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LCR Circuits

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 8.) 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 2.3, 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

$\displaystyle L\,\dot{I} + R\,I + Q/C=0.$ (90)

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

$\displaystyle \ddot{I} + \nu\,\dot{I} + \omega_0^{\,2}\,I=0,$ (91)

where

$\displaystyle \omega_0$ $\displaystyle = \frac{1}{\sqrt{L C}},$ (92)
$\displaystyle \nu$ $\displaystyle = \frac{R}{L}.$ (93)

Comparison with Equation (63) reveals that Equation (91) 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 (72)]

$\displaystyle I(t) = I_0 {\rm e}^{-\nu t/2} \cos(\omega_1 t-\phi),$ (94)

where $ I_0$ 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 8: An LCR circuit.
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Multiplying Equation (90) by $ I$ , and making use of the fact that $ I = \dot{Q}$ , we obtain

$\displaystyle L\,\dot{I}\,I + R\,I^{\,2}+\dot{Q}\,Q/C = 0,$ (95)

which can be rearranged to give

$\displaystyle \frac{dE}{dt} = - R\,I^{\,2},$ (96)

where

$\displaystyle E = \frac{1}{2}\,L\,I^{\,2} + \frac{1}{2}\,\frac{Q^{\,2}}{C}.$ (97)

Here, $ E$ is the circuit energy: that is, the sum of the energies stored in the inductor and the capacitor. Moreover, according to Equation (96), 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 (89), (92), and (93) reveals that the quality factor of an LCR circuit is

$\displaystyle Q_f = \frac{\sqrt{L/C}}{R}.$ (98)


next up previous
Next: Driven Damped Harmonic Oscillation Up: Damped and Driven Harmonic Previous: Quality Factor
Richard Fitzpatrick 2013-04-08