Summarizing RLC Series AC Circuits

Summary

• The AC analogy to resistance is impedance \(Z\), the combined effect of resistors, inductors, and capacitors, defined by the AC version of Ohm’s law:

  \({I}_{0}=\cfrac{{V}_{0}}{Z}\phantom{\rule{0.25em}{0ex}}\text{or}\phantom{\rule{0.25em}{0ex}}{I}_{\text{rms}}=\cfrac{{V}_{\text{rms}}}{Z},\)

  where \({I}_{0}\) is the peak current and \({V}_{0}\) is the peak source voltage.
• Impedance has units of ohms and is given by \(Z=\sqrt{{R}^{2}+({X}_{L}-{X}_{C}{)}^{2}}\).
• The resonant frequency \({f}_{0}\), at which \({X}_{L}={X}_{C}\), is

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

• In an AC circuit, there is a phase angle \(\varphi \) between source voltage \(V\) and the current \(I\), which can be found from

  \(\text{cos}\phantom{\rule{0.25em}{0ex}}\varphi =\cfrac{R}{Z}\text{,}\)

• \(\varphi =0º\) for a purely resistive circuit or an RLC circuit at resonance.
• The average power delivered to an RLC circuit is affected by the phase angle and is given by

  \({P}_{\text{ave}}={I}_{\text{rms}}{V}_{\text{rms}}\phantom{\rule{0.25em}{0ex}}\text{cos}\phantom{\rule{0.25em}{0ex}}\varphi \text{,}\)

  \(\text{cos}\phantom{\rule{0.25em}{0ex}}\varphi \) is called the power factor, which ranges from 0 to 1.

Glossary

impedance

the AC analogue to resistance in a DC circuit; it is the combined effect of resistance, inductive reactance, and capacitive reactance in the form \(Z=\sqrt{{R}^{2}+({X}_{L}-{X}_{C}{)}^{2}}\)

resonant frequency

the frequency at which the impedance in a circuit is at a minimum, and also the frequency at which the circuit would oscillate if not driven by a voltage source; calculated by \({f}_{0}=\cfrac{1}{2\pi \sqrt{\text{LC}}}\)

phase angle

denoted by \(\varphi \), the amount by which the voltage and current are out of phase with each other in a circuit

power factor

the amount by which the power delivered in the circuit is less than the theoretical maximum of the circuit due to voltage and current being out of phase; calculated by \(\text{cos}\phantom{\rule{0.25em}{0ex}}\varphi \)