Alternating Current

The electricity in your walls doesn’t flow one way: it surges back and forth fifty or sixty times a second. That alternating current comes straight from Faraday’s law applied to a spinning coil.

Alternating current#

A battery drives current one way at a nearly steady size. A current whose direction keeps reversing is an alternating current (AC). We study the sinusoidal kind:

The current is positive for half of each period and negative for the other half. To drive it we need a source of sinusoidal emf: an AC generator (AC dynamo), which turns mechanical energy into electrical energy.

How a generator works#

• Magnet: permanent or electro-, with facing poles giving a strong uniform field B.
• Armature: a coil, usually on a soft-iron core to strengthen the field, turning about an axis in its own plane and perpendicular to B. Its ends go to two slip rings.
• Brushes: two graphite brushes press on the rings and connect to the outer terminals, keeping contact as the rings turn.

Let the coil (N turns, area A) turn at constant ω, with its plane perpendicular to B at t = 0. The flux through each turn is BA cos ωt, so each turn has emf −dΦ/dt = BAω sin ωt, and the whole coil:

An alternating current averages to zero, yet it lights bulbs perfectly well. The right way to describe its size is the root-mean-square value.

The average is zero#

For i = i₀ sin(ωt + φ), the average over a time t is

Over a whole period this is exactly zero, and over a long time it tends to zero (the bracket stays between −2 and 2 while ωt grows).

Root mean square#

But i² is never negative. Its average over a period:

Why rms is the useful value#

A current i = i₀ sin(ωt + φ) through a resistor R produces heat i²R dt in each instant; over a period,

So a steady current equal to i_rms heats a resistor exactly as much as the alternating current does. That’s why AC voltages and currents are quoted as rms values: “230 V AC” means V_rms = 230 V, and the peak is 230√2 ≈ 325 V. (For 220 V supplies the peak is 311 V.)

Put a resistor, a capacitor or an inductor alone across an AC source. Each limits the current in its own way, and each shifts the current’s timing relative to the emf differently.

Resistor only#

With ℰ = ℰ₀ sin ωt across R, the loop law gives ℰ₀ sin ωt = Ri:

Capacitor only#

With no resistance, the capacitor’s voltage equals the source’s: q/C = ℰ₀ sin ωt, so q = Cℰ₀ sin ωt and

• On a battery a capacitor takes current only briefly while charging. On AC the charge must keep changing, so current flows all the time: a capacitor blocks DC but passes AC.
• X_C, the capacitive reactance, acts like a resistance (in ohms). It is large at low frequency and small at high frequency; for DC (ω = 0) it is infinite.
• When the emf is zero the current is at its largest, and vice versa.

Inductor only#

For a pure inductor, ℰ₀ sin ωt − L di/dt = 0, so di/dt = (ℰ₀/L) sin ωt and i = −(ℰ₀/ωL) cos ωt + c. A sinusoidal current must average to zero, so c = 0:

The inductive reactance X_L is zero for DC and grows with frequency.

Impedance and phase#

In all three cases the peak current is the peak emf divided by an effective resistance, the impedance Z:

Summary

With several elements in series, a neat vector trick gives the impedance and phase at once. It reveals resonance: the frequency at which a circuit responds most strongly, the principle behind every radio tuner.

The vector (phasor) method#

For a series circuit with ℰ = ℰ₀ sin ωt:

1. Draw R as a vector along the x-axis.
2. Draw the capacitive reactance 1/ωC along +y (π/2 ahead) and the inductive reactance ωL along −y (π/2 behind).
3. Add them. The resultant’s length is the impedance Z; its angle with the x-axis is the phase factor φ.

On switching on, the current behaves in a complicated way, but soon settles into this steady state at the source’s frequency.

CR and LR circuits

CR: Z = √(R² + 1/ω²C²), tan φ = 1/ωCR, current leads. LR: Z = √(R² + ω²L²), tan φ = ωL/R, current lags.

The LCR circuit#

The two reactances point in opposite directions, so the net reactance is X = 1/ωC − ωL:

• X_C > X_L: φ > 0, current leads (capacitive).
• X_C < X_L: φ < 0, current lags (inductive).
• X_C = X_L: net reactance zero, current in phase, Z = R.

Resonance#

Vary ω and the peak current i₀ = ℰ₀/Z is greatest when 1/ωC = ωL:

A small R makes the resonance sharp: the current is large only very close to ν₀. Such a circuit used near ν₀ is a resonant circuit.

