Chapter 15. Alternating-Current Circuits
15.5 Resonance in an AC Circuit
Learning Objectives
By the end of the section, you will be able to:
- 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 RLC series circuit of Figure 15.11, the current amplitude is, from Equation 15.10,
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 [latex]{I}_{0}[/latex] versus [latex]\text{ω}[/latex] is shown in Figure 15.17.
In Oscillations, we encountered a similar graph where the amplitude of a damped harmonic oscillator was plotted against the angular frequency of a sinusoidal driving force (see Forced Oscillations). This similarity is more than just a coincidence, as shown earlier by the application of Kirchhoff’s loop rule to the circuit of Figure 15.11. This yields
or
where we substituted dq(t)/dt for i(t). A comparison of Equation 15.16 and, from Oscillations, Damped Oscillations for damped harmonic motion clearly demonstrates that the driven RLC series circuit is the electrical analog of the driven damped harmonic oscillator.
The resonant frequency [latex]{f}_{0}[/latex] of the RLC 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 [latex]{\text{ω}}_{0}=2\pi {f}_{0}[/latex] at which the impedance Z in Equation 15.15 is a minimum, or when
and
This is the resonant angular frequency of the circuit. Substituting [latex]{\text{ω}}_{0}[/latex] into Equation 15.9, Equation 15.10, and Equation 15.11, we find that at resonance,
Therefore, at resonance, an RLC circuit is purely resistive, with the applied emf and current in phase.
What happens to the power at resonance? Equation 15.14 tells us how the average power transferred from an ac generator to the RLC combination varies with frequency. In addition, [latex]{P}_{\text{ave}}[/latex] reaches a maximum when Z, which depends on the frequency, is a minimum, that is, when [latex]{X}_{L}={X}_{C}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}Z=R.[/latex] Thus, at resonance, the average power output of the source in an RLC series circuit is a maximum. From Equation 15.14, this maximum is [latex]{V}_{\text{rms}}^{2}\text{/}R.[/latex]
Figure 15.18 is a typical plot of [latex]{P}_{\text{ave}}[/latex] versus [latex]\text{ω}[/latex] in the region of maximum power output. The bandwidth [latex]\text{Δ}\text{ω}[/latex] of the resonance peak is defined as the range of angular frequencies [latex]\text{ω}[/latex] over which the average power [latex]{P}_{\text{ave}}[/latex] is greater than one-half the maximum value of [latex]{P}_{\text{ave}}.[/latex] The sharpness of the peak is described by a dimensionless