LCR Circuit

A typical example of driven harmonic oscillator is a series LCR circuit in which oscillations are sustained by an alternating emf of frequency ω, ε = ε₀ sin ω t.

Consider a capacitor C, an inductor L, and a resistor R in series with an emf ε. According to Kirchoff’s law:

which is similar to typical forced mechanical oscillator equation:





Q = Q₀ sin (ω t + Ø)





The instantaneous current in the circuit i = dQ/dt corresponds to velocity of mechanical oscillator:

i = Q₀ ω cos (ω t + Ø) = i₀ cos (ω t + Ø)



i = i₀ sin (ω t + Ø) , θ = Ø + π/2

where, θ now refers to phase difference between current i and emf ε.

Note that resistance R in electrical impedance plays the same role as resistive part ‘a’ in mechanical impedance. The reactive part of electrical impedance is X = ω L – 1/C ω

The inductive reactance ω L is equivalent to inertia term m ω and capacitative reactance 1/ω C corresponds to elasticity of spring k/ω of mechanical oscillator. Inductance produces inertia while (inverse of) capacitance produces springiness. Power: The instantaneous power input in driven LCR circuit is

P ( t ) = ε_i = ε₀ i₀ sin ω t sin (ω t + θ)

Average power input per cycle is



Average power input is maximum when cos θ = 1; that is, power resonance occurs when ω = ω_r, such that



The Q of the circuit is given by

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