Derive an expression for the impedance of a series LCR circuit connected to an AC supply of variable frequency.

Plot a graph showing variation of current with the frequency of the applied voltage.

Explain briefly how the phenomenon of resonance in the circuit can be used in the

Tuning mechanism of a radio or a TV set.

a)
Expression for impedance in LCR series circuit:

Suppose a resistance R, inductance L and capacitance C are connected to series and an alternating voltage V =  is applied across it. 

Since L, C and R are connected in series, current flowing through them is the same. The voltage across R is V_R, inductance across L is V_L and across capacitance is V_C. 
The voltage V_R and current i are in the same phase, the voltage V_L will lead the current by angle 90° while the voltage V_C will lag behind the current by 90°. 
Clearly V_C and V_L are in opposite directions, therefore their resultant potential difference =V_C -V_L (if V_C >V_C).

Thus, V_R and (V_C -V_L) are mutually perpendicular and the phase difference between them is 90°. As seen in the fig, we can say that, as applied voltage across the circuit is V, the resultant of V_R and (V_C -V_L) will also be V. 
So, 
 

Therefore, Impedance of the circuit is given by, 

This is the impedence of the LCR series circuit.

b)
The below graph shows the variation of current with the frequency of the applied voltage. 
 
c)
A radio or a TV set has a L-C circuit with capacitor of variable capacitance C. The circuit remains connected with an aerial coil through the phenomenon of mutual inductance. Suppose a radio or TV station is transmitted a programme at frequency f, then waves produce alternating voltage of frequency f, in area, due to which an emf of same frequency is induced in LC circuit, When capacitor C is in circuit is varied then for a particular value of capacitance, , the resonance occurs and maximum current flows in the circuit; so the radio or TV gets tuned.