Alternating Current

Alternating Current (AC) is electric current whose magnitude and direction reverse periodically in time according to a sinusoidal function.

1. The Simple AC Generator (Alternator)

Consider a planar rectangular armature coil of $N$ turns and area $A$ rotated with constant angular velocity $\omega$ in a uniform magnetic field $\vec{B}$. The instantaneous angle between the coil normal and the field is $\theta(t) = \omega t$. The magnetic flux through the coil is: $$\Phi_B(t) = B A \cos(\omega t)$$ By Faraday's law of electromagnetic induction, the induced EMF is: $$\mathcal{E}(t) = -N \frac{d\Phi_B}{dt} = -N B A \frac{d}{dt}[\cos(\omega t)] = N B A \omega \sin(\omega t)$$ $$\mathcal{E}(t) = \mathcal{E}_0 \sin(\omega t)$$ where $\mathcal{E}_0 = N B A \omega$ is the **peak voltage amplitude** (Volts).

2. Root-Mean-Square (R.M.S.) and Average Values

Consider a sinusoidal voltage $v(t) = V_0 \sin(\omega t)$ with period $T = 2\pi / \omega$:

• Full-Cycle Average: $$\bar{V} = \frac{1}{T}\int_0^T V_0 \sin(\omega t) dt = 0$$ The average over a complete cycle vanishes because positive and negative half-cycles cancel identically.
• Half-Cycle Average: $$\bar{V}_{1/2} = \frac{2}{T}\int_0^{T/2} V_0 \sin(\omega t) dt = \frac{2 V_0}{\pi} \approx 0.6366 V_0$$
• Root-Mean-Square (R.M.S. / Effective) Value: The equivalent DC voltage that produces the identical average heating power in a pure resistor: $$V_{\text{rms}} = \sqrt{ \frac{1}{T}\int_0^T [V_0 \sin(\omega t)]^2 dt } = \sqrt{ \frac{V_0^2}{T}\int_0^T \left(\frac{1 - \cos(2\omega t)}{2}\right) dt } = \frac{V_0}{\sqrt{2}} \approx 0.7071 V_0$$ $$I_{\text{rms}} = \frac{I_0}{\sqrt{2}} \approx 0.7071 I_0$$ (Standard household mains 230 V or 120 V AC are R.M.S. values; the peak amplitude is $V_0 = 230\sqrt{2} \approx 325\text{ V}$).

When sinusoidal voltage $v(t) = V_0 \sin(\omega t)$ is applied individually to ideal passive elements, distinct phase relationships emerge.

1. Pure Resistive Circuit ($R$)

Applying Ohm's law: $$i_R(t) = \frac{v(t)}{R} = \frac{V_0}{R} \sin(\omega t) = I_0 \sin(\omega t)$$ Phase: Current and voltage are strictly **in phase** ($\phi = 0$).

2. Pure Inductive Circuit ($L$)

By Faraday's law: $v(t) = L \frac{di}{dt} \implies di = \frac{V_0}{L} \sin(\omega t) dt$. Integrating: $$i_L(t) = -\frac{V_0}{\omega L} \cos(\omega t) = \frac{V_0}{\omega L} \sin\left( \omega t - \frac{\pi}{2} \right) = I_0 \sin\left( \omega t - \frac{\pi}{2} \right)$$ where $X_L = \omega L = 2\pi f L$ is the Inductive Reactance ($\Omega$). Phase: In an inductor, current **lags voltage by 90° ($\pi/2$ radians)**. An inductor opposes high frequencies ($X_L \propto f$).

3. Pure Capacitive Circuit ($C$)

By charge-voltage relation: $q(t) = C v(t) = C V_0 \sin(\omega t)$. Current is $i(t) = \frac{dq}{dt}$: $$i_C(t) = \omega C V_0 \cos(\omega t) = \frac{V_0}{1/(\omega C)} \sin\left( \omega t + \frac{\pi}{2} \right) = I_0 \sin\left( \omega t + \frac{\pi}{2} \right)$$ where $X_C = \frac{1}{\omega C} = \frac{1}{2\pi f C}$ is the Capacitive Reactance ($\Omega$). Phase: In a capacitor, current **leads voltage by 90° ($\pi/2$ radians)**. A capacitor blocks DC ($X_C \to \infty$ as $f \to 0$) and passes high frequencies ($X_C \to 0$). Mnemonic: **ELI the ICE man** (in $L$, $E$ leads $I$; in $C$, $I$ leads $E$).

Connecting a resistor, inductor, and capacitor in series across an AC source $v(t) = V_0 \sin(\omega t)$ establishes a driven harmonic system.

