Damped harmonic oscillators - LRC circuits

The basic laws of circuits containing inductors (inductance L in Henries, H), resistors (resistance R in Ohms, $\Omega$) and capacitors (capacitance C in Farads, F) relate the current i in these elements to the voltage V across them,

$$V_{R}=iR;\ \ \ V_{L}=-L{\frac{di}{dt}};\ \ \ V_{C}={\frac{Q}{C}}$$ (4)

where the charge $Q=\int_{0}^{t}idt^{\prime }$ is the integral of the current. If these three elements are combined in series in a ciruit, and are also driven by an alternating voltage source $V_{s}=V_{0}\sin \omega t$, then the equation for the current in the circuit as a function of time is:

V_s+V_R+V_C+V_L=0. (5)

Taking a derivative of this equation leads to the damped harmonic oscillator equation,

$$V_{0}\omega \cos \omega t+R{\frac{di}{dt}}+{\frac{i}{C}}-L{\frac{di^{2}}{% dt^{2}}}=0.$$ (6)

The damped harmonic oscillator equation also applies to damped pendulums and to damped mass-spring systems. In these problems is is important to distinguish between transients and steady state response. If there is no driving term then the behavior is only transient and is classified as underdamped or overdamped. If there is no damping then there is a natural frequency $\omega _{0}=1/\sqrt{LC}$. When the system is ``driven'' and the damping is not too large there is a special drive frequency (e.g. $\omega =\omega _{0}$ for no damping) at which the amplitude of the response gets large. This is the phenomenon of resonance. In the driven case there are also transients but they are rather complex and are not treated in most undergraduate courses.