If we can vary the frequency of the ac generator while keeping the amplitude of its output voltage constant, then the current changes accordingly. A plot of
Figure 12.5.1At an circuit’s resonant frequency, the current amplitude is at its maximum value.
Figure 12.5.1 has a similar appearance to the plot of a damped harmonic oscillator’s variation in amplitude with respect to the angular frequency of a sinusoidal driving force. This similarity is more than just coincidence, as shown by the application of Kirchhoff’s loop rule to the circuit of Figure 12.3.1. This yields
(12.5.2)
or
where we substituted
for
Equation 12.5.2 has the general form of the differential equation for damped harmonic motion, demonstrating that the driven
series circuit is the electrical analog of the driven damped harmonic oscillator.
The resonant frequency
of the
circuit is the frequency at which the amplitude of the current is a maximum and the circuit would oscillate if not driven by a voltage source. By inspection, this corresponds to the angular frequency
in the region of maximum power output. The bandwidth
of the resonance peak is defined as the range of angular frequencies
over which the average power
is greater than one-half the maximum value of
The sharpness of the peak is described by a dimensionless quantity known as the quality factor
of the circuit. By definition,
(12.5.4)
where
is the resonant angular frequency. A high
indicates a sharp resonance peak. We can give
in terms of the circuit parameters as
(12.5.5)
Figure 12.5.2Like the current, the average power transferred from an ac generator to an RLC circuit peaks at the resonant frequency.
Resonant circuits are commonly used to pass or reject selected frequency ranges. This is done by adjusting the value of one of the elements and hence “tuning” the circuit to a particular resonant frequency. For example, in radios, the receiver is tuned to the desired station by adjusting the resonant frequency of its circuitry to match the frequency of the station. If the tuning circuit has a high
it will have a small bandwidth, so signals from other stations at frequencies even slightly different from the resonant frequency encounter a high impedance and are not passed by the circuit. Cell phones work in a similar fashion, communicating with signals of around
that are tuned by an inductor-capacitor circuit. One of the most common applications of capacitors is their use in ac-timing circuits, based on attaining a resonant frequency. A metal detector also uses a shift in resonance frequency in detecting metals (Figure 12.5.3).
(Figure 12.5.3)
Figure 12.5.3When a metal detector comes near a piece of metal, the self-inductance of one of its coils changes. This causes a shift in the resonant frequency of a circuit containing the coil. That shift is detected by the circuitry and transmitted to the diver by means of the headphones.
EXAMPLE 12.5.1
Resonance in an Series Circuit
(a) What is the resonant frequency of the circuit of Example 12.2.1? (b) If the ac generator is set to this frequency without changing the amplitude of the output voltage, what is the amplitude of the current?
Strategy
The resonant frequency for a
circuit is calculated from Equation 12.5.3, which comes from a balance between the reactances of the capacitor and the inductor. Since the circuit is at resonance, the impedance is equal to the resistor. Then, the peak current is calculated by the voltage divided by the resistance.
Solution
a. The resonant frequency is found from Equation 12.5.3:
b. At resonance, the impedance of the circuit is purely resistive, and the current amplitude is
Significance
If the circuit were not set to the resonant frequency, we would need the impedance of the entire circuit to calculate the current.
EXAMPLE 12.5.2
Power Transfer in an Series Circuit at Resonance
(a) What is the resonant angular frequency of an
circuit with
and
?
(b) If an ac source of constant amplitude
is set to this frequency, what is the average power transferred to the circuit? (c) Determine
and the bandwidth of this circuit.
Strategy
The resonant angular frequency is calculated from Equation 12.5.3. The average power is calculated from the rms voltage and the resistance in the circuit. The quality factor is calculated from Equation 12.5.5 and by knowing the resonant frequency. The bandwidth is calculated from Equation 12.5.4 and by knowing the quality factor.
Solution
a. The resonant angular frequency is
b. At this frequency, the average power transferred to the circuit is a maximum. It is
c. The quality factor of the circuit is
We then find for the bandwidth
Significance
If a narrower bandwidth is desired, a lower resistance or higher inductance would help. However, a lower resistance increases the power transferred to the circuit, which may not be desirable, depending on the maximum power that could possibly be transferred.
(a) What is the resonant frequency? (b) What is the impedance of the circuit at resonance? (c) If the voltage amplitude is
what is
at resonance? (d) The frequency of the AC generator is now changed to
Calculate the phase difference between the current and the emf of the generator.
CHECK YOUR UNDERSTANDING 12.7
What happens to the resonant frequency of an
series circuit when the following quantities are increased by a factor of
: (a) the capacitance, (b) the self-inductance, and (c) the resistance?
CHECK YOUR UNDERSTANDING 12.8
The resonant angular frequency of an
series circuit is
An ac source operating at this frequency transfers an average power of
to the circuit. The resistance of the circuit is
Write an expression for the emf of the source.
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