Transformers

How a transformer works

A transformer changes the voltage of an alternating supply. It consists of two coils wound on a shared soft-iron core:

• The primary coil is connected to the alternating supply.
• The alternating current produces a continually changing magnetic flux in the core.
• The soft-iron core channels almost all that flux through the secondary coil.
• By Faraday's law, the changing flux through the secondary induces an alternating emf in it.

$$\frac{V_s}{V_p} = \frac{N_s}{N_p}$$

• $V_p, V_s$ — primary and secondary voltages (V)

• $N_p, N_s$ — number of turns on the primary and secondary coils

• Step-up transformer: $N_s > N_p$, so $V_s > V_p$.

• Step-down transformer: $N_s < N_p$, so $V_s < V_p$.

Why a transformer only works on AC

This is examined frequently and the reasoning must be explicit:

• Induction requires a changing magnetic flux.
• An alternating current continually changes direction and magnitude, so it produces a continually changing flux and a continuous induced emf in the secondary.
• A direct current produces a constant flux once it is established. With $\Delta\Phi/\Delta t = 0$, no emf is induced and the secondary output is zero.
  • The only exception is the brief instant of switching on or off, when the flux is changing — which is why a DC-fed transformer gives a single pulse and then nothing.

Power and current

For an ideal transformer, energy is conserved and no power is lost:

$$P_p = P_s \qquad \Rightarrow \qquad V_pI_p = V_sI_s$$

• So stepping the voltage up steps the current down, in the same ratio, and vice versa.
• A transformer cannot create energy. Any voltage gain is paid for by an equal proportional loss of current.

Real transformers and their losses

Real transformers are typically $95$–$99\%$ efficient. The losses are:

• Resistive (copper) losses — the coils have resistance, so $I^2R$ heating occurs. Reduced by using thick copper wire.
• Eddy currents — the changing flux induces circulating currents in the iron core itself, dissipating energy as heat. Reduced by laminating the core: building it from thin insulated sheets that break up the eddy current paths.
• Hysteresis losses — energy is used repeatedly re-magnetising the core each cycle. Reduced by using soft iron, which magnetises and demagnetises easily.
• Flux leakage — not all the flux from the primary reaches the secondary. Reduced by good core design that provides a complete magnetic circuit.

Why the grid uses transformers

• Power is transmitted at very high voltage to keep the current low, because transmission losses are $P = I^2R$ in the cables.
• Halving the current quarters the loss, so raising the voltage by a factor of 20 reduces losses by a factor of 400.
• Transformers make this possible: step-up at the power station, transmit at high voltage, and step-down near the consumer to a safe $230$ V.
• This is the fundamental reason mains electricity is AC rather than DC — transformers do not work on DC, and no comparably simple and efficient way of changing DC voltage existed when the grid was built.