The previous parts dealt almost entirely in direct current, a steady flow in one direction. The grid does not work that way. Its voltage reverses many times a second and traces a sine wave, and almost every machine, transformer, and outlet in the world runs on it. This part is about why that shape appears, why the world settled on it, and the handful of tools needed to work with a voltage that never holds still.
The shape of alternating current
Direct current flows one way at a steady level, the kind a battery delivers. Alternating current does the opposite: it reverses direction over and over, many times a second, and its voltage traces a smooth sine wave, rising to a peak, falling back through zero, dropping to an equal negative peak, and returning. One full there-and-back is a cycle. The standard rate is 50 cycles per second, written 50 hertz (50 Hz), across most of the world; the Americas standardise on 60 Hz instead. Nothing physical prefers one number over the other; each grid fixed a rate early so that every appliance and meter built for it would interoperate.
Where the sine comes from
The sine is not a design choice. It falls straight out of how alternating current is generated.
Start with the one fact a generator rests on: a wire moving through a magnetic field has a voltage pushed along it, but only when it cuts across the field lines. The rule behind this is the force a magnetic field puts on a moving charge,
with \(v\) the charge's speed, \(B\) the field strength, and \(\theta\) the angle between the motion and the field. Two things in it matter. First, the \(\sin\theta\): a magnetic field acts only on the part of a motion that crosses it. Drag a charge along the field lines and the field is blind to it, zero force; drag it across them and it feels the full push. Second, the force comes out perpendicular to both the motion and the field at once.
So there is one force and two ingredients. You supply the motion by moving the wire; the magnet supplies the field; the two together produce the single force that pushes the electrons. The motion and the field are not themselves pushes on the charge, they are what combine to create the push:
move the wire (motion) ─┐ ├──► force on the electrons ──► current magnet (field) ─────────┘
For that force to drive current, it has to point along the wire's length. That fixes the geometry: the wire, its motion, and the field must sit mutually at right angles. Then the field's sideways shove lands straight down the wire and drives the most current. Line any two of them up instead and it collapses. Move the wire along the field and there is no force at all. Run the field along the wire and the force comes out sideways, pressing electrons against the wall of the wire rather than along it. Either way nothing flows.
Now make the wire a loop and spin it. As the loop turns, each side sweeps around in a circle, so the angle at which it cuts the field lines changes continuously. At one point in the turn a side is moving straight across the lines, cutting hardest, and the voltage peaks. A quarter-turn later it is moving along them, cutting nothing, and the voltage is zero. The smooth slide between those two, repeated every revolution, is the sine. Spin the loop 50 times a second and you have 50 Hz.
A single loop gives a small voltage. Wind it into a coil of many turns and each turn is another wire cutting the same lines, their voltages adding in series; a hundred turns give a hundred times the voltage. That is all a coil buys.
Two things follow about the energy and the field:
- The energy is neither free nor from the field. The field only converts; the work comes from whatever spins the shaft, the turbine or the engine. A generator with nothing connected spins easily, but draw current and it fights back, harder to turn, because the electrical energy you take out is paid for in mechanical effort going in.
- You always need a field, but not always a permanent magnet. Large generators make the field with an electromagnet, a coil carrying current, so the output can be tuned by adjusting it. A spinning coil is only one kind of source; batteries, solar cells, and thermocouples make electricity with no magnet at all.
Direct current from the same machine
A spinning coil can only make alternating current; the cutting reverses every half turn, so the raw output always alternates. A DC generator is the same machine with one addition. A commutator, a split ring with sliding contacts, flips the connection to the outside world every half turn, just as the voltage is about to reverse, so the terminals always see the same polarity. The result is a bumpy one-directional voltage, smoothed toward flat by using many coils set at staggered angles. The AC generator and the DC generator share identical physics; they differ only in how the output is tapped. Modern machines often generate AC and convert it to DC with diodes instead, skipping the wearing brushes.
