# 11.6 RLC Series Circuits

#### LEARNING OBJECTIVES

By the end of this section, you will be able to:

• Determine the angular frequency of oscillation for a resistor, inductor, capacitor ( ) series circuit
• Relate the circuit to a damped spring oscillation

When the switch is closed in the circuit of Figure 11.6.1(a), the capacitor begins to discharge and electromagnetic energy is dissipated by the resistor at a rate . With given by Equation 11.3.2, we have

(11.6.1) where and are time-dependent functions. This reduces to

(11.6.2) (Figure 11.6.1)  Figure 11.6.1 (a) An circuit. Electromagnetic oscillations begin when the switch is closed. The capacitor is fully charged initially. (b) Damped oscillations of the capacitor charge are shown in this curve of charge versus time, or versus . The capacitor contains a charge before the switch is closed.

This equation is analogous to which is the equation of motion for a damped mass-spring system. As we saw in that chapter, it can be shown that the solution to this differential equation takes three forms, depending on whether the angular frequency of the undamped spring is greater than, equal to, or less than . Therefore, the result can be underdamped ( ), critically damped ( ), or overdamped ( ). By analogy, the solution to the differential equation has the same feature. Here we look only at the case of under-damping. By replacing by , by , by , and by in Equation 11.6.2, and assuming , we obtain

(11.6.3) where the angular frequency of the oscillations is given by

(11.6.4) This underdamped solution is shown in Figure 11.6.1(b). Notice that the amplitude of the oscillations decreases as energy is dissipated in the resistor. Equation 11.6.3 can be confirmed experimentally by measuring the voltage across the capacitor as a function of time. This voltage, multiplied by the capacitance of the capacitor, then gives .

### INTERACTIVE

Try an interactive circuit construction kit that allows you to graph current and voltage as a function of time. You can add inductors and capacitors to work with any combination of , , and circuits with both dc and ac sources.

### INTERACTIVE

Try out a circuit-based java applet website that has many problems with both dc and ac sources that will help you practice circuit problems.

In an circuit, , , and . (a) Is the circuit underdamped, critically damped, or overdamped? (b) If the circuit starts oscillating with a charge of on the capacitor, how much energy has been dissipated in the resistor by the time the oscillations cease? 