Given an electric circuit composed of an inductor with inductance L, a resistor with resistance R, and a capacitor with capacitance C, the Kirchhoff loop rule requires that the sum of the changes in potential around the circuit must be zero, so

where I is the current through the inductor,resistor and Q is the charge on the capacitor, and t is the elapsed time. Differentiating gives

which can be rewritten

Now define the variables


to write the differential equation in the standard form

Here is the differential operator.
As in any simple harmonic motion,there are three classes of solution depending on the sign of : underdamped, critically damped, and overdamped. For an underdamped circuit,

which is equivalent to

and the simple harmonic motion solution is

where

and


For a critically damped circuit

which is equivalent to

and the solution is

where


For an overdamped circuit,

or

The solution is

where

giving


and the constants are given by


For a sinusoidally driven CLR circuit, define





where is the permittivity of free space. The equations are




The problem can be solved by solving the differential equation. However, it is much simpler to assume a harmonic solution

Use complex impedance to find the solution of this type, if it exists



So the harmonic solution is

where


Dividing by CL,




The homogeneous solution will be underdamped, critically, or overdamped depending on the initial conditions. However, it will be exponentially decaying, so the steady state solution is given be the solution above. The maximum of

occurs at

Furthermore, at ,, so



]
but

so






Both and are maximum at , so

To find the half-power points, solve







The full width at half maximum is

The amplitude decay time is

The energy stored in the system is


so




Q is a minimum when



Note that near resonance,