Circuits & Electrical Power

Ohm's law is the smallest true sentence in electrical engineering, and you will use it more than everything else in this course combined. V = I R, read aloud V equals I R.

A current is not a substance. It is a rate — the rate at which charge crosses a section of conductor — and once you hold that, Q = I t stops being a formula to memorise and becomes a definition to read.

Power is the rate at which electrical energy becomes something else — heat in a resistor, torque in a motor, light in a lamp.

Two resistors can be wired two ways, and the difference is not cosmetic. In series they sit end to end, so the same current must cross both in turn.

Series and parallel pairs turn up so often that both have a shortcut worth knowing cold. They look almost identical on the page, and one of them crosses over.

So far a resistance has been a number on a component.

Ohm's law tells you what ONE resistor does. Kirchhoff's two laws tell you what a whole network does, and every technique later in this chapter — Thevenin, Norton, delta-wye — is built out of them.

The second law is the loop rule's twin, and it is conservation of CHARGE. Charge is neither created nor destroyed at a wire junction, and a junction has nowhere to store any: whatever arrives in a second must leave in the same second.

Real analysis never uses one law at a time. A single board hands you a series string to walk with the loop rule and a junction to balance with the node rule, and the skill being examined is picking the right one for the line you are on.

Thevenin's theorem is the largest idea in this chapter, and it is startlingly generous: any network of sources and resistances, however tangled, behaves at one pair of terminals exactly like a single voltage source in series with a single…

Norton's theorem says the identical thing as Thevenin's in the other dialect: the box is a current source in parallel with a resistance.

Sooner or later a network arrives that will not reduce. Look at a bridge: five resistors, and no two of them are cleanly in series or cleanly in parallel.

Everything so far has analysed a circuit somebody else drew. This lesson designs one, and the answer is not a number — it is a component you could order.

A capacitor is two plates that do not touch. Push charge onto one and an equal charge leaves the other, and a voltage appears between them.

You have spent a chapter adding resistors in series and taking product-over-sum in parallel. Capacitors do both of those things, in the opposite places.

Connect a charged capacitor to a resistor and it empties — not steadily, but fast at first and slower and slower after.

Last lesson a full capacitor emptied through a resistor. Now run the film backwards: an EMPTY capacitor is switched onto a DC supply through a resistor, and its voltage climbs.

An inductor is a coil that resists a CHANGE in current.

Capacitors ran the combining rules backwards.

An AC voltage is never one number. It sweeps from zero up to a crest, back through zero to an equal trough, and round again, fifty or sixty times a second.

A resistor opposes current by the same number of ohms whether the current is steady or racing. Coils and capacitors do not.

A resistor and a reactance both oppose current in ohms, and it is tempting to add them.

Last lesson's triangle worked because a pure resistance and a pure reactance sit exactly 90° apart. Real loads are rarely that tidy.

Inductive reactance climbs with frequency and capacitive reactance falls. Put both in one circuit and there is exactly one frequency where they are equal — and being opposite in sign, they cancel completely.

A resistor with a capacitor, or a resistor with a coil, makes a filter: some frequencies get through and some do not.

On direct current, volts times amps is the whole story.

Draw the three powers as a right triangle and the whole subject becomes geometry. P is the real power in kilowatts, along the base — the part that turns shafts and makes heat. Q is the reactive power in kilovars (kvar), vertical — the part…

The subscript L means LINE throughout — outside the machine — as opposed to phase, which is inside it.

Fix the naming first, because everything here depends on it: line quantities, subscript L, are measured outside the machine — between two supply conductors, or in one of them.

Motors draw lagging vars; capacitors supply leading ones.

A utility does not sell power. It sells energy — power multiplied by the time you kept drawing it. E = P t, read aloud E equals P t: E is the energy, P the constant power draw and t the running time.

Last lesson's bill had one line: kilowatt-hours times a rate. A house is billed that way. A plant, a school or a supermarket is not.

That leading 2 is the named trap of this lesson, and it is not a fudge factor: the current goes out along one conductor and comes back along another, and both of them drop volts.

A battery's capacity is quoted in amp-hours, and an amp-hour is a charge: Q = I t, read aloud Q equals I t, with Q the charge, I the steady current in amperes and t the time.

Feed a three-phase stator and the windings build a magnetic field that rotates.

A motor nameplate sells you power. The coupling, the gearbox and the shaft key all care about torque.

Spin a coil in a magnetic field and it generates a voltage, whether or not you meant it to.

A motor is a converter, and no converter is free. Some of what comes in the terminals leaves as heat in the copper, iron losses in the core, windage and friction at the bearings.

A motor nameplate quotes a shaft rating in kilowatts.

Two coils on one iron core, and a changing current in the first induces a voltage in the second.

The ideal transformer of the last lesson passes every watt through. A real one keeps a little, as heat, in two places that behave completely differently.

Every transformer has impedance in its own windings, and the nameplate states it as percent impedance, %Z.

Add up every load on a standby panel and you get a number that will never happen. The lift is not running while the boiler is firing while every pump is at full flow.

In this lesson the two charges and the separation are always given, and F is what you are solving for.

A magnetic field does nothing to a charge that is sitting still. Set that charge moving and the field pushes on it — sideways, at right angles to both the motion and the field.

The last lesson took the magnetic field as given and asked what it does to a moving charge. Now turn the question round: where do fields come from?

A current is nothing but charge on the move, so a wire in a magnetic field feels the same push a single charge would — added up over every carrier in it.

Wind a wire into a long tight helix, push a current through it, and the field inside comes out remarkably uniform — that is a solenoid, and it is the working heart of every relay, contactor and solenoid valve you will ever meet.

A field on its own induces nothing. What generates voltage is a field CHANGING through a loop — and to talk about that you first need a word for how much field a loop catches.

Faraday's law does not care where a changing flux comes from. Put a second coil near the first, and some of the first coil's flux threads the second.

Take Faraday's law and hand it the simplest possible machine: a straight conducting bar sliding along two rails through a magnetic field.