32.10:
Resonance in an AC Circuit
Let a resistor, an inductor, and a capacitor in series be supplied with a sinusoidal voltage. The circuit's impedance depends on the source's angular frequency. The inductor's reactance increases with the frequency, whereas the capacitor's reactance decreases.
The current is maximum if these reactances are equal. On equating them, the source's frequency which permits this is obtained.
This phenomenon is called the resonance in a series R–L–C circuit. At this frequency, the phase difference between the current and voltage, proportional to the difference between the inductor and capacitor's reactances, is zero.
At the resonance frequency, the impedance is simply the resistance. Hence, the current peaks sharply if the resistance is lower.
The resonance bandwidth is defined as the range of angular frequencies over which the average power is more than half of the maximum value, which is the ratio R over L.
The quality factor is defined as the ratio of the resonance frequency and the bandwidth. The higher the quality factor, the sharper the resonance.
32.10:
Resonance in an AC Circuit
The property of an inductor makes it resist any change in the current passing through it, while the property of a capacitor is to build up the charge across its terminals. Hence, if an inductor and capacitor are connected in series, they have opposite effects on the relative phase between current and voltage. The current through the circuit undergoes forced oscillation at the frequency of the source. The resistance term in an R-L-C circuit acts as a damping term because power is dissipated across it during every cycle in the form thermal energy, or heat.
If the source's frequency is high, the capacitor hardly offers any reactance to the current. However, the inductive reactance is more. On the other hand, at low frequencies, the inductor hardly offers any reactance to the alternating current; however. the capacitor, which tends to build and store charge across it, offers high reactance. At an intermediate frequency, the capacitive and inductive reactances are equal. This frequency is the resonance frequency.
At the resonance frequency, the circuit's impedance is equal to the resistance of the R-L-C circuit because the reactance terms from the L and C parts cancel, which also implies that the current and voltage are exactly in phase. Thus, it is possible to think of the resonance frequency as the natural frequency of oscillation of the circuit.
At higher frequencies, inductive reactance is more than capacitive reactance. The phase difference is positive, implying that the voltage leads the current. At frequencies lesser than the resonance frequency, the capacitive reactance is greater than the inductive reactance. Hence, the phase difference is negative, implying that the current leads the voltage.
The greater the resistance, the lower is the current's amplitude at the resonance frequency. Moreover, the peak of the power versus the frequency curve is more peaked if the resistance is lower. This phenomenon is described by the bandwidth and quality factor of the circuit. A higher bandwidth implies a greater spread of power around the resonance frequency and a less sharp peak. Hence, the quality factor is lower. A higher quality factor follows from a lower bandwidth, implying a sharper peak.
By adjusting the values of the capacitance, inductance, and resistance, it is thus possible to tune the amount of power dissipated at different frequencies. Applications of this possibility include, for example, radio frequency transmission and reception.
Suggested Reading
- OpenStax. (2019). University Physics Vol. 2. [Web version]. Retrieved from 15.5 Resonance in an AC Circuit – University Physics Volume 2 | OpenStax; p 679