The turns-ratio equation

Definition

Turns ratio

The turns ratio of a transformer is the number of turns on its primary coil divided by the number of turns on its secondary coil.

Definition

Step-up transformer

A step-up transformer is a transformer with more turns on its secondary coil than on its primary coil, so the output potential difference is larger than the input potential difference.

Definition

Step-down transformer

A step-down transformer is a transformer with fewer turns on its secondary coil than on its primary coil, so the output potential difference is smaller than the input potential difference.

The potential differences and the numbers of turns of a transformer are linked by $VpVs=NpNs\dfrac{V_{\mathrm{p}}}{V_{\mathrm{s}}}=\dfrac{N_{\mathrm{p}}}{N_{\mathrm{s}}}$ .
In that equation $VpV_{\mathrm{p}}$ is the primary potential difference in volts, $VsV_{\mathrm{s}}$ is the secondary potential difference in volts, $NpN_{\mathrm{p}}$ is the number of turns on the primary coil and $NsN_{\mathrm{s}}$ is the number of turns on the secondary coil.
The ratio of the potential differences matches the ratio of the turns because the same changing magnetic field passes through every turn of both coils, so each turn has the same potential difference induced across it.
A coil with more turns therefore has more of those equal contributions added in series, which is why more turns means a larger potential difference.
A step-up transformer has $Ns>NpN_{\mathrm{s}}>N_{\mathrm{p}}$ , so $Vs>VpV_{\mathrm{s}}>V_{\mathrm{p}}$ and the potential difference is raised.
A step-down transformer has $Ns<NpN_{\mathrm{s}}<N_{\mathrm{p}}$ , so $Vs<VpV_{\mathrm{s}}<V_{\mathrm{p}}$ and the potential difference is lowered.
Equal numbers of turns give $Vs=VpV_{\mathrm{s}}=V_{\mathrm{p}}$ , which is useful when two circuits must be kept electrically separate without changing the potential difference.
The turns ratio $NpNs\dfrac{N_{\mathrm{p}}}{N_{\mathrm{s}}}$ is a pure number with no unit, so a transformer described as $10 : 1$ steps the potential difference down by a factor of ten.

Using and rearranging the equation

The rearranged forms are $Vs=VpNsNpV_{\mathrm{s}}=\dfrac{V_{\mathrm{p}}N_{\mathrm{s}}}{N_{\mathrm{p}}}$ , $Vp=VsNpNsV_{\mathrm{p}}=\dfrac{V_{\mathrm{s}}N_{\mathrm{p}}}{N_{\mathrm{s}}}$ , $Ns=NpVsVpN_{\mathrm{s}}=\dfrac{N_{\mathrm{p}}V_{\mathrm{s}}}{V_{\mathrm{p}}}$ and $Np=NsVpVsN_{\mathrm{p}}=\dfrac{N_{\mathrm{s}}V_{\mathrm{p}}}{V_{\mathrm{s}}}$ .
List the four quantities and label each one as primary or secondary before substituting, because swapping a pair turns a step-up answer into a step-down answer.
Convert every potential difference into the same unit first, so $kV132\ \text{kV}$ becomes $V132\,000\ \text{V}$ and $mV25\ \text{mV}$ becomes $V0.025\ \text{V}$ .
Numbers of turns are counted, so they never carry a unit and must never be rounded to a fraction of a turn.
A useful shortcut is to work out the ratio first: if the secondary has $120\dfrac{1}{20}$ of the turns, the secondary potential difference is $120\dfrac{1}{20}$ of the primary potential difference.
Check the finished answer against the turns: the coil with more turns must always be the one with the larger potential difference.

Example

Finding an output potential difference

A transformer has $1150$ turns on its primary coil and $50$ turns on its secondary coil, and its primary is connected to a $V230\ \text{V}$ alternating supply.
Rearranging the turns-ratio equation gives $Vs=VpNsNpV_{\mathrm{s}}=\dfrac{V_{\mathrm{p}}N_{\mathrm{s}}}{N_{\mathrm{p}}}$ .
Substituting gives $Vs=230×501150V_{\mathrm{s}}=\dfrac{230\times50}{1150}$ .
The secondary potential difference is $VV_{\mathrm{s}}=10\ \text{V}$ .
The secondary has fewer turns than the primary, so this is a step-down transformer and the smaller output is the expected result.

