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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: U UC U L 2 Q 2 U 0.5 L I 2 C 2 Q • The energy stored in the capacitor is and 2 C the energy stored in the inductor is 0.5·L·I2. • 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 I2·R, we have: dU 2 dt I R – 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: Q2 L I2 d d 2 C 2 dU dU 2 I R dt dt dt dt 2 2 dU 1 d Q L d I dt 2 C dt 2 dt dU 1 dQ L dI 2 Q 2 I dt 2 C dt 2 dt dU Q dQ dI L I dt C dt dt Q dQ dI 2 I R L I C dt dt • Using the fact that dQ I dt and 2 dI d Q 2 dt dt Q dQ dI I R L I C dt dt 2 2 Q d Q I R I L I 2 C dt 2 • Factor out an I and set up the resulting quadratic equation: 2 Q d Q I R I L I 2 C dt 2 Q d Q I ( I R ) I L C dt 2 L L 2 d Q 2 dt 2 d Q dt 2 Q I R 0 C dQ Q R 0 dt C dQ I dt • The RLC circuit is analogous to the damped harmonic oscillator. • The equation of motion for the damped harmonic oscillator is: 2 m d x dt 2 dx b k x 0 dt • Comparing the two equations: L d 2Q dt 2 dQ Q R 0 dt C – 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. d 2Q dQ Q • When R = 0 , L reduces to a R 0 dt C dt 2 simple LC circuit and the charge and current oscillate sinusoidally in time. • When R is small, the solution is: Q Qm R t e 2L cos d t 1 2 2 1 R where wd L C 2 L • 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 Rc above which no oscillations occur. – The critical value is given by 4L Rc C • A system with R = Rc is said to be critically damped. • When R exceeds Rc, the system is said to be overdamped. • The graph of Q vs. t for an overdamped RLC circuit, which occurs when the value of 4L R C