Why alternating current won
The grid runs on AC for one main reason: the transformer, which responds only to changing current and so needs AC. It lets voltage be traded against current while the power is held fixed, since
A fixed power can be shipped as a low voltage with a large current, or a high voltage with a small one. The reason to pick high voltage is the loss in the transmission line itself,
The loss depends on the current squared, and on nothing about the voltage directly. Cut the current to a tenth, by stepping the voltage up tenfold, and the loss falls to a hundredth. That squaring is the entire case for high-voltage transmission. The transformer steps the voltage up at the sending end for the long haul, then back down to safe levels for use. DC could not be transformed easily when the grid was built, so AC won the wiring of the world.
One intuition to retire along the way: high voltage does not mean fast-moving electrons. Current counts the charge passing a point each second, not the speed of any one electron, and a wire holds so vast a number of free electrons that an ordinary current creeps them along at well under a millimetre a second. The lights still come on the instant you flip the switch, because the push, the electric field, travels down the already-full wire at close to the speed of light and sets every electron moving at once, while the electrons themselves barely budge. In AC they do not even progress; they rock a tiny distance back and forth fifty times a second and go nowhere on average, while the energy rides the field straight past them.
Describing the wave
Knowing the voltage is a sine, we need a vocabulary to pin down which sine. Any sinusoid is fixed by three numbers: how tall it is, how fast it repeats, and where it starts.
The peak value Vp is the height from the centre line to the crest, the most the voltage ever reaches in either direction. The full swing from trough to crest is the peak-to-peak value, Vpp = 2·Vp.
The period T is the time for one complete cycle, in seconds. The frequency f is how many cycles fit in a second, in hertz. They are reciprocals, two views of the same thing:
For 50 Hz mains, T = 1/50 = 20 ms, and the whole pattern repeats every 20 ms.
The third measure of rate looks unfamiliar but earns its place. A sine wave is the height of a point going around a circle, the same spinning picture that generates it, so its progress can be measured as an angle rather than a count of cycles. One complete cycle is one full turn, which is 2π radians. Measuring radians per second instead of cycles per second gives the angular frequency ω:
The 2π/T is read directly: 2π radians swept in one turn, over the T seconds that turn takes. The 2πf is the same number unit by unit: 2π radians per cycle times f cycles per second. For 50 Hz, ω = 2π·50 ≈ 314 rad/s, a large number because the point whips around fifty times a second. Watch the reciprocal: dividing by T means multiplying by f, so ω lands in the hundreds, not fractions.
Angular frequency exists because the sine function takes an angle, not a time. To write the wave as a function of time, ω converts elapsed seconds into swept angle, angle = ω·t:
At time t the rotating point has swept through ω·t radians, and the voltage is Vp times the sine of that angle. That equation is the AC voltage.
The last number, the phase φ, says where the wave sits in its cycle when the clock starts:
A single wave's phase is arbitrary; it only reflects when you began counting. What is physical is the phase difference between two waves of the same frequency: two clean sines whose crests do not line up in time, one slid sideways from the other. Whichever reaches a given point in the cycle first leads; the other lags. A quarter-cycle head start is 90°, or π/2 radians. Nothing here is random; both waves are smooth and predictable, one simply ahead of the other.
That lead and lag between the voltage across a component and the current through it is the entire behaviour of capacitors and inductors in AC, and it is where the next concepts head. So the full description of any AC voltage is three numbers: Vp, the rate as any of f, T, or ω, and the phase φ.
The honest average: RMS
What single number do you put on an AC voltage? Not the peak, which is touched for only an instant, and not the plain average, which is zero: over a full cycle the positive and negative halves cancel exactly. Yet AC clearly does work, so we need a number that captures it.
The number that matters is the one that gets the power right, so start from power. A resistor dissipates P = v²/R at every instant. Since the voltage varies, so does the power, and what we care about is its average over a cycle:
That is just the instantaneous power v²/R averaged, with the constant R taken outside. The whole result rests on one rule of order: you must square first, then average. Squaring at each instant is what makes the negative half of the cycle count instead of cancelling; averaging the raw voltage first would wrongly give zero.
