A circuit with an inductor, a capacitor, and a resistor, in series or parallel or any combination. Closely related to the R-C circuit, the series LCR circuit is a lovely demonstration of the calculus involved in oscillations.

Briefly, harmonic oscillations are the solutions to differential equations of the form x''+ax'+bx=0, where primes indicate derivatives and x is some parameter or variable. Since the voltage drop across a inductor is proportional to the rate of change of the current, and the drop across a resistor is proportional to the current, and the drop across a capacitor is proportional to the charge, and the current is the rate of change of the charge, the Kirkhoff loop across and LCR circuit is of this form with x=q, the charge. Solutions to x''+x=0 are sinusiodal, while solutions of the LCR equation are sinuisoidal with a decay envelope superimposed, the speed of the devay depending on R, mostly. Analyses of these systems involve things like Q factors and damping rates, and have great application to everything, including quantum mechanics, car suspensions, and subwoofers.

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