Impedance, Phasors & RLC Resonance Reference Cheat Sheet

Key Facts

• Inductive reactance is $X_L = \omega L = 2\pi fL$, so an inductor opposes AC more strongly as frequency increases.
• Capacitive reactance is $X_C = \frac{1}{\omega C} = \frac{1}{2\pi fC}$, so a capacitor opposes AC less strongly as frequency increases.
• For a series RLC circuit, the impedance is $Z = R + j(X_L - X_C)$ and the magnitude is $|Z| = \sqrt{R^2 + (X_L - X_C)^2}$.
• Ohm’s law for AC rms values is $V_{\text{rms}} = I_{\text{rms}} |Z|$.
• The phase angle in a series RLC circuit satisfies $\tan\phi = \frac{X_L - X_C}{R}$.
• Series resonance occurs when $X_L = X_C$, giving $\omega_0 = \frac{1}{\sqrt{LC}}$ and $f_0 = \frac{1}{2\pi\sqrt{LC}}$.
• Average power in an AC circuit is $P = V_{\text{rms}} I_{\text{rms}} \cos\phi = I_{\text{rms}}^2 R$ for a series RLC circuit.
• The quality factor for a series RLC circuit can be written as $Q = \frac{\omega_0 L}{R} = \frac{1}{\omega_0 C R}$.

Vocabulary

Impedance

Impedance $Z$ is the total opposition an AC circuit gives to current, including both resistance and reactance.

Reactance

Reactance is the frequency-dependent opposition to AC current caused by inductors or capacitors.

Phasor

A phasor is a rotating vector used to represent the magnitude and phase of a sinusoidal voltage or current.

Resonance

Resonance is the condition in an RLC circuit where $X_L = X_C$ and the net reactance is zero.

Phase Angle

The phase angle $\phi$ measures how far voltage and current are shifted from each other in an AC circuit.

Power Factor

The power factor is $\cos\phi$, the fraction of apparent power converted into average real power.