .

The potential difference across the capacitor –

V_C = I X_C ……… (2)

This potential difference or voltage lags behind the current ( I ) by an angle \left ( \frac {\pi}{2} \right ) . It is represented by OF perpendicular to the direction of ( I ) .

The potential difference across resistor –

V_R = I R ……… (3)

This potential difference or voltage is in same phase with the current ( I ) . It is represented by OA in the direction of ( I ) .

Since ( V_L ) and ( V_C ) are in opposite phases, so their resultant obtained by vector sum, \left ( V_L - V_C \right ) is represented by OD . Here, we have assumed that \quad \left ( V_L \ > \ V_C \right ) .

The resultant of ( V_R ) and \left ( V_L - V_C \right ) is represented by OH .

The magnitude of OH is given by –

OH = \sqrt {\left ( OH \right )^2 + \left ( OD \right )^2}

Or, \quad OH = \sqrt { V_R^2 + \left (V_L - V_C \right )^2 } .

Therefore, \quad V = \sqrt { V_R^2 + \left (V_L - V_C \right )^2 } ……… (4)

### Impedance

*Effective opposition offered by an LCR circuit is called impedance of the circuit.*

Impedance is similar to the effective resistance of a circuit.

For an LCR circuit in which elements are connected in series to an AC supply, we have –

V = \sqrt { V_R^2 + \left (V_L - V_C \right )^2 }

We have the relations –

- For
*Inductance*( V_L = I X_L ) ……… (1) - For
*Capacitance*( V_C = I X_C ) ……… (2) - For
*Resistance*( V_R = I R ) ……… (3)

Therefore, from equations (1), (2) and (3) we get –

V = \sqrt { I^2 R^2 + \left (I X_L - I X_C \right )^2 }

= I \sqrt { R^2 + \left (X_L - X_C \right )^2 } .

So, \quad \left ( \frac {V}{I} \right ) = \sqrt { R^2 + \left (X_L - X_C \right )^2 } .

But \quad \left ( \frac {V}{I} \right ) = Z .

*Where ( Z ) is the effective opposition offered by the LCR circuit. It is called impedance of the circuit.*

Therefore, Impedance for an LCR circuit is given by –

Z = \sqrt { R^2 + \left (X_L - X_C \right )^2 } …….. (4)

Also, \quad \left ( \frac {V}{I} \right ) = Z

Therefore, \quad I = \left ( \frac {V}{Z} \right )

= \left ( \frac {V}{\sqrt { R^2 + \left (X_L - X_C \right )^2 }} \right )

= \left [ \frac {V}{\sqrt { R^2 + \left (L \omega - \frac {1}{C \omega} \right )^2 }} \right ] …….. (5)

Let ( \phi ) is the angle between ( V ) and ( I ) . Then from geometry of the figure –

\tan \phi = \left ( \frac {AH}{OA} \right ) = \left ( \frac {V_L - V_C}{V_R} \right )

= \left ( \frac {I X_L - I X_C}{I R} \right ) = \left ( \frac {X_L - X_C}{R} \right )

= \left [ \frac {\left ( L \omega - \frac {1}{C \omega } \right )}{R} \right ] ……. (6)

## Power Factor of an LCR circuit

*Ratio of true power and apparent power in an A.C circuit is called as power factor of the circuit.*

\text {Power Factor} = \left ( \frac {\text {True Power}}{\text {Apparent Power}} \right )

Therefore, \quad PF = \left ( \frac {I V_R}{I V} \right )

= \left ( \frac {I^2 R}{I^2 Z} \right )

= \left ( \frac {R}{Z} \right )

In general, power factor of a circuit is represented by the cosine of angle ( \phi ) which is the *phase difference* between the voltage and the current.

From geometry of phasor diagram, we have –

\cos \phi = \left ( \frac {OA}{OH} \right )

= \left ( \frac {V_R}{V} \right )

= \left ( \frac {I R}{I Z} \right ) = \left ( \frac {R}{Z} \right )

Therefore, Power factor is given by –

\cos \phi = \left ( \frac {R}{Z} \right )

= \left [ \frac {R}{\sqrt {R^2 + \left ( X_L - X_C \right )^2}} \right ]

= \left [ \frac {R}{\sqrt {R^2 + \left ( L \omega - \frac {1}{C \omega} \right )^2}} \right ]

**TO BE NOTED –**

Power factor of a circuit is always positive and lies between 0 and 1 .

- For a pure resistive circuit, power factor ( \cos \phi = 1 )
- For a pure inductive circuit, power factor ( \cos \phi = 0 )
- For a pure capacitive circuit, power factor ( \cos \phi = 0 )
- For other type of circuits ( 1 \ > \ \cos \phi \ > \ 0 )