Transformers and power systems
Electricity is generated in power stations, stepped up for efficient transmission, and stepped down for use. Transformers make this possible, and three-phase systems make it efficient.
The ideal transformer
A transformer has two coils, primary and secondary, wound on a shared magnetic core. An alternating current in the primary creates a changing flux that induces a voltage in the secondary. For an ideal transformer, \(\frac{V_s}{V_p} = \frac{N_s}{N_p}\) and, conserving power, \(\frac{I_s}{I_p} = \frac{N_p}{N_s}\). A 10:1 step-down transformer turns 230 V into 23 V while its secondary can deliver ten times the primary current.
Why transmit at high voltage
Power lost in transmission lines is \(I^2R\). For a given power \(P = VI\), raising the voltage lowers the current proportionally and cuts losses by the square: increasing voltage tenfold reduces resistive losses a hundredfold. That is why transmission lines operate at hundreds of kilovolts, while homes receive about 230 V or 120 V for safety.
Real transformers
Real transformers have winding resistance (copper losses), core losses from eddy currents and hysteresis, and leakage flux that does not link both coils. Large power transformers nonetheless exceed 99% efficiency. Transformers work only with AC, since steady DC creates no changing flux, a key reason AC won over DC for power distribution in the late nineteenth century.
The war of the currents
In the 1880s and 1890s, Thomas Edison promoted DC distribution, while George Westinghouse and Nikola Tesla championed AC. AC's transformers allowed efficient long-distance transmission, and the 1896 Niagara Falls hydroelectric plant supplying Buffalo, New York, demonstrated it. Today, high-voltage DC (HVDC) links, made practical by power electronics, are used for very long distances and undersea cables.
Three-phase power
Grids generate and transmit three-phase power: three sinusoidal voltages offset by 120 degrees. Three-phase systems deliver smooth, constant total power, need less conductor material than single-phase for the same power, and naturally create rotating magnetic fields for motors. Line-to-line voltage is root 3 times phase voltage: a 230 V phase supply has 400 V between lines.
The grid
The electricity grid links generators, transmission networks, substations, distribution networks, and consumers, and must balance supply and demand every second. If demand exceeds supply, frequency falls below its nominal 50 or 60 Hz; operators adjust generation and use reserves to correct it. Large interconnected grids, such as continental Europe's, span many countries and synchronise at one frequency.
Generation mix
Electricity comes from fossil fuels (coal, gas, oil), nuclear, hydro, wind, solar, and other renewables. Thermal and hydro plants use spinning synchronous generators; wind and solar connect through power-electronic inverters. Global electricity generation has been shifting towards renewables, which supplied about 30% of the world's electricity in 2023 according to energy think tank Ember, raising new challenges in balancing variable output.
Worked example
A wind farm sends 10 MW through a line with total resistance 5 \(\Omega\). At 33 kV, the current is \(\frac{10{,}000{,}000}{33{,}000} \approx 303\) A, and losses are \(303^2 \times 5 \approx 459\) kW, about 4.6%. Stepped up to 132 kV, the current is about 76 A and losses about \(76^2 \times 5 \approx 29\) kW, under 0.3%. Quadrupling the voltage cut losses roughly sixteenfold.
Key takeaways
Ideal transformers scale voltage by the turns ratio and current inversely, conserving power; real ones exceed 99% efficiency despite copper and core losses, and work only with AC. Transmission at high voltage cuts I^2 R losses by the square of the voltage ratio. AC won the 1880s-1890s war of the currents, shown at Niagara Falls in 1896; HVDC now serves long and undersea links. Three-phase power, 120 degrees apart, gives constant power and rotating fields; line voltage is root 3 times phase voltage. Grids balance supply and demand, with frequency as the signal. Renewables supplied about 30% of world electricity in 2023.
Flashcard
Coupled coils on a shared core
Check your understanding
Why do power companies transmit electricity at very high voltages?