14.1 Alternating Current: RMS Values, Reactance, Impedance, Resonance & Power

Alternating Current: RMS Values, Reactance, Impedance, Resonance & Power

An alternating current (AC) reverses direction periodically. The AC generator of Section 13.3 produces a sinusoidal emf, and almost all power distribution uses AC because transformers can step its voltage up or down efficiently. This section builds the circuit rules for AC.

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1. Describing an Alternating Voltage

$V = V_0\sin\omega t, \qquad \omega = 2\pi f$

• $V_0$ is the peak value; the peak-to-peak value is $2V_0$.
• $f$ is the frequency (50 Hz in Pakistan) and $T = \frac{1}{f}$ the period (20 ms at 50 Hz).
• The average of a full sine wave is zero, so AC is described by its root-mean-square (rms) value, the steady DC value that produces the same heating in a resistor:

$V_{\text{rms}} = \frac{V_0}{\sqrt{2}} \approx 0.707V_0, \qquad I_{\text{rms}} = \frac{I_0}{\sqrt{2}}$

Mains voltage quoted as 220 V is an rms value, so its peak is $220\sqrt{2} \approx 311\text{ V}$. AC meters normally read rms values.

The phase of an AC quantity is the angle $\omega t + \phi$ that fixes its position in the cycle. Two quantities with the same frequency can be represented by rotating vectors (phasors) whose angle between them is the phase difference.

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2. Resistor, Inductor and Capacitor in AC

Memory aid: in an inductor ($L$) the emf ($E$) comes before the current ($I$): ELI; in a capacitor ($C$) the current comes first: ICE.

• $X_L$ increases with frequency: an inductor blocks high frequencies. For DC ($f = 0$), $X_L = 0$.
• $X_C$ decreases with frequency: a capacitor passes high frequencies and blocks DC ($X_C \to \infty$ as $f \to 0$).

Example: A $0.1\text{ H}$ inductor on 50 Hz mains has $X_L = 2\pi(50)(0.1) \approx 31.4\,\Omega$. A $100\,\mu\text{F}$ capacitor has $X_C = \frac{1}{2\pi(50)(100 \times 10^{-6})} \approx 31.8\,\Omega$.

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3. Impedance of Series Circuits

When elements are in series, the same current flows through each, but their voltages are out of phase, so they add as phasors, not as plain numbers.

Ohm's law becomes $V_{\text{rms}} = I_{\text{rms}}Z$. The rms voltages across the separate elements can add up to more than the supply voltage; only their phasor sum equals it.

Worked example: $R = 30\,\Omega$, $X_L = 80\,\Omega$, $X_C = 40\,\Omega$ in series on a 100 V rms supply.

$Z = \sqrt{30^2 + (80 - 40)^2} = \sqrt{900 + 1600} = 50\,\Omega, \qquad I = \frac{100}{50} = 2\text{ A}$

$V_R = 60\text{ V}$, $V_L = 160\text{ V}$, $V_C = 80\text{ V}$; check: $\sqrt{60^2 + (160 - 80)^2} = 100\text{ V}$. The power factor is $\cos\phi = \frac{R}{Z} = 0.6$, and the circuit is inductive (current lags).

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4. Resonance

Series Resonance

As frequency changes, $X_L$ rises and $X_C$ falls. At the resonant frequency they are equal:

$\omega_0 L = \frac{1}{\omega_0 C} \implies f_0 = \frac{1}{2\pi\sqrt{LC}}$

At series resonance: $Z = R$ (minimum), the current is maximum, $V$ and $I$ are in phase, and the power factor is 1. The voltage across $L$ or $C$ alone can be much larger than the supply voltage. Tuning a radio receiver adjusts $C$ until $f_0$ matches the wanted station.

Parallel Resonance

For $L$ and $C$ in parallel, the same resonant frequency applies, but now the circuit's impedance is maximum and the current drawn from the supply is minimum. Such a circuit (a "rejector" or tank circuit) is used to select or reject one frequency.

Example: $L = 10\text{ mH}$ and $C = 1\,\mu\text{F}$ give $f_0 = \frac{1}{2\pi\sqrt{10^{-2} \times 10^{-6}}} = \frac{1}{2\pi \times 10^{-4}} \approx 1.6\text{ kHz}$.

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5. Power in AC Circuits

$P_{\text{avg}} = V_{\text{rms}}I_{\text{rms}}\cos\phi$

• $\cos\phi$ is the power factor. It is 1 for a pure resistor and 0 for a pure inductor or capacitor.
• A choke (an inductor with low resistance) can reduce the current in a circuit, for example in fluorescent lamps, while wasting very little energy, unlike a series resistor that dissipates $I^2R$ as heat.
• Utilities prefer loads with a power factor close to 1, because a low power factor means a larger current (and larger line losses) for the same useful power.

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6. Three-Phase Supply and AC Applications

• A three-phase generator has three coils $120^\circ$ apart, giving three voltages that are $120^\circ$ out of phase. For a star connection, the line voltage is $\sqrt{3}$ times the phase voltage: $230\text{ V}$ phase gives about $400\text{ V}$ between lines. Three-phase supply delivers smoother power and drives industrial motors.
• Metal detectors use a resonant LC circuit; a metal object near the coil changes its inductance and shifts the resonant frequency, which triggers an alarm.

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7. Electromagnetic Waves and Modulation

Maxwell showed that a changing electric field produces a magnetic field and vice versa, so oscillating charges radiate electromagnetic waves travelling at

$c = \frac{1}{\sqrt{\mu_0\varepsilon_0}} \approx 3 \times 10^8\text{ m/s}$

with $\mathbf{E}$ and $\mathbf{B}$ perpendicular to each other and to the direction of travel. An LC oscillator connected to an antenna generates radio waves.

To carry information, a high-frequency carrier wave is modulated by the signal: