Derive an expression for the impedance of a series LCR circuit connected to an AC supply of variable frequency. Plot a graph showing variations 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 resistance R, inductance L and capacitance C are connected in series and an alternating source of voltage V =V0sinω is applied across it. (fig. a) On account of being in series, the current (i ) flowing through all of them is the same.
Suppose the voltage across resistance R is VR, voltage across inductance L is VL and voltage across capacitance C is VC. The voltage VR and current i are in the same phase, the voltage VL will lead the current by angle 90° while the voltage VC will lag behind the current by angle 90° (fig. b). Clearly VC and VL are in opposite directions, therefore their resultant potential difference =VC -VL (if VC >VC).
Thus VR and (VC - VL) are mutually perpendicular and the phase difference between them is 90°. As applied voltage across the circuit is V, the resultant of VR and (VC - VL ) will also be V. From fig.
The phase difference (ϕ) between current and voltage ϕ is given by
Graph of variation of current (i) with frequency (f):
The resonance in series LCR circuit occurs when inductive reactance is equal to capacitance reactance i.e., XL = XC. Alternatively when the frequency of applied a.c. source becomes
Graph is shown in fig.
Use of phenomenon of resonance in tuning a ratio or TV set:
A radio or a TV set has a L-C circuit with capacitor of variable capacitance C. The circuit remains connected with an areal 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 areal, 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 C, f = 1/2π√(LC), the resonance occurs and maximum current flows in the circuit; so the radio or TV gets tunned.