1. Phasor Addition and Total Impedance ($Z$)

Because elements are in series, the common current is $i(t) = I_0 \sin(\omega t - \phi)$. The voltage drops across each element are: $$V_R = I_0 R, \quad V_L = I_0 X_L, \quad V_C = I_0 X_C$$ On the phasor plane, $V_L$ leads $I$ by $+90^\circ$ and $V_C$ lags $I$ by $-90^\circ$. The net reactive voltage is $V_L - V_C$. By the Pythagorean theorem: $$V_0 = \sqrt{ V_R^2 + (V_L - V_C)^2 } = I_0 \sqrt{ R^2 + (X_L - X_C)^2 } = I_0 Z$$ The Impedance ($Z$) of the series LCR circuit is: $$Z = \sqrt{ R^2 + (\omega L - \frac{1}{\omega C})^2 }$$ The phase angle $\phi$ of voltage relative to current is: $$\tan\phi = \frac{X_L - X_C}{R} = \frac{\omega L - 1/(\omega C)}{R}$$

• If $X_L > X_C$: $\phi > 0$ (circuit is inductive, voltage leads current).
• If $X_L < X_C$: $\phi < 0$ (circuit is capacitive, current leads voltage).
• If $X_L = X_C$: $\phi = 0$ (circuit is purely resistive).

2. Series Electrical Resonance

When the applied frequency makes inductive reactance balance capacitive reactance: $$X_L = X_C \implies \omega_0 L = \frac{1}{\omega_0 C} \implies \omega_0 = \frac{1}{\sqrt{LC}}, \quad f_0 = \frac{1}{2\pi\sqrt{LC}}$$ At the resonant frequency $\omega_0$:

1. The total impedance drops to its absolute theoretical minimum: $Z_{\min} = R$.
2. The current reaches its absolute theoretical maximum: $I_{\max} = V_0 / R$.
3. Current and voltage are perfectly in phase ($\phi = 0$, power factor $\cos\phi = 1$).

3. Quality Factor ($Q$) and Bandwidth

The Quality Factor $Q$ quantifies the sharpness and selectivity of resonance: $$Q = \frac{\omega_0 L}{R} = \frac{1}{\omega_0 C R} = \frac{1}{R}\sqrt{\frac{L}{C}}$$ The half-power bandwidth is $\Delta\omega = \omega_2 - \omega_1 = \frac{R}{L} = \frac{\omega_0}{Q}$. High $Q$ produces extreme selectivity, essential for radio tuning circuits.

Unlike DC circuits where power is simply $P = V I$, AC power calculations must account for the phase angle between voltage and current.

1. Instantaneous and Real Average Power

Let $v(t) = V_0 \sin(\omega t)$ and $i(t) = I_0 \sin(\omega t - \phi)$. The instantaneous power is: $$p(t) = v(t) i(t) = V_0 I_0 \sin(\omega t) [\sin(\omega t)\cos\phi - \cos(\omega t)\sin\phi]$$ $$p(t) = V_0 I_0 \sin^2(\omega t)\cos\phi - \frac{1}{2} V_0 I_0 \sin(2\omega t)\sin\phi$$ Averaging over a complete cycle ($\langle \sin^2\omega t \rangle = 1/2$, $\langle \sin 2\omega t \rangle = 0$): $$\langle P \rangle = \frac{1}{2} V_0 I_0 \cos\phi = \left(\frac{V_0}{\sqrt{2}}\right) \left(\frac{I_0}{\sqrt{2}}\right) \cos\phi$$ $$\langle P \rangle = V_{\text{rms}} I_{\text{rms}} \cos\phi$$ where:

• $P$: Real (Active) Power dissipated as heat or mechanical work (Watts, W). Pure inductors and capacitors consume zero average real power!
• $S = V_{\text{rms}} I_{\text{rms}}$: Apparent Power (Volt-Amperes, VA).
• $Q_{\text{react}} = V_{\text{rms}} I_{\text{rms}} \sin\phi$: Reactive Power surging back and forth between source and reactive fields (Volt-Amperes Reactive, VAR).
• $\cos\phi = \frac{R}{Z}$: Power Factor ($0 \le \cos\phi \le 1$). Power companies mandate $\cos\phi \ge 0.95$ using power factor correction capacitors to minimize transmission line $I^2 R$ heat losses.

2. The Ideal Transformer

A transformer consists of two coils (primary of $N_p$ turns and secondary of $N_s$ turns) wound around a common laminated ferromagnetic core. Assuming zero flux leakage ($k = 1$) and zero resistance: $$\mathcal{E}_p = -N_p \frac{d\Phi_B}{dt}, \quad \mathcal{E}_s = -N_s \frac{d\Phi_B}{dt}$$ Dividing: $$\frac{V_s}{V_p} = \frac{N_s}{N_p} = a \quad (\text{Transformation Ratio})$$ By energy conservation ($P_{\text{in}} = P_{\text{out}} \implies V_p I_p = V_s I_s$): $$\frac{I_p}{I_s} = \frac{V_s}{V_p} = \frac{N_s}{N_p} = a \implies I_s = \frac{I_p}{a}$$ Impedance reflection: A load $R_L$ connected across the secondary reflects back to the primary as an equivalent impedance: $$R_{\text{in}} = \frac{V_p}{I_p} = \frac{V_s / a}{a I_s} = \frac{1}{a^2}\left(\frac{V_s}{I_s}\right) = \frac{R_L}{a^2} = \left(\frac{N_p}{N_s}\right)^2 R_L$$ This principle allows audio and RF engineers to achieve impedance matching for maximum power transfer.