Inductance & AC Circuits

Self-inductance. When the current in a coil changes, the flux it produces through itself changes, inducing a back-EMF that opposes the change. This property is self-inductance $L$, defined by $\varepsilon = -L\frac{dI}{dt}$. Its SI unit is the henry (H). A coil with large $L$ resists rapid changes in current — it behaves like electrical inertia. The energy stored in an inductor carrying current $I$ is $U = \frac{1}{2}LI^2$.

Mutual inductance. When two coils are close, a changing current in one (primary) induces an EMF in the other (secondary): $\varepsilon_2 = -M\frac{dI_1}{dt}$, where $M$ is the mutual inductance (also in henry). Mutual inductance is the working principle of the transformer.

Alternating current and RMS values. An alternating current reverses direction periodically: $I = I_0\sin\omega t$, where $I_0$ is the peak value and $\omega = 2\pi f$. Since AC averages to zero over a cycle, we use the root-mean-square (RMS) value, which gives the same heating as an equivalent DC: $I_{rms} = \frac{I_0}{\sqrt2} \approx 0.707\,I_0$ and similarly $V_{rms} = \frac{V_0}{\sqrt2}$. The $220\,\text{V}$ mains is an RMS value (peak $\approx 311\,\text{V}$).

AC through a resistor, inductor and capacitor.

• Pure resistor (R): voltage and current are in phase. $I_{rms} = V_{rms}/R$.
• Pure inductor (L): current lags the voltage by $90^\circ$. The opposition is the inductive reactance $X_L = \omega L$, which increases with frequency.
• Pure capacitor (C): current leads the voltage by $90^\circ$. The opposition is the capacitive reactance $X_C = \frac{1}{\omega C}$, which decreases with frequency.

Series LCR circuit and impedance. When R, L and C are in series, the net opposition is the impedance $Z = \sqrt{R^2 + (X_L - X_C)^2}$, and $I_{rms} = V_{rms}/Z$. The current and voltage differ in phase by $\phi$, where $\tan\phi = \frac{X_L - X_C}{R}$. A phasor diagram represents $V_R$, $V_L$ and $V_C$ as rotating vectors and adds them to find the resultant.

Resonance. When $X_L = X_C$, they cancel, so $Z = R$ is minimum and the current is maximum. This is series resonance, occurring at the resonant frequency $\omega_0 = \frac{1}{\sqrt{LC}}$ (i.e. $f_0 = \frac{1}{2\pi\sqrt{LC}}$). Resonant circuits are used to tune radios and TVs to a chosen station.

Power and power factor. The average power in an AC circuit is $P = V_{rms}I_{rms}\cos\phi$, where $\cos\phi$ is the power factor. For a pure resistor $\cos\phi = 1$ (maximum power), while for a pure inductor or capacitor $\cos\phi = 0$, so no power is consumed — the current that flows without dissipating energy is called wattless current.