In this explainer, we will learn how to calculate the change in the potential difference and current produced by a transformer.
We recall that electromagnetic induction is the term for the production of an electric current in a conductor when the conductor is moving near a magnet.
An example of electromagnetic induction using a moving bar magnet and a coil of conducting wire is shown in the following figure.
Only one line of the magnetic field of the bar magnet is shown. The field lines actually radiate from the north pole of the magnet to the south pole symmetrically in all directions.
Moving the bar magnet toward or away from the face of the coil produces a change of magnetic field strength through the coil. The current produced is the result of the change of magnetic field strength.
Electromagnetic induction does not only involve producing current using a changing magnetic field, but also involves inducing a magnetic field using a changing current. For example, when the current in a coil of wire changes, this induces a magnetic field in the coil.
When the current is varied in a coil of wire with many circular turns, a magnetic field is induced that is a very similar shape to that of a bar magnet. This can be seen in the following figure.
A coil of wire with this shape is often called a solenoid.
Let us consider two solenoids that are placed next to each other. Initially, neither solenoid has a current in it.
We then increase, from zero, the current in one solenoid, which we call the primary coil. The effect of this is shown in the following figure.
We see that the changing current in the primary coil induces a magnetic field. This field contains the secondary coil.
A changing magnetic field through the secondary coil produces a current in the secondary coil.
The magnitude of the current depends on the rate of change of the magnetic field through the secondary coil.
The following figure shows how the magnetic field strength in the region within the primary coil compares to that within the secondary coil.
We see that the magnetic field within the secondary coil is much less strong than that within the primary coil. This means that the current produced in the secondary coil will have a much smaller magnitude than the current in the primary coil.
We see then that solenoids are inefficient at transferring electrical energy. The inefficiency is due to the shape of the magnetic field of the primary coil. If the shape of the field is changed, more efficient transfer of energy from the primary coil to the secondary coil is possible.
The magnetic field between two solenoids can be redirected by linking the solenoids with a common core of a magnetizable substance such as iron. This is shown in the following figure.
Two solenoids that share a core form an object that is called a transformer.
The magnitude of the magnetic field produced by the primary coil within the core material is much greater than that in air.
The following figure compares the magnetic field line density within the transformer core and outside the core.
Only some of the field lines within the core are shown completely. We see that these lines are much closer together than the lines outside the core, and from this that, the magnetic field strength in the core is much greater than outside the core.
A transformer can transfer energy between solenoids with great efficiency. It is a reasonable approximation to model a transformer as transferring energy between solenoids with efficiency.
Energy can be transferred between solenoids without the use of electromagnetic induction, simply by connecting solenoids in an electric circuit.
Electromagnetic induction is used to transfer energy between solenoids, as this makes it possible to have different values for the current and potential difference in the solenoid that energy is transferred to and the solenoid that energy is transferred from.
Having unequal values of current and potential difference for the primary and secondary coils of a transformer requires that the primary coil and secondary coil be of unequal lengths, as shown in the following figure.
For a perfectly efficient transformer, the electrical energy transferred by the primary coil must equal the electrical energy transferred to the secondary coil.
The energy transfer between the coils occurs in a time interval. The energy transfer in that time interval is equal to the electrical power input from the primary coil and also equal to the electrical power output from the secondary coil.
Electrical power, , is given by the formula where is the potential difference across a coil and is the current in a coil.
It must be the case that so it must be the case that
The following figure shows the input and output current and potential difference for a transformer.
We see that
The turns of a solenoid can be considered to be in series with each other. Each turn will have the same potential difference across it. The sum of the potential differences across the turns will equal the potential difference across the solenoid.
For the transformer shown, the primary coil has 6 turns and the secondary coil has 2 turns, as shown in the following figure.
We see then that
We can rearrange this to determine :
As it is true that we see that
We can rearrange this to determine :
The output of this transformer increases the input current and decreases the input potential difference.
Transformers are termed for their effect on the input potential difference, so a transformer of this design is called a step-down transformer.
A transformer that increases input potential difference is called a step-up transformer, as shown in the following figure.
What makes a transformer a step-up or step-down transformer is the ratio of the number of turns of the primary coil to the number of turns of the secondary coil. Assuming that each turn of each coil is the same length, the ratio of turns is equal to the ratio of the potential differences across the coils. This can be written as
Transformers are used in the transmission of electrical power over great distances.
When a wire carries a current, the resistance of the wire dissipates the energy of the current. The greater the current, the greater the energy dissipated by the wire.
By using a step-up transformer, electrical power can be transmitted through wires carrying very small currents and at great potential differences. The power dissipated in such wires is then reduced.
A step-down transformer can increase the value of the current of transmitted electricity when it arrives at circuits where it is needed to do work.
Let us now look at some examples involving transformers.
Example 1: Finding the Output Potential Difference of a Transformer
A transformer has 200 turns on its primary coil and 50 turns on its secondary coil. If the input potential difference is 20 V, what is the output potential difference?
Answer
The ratio of the number of turns, , in the input and output coils of a transformer is the same as the ratio of the potential difference, , across these coils.
The ratio of turns in the coils is given by
It must then be the case that
The question states that the potential difference across the input coil is 20 V.
We see then that
We can rearrange this to make the subject.
Example 2: Finding the Output Current of a Transformer
A efficient transformer has 5 times as many turns on its secondary coil as it does on its primary. If the current through the primary coil is 20 A, what is the current through the secondary coil?
Answer
The ratio of the number of turns, , in the input and output coils of a transformer is the same as the ratio of the potential difference, , across these coils.
The transformer is stated to have 5 times as many turns on its secondary coil as on its primary coil. We can express this as
From this, we see that where is the potential difference across a coil.
This can be written as
The power, , in each coil is equal and is given by where is the current in the coil.
This means that
Substituting the expression for into this equation, we have
We can divide both sides of this equation by . We then have
The question states that the current in the input coil is 20 A.
The current in the output coil is then given by
Example 3: Finding the Number of Turns on the Primary Coil of a Transformer
A step-down transformer changes the potential difference of an alternating current from 10 000 V to 250 V. If it has 25 turns on its secondary coil, how many turns does it have on its primary coil?
Answer
The ratio of the number of turns, , in the input and output coils of a transformer is the same as the ratio of the potential difference, , across these coils. This means that
The potential difference across each coil is stated in the question. The ratio of these potential differences can be determined:
We see then that
The question states that is 25. We see then that
can be made the subject of this equation, so we see that
Let us now summarize what has been learned in this explainer.
Key Points
- A transformer uses electromagnetic induction to transfer energy between solenoids that are not connected to each other in an electric circuit.
- A transformer consists of two solenoids that are linked by a core made of a magnetizable substance.
- A changing current in one of the solenoids (called the input coil) in a transformer produces a current in the other solenoid (called the output coil).
- For a perfectly efficient transformer, the power in the two solenoids is equal. Therefore, where and are potential differences and currents in the input and output coils of the transformer.
- For a perfectly efficient transformer, the ratio of the number of turns, , in the input and output coils of a transformer is the same as the ratio of the potential difference, , across these coils. Therefore,
- A transformer for which is called a step-up transformer.
- A transformer for which is called a step-down transformer.
- When electricity is transmitted over great distances through a wire, step-up transformers can be used to reduce the current that is transmitted. The smaller the current transmitted, the less energy dissipated by the wire.
