RLC circuit

A circuit containing resistance(R), inductance(L), and capacitance(C) is called an RLC circuit. I shall here discuss only series simple RLC circuits(only one R,L,C). With a resistance R present,the total electromagnetic energy U of the circuit(the sum of the electric energy and magnetic energy)is no longer constant: instead, it decreases with time as energy is transferred to thermal energy in the resistance. Because of this loss of energy, the oscillations of charge, current and potential difference continuously decrease in amplitude, and the oscillations are said to be damped. As you will see, they are damped in exactly the same way as those of the damped block-spring oscillator. To analyze the oscillations of this circuit, we write an equation for the total electromagnetic energy U in the circuit at any instant. Because the resistance does not store electromagnetic energy.

$U = U_B + U_E = \frac{Li^2}{2} + \frac{q^2}{2C}$ =>Eq.1

However, this total energy decreases as energy is transferred to thermal energy.

$\frac{dU}{dt} = -{i^2}R$ =>Eq.2

where the minus sign indicates that U decreases. By differentiating Eq.1 with respect to time and then substituting the result in Eq.2, we obtain

$\frac{dU}{dt} = {Li}\frac{di}{dt} + \frac{q}{c}\frac{dq}{dt} = -{i^2}{R}$

Substituting $dq/dt$ for $i$ and $d^2/dt^2$ for $di/dt$, we obtain

${L}\frac{d^2q}{dt^2} + {R}\frac{dq}{dt} + \frac{q}{C} = 0$ (RLC circuit)