The averaged square, \overline{v^2}, is in volts-squared. Take its square root to get back to volts, and you have the root mean square value, the recipe read right to left: root of the mean of the square.
This is the equivalent DC voltage: the steady voltage that would deliver the same average power to a resistor. With it, the power formula regains its DC form, P_avg = Vrms²/R. For a sine the average of sin² is 1/2, so the result is clean:
The 230 V at a European outlet is this RMS value; its peak is 230·√2 ≈ 325 V. Current has its own RMS the same way, Irms = Ip/√2, and with RMS values every DC formula returns unchanged:
When anyone quotes an AC voltage or current without saying "peak," it is RMS. The peak matters mainly for insulation and for the diodes in a power supply.
Phasors: freezing the spin
Working with sines as sin(ωt + φ) means dragging trig through every calculation. Phasors make it algebra instead, and they rest on one observation: in a single circuit, every voltage and current oscillates at the same frequency, the one the source sets. Since the frequency is common to all of them, it tells you nothing about how they differ. Only two things distinguish one sinusoid from another: its size and its phase.
So drop the frequency and keep just those two as an arrow, a phasor:
- its length is the size (peak or RMS),
- its angle is the phase.
This is the rotating point from the generator, frozen. Every sinusoid is a point going around a circle at ω; since they all turn together at the same rate, you lose nothing by stopping the rotation and looking at the snapshot. What was a quarter-cycle lead in time becomes an arrow sitting 90° around from another.
The payoff is that adding or comparing sinusoids becomes adding or comparing arrows, plain vectors, with no trig. Two voltages in a loop add tip-to-tail. A current lagging its voltage by 90° is just an arrow a quarter-turn behind. This is the tool that makes the next concepts, reactance and impedance, tractable: each component becomes an arrow of a certain length at a certain angle, and the whole circuit is solved by adding arrows.
Reactance: opposition that stores instead of burning
A resistor opposes current by turning energy into heat, and its opposition R is the same at any frequency. Capacitors and inductors also oppose AC current, but they store the energy and give it back instead of burning it, and their opposition depends on frequency. That frequency-dependent opposition is reactance, symbol X, measured in ohms like resistance, so it slots into Ohm's law as V = I·X.
The two components are mirror images.
The capacitor passes fast, blocks slow:
High frequency gives a small Xc and easy current; at DC (ω = 0) it is infinite and blocks completely. At the atomic level, nothing crosses the gap between the plates; electrons pile onto one plate and their field pushes electrons off the other, exactly as on DC. The difference is that the source reverses before the plate fills. Electrons rush onto and off the plate, back and forth, and that rushing in the wires is the current. The faster the reversal, the less charge accumulates before it is yanked back, so the plate's own counter-field never builds up to oppose the source, and a large current flows. Slow it down and the plate has time to fill, its counter-field chokes the current, and at DC it fills completely and stops.
The inductor passes slow, blocks fast:
High frequency gives a large Xl and chokes the current; at DC it is zero, a plain wire. The mechanism is the coil's magnetic field. A changing current means a changing field, and a changing field induces a back-voltage that opposes the change (Lenz's law, from an earlier part). On DC, once the current is steady the field is constant, nothing changes, and there is no opposition beyond the wire's small resistance. On AC the current never holds still, so the field is always building and collapsing, always inducing a back-voltage. The faster the current changes, the larger that opposing voltage, so opposition climbs with frequency.
Both store rather than burn. The capacitor's electric field and the inductor's magnetic field push back while energy flows in, then push the energy back out when the cycle reverses; over a full cycle an ideal one dissipates no heat. Reactance limits how much current flows for a given voltage, it does not consume the energy, which is why it is called reactance and not resistance. This storing and returning is also what throws current and voltage out of phase, the next concept.
A quick summary: a capacitor blocks DC and passes AC; an inductor passes DC and blocks AC. Pairing a resistor with one of them is the basis of every filter, picking which frequencies get through.