The transformer equation

Definition

Potential difference

The energy transferred per unit charge as the charge passes between two points in a circuit.

Definition

Turns ratio

The number of turns on the primary coil of a transformer compared with the number of turns on its secondary coil.

The potential differences across the two coils of a transformer are linked to their numbers of turns by $VpVs=NpNs\dfrac{V_p}{V_s}=\dfrac{N_p}{N_s}$ .
In words, $coil\dfrac{\text{potential difference across primary coil}}{\text{potential difference across secondary coil}}=\dfrac{\text{number of turns in primary coil}}{\text{number of turns in secondary coil}}$ .
$V_p$ is the potential difference across the primary coil and $V_s$ is the potential difference across the secondary coil, both measured in volts ( $V\text{V}$ ).
$N_p$ is the number of turns on the primary coil and $N_s$ is the number of turns on the secondary coil.
Numbers of turns are pure numbers with no unit, so the turns ratio $NpNs\dfrac{N_p}{N_s}$ has no unit either.
Both potential differences must be in the same unit, so convert kilovolts to volts, or volts to kilovolts, before substituting.
Turns ratios are often written in the form primary : secondary, so a ratio of $1 : 20$ means the secondary coil has $20$ times as many turns as the primary coil.
The equation assumes an ideal transformer, which is the assumption made in every calculation of this kind.

Rearranging the equation

The equation can be written the other way up, as $VsVp=NsNp\dfrac{V_s}{V_p}=\dfrac{N_s}{N_p}$ , which is often easier when the secondary is the unknown.
Whichever way up you write it, both primary quantities must be on the top or both must be on the bottom.
To find the secondary potential difference, use $Vs=Vp×NsNpV_s=V_p\times\dfrac{N_s}{N_p}$ .
To find the primary potential difference, use $Vp=Vs×NpNsV_p=V_s\times\dfrac{N_p}{N_s}$ .
To find the number of turns on the secondary coil, use $Ns=Np×VsVpN_s=N_p\times\dfrac{V_s}{V_p}$ .
To find the number of turns on the primary coil, use $Np=Ns×VpVsN_p=N_s\times\dfrac{V_p}{V_s}$ .
A quick check catches most mistakes: if the secondary coil has more turns than the primary coil, the secondary potential difference must be the larger of the two.

Example

Finding the output potential difference

A transformer has $500$ turns on its primary coil and $25$ turns on its secondary coil.
The primary coil is connected to the $V230\ \text{V}$ mains.
Write the equation with the secondary quantities on top:

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

Rearrange and substitute:

V_s=230\times\frac{25}{500}

V_s=11.5\ \text{V}

The secondary coil has fewer turns, so a smaller output is expected, which matches the answer.

Example

Finding the number of turns

A step-up transformer changes $V12\ \text{V}$ into $kV3.0\ \text{kV}$ , and its primary coil has $40$ turns.
Convert the output into volts: $V3.0\ \text{kV}=3000\ \text{V}$ .
Rearrange to make $N_s$ the subject and substitute:

N_s=N_p\times\frac{V_s}{V_p}=40\times\frac{3000}{12}

N_s=10\,000\ \text{turns}

Example

Using a turns ratio

A transformer has a turns ratio of $20 : 1$ , primary to secondary, and gives an output of $V6.0\ \text{V}$ .
The primary coil has $20$ times as many turns as the secondary coil, so its potential difference is $20$ times larger:

\frac{V_p}{V_s}=\frac{N_p}{N_s}=\frac{20}{1}

V_p=20\times6.0=120\ \text{V}

Common Mistake

Do not write the equation with $V_p$ on top on one side and $N_s$ on top on the other, because the primary quantities must be on the same level of both fractions.
Do not mix kilovolts and volts in the same calculation, because the answer will be wrong by a factor of $1000$ .

Exam technique

Transformer calculations

This equation is one you select and apply rather than recall, so the marks come from rearranging it correctly and putting each value in the right place.
Label your values $V_p$ , $V_s$ , $N_p$ and $N_s$ before substituting, because swapping the primary and secondary values is the most common way to lose the marks.
Give a number of turns as a whole number with no unit, and give a potential difference with the unit $V\text{V}$ .
Check the size of your answer against the type of transformer before moving on.

Self review

Write down the equation that links the potential differences and the numbers of turns in a transformer.
A transformer has $100$ turns on its primary coil and $2000$ turns on its secondary coil, and the input is $V12\ \text{V}$ ; calculate the output potential difference.
Rearrange the equation to make $N_s$ the subject.
Is a transformer with a turns ratio of $1 : 5$ , primary to secondary, a step-up or a step-down transformer?
Why must both potential differences be in the same unit before you substitute them?

The transformer equation

Definition

Potential difference

The energy transferred per unit charge as the charge passes between two points in a circuit.

Definition

Turns ratio

The number of turns on the primary coil of a transformer compared with the number of turns on its secondary coil.

