The current, I, in an RLC series circuit is given by
I = V/Z = V/√R2 + (XL – XC)2
Where XL = 2πfL and XC = 1/2πfC. The maximum current is obtained in the circuit when the impedance Z, in the above equation is minimum. This happens when
XL = XC or 2πfL = 1/2πfC
Resonance is said to occur in a.c. series circuit when the maximum current is obtained from such a circuit.
The frequency at which this resonance occurs is called the resonance frequency (f0). This is the frequency at which XL = XC or 2πf0L = 1/2πf0C.
Hence solving the above equation we obtain that f0 is given by
f0 = 1/2π√LC
or since ω = 2πf, we can write the condition of resonance as:
ω0 = 1/√LC
The variation of current I, and frequency f, in an RLC series circuit
Resonance in RLC Series Circuit
Application of Resonance
The resonant circuit finds applications in electronics. It is used to tune radios and TVs. Its great advantage is that it responds strongly to one particular frequency. The other frequencies are very little effect.
Hence such a resonant circuit can select one signal of a definite frequency from a jumble of other signals available to it. That resonance frequency corresponds to that of a particular incoming radio signal. When this happens, maximum current is obtained and the distant radio station is loudly and clearly heard.
An a.c. voltage of amplitude 2.0 volts is connected to an RLC series circuit. If the resistance in the circuit is 5 ohms, and the inductance and capacitance are 3 mH and 0.05μF respectively, calculate
- the resonance frequency f0
- the maximum a.c. current at resonance
f0 = 1/2π√LC
= 1/2π√3 x 10-3 x 5 x 10-8 = 1/2π√15 x 10-11
= 1299.545 Hz
= 13 KHz
At resonance Z = R since XL = XC
I0 = V0/R = 2/5 = 0.4 Amps