12.5 Resonance in an AC Circuit
LEARNING OBJECTIVES
- Determine the peak ac resonant angular frequency for a RLC circuit
- Explain the width of the average power versus angular frequency curve and its significance using terms like bandwidth and quality factor
In the series circuit of Figure 12.3.1, the current amplitude is, from Equation 12.3.2,
(12.5.1)
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 versus is shown in Figure 12.5.1.
(Figure 12.5.1)
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 at which the impedance in Equation 12.5.1 is a minimum, or when
and
This is the resonant angular frequency of the circuit. Substituting ω0ω0 into Equation 12.3.1, Equation 12.3.2, and Equation 12.3.3, we find that at resonance,
Therefore, at resonance, an circuit is purely resistive, with the applied emf and current in phase.
What happens to the power at resonance? Equation 12.4.3 tells us how the average power transferred from an ac generator to the combination varies with frequency. In addition, reaches a maximum when which depends on the frequency, is a minimum, that is, when and Thus, at resonance, the average power output of the source in an series circuit is a maximum. From Equation 12.4.3, this maximum is
Figure 12.5.2 is a typical plot of versus 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,
where is the resonant angular frequency. A high indicates a sharp resonance peak. We can give in terms of the circuit parameters as
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