Transformer Equations

Transformer Basics

A transformer is an electrical device used to change the voltage of an alternating current (AC). It consists of two coils of wire, called the primary coil and the secondary coil, wrapped around a soft iron core.

• The primary coil is connected to the input voltage supply.
• The secondary coil provides the output voltage.

Transformers only work with AC because the changing current produces a changing magnetic flux in the iron core, which induces a voltage in the secondary coil according to Faraday's law.

There are two main types of transformers:

• Step-up transformer: increases voltage from primary to secondary coil by having more turns on the secondary coil.
• Step-down transformer: decreases voltage by having fewer turns on the secondary coil.

Transformer Equations

The relationship between the voltages and the number of turns in the coils is given by the transformer equation:

Voltage ratio = turns ratio

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

• $V_p$ = potential difference (voltage) across the primary coil (volts, V)
• $V_s$ = potential difference across the secondary coil (volts, V)
• $N_p$ = number of turns on the primary coil
• $N_s$ = number of turns on the secondary coil

This means the voltage ratio between the coils is directly proportional to the ratio of the number of turns in the coils.

Transformers are designed to be very efficient, so the power input to the primary coil is approximately equal to the power output from the secondary coil (ignoring small energy losses). Real transformers typically have efficiencies between 95% and 99%:

$$V_p I_p \approx V_s I_s$$

• $I_p$ = current in the primary coil (amps, A)
• $I_s$ = current in the secondary coil (amps, A)

Voltage and Turns Ratio

The number of turns in each coil determines the voltage produced:

• More turns in the coil means a higher voltage.
• A step-up transformer has $N_s > N_p$, so $V_s > V_p$.
• A step-down transformer has $N_s < N_p$, so $V_s < V_p$.

For example, if the secondary coil has twice as many turns as the primary coil, the output voltage will be twice the input voltage.

For instance, if a transformer has 100 turns on the primary coil and 200 turns on the secondary coil, and the input voltage is 230 V, the output voltage is:

$$V_s = V_p \times \frac{N_s}{N_p} = 230 \times \frac{200}{100} = 460 \text{ V}$$

Current and Power in Transformers

Since transformers ideally conserve power (ignoring losses), when the voltage increases, the current decreases, and vice versa. This means current and voltage are inversely proportional:

$$\frac{I_p}{I_s} = \frac{N_s}{N_p}$$

This relationship shows that:

• In a step-up transformer (increasing voltage), the current in the secondary coil is less than in the primary coil.
• In a step-down transformer (decreasing voltage), the current in the secondary coil is greater than in the primary coil.

This is important for the National Grid, where high voltages and low currents are used to reduce energy loss during transmission.

As a quick example, if the primary coil current is 4 A and the transformer is step-up with twice as many turns on the secondary coil, the secondary current is:

$$I_s = I_p \times \frac{N_p}{N_s} = 4 \times \frac{1}{2} = 2 \text{ A}$$