Summary

Peak and rms values. For $i=i_0\sin\omega t$, the rms value is $I_{rms}=I_0/\sqrt2\approx0.707\,I_0$ (similarly for voltage); ordinary AC ratings (e.g. household $220\ \text{V}$) are rms values.

Phasors represent a sinusoidal quantity as a rotating arrow of length equal to its peak value, whose projection on a reference axis gives its instantaneous value; the angle between two phasors rotating together gives their phase difference.

Pure $R$: current in phase with voltage, $Z=R$. Pure $L$: current lags by $90^\circ$, reactance $X_L=\omega L$ (grows with frequency). Pure $C$: current leads by $90^\circ$, reactance $X_C=1/(\omega C)$ (shrinks with frequency).

Series LR: $Z=\sqrt{R^2+X_L^2}$, $\tan\phi=X_L/R$ (current lags). Series CR: $Z=\sqrt{R^2+X_C^2}$, $\tan\phi=X_C/R$ (current leads). Series LCR: $Z=\sqrt{R^2+(X_L-X_C)^2}$, $\tan\phi=(X_L-X_C)/R$ -- inductive if $X_L>X_C$, capacitive if $X_C>X_L$.

Resonance occurs when $X_L=X_C$, i.e. at $\omega_0=1/\sqrt{LC}$, where $Z$ is minimum ($=R$) and current is maximum ($=V_{rms}/R$). Quality factor $Q=\omega_0L/R=\dfrac1R\sqrt{L/C}=\omega_0/\Delta\omega$: a large $Q$ gives a tall, narrow (sharply selective) resonance peak.

LC oscillations (qualitative): a resistance-free LC loop exchanges energy repeatedly between the capacitor's electric field and the inductor's magnetic field at angular frequency $\omega_0=1/\sqrt{LC}$, exactly analogous to a frictionless spring-mass system; real (resistive) oscillations damp out unless replenished by an active feedback circuit.

Average power: $P_{avg}=V_{rms}I_{rms}\cos\phi$, where $\cos\phi$ is the power factor ($=1$ for pure $R$, $=0$ for pure $L$ or $C$). The current component $90^\circ$ out of phase with the voltage delivers no average power and is called the wattless current. …