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Consider an circuit consisting of an inductor, , a capacitor, , and a resistor, , connected
in series with an emf of voltage . See Figure 10. Let
be the instantaneous current flowing around the circuit.
Now, according to
Kirchhoff's second circuital law, the sum of the potential drops across the
various components of a closed circuit loop is equal to zero. Thus, since the
potential drop across an emf is minus the associated voltage,
we obtain [cf., Equation (83)]

(114) 
where . Suppose that the emf is such that its voltage oscillates
sinusoidally at the angular frequency , with the peak value , so that

(115) 
Dividing Equation (114) by , and differentiating with respect to time,
we obtain [cf., Equation (84)]

(116) 
where
and . Comparison with Equation (94) reveals that this is a driven damped harmonic oscillator equation.
It follows, by comparison with the analysis contained in the previous section, that
the current driven in the circuit by the oscillating emf is of the form

(117) 
where
In the immediate vicinity of the resonance (i.e.,
) these expression simplify to
Now, the circuit's mean power absorption from the emf is written
so that

(123) 
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: i.e.,
.
circuits are often employed as analogue radio tuners. In this application,
the emf represents the analogue signal pickedup by a radio antenna. It is clear, from the
above analysis, that 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, whilst essentially ignoring the signals from other stations
with different carrier frequencies. In practice, the values of and are fixed,
whilst the value of is varied (by turning a knob which adjusts the degree of overlap
between two sets of parallel semicircular conducting plates) until the signal from the desired radio
station is found.
Next: Transient Oscillator Response
Up: Damped and Driven Harmonic
Previous: Driven Damped Harmonic Oscillation
Richard Fitzpatrick
20101011