LC Circuits

Dividing by , and differentiating with respect to , we obtain

where

Comparison with Equation (17) reveals that Equation (34) is a simple harmonic oscillator equation with the associated angular oscillation frequency . We conclude that the current in an LC circuit executes simple harmonic oscillations of the form

where and are constants. According to Equation (33), the potential drop, , across the capacitor is minus that across the inductor, so that , giving

Here, use has been made of the trigonometric identity . (See Appendix B.) It follows that the voltage in an LC circuit oscillates at the same frequency as the current, but with a phase shift of radians. In other words, the voltage is maximal when the current is zero, and vice versa. The amplitude of the voltage oscillation is that of the current oscillation multiplied by . Thus, we can also write

Comparing with Equation (22), we deduce that

(39) |

is a conserved quantity. However, , and . Thus, multiplying the preceding expression by , we obtain

The first and second terms on the right-hand side of the preceding expression can be recognized as the instantaneous energies stored in the capacitor and the inductor, respectively (Fitzpatrick 2008). The former energy is stored in the electric field generated when the capacitor is charged, whereas the latter is stored in the magnetic field induced when current flows through the inductor. It follows that that the quantity in Equation (40) is the total energy of the circuit, and that this energy is a conserved quantity. The oscillations of an LC circuit can, thus, be understood as a cyclic interchange between electric energy stored in the capacitor, and magnetic energy stored in the inductor.

Suppose that at the capacitor is charged to a voltage , and there is zero current flowing through the inductor. In other words, the initial state is one in which all of the circuit energy resides in the capacitor. The initial conditions are and . In this case, it can be shown that the current evolves in time as

(41) |

Suppose that at the capacitor is fully discharged, and there is a current flowing through the inductor. In other words, the initial state is one in which all of the circuit energy resides in the inductor. The initial conditions are and . In this case, it can be demonstrated that the current evolves in time as

(42) |

Suppose, finally, that at the capacitor is charged to a voltage , and the current flowing through the inductor is . Since the solutions of the simple harmonic oscillator equation are superposable, it follows that the current evolves in time as

Furthermore, according to Equation (38), the voltage evolves in time as

(44) |

or

Here, use has been made of the trigonometric identities and . (See Appendix B.)

The instantaneous electrical power absorption by the capacitor, which can be shown to be minus the instantaneous power absorption by the inductor, is (Fitzpatrick 2008)

(46) |

where use has been made of Equations (43) and (45), as well as the trigonometric identities and . (See Appendix B.) Hence, the average power absorption during a cycle of the oscillation,

(47) |

is zero, because it is readily demonstrated that . In other words, any energy that the capacitor absorbs from the circuit during one half of the oscillation cycle is returned to the circuit, without loss, during the another. The same goes for the inductor.