5.8.9. Transformers

Table of Contents

Basic idea

A transformer is a device that changes an alternating voltage from one value to another. It can increase the voltage, called a step up transformer, or decrease the voltage, called a step down transformer. Transformers work with alternating current because they rely on a changing magnetic field.

A transformer usually has two coils of wire wound around a common iron core. The coil connected to the input is the primary coil. The coil connected to the output is the secondary coil. When alternating current flows in the primary coil, it creates a changing magnetic flux in the core. This changing flux produces an induced voltage in the secondary coil.

Rendering scientific sketch…

Basic transformer structure

How a transformer works

The operation of a transformer follows from electromagnetic induction. A changing current in the primary coil produces a changing magnetic field. This changing field passes through the secondary coil and induces an electromotive force in it.

The amount of induced voltage depends on the number of turns in each coil. If the secondary coil has more turns than the primary coil, the output voltage is larger. If it has fewer turns, the output voltage is smaller.

For an ideal transformer, the voltage ratio equals the turns ratio:
$$\frac{V_s}{V_p} = \frac{N_s}{N_p}$$
where $V_p$ and $V_s$ are the primary and secondary voltages, and $N_p$ and $N_s$ are the numbers of turns.

This means that doubling the number of turns in the secondary coil doubles the output voltage, in the ideal case.

Step up and step down transformers

A step up transformer has more turns in the secondary than in the primary, so $N_s > N_p$. Therefore, $V_s > V_p$. It increases voltage.

A step down transformer has fewer turns in the secondary than in the primary, so $N_s < N_p$. Therefore, $V_s < V_p$. It decreases voltage.

Transformer type	Turns relation	Voltage relation	Main effect
Step up	$N_s > N_p$	$V_s > V_p$	Increases voltage
Step down	$N_s < N_p$	$V_s < V_p$	Decreases voltage
Equal turns	$N_s = N_p$	$V_s = V_p$	No voltage change

Current relationship

In an ideal transformer, power is conserved. The electrical power entering the primary equals the electrical power leaving the secondary.

$$P_p = P_s$$

Since electrical power is $P = VI$, we get

$$V_p I_p = V_s I_s$$

Using the voltage ratio, we find the current ratio:

For an ideal transformer,
$$\frac{I_s}{I_p} = \frac{N_p}{N_s}$$
So when voltage increases, current decreases, and when voltage decreases, current increases.

This opposite behavior of voltage and current is very important. A transformer does not create energy. It trades voltage for current, or current for voltage.

Ideal and real transformers

The ideal transformer is a simplified model. In a real transformer, some energy is lost. The output power is slightly smaller than the input power.

The efficiency is

$$\eta = \frac{P_s}{P_p} \times 100\%$$

Real transformers can be highly efficient, often above $90\%$, but never perfectly efficient.

Common causes of energy loss include resistance in the wires, heating of the coils, and energy losses in the core. Some magnetic flux may also fail to link both coils. This is called flux leakage.

In a real transformer,
$$P_s < P_p$$
and
$$\eta < 100\%$$

Why transformers use AC

A transformer requires a changing magnetic flux. Direct current produces a steady current after the circuit settles, so the magnetic flux becomes constant. A constant flux does not induce a voltage in the secondary coil.

This is why ordinary transformers work with alternating current, not steady direct current.

The role of the core

The iron core helps guide the magnetic flux from the primary coil to the secondary coil. This makes the magnetic coupling strong and improves efficiency. Without a core, much more flux would spread into space and the transformer would work poorly.

In many transformers, the core is made of thin laminated sheets rather than one solid block. This reduces unwanted currents in the core and lowers energy loss.

Everyday use of transformers

Transformers are used wherever AC voltage must be changed. In electric power systems, power is sent over long distances at high voltage and low current. This reduces energy loss in transmission lines. Near homes and buildings, transformers reduce the voltage to safer and more useful values.

They are also found in chargers, adapters, audio systems, and many electrical devices.

Rendering scientific sketch…

Step up and step down idea

A simple example

Suppose a transformer has $N_p = 100$ turns and $N_s = 500$ turns. If the primary voltage is $V_p = 12 \, \text{V}$, then

$$\frac{V_s}{12} = \frac{500}{100} = 5$$

$$V_s = 60 \, \text{V}$$

This is a step up transformer.

If the transformer is ideal and the primary current is $I_p = 2.0 \, \text{A}$, then

$$V_p I_p = V_s I_s$$

$$12 \times 2.0 = 60 \times I_s$$

$$I_s = 0.40 \, \text{A}$$

The voltage became larger, and the current became smaller.

Key formulas

Important transformer relations for an ideal transformer:
$$\frac{V_s}{V_p} = \frac{N_s}{N_p}$$
$$V_p I_p = V_s I_s$$
$$\frac{I_s}{I_p} = \frac{N_p}{N_s}$$

These equations summarize the essential behavior of transformers. They show how turns, voltage, current, and power are connected.

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