Driven LCR Circuits

where . Suppose that the emf is such that its voltage oscillates sinusoidally at the angular frequency , with the peak value , so that

(122) |

Dividing Equation (121) by , and differentiating with respect to time, we obtain [cf., Equation (91)]

(123) |

where and . Comparison with Equation (101) reveals that this is a driven damped harmonic oscillator equation. It follows, by analogy with the analysis contained in the previous section, that the current driven in the circuit by the oscillating emf is of the form

(124) |

where

(125) | ||

(126) |

In the immediate vicinity of the resonance (i.e., ), these expression simplify to

(127) | ||

(128) |

The circuit's mean power absorption from the emf is written

(129) |

so that

(130) |

close to the resonance. It follows that the peak power absorption, , takes place when the emf oscillates at the resonant frequency, . Moreover, the power absorption falls to half of this peak value at the edges of the resonant peak: that is, .

LCR circuits can be used as *analog radio tuners*. In this application,
the emf represents the analog signal picked-up by a radio antenna. According to the
previous analysis, the circuit only has a strong response (i.e., it only absorbs significant energy) when
the signal oscillates in the angular frequency range
, which corresponds
to
. Thus, if the values of
,
, and
are
properly chosen then the circuit can be made to strongly absorb the signal from a
particular radio station, which has a given carrier frequency and bandwidth (see Section 9.3), while essentially ignoring the signals from other stations
with different carrier frequencies. In practice, the values of
and
are fixed,
while the value of
is varied (by turning a knob that adjusts the degree of overlap
between two sets of parallel semicircular conducting plates) until the signal from the desired radio
station is found.