Electrical Science Revision: Ohm's Law to Three-Phase

Worked Example 1

A 3kW immersion heater is connected to a 230V supply. What current does it draw?

First, convert 3kW to watts: 3,000W

Using P = IV, rearranged to I = P / V:

I = 3,000 / 230 = 13.04A

This tells you the heater draws approximately 13A — important for selecting the correct cable size and protective device.

Worked Example 2

A cable has a resistance of 0.5 ohms and carries 20A. What is the voltage drop?

Using V = IR:

V = 20 x 0.5 = 10V

A 10V drop on a 230V supply means the load receives 220V. BS 7671 limits voltage drop to 3% for lighting (6.9V) and 5% for other circuits (11.5V), so this cable may be borderline for some installations.

Worked Example

An electric shower is rated at 10.5kW on a 230V supply. What current does it draw, and what is the resistance of the heating element?

Current: I = P / V = 10,500 / 230 = 45.65A

Resistance: R = V / I = 230 / 45.65 = 5.04 ohms

This is why electric showers need a dedicated circuit with a high-rated cable (typically 10mm² or 16mm²) and a 45A or 50A MCB.

Worked Example — Kirchhoff's Voltage Law

A 230V supply feeds a cable with 0.3 ohms resistance, connected to a heater drawing 20A. What voltage does the heater actually receive?

Voltage drop in cable: V = IR = 20 x 0.3 = 6V

Voltage at heater: 230 - 6 = 224V

By KVL, the supply voltage (230V) equals the cable voltage drop (6V) plus the voltage across the heater (224V). This is exactly how voltage drop calculations work in practice.

Worked Example — Parallel Resistance

Three 30-ohm resistors are connected in parallel. What is the total resistance?

1/Rtotal = 1/30 + 1/30 + 1/30 = 3/30 = 1/10

Rtotal = 10 ohms

Three identical resistors in parallel give one-third the resistance of one. This principle applies to ring circuits — the R1+R2 value of a ring is one-quarter the end-to-end resistance because you have two parallel paths.

• Reactance (X) — the opposition to current flow caused by inductors and capacitors in AC circuits. Inductive reactance (XL = 2πfL) increases with frequency — coils and motors resist changes in current. Capacitive reactance (XC = 1/(2πfC)) decreases with frequency — capacitors pass AC more easily at higher frequencies. Measured in ohms.
• Impedance (Z) — the total opposition to current flow in an AC circuit, combining resistance and reactance. Z = √(R² + X²). In a purely resistive circuit (kettle, heater), Z = R. In circuits with motors or transformers, Z is greater than R because of the reactive component.
• Power factor (cos φ) — the ratio of real power (watts) to apparent power (VA). Power factor = P / S = cosφ, where φ is the phase angle between voltage and current. A power factor of 1.0 means voltage and current are in phase (purely resistive). A power factor of 0.8 means 80% of the apparent power is doing useful work.

Worked Example — Power Factor

A motor has a real power of 2,000W and an apparent power of 2,500VA. What is the power factor, and what current does it draw on a 230V supply?

Power factor: PF = P / S = 2,000 / 2,500 = 0.8

Current: I = S / V = 2,500 / 230 = 10.87A

Notice that the current is based on the apparent power (2,500VA), not the real power (2,000W). The motor draws more current than a purely resistive 2,000W load would (which would draw 8.7A). This extra current is the reactive component — it does no useful work but still heats the cables and loads the supply.

Worked Example — Three-Phase Power

A balanced three-phase load draws 20A per phase from a 400V supply with a power factor of 0.85. What is the total power?

Three-phase power: P = √3 x VL x IL x PF

P = 1.732 x 400 x 20 x 0.85 = 11,777W (approximately 11.8kW)

This formula appears frequently in exam questions and in real commercial/industrial design work. Memorise it: P = √3 x VL x IL x PF.

• How they work — an AC current in the primary winding creates a changing magnetic field in the iron core. This changing magnetic field induces a voltage in the secondary winding. The voltage ratio equals the turns ratio: V1/V2 = N1/N2. No electrical connection between primary and secondary — energy is transferred magnetically.
• Step-up and step-down — a step-down transformer has more primary turns than secondary turns, reducing the voltage. A step-up transformer has more secondary turns, increasing the voltage. The power is (approximately) the same on both sides: if voltage goes down, current goes up, and vice versa.
• On-site examples — 110V centre-tapped transformers for construction site power tools (reduces the voltage to earth to 55V for safety), bell transformers for doorbells (230V to 8V or 12V), and distribution transformers that step down 11kV to 400V/230V for domestic supplies.

Worked Example

A transformer has 460 primary turns and 20 secondary turns. If the primary voltage is 230V, what is the secondary voltage?

V2 = V1 x (N2 / N1) = 230 x (20 / 460) = 10V

This is a step-down transformer with a 23:1 turns ratio, commonly used for low-voltage lighting or bell circuits.

• How motors work — a current-carrying conductor in a magnetic field experiences a force (Fleming's left-hand rule). In a motor, the stator creates a magnetic field, and the rotor carries current — the interaction produces rotation. The direction of rotation depends on the direction of current and the magnetic field.
• Starting current — when a motor starts, it draws a significantly higher current than its running current (typically 5 to 8 times the full-load current). This is called the starting current or inrush current. A motor rated at 10A full-load may draw 50 to 80A for a fraction of a second on starting. This is why motor circuits use Type C or Type D MCBs — they tolerate the brief inrush without tripping.
• Single-phase vs three-phase motors — single-phase motors are common in domestic appliances and small commercial equipment. Three-phase motors are more efficient, produce smoother torque, and are standard for industrial machinery. A three-phase motor runs directly from the three-phase supply; a single-phase motor needs a capacitor or other starting mechanism to create the rotating field.
• Back EMF — as a motor spins, it generates a voltage that opposes the supply voltage (back electromotive force). This is why running current is lower than starting current — the back EMF reduces the net voltage across the windings, limiting the current. If a motor stalls (stops turning), back EMF drops to zero and current rises to the starting level, which can overheat the windings if sustained.