In real LC circuits, there is always some resistance, and in this type of circuits, the energy is also transferred by radiation. Here we deal with the real case, that is including resistance. An electric circuit that consists of inductor, capacitor and resistor connected in series is called LRC or RLC series circuit.
In the ideal case of zero resistance, the oscillations never die out but with resistance, the oscillations die out after some time. The energy is used up in heating and radiation. To do further analysis of LRC circuit, we consider an electric circuit where inductor of inductance \(L\), resistor of resistance \(R\) and capacitor of capacitance \(C\) are connected in series as shown in Figure 1.
In Figure 1, first we charge the capacitor alone by closing the switch \(S_1\) and opening the switch \(S_2\). Once the capacitor is fully charged we let the capacitor discharge through inductor and resistance by opening the switch \(S_1\) and closing the switch \(S_2\). As the capacitor starts to discharge, the oscillations begin but now we also have the resistance, so the oscillations die out after some time.
To analyze this circuit further we apply Kirchhoff's voltage law (loop rule) in the lower loop in Figure 1. For now we move clockwise in the lower loop and find
\[\frac{q}{C} + L\frac{di}{dt} - iR = 0\]
where \(q\) and \(i\) are the charge and current at any time. You know that \(di/dt = d^2q/dt^2\), so you can rewrite the above equation in the form
\[\frac{d^2q}{dt^2} + \frac{R}{L}\frac{dq}{dt} + \frac{1}{LC}q = 0\]
The solution of the above differential equation for the small value of resistance, that is for low damping or underdamped oscillation is (similar to mechanical damped oscillation of spring-mass system)
\[q = Q_0e^{-Rt/2L}\cos(\omega\,t + \theta) \]
The above equation is analogous to the equation of mechanical damped oscillation. We considered low value of \(R\) to solve the equation, that is when \(R < \sqrt{4L/C}\) because the solution has different forms for small and large values of \(R\). The above equation is for the underdamped case which is shown in Figure 2.
Note that the amplitude \(Q' = Q_0e^{-Rt/2L}\) decreases exponentially with time. The angular frequency of this oscillation is
\[\omega = \sqrt{\frac{1}{LC} - \frac{R^2}{4L^2}}\]
You can see that if there is no resistance \(R\), that is if \(R = 0\), the angular frequency of the oscillation is the same as that of LC-circuit. Similar to mechanical damped oscillation of spring-mass system, when \(\omega = 0\), we get
\[R = \sqrt{4L/C}\]
If the resistance is \(R = \sqrt{4L/C}\) at which the angular frequency becomes zero, there is no oscillation and such damping is called critical damping and the system is said to be critically damped. If \(R > \sqrt{4L/C}\), the system is overdamped.