LC oscillations#

With R = 0 the current at resonance would be infinite for any emf: a finite current can flow with no source at all. Connect a charged capacitor to a pure inductor: it discharges through the inductor, whose emf then recharges it the other way, and so on, at ν = 1/2π√(LC). Energy swaps between the capacitor’s electric field and the inductor’s magnetic field.

In an AC circuit, current times voltage isn’t the whole story: if the two are out of step, some energy just sloshes back and forth. The power factor measures how much is really used.

Average power#

With ℰ = ℰ₀ sin ωt and i = i₀ sin(ωt + φ), the work done in time dt is ℰi dt = ℰ₀i₀(sin²ωt cos φ + sin ωt cos ωt sin φ) dt. Over a full cycle the second term averages to zero:

• Purely resistive: φ = 0, cos φ = 1, P = ℰ_rmsi_rms.
• Purely reactive (only L and/or C): φ = ±π/2, cos φ = 0, no power is absorbed, even though current flows.

The choke coil#

A choke is a coil of large inductance and small resistance. Old fluorescent tube fittings put one in series with the tube to cut down the voltage across it. Treating the tube as a resistance R and the choke as an ideal inductor, the circuit is an LR circuit with Z = √(R² + ω²L²), and the tube’s rms voltage is

The advantage over a series resistor: an ideal inductor consumes no power (its voltage and current are 90° apart), so no energy is wasted as heat.

Hot-wire instruments#

A moving-coil meter can’t read AC: its torque reverses every half cycle and averages to zero. AC meters use an effect that depends on i², which is never negative. In a hot-wire ammeter, current heats a platinum–iridium wire AB stretched between fixed points; it expands and sags, and a spring pulls a thread wound round a small drum, turning a pointer. The rise in temperature, and so the deflection, depends on i_rms²; the scale is marked to read i_rms directly. A small shunt keeps the meter’s resistance low. A hot-wire voltmeter is the same but with a large series resistance instead of a shunt.

Three machines built on induction and magnetic force: the DC dynamo, the DC motor and the transformer, which together make it possible to generate, use and ship electrical power across a country.

The DC dynamo#

Same magnet and armature as an AC dynamo, but the slip rings are replaced by a split ring: a cylinder cut into two halves C₁ and C₂, one joined to each end of the coil, with carbon brushes B₁ and B₂ pressing on them. Every half turn the gaps pass under the brushes and the connections to the outside swap. Arranged so that the swap happens just when the emf is zero, the external current always flows the same way.

This split ring with its brushes is the commutator. The output is one-way but pulsing; a second coil at right angles (its emf peaks when the first’s is zero) smooths it, and more coils smooth it further.

The DC motor#

A DC motor is built just like a DC dynamo, but run in reverse: a battery sends current through the armature via the brushes, the field exerts a torque on the coil (largest when the coil’s plane lies along the field, zero when it faces it), and the commutator reverses the current every half turn so the torque keeps the same sense. The shaft turns the load.

As the coil turns it also acts as a generator, producing a back emf e that opposes the applied emf ℰ:

The transformer#

Two coils, a primary (N₁ turns, connected to the AC source) and a secondary (N₂ turns, feeding the load), wound on one laminated soft-iron core. The core’s magnetization dominates the field, so the same flux Φ passes through every turn of both coils.

Neglecting the primary’s resistance, ℰ₁ = N₁ dΦ/dt; the secondary has ℰ₂ = −N₂ dΦ/dt. Dividing:

Power transfer

With the secondary open, the primary is a pure inductor: a small current i_s flows, 90° out of phase with ℰ₁, and no power is drawn. Connect a load and a secondary current i₂ flows. It would change the core’s flux, so an extra primary current i₁ appears that exactly cancels its effect; i₁ is in phase with ℰ₁, and the source now delivers power ℰ₁i₁. With no losses this all reaches the load:

• Step-up (N₂ > N₁): higher voltage, lower current out; the primary is wound from thick wire.
• Step-down (N₂ < N₁): lower voltage, higher current out; the secondary needs thick wire.
• Efficiency η = output power/input power is lowered by coil resistance, hysteresis and eddy currents in the core, but 99 % is achievable.

Long-distance transmission

A power station supplies a fixed power P = Vi. Sent through lines of resistance R, it loses i²R as heat. At high voltage the current, and so the loss, is small. So voltages are stepped up at the station, sent across country, and stepped down near the users.