EC-12.1 AC Power and Three-Phase Systems

What this is and why it exists

Circuit analysis already covers the sinusoidal steady state, and it stops exactly where power engineering begins.

This topic is the bridge. The same phasors, now used to ask a different question. How much energy actually moves, and how much merely sloshes back and forth without doing anything.

Three-phase follows from that, and it is the arrangement almost all generated and industrial power actually uses. It is worth meeting properly rather than as a footnote.

The vocabulary

• Instantaneous power — the product of voltage and current at one moment.
• Real power — the average power, which does work. Measured in watts.
• Reactive power — power that moves back and forth without doing work.
• Apparent power — the product of the measured voltage and current magnitudes.
• Power factor — the ratio of real power to apparent power.
• Balanced load — a three-phase load drawing equal current in each phase.
• Star connection — three loads joined at a common point.
• Delta connection — three loads joined end to end in a closed loop.

The mental model

Take a sinusoidal voltage across a load and the current that flows. Multiply them instant by instant.

The product varies at twice the supply frequency, and for part of every cycle it is negative. Negative means energy flowing back from the load to the source. Only the average over a whole cycle represents energy actually delivered, and that average is the real power.

Where the current is in step with the voltage, the product is never negative and all the energy delivered stays delivered. Where the current is a quarter cycle out of step, the average is exactly zero: energy goes out and comes back, every cycle, forever.

Real loads are somewhere between. That is the whole content of the three powers.

Real power does work and is what an electricity meter records. Reactive power is the part that goes back and forth. Apparent power is what you get by multiplying the measured voltage and current, ignoring the phase between them.

The three form a right-angled relationship, with apparent power as the hypotenuse. Drawing that triangle once makes the arithmetic obvious and worth remembering.

Power factor is the ratio of real to apparent power, and the reason a supplier cares is physical rather than commercial. Conductors, transformers and switchgear must be sized for the current that flows, and reactive current flows without doing any work.

A factory drawing a hundred kilowatts at a power factor of one half needs equipment sized for two hundred kilovolt-amperes. The supplier built that equipment and charges accordingly, which is why a poor power factor appears on a bill.

Correcting it is arithmetic on the reactive part alone. Most industrial load is inductive, drawing current that lags. A capacitor draws current that leads. Placing capacitors at the load supplies the reactive part locally, so it no longer travels back to the source through every conductor on the way.

Sizing the capacitor means finding the reactive power at the present power factor, the reactive power at the target, and supplying the difference.

Now three phases, and the question of what they buy. Three supplies a third of a cycle apart, feeding a balanced load, deliver constant total power rather than power that pulsates twice per cycle. A motor fed that way produces steady torque rather than a vibration.

They also need less conductor for the same energy delivered, because the return currents in a balanced system cancel. And three windings fed this way produce a magnetic field that rotates, which is the single fact that makes the induction motor possible.

Two connections and two ratios cover the arithmetic. In star, the three loads meet at a common point. The voltage between two lines exceeds the voltage across one load by the square root of three. In delta, the loads form a closed loop, and the current in a line exceeds the current in one load by that same factor.

For a balanced load, a three-phase problem reduces to a single-phase one. Solve one phase, multiply the power by three, and remember which ratio applies. That is the practical technique, and it is why balanced problems are tractable by hand.

What you should now be able to explain or do

• Explain why instantaneous power goes negative and why only the average is delivered.
• Separate real, reactive and apparent power and draw the relationship between them.
• Say what power factor means physically and why a supplier charges for it.
• Size a capacitor to correct a given load to a target power factor.
• List three things three-phase supply buys over single-phase.
• Analyse a balanced star or delta load by reducing it to one phase.

Check yourself