The RLC Circuit AP Physics C Montwood High School R. Casao

A more realistic circuit consists of an inductor, a capacitor, and a resistor connected in series. Assume that the capacitor has an initial charge Qm before the switch is closed. Once the switch is closed and a current is established, the total energy stored in the circuit at any time is given by:

The energy stored in the capacitor is and the energy stored in the inductor is 0.5·L·I 2. However, the total energy is no longer constant, as it was in the LC circuit, because of the presence of the resistor, which dissipates energy as heat. Since the rate of energy dissipation through the resistor is I 2 ·R, we have: –the negative sign signifies that U is decreasing in time.

Substituting this equation into the time derivative of the total energy stored in the LC circuit equation:

Using the fact that

Factor out an I and set up the resulting quadratic equation:

The RLC circuit is analogous to the damped harmonic oscillator. The equation of motion for the damped harmonic oscillator is: Comparing the two equations: –Q corresponds to x; L corresponds to m; R corresponds to the damping constant b; and 1/C corresponds to 1/k, where k is the force constant of the spring. The quantitative solution for the quadratic equation involves more knowledge of differential equations than we possess, so we will stick with the qualitative description of the circuit behavior.

When R = 0 , reduces to a simple LC circuit and the charge and current oscillate sinusoidally in time. When R is small, the solution is: The charge will oscillate with damped harmonic motion in analogy with a mass-spring system moving in a viscous medium.

The graph of charge vs. time for a damped RLC circuit. For large values of R, the oscillations damp out more rapidly; in fact, there is a critical resistance value R c above which no oscillations occur. –The critical value is given by

A system with R = R c is said to be critically damped. When R exceeds R c, the system is said to be overdamped. The graph of Q vs. t for an overdamped RLC circuit, which occurs when the value of