LCR Circuit

Consider an electrical circuit consisting of an inductor, of inductance $L$, connected in series with a capacitor, of capacitance $C$, and a resistor, of resistance $R$. See Figure 2.2. Such a circuit is known as an LCR circuit, for obvious reasons. Suppose that $I(t)$ is the instantaneous current flowing around the circuit. As we saw in Section 1.4, the potential drops across the inductor and the capacitor are $L\,\dot{I}$ and $Q/C$, respectively. Here, $Q$ is the charge on the capacitor's positive plate, and $I = \dot{Q}$. Moreover, from Ohm's law, the potential drop across the resistor is $V=I\,R$ (Fitzpatrick 2008). Kirchhoff's second circuital law states that the sum of the potential drops across the various components of a closed circuit loop is zero. It follows that

$$L\,\dot{I} + R\,I + Q/C=0.$$ (2.32)

Dividing by $L$, and differentiating with respect to time, we obtain

$$\ddot{I} + \nu\,\dot{I} + \omega_0^{\,2}\,I=0,$$ (2.33)

where

$$\omega_0 = \frac{1}{\sqrt{L\,C}},$$ (2.34)

$$\nu = \frac{R}{L}.$$ (2.35)

Comparison with Equation (2.2) reveals that Equation (2.33) is a damped harmonic oscillator equation. Thus, provided that the resistance is not too high (i.e., provided $\nu< 2\,\omega_0$, which is equivalent to $R<2\sqrt{L/C}$), the current in the circuit executes damped harmonic oscillations of the form [cf., Equation (2.12)]

$$I(t) = \hat{I}\,{\rm e}^{-\nu\,t/2}\,\cos(\omega_1\,t-\phi),$$ (2.36)

where $\hat{I}$ and $\phi$ are constants, and $\omega_1=(\omega_0^{\,2}-\nu^{\,2}/4)^{1/2}$. We conclude that if a small amount of resistance is introduced into an LC circuit then the characteristic oscillations in the current damp away exponentially at a rate proportional to the resistance.

Figure 2.2: An LCR circuit.

Multiplying Equation (2.32) by $I$, and making use of the fact that $I = \dot{Q}$, we obtain

$$L\,\dot{I}\,I + R\,I^{\,2}+\dot{Q}\,Q/C = 0,$$ (2.37)

which can be rearranged to give

$$\frac{dE}{dt} = - R\,I^{\,2},$$ (2.38)

where

$$E = \frac{1}{2}\,L\,I^{\,2} + \frac{1}{2}\,\frac{Q^{\,2}}{C}.$$ (2.39)

Here, $E$ is the circuit energy; that is, the sum of the energies stored in the inductor and the capacitor. Moreover, according to Equation (2.38), the circuit energy decays in time due to the power $R\,I^{\,2}$ dissipated via Joule heating in the resistor (Fitzpatrick 2008). Of course, the dissipated power is always positive. In other words, the circuit never gains energy from the resistor.

Finally, a comparison of Equations (2.31), (2.34), and (2.35) reveals that the quality factor of an LCR circuit is

$$Q_f = \frac{\sqrt{L/C}}{R}.$$ (2.40)