Practical electrical theory

The electrical theory used every day in a control panel fits in a few formulas: Ohm's law, power, the difference between phase and line voltage, and power factor. Using them fluently lets you check whether a measurement makes sense, choose a protective device or explain to a customer how much it costs to keep a motor running. This topic goes over them with workshop examples.

Basic quantities

Ohm's law and power

U = I × R  ·  P = U × I (in DC and with resistive loads)

Example: a 2 kW heating element on 230 V draws I = 2000 / 230 ≈ 8.7 A. A 24 V relay coil of 480 Ω draws 24 / 480 = 0.05 A. If you want to practise, the Ohm's law calculator solves any combination.

Resistors in series and in parallel

• In series, resistances add up: R = R1 + R2 + … The same current flows through all of them.
• In parallel, the reciprocals add up: 1/R = 1/R1 + 1/R2 + … They all have the same voltage. Two equal resistors in parallel give half the value of one.

This explains a measurement that confuses many people: if you measure a resistor without disconnecting it and there is another one in parallel, the meter reads less than expected.

The heat of the current

A conductor of resistance R carrying a current I heats up with a power of I² × R (Joule heating). Note the square: if the current doubles, the heat is multiplied by four. That is why cables are sized for the current they will carry, why a loose terminal (more resistance) gets hot, and why protective devices interrupt overloads.

Alternating current

The supply completes 50 cycles per second (50 Hz in Europe and much of the world) or 60 (60 Hz in North America), and in each cycle the current reverses twice. When we say «230 V» we mean its RMS value, the one that produces the same heat as 230 V DC. The peak value is √2 times larger: about 325 V. Multimeters measure RMS (True RMS meters do it correctly even with non-sinusoidal waveforms).

Phase voltage and line voltage

In a three-phase supply, the voltage between two phases (line voltage) is √3 times the voltage between phase and neutral (phase voltage): 400 V ≈ √3 × 230 V. In North America you find the same ratio in 208 V ≈ √3 × 120 V and 480 V ≈ √3 × 277 V. That is why in a European panel you measure about 400 V between phases and about 230 V between phase and neutral.

AC power and power factor

With motors, transformers and coils, the current is out of phase with the voltage, and three powers appear:

• Active (P, in W): the power turned into useful work or heat.
• Reactive (Q, in var): the power that flows back and forth to build the magnetic fields; it does no work, but it flows through the cables.
• Apparent (S, in VA): the combination of the two, what the installation really «sees».

The active power formulas:

• Single-phase: P = U × I × cos φ
• Three-phase: P = √3 × U × I × cos φ (with U the voltage between phases)

Example: a three-phase motor on 400 V drawing 10 A with cos φ = 0.85 takes P = 1.732 × 400 × 10 × 0.85 ≈ 5.9 kW from the supply.

Energy and cost

energy (kWh) = input power (kW) × hours

Example: a motor with 4 kW of output power and 88 % efficiency takes about 4.55 kW. Running 8 hours a day for 220 days a year it uses 4.55 × 8 × 220 ≈ 8,000 kWh. At 0.15 per kWh (in euros or dollars), that is about 1,200 a year. That is why a more efficient motor, or a drive on a pump that does not always need full speed, can pay for itself quickly.

Overload and short circuit

• Overload: a little more current than intended flows for some time, for example a motor under strain. The cable heats up slowly. It is covered by the thermal part of the circuit breaker or by the overload relay.
• Short circuit: two points at different voltages are joined with almost no resistance and the current shoots up to hundreds or thousands of amps. It must be interrupted in milliseconds: the magnetic part of the breaker or the fuses do it.

Common mistakes

• Applying P = U × I to a motor and forgetting the power factor and the efficiency.
• Using the phase voltage in the three-phase formula, or the line voltage in the single-phase one.
• Confusing the output power on the nameplate with the power the motor draws from the supply.
• Thinking that twice the current means twice the heat: it means four times as much.
• Measuring a resistor in circuit and ignoring what is in parallel with it.

Summary

• U = I × R and, with resistive loads, P = U × I.
• Series: resistances add up. Parallel: their reciprocals add up.
• The heat in a conductor is I² × R: twice the current, four times the heat.
• 230 V is the RMS value; the peak is about 325 V. Between phases, 400 V ≈ √3 × 230 V.
• P = √3 × U × I × cos φ in three-phase. A low power factor means more current for the same work.
• Energy = input power × hours; the cost is the energy times the price per kWh.