The potential differences across the two coils of a transformer are linked to their numbers of turns by $VpVs=NpNs\dfrac{V_p}{V_s}=\dfrac{N_p}{N_s}$ .
In words, $coil\dfrac{\text{potential difference across primary coil}}{\text{potential difference across secondary coil}}=\dfrac{\text{number of turns in primary coil}}{\text{number of turns in secondary coil}}$ .
$V_p$ is the potential difference across the primary coil and $V_s$ is the potential difference across the secondary coil, both measured in volts ( $V\text{V}$ ).
$N_p$ is the number of turns on the primary coil and $N_s$ is the number of turns on the secondary coil.
Numbers of turns are pure numbers with no unit, so the turns ratio $NpNs\dfrac{N_p}{N_s}$ has no unit either.
Both potential differences must be in the same unit, so convert kilovolts to volts, or volts to kilovolts, before substituting.
Turns ratios are often written in the form primary : secondary, so a ratio of $1 : 20$ means the secondary coil has $20$ times as many turns as the primary coil.
The equation assumes an ideal transformer, which is the assumption made in every calculation of this kind.

Rearranging the equation

The equation can be written the other way up, as $VsVp=NsNp\dfrac{V_s}{V_p}=\dfrac{N_s}{N_p}$ , which is often easier when the secondary is the unknown.
Whichever way up you write it, both primary quantities must be on the top or both must be on the bottom.
To find the secondary potential difference, use $Vs=Vp×NsNpV_s=V_p\times\dfrac{N_s}{N_p}$ .
To find the primary potential difference, use $Vp=Vs×NpNsV_p=V_s\times\dfrac{N_p}{N_s}$ .
To find the number of turns on the secondary coil, use $Ns=Np×VsVpN_s=N_p\times\dfrac{V_s}{V_p}$ .
To find the number of turns on the primary coil, use $Np=Ns×VpVsN_p=N_s\times\dfrac{V_p}{V_s}$ .
A quick check catches most mistakes: if the secondary coil has more turns than the primary coil, the secondary potential difference must be the larger of the two.

Example

Finding the output potential difference

A transformer has $500$ turns on its primary coil and $25$ turns on its secondary coil.
The primary coil is connected to the $V230\ \text{V}$ mains.
Write the equation with the secondary quantities on top:

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

Rearrange and substitute:

V_s=230\times\frac{25}{500}

V_s=11.5\ \text{V}

The secondary coil has fewer turns, so a smaller output is expected, which matches the answer.

Example

Finding the number of turns

A step-up transformer changes $V12\ \text{V}$ into $kV3.0\ \text{kV}$ , and its primary coil has $40$ turns.
Convert the output into volts: $V3.0\ \text{kV}=3000\ \text{V}$ .
Rearrange to make $N_s$ the subject and substitute:

N_s=N_p\times\frac{V_s}{V_p}=40\times\frac{3000}{12}

N_s=10\,000\ \text{turns}

Example

Using a turns ratio

A transformer has a turns ratio of $20 : 1$ , primary to secondary, and gives an output of $V6.0\ \text{V}$ .
The primary coil has $20$ times as many turns as the secondary coil, so its potential difference is $20$ times larger:

\frac{V_p}{V_s}=\frac{N_p}{N_s}=\frac{20}{1}

V_p=20\times6.0=120\ \text{V}

Common Mistake

Do not write the equation with $V_p$ on top on one side and $N_s$ on top on the other, because the primary quantities must be on the same level of both fractions.
Do not mix kilovolts and volts in the same calculation, because the answer will be wrong by a factor of $1000$ .

Exam technique

Transformer calculations

This equation is one you select and apply rather than recall, so the marks come from rearranging it correctly and putting each value in the right place.
Label your values $V_p$ , $V_s$ , $N_p$ and $N_s$ before substituting, because swapping the primary and secondary values is the most common way to lose the marks.
Give a number of turns as a whole number with no unit, and give a potential difference with the unit $V\text{V}$ .
Check the size of your answer against the type of transformer before moving on.

Self review

Write down the equation that links the potential differences and the numbers of turns in a transformer.
A transformer has $100$ turns on its primary coil and $2000$ turns on its secondary coil, and the input is $V12\ \text{V}$ ; calculate the output potential difference.
Rearrange the equation to make $N_s$ the subject.
Is a transformer with a turns ratio of $1 : 5$ , primary to secondary, a step-up or a step-down transformer?
Why must both potential differences be in the same unit before you substitute them?

4.2.9 Potential differences and turns in transformers

The transformer equation

Rearranging the equation

Finding the output potential difference

Finding the number of turns

Using a turns ratio

Transformer calculations

Review 4.2.9 Potential differences and turns in transformers by teaching Genie

4.2.9 Potential differences and turns in transformers

The transformer equation

Rearranging the equation

Finding the output potential difference

Finding the number of turns

Using a turns ratio

Transformer calculations

Review 4.2.9 Potential differences and turns in transformers by teaching Genie

4.2.9 Potential differences and turns in transformers

The transformer equation

Rearranging the equation

Finding the output potential difference

Finding the number of turns

Using a turns ratio

Transformer calculations

Review 4.2.9 Potential differences and turns in transformers by teaching Genie

4.2.9 Potential differences and turns in transformers Revision Guide

4.2.9 Potential differences and turns in transformers

The transformer equation

Rearranging the equation

Finding the output potential difference

Finding the number of turns

Using a turns ratio

Transformer calculations

Review 4.2.9 Potential differences and turns in transformers by teaching Genie

4.2.9 Potential differences and turns in transformers Revision Guide