# Alternating Current

Voltage, current, and phase angle.  In general, the instantaneous voltage $v=V\cos(\omega t+\phi)$ between two points in an ac circuit is not in phase with the instantaneous current $i=I\cos\omega t$ passing through those points.

Resistance and reactance.  The voltage across a resistor is in phase with the current, $V_R=IR$.  The voltage across an inductor leads the the current by $\frac\pi 2$, $V_L=IX_L$, inductive reactance $X_L=\omega L$.  The voltage across a capacitor lags the the current by $\frac\pi 2$, $V_C=IX_C$, capacitive reactance $X_C=\frac1{\omega C}$.

Impedance and the L-R-C series circuit.  In general ac circuit, the voltage and current amplitutes are related by the circuit impedance $Z$, $V=IZ$.  In an L-R-C series circuit, $Z=\sqrt{R^2+(\omega L-\frac1{\omega C})^2}$, $\tan\phi=\frac{\omega L-\frac1{\omega C}}R$.

Power in ac circuits.  The average power input to an ac circuit: $P_{av}=\frac12VI\cos\phi=V_{\mathrm{rms}}I_{\mathrm{rms}}\cos\phi$, where $\phi$ is the phase angle of the voltage relative to the current.  The factor $\cos\phi$ is called the power factor of the circuit.

Resonance angular frequencey.  $\omega_0=\frac1{\sqrt{LC}}$.

Transformers.  $\frac{V_2}{V_1}=\frac{N_2}{N_1}$, $V_1I_1=V_2I_2$.