14.6 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 $(R L C)$ series circuit
Relate the $R L C$ circuit to a damped spring oscillation

When the switch is closed in the RLC circuit of Figure 14.17(a), the capacitor begins to discharge and electromagnetic energy is dissipated by the resistor at a rate $i^{2} R$ . With U given by Equation 14.37, we have

\frac{d U}{d t} = \frac{q}{C} \frac{d q}{d t} + L i \frac{d i}{d t} = − i^{2} R

14.43

where i and q are time-dependent functions. This reduces to

L \frac{d^{2} q}{d t^{2}} + R \frac{d q}{d t} + \frac{1}{C} q = 0 .

14.44

Figure a is a circuit with a capacitor, an inductor and a resistor in series with each other. They are also in series with a switch, which is open. Figure b shows the graph of charge versus time. The charge is at maximum value, q0, at t=0. The curve is similar to a sine wave that reduces in amplitude till it becomes zero.

Figure 14.17 (a) An RLC 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 q versus t. The capacitor contains a charge

q_{0}

before the switch is closed.

This equation is analogous to

m \frac{d^{2} x}{d t^{2}} + b \frac{d x}{d t} + k x = 0,

which is the equation of motion for a damped mass-spring system (you first encountered this equation in Oscillations). 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 b/2m. Therefore, the result can be underdamped $(\sqrt{k / m} > b / 2 m)$ , critically damped $(\sqrt{k / m} = b / 2 m)$ , or overdamped $(\sqrt{k / m} < b / 2 m)$ . By analogy, the solution q(t) to the RLC differential equation has the same feature. Here we look only at the case of under-damping. By replacing m by L, b by R, k by 1/C, and x by q in Equation 14.44, and assuming $\sqrt{1 / L C} > R / 2 L$ , we obtain

q (t) = q_{0} e^{− R t / 2 L} cos (ω ′ t + ϕ)

14.45

where the angular frequency of the oscillations is given by

ω^{'} = \sqrt{\frac{1}{L C} - {(\frac{R}{2 L})}^{2}}

14.46

This underdamped solution is shown in Figure 14.17(b). Notice that the amplitude of the oscillations decreases as energy is dissipated in the resistor. Equation 14.45 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 q(t).

Interactive

In the interactive circuit construction kit simulation below, you can add inductors and capacitors to work with any combination of R, L, and C circuits with both dc and ac sources. Then you can measure and evaluate current and voltage, and graph them as a function of time.

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Interactive

With the interactive DC circuit construction kit simulation below, you can add inductors and capacitors to work with any combination of R, L, and C circuits with both dc and ac sources. Then you can measure and evaluate current and voltage, and graph them as a function of time.

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Check Your Understanding 14.11

In an RLC circuit, $L = 5.0 mH, C = 6.0 μ F, and R = 200 Ω .$ (a) Is the circuit underdamped, critically damped, or overdamped? (b) If the circuit starts oscillating with a charge of $3.0 \times 10^{−3} C$ on the capacitor, how much energy has been dissipated in the resistor by the time the oscillations cease?