Resonance in <em>RLC</em> Series AC Circuits

Resonance in RLC Series AC Circuits

How does an RLC circuit behave as a function of the frequency of the driving voltage source? Combining Ohm’s law, \({I}_{\text{rms}}={V}_{\text{rms}}/Z\), and the expression for impedance \(Z\) from \(Z=\sqrt{{R}^{2}+({X}_{L}-{X}_{C}{)}^{2}}\) gives

\({I}_{\text{rms}}=\cfrac{{V}_{\text{rms}}}{\sqrt{{R}^{2}+({X}_{L}-{X}_{C}{)}^{2}}}\text{.}\)

The reactances vary with frequency, with \({X}_{L}\) large at high frequencies and \({X}_{C}\) large at low frequencies, as we have seen in three previous examples. At some intermediate frequency \({f}_{0}\), the reactances will be equal and cancel, giving \(Z=R\) —this is a minimum value for impedance, and a maximum value for \({I}_{\text{rms}}\) results. We can get an expression for \({f}_{0}\) by taking

\({X}_{L}={X}_{C}\text{.}\)

Substituting the definitions of \({X}_{L}\) and \({X}_{C}\),

\(2{\mathrm{\pi f}}_{0}L=\cfrac{1}{2{\mathrm{\pi f}}_{0}C}\text{.}\)

Solving this expression for \({f}_{0}\) yields

\({f}_{0}=\cfrac{1}{2\pi \sqrt{\text{LC}}}\text{,}\)

where \({f}_{0}\) is the resonant frequency of an RLC series circuit. This is also the natural frequency at which the circuit would oscillate if not driven by the voltage source. At \({f}_{0}\), the effects of the inductor and capacitor cancel, so that \(Z=R\), and \({I}_{\text{rms}}\) is a maximum.

Resonance in AC circuits is analogous to mechanical resonance, where resonance is defined to be a forced oscillation—in this case, forced by the voltage source—at the natural frequency of the system. The receiver in a radio is an RLC circuit that oscillates best at its \({f}_{0}\). A variable capacitor is often used to adjust \({f}_{0}\) to receive a desired frequency and to reject others. This figure is a graph of current as a function of frequency, illustrating a resonant peak in \({I}_{\text{rms}}\) at \({f}_{0}\). The two curves are for two different circuits, which differ only in the amount of resistance in them. The peak is lower and broader for the higher-resistance circuit. Thus the higher-resistance circuit does not resonate as strongly and would not be as selective in a radio receiver, for example.

The figure describes a graph of current I versus frequency f. Current I r m s is plotted along Y axis and frequency f is plotted along X axis. Two curves are shown. The upper curve is for small resistance and lower curve is for large resistance. Both the curves have a smooth rise and a fall. The peaks are marked for frequency f zero. The curve for smaller resistance has a higher value of peak than the curve for large resistance.

A graph of current versus frequency for two RLC series circuits differing only in the amount of resistance. Both have a resonance at \({f}_{0}\), but that for the higher resistance is lower and broader. The driving AC voltage source has a fixed amplitude \({V}_{0}\).

Example: Calculating Resonant Frequency and Current

For the same RLC series circuit having a \(\text{40.0 Ω}\) resistor, a 3.00 mH inductor, and a F\(\text{5.00 μF}\) capacitor: (a) Find the resonant frequency. (b) Calculate \({I}_{\text{rms}}\) at resonance if \({V}_{\text{rms}}\) is 120 V.

Strategy

The resonant frequency is found by using the expression in \({f}_{0}=\cfrac{1}{2\pi \sqrt{\text{LC}}}\). The current at that frequency is the same as if the resistor alone were in the circuit.

Solution for (a)

Entering the given values for \(L\) and \(C\) into the expression given for \({f}_{0}\) in \({f}_{0}=\cfrac{1}{2\pi \sqrt{\text{LC}}}\) yields

\(\begin{array}{lll}{f}_{0}& =& \cfrac{1}{2\pi \sqrt{\text{LC}}}\\ & =& \cfrac{1}{2\pi \sqrt{(3\text{.}\text{00}×{\text{10}}^{-3}\phantom{\rule{0.25em}{0ex}}\text{H})(5\text{.}\text{00}×{\text{10}}^{-6}\phantom{\rule{0.25em}{0ex}}\text{F})}}=1\text{.}\text{30}\phantom{\rule{0.25em}{0ex}}\text{kHz}\text{.}\end{array}\)

Discussion for (a)

We see that the resonant frequency is between 60.0 Hz and 10.0 kHz, the two frequencies chosen in earlier examples. This was to be expected, since the capacitor dominated at the low frequency and the inductor dominated at the high frequency. Their effects are the same at this intermediate frequency.

Solution for (b)

The current is given by Ohm’s law. At resonance, the two reactances are equal and cancel, so that the impedance equals the resistance alone. Thus,

\({I}_{\text{rms}}=\cfrac{{V}_{\text{rms}}}{Z}=\cfrac{\text{120}\phantom{\rule{0.25em}{0ex}}\text{V}}{\text{40}\text{.}\text{0}\phantom{\rule{0.25em}{0ex}}\Omega }=3\text{.}\text{00}\phantom{\rule{0.25em}{0ex}}\text{A.}\)

Discussion for (b)

At resonance, the current is greater than at the higher and lower frequencies considered for the same circuit in the preceding example.

Resonance in <em>RLC</em> Series AC Circuits

Resonance in RLC Series AC Circuits

Example: Calculating Resonant Frequency and Current

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