# 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 $$\text{cos}\phantom{\rule{0.25em}{0ex}}\varphi$$