Example

Finding a number of turns

A power station transformer raises an alternating potential difference from $kV25\ \text{kV}$ to $kV400\ \text{kV}$ , and its primary coil has $500$ turns.
Both potential differences are converted to volts, giving $VV_{\mathrm{p}}=25\,000\ \text{V}$ and $VV_{\mathrm{s}}=400\,000\ \text{V}$ .
Rearranging the turns-ratio equation gives $Ns=NpVsVpN_{\mathrm{s}}=\dfrac{N_{\mathrm{p}}V_{\mathrm{s}}}{V_{\mathrm{p}}}$ .
Substituting gives $000N_{\mathrm{s}}=\dfrac{500\times400\,000}{25\,000}$ .
The secondary coil needs $Ns=8000N_{\mathrm{s}}=8000$ turns.
The potential difference has been raised by a factor of $16$ , and the number of turns has risen by the same factor of $16$ , which confirms the arithmetic.

Practical

Measuring the turns ratio of a demountable transformer

Aim: to test whether the ratio of the potential differences across two coils equals the ratio of their numbers of turns.
Apparatus: demountable transformer kit with a laminated C-core and clamping yoke, a set of coils of known turns such as $300$ , $600$ and $1200$ turns, low-voltage alternating supply, two multimeters or a data logger with two voltage sensors, and connecting leads.
Variables: the number of turns on the secondary coil is the independent variable, the secondary potential difference is the dependent variable, and the primary coil, the primary potential difference, the core, the yoke tightness and the supply frequency are controlled.
Method, setting up:
- Slide the $600$ turn coil onto one arm of the C-core to act as the primary, and a coil of known turns onto the other arm to act as the secondary.
- Clamp the yoke firmly across the open ends of the core so the magnetic circuit is closed, and use the same tightness for every run.
- Connect the primary to the low-voltage alternating supply set to about $V4\ \text{V}$ , and connect a meter across each coil so both potential differences are read at the same moment.
- Record the number of turns marked on each coil before switching on.
Method, taking the readings:
- Switch on, read the primary and secondary potential differences together, then switch off promptly so the coils do not warm up.
- Repeat the reading three times, switching off between runs, and calculate a mean for each potential difference.
- Swap the secondary for a coil with a different number of turns without moving the primary or altering the supply, and repeat the readings.
- Collect results for at least four different secondary coils, including one with fewer turns than the primary and one with more.
- Reverse the roles of the two coils for one pair, so the step-down arrangement becomes a step-up arrangement, and check that the ratio inverts.
Measurements and processing: tabulate $NpN_{\mathrm{p}}$ , $NsN_{\mathrm{s}}$ , $VpV_{\mathrm{p}}$ and $VsV_{\mathrm{s}}$ with the mean values, then calculate $NpNs\dfrac{N_{\mathrm{p}}}{N_{\mathrm{s}}}$ and $VpVs\dfrac{V_{\mathrm{p}}}{V_{\mathrm{s}}}$ for each pair and compare the two columns.
Graph: plot $VsV_{\mathrm{s}}$ against $NsN_{\mathrm{s}}$ for a fixed primary, which should give a straight line through the origin.
Expected pattern: the two ratios agree closely, and the measured secondary potential difference is usually a little lower than the predicted value because the transformer is not perfectly efficient.
Sources of uncertainty: an air gap left by a loose yoke reduces the field reaching the secondary, the coils warm up and their resistance rises during a run, meters have a limited resolution on low readings, and the supply potential difference may drift.
Improvements: clamp the yoke to a fixed tightness every time, monitor the primary potential difference throughout so any drift is spotted, keep each run short, and use a data logger so both readings are captured at the same instant.
Safety: use only the low-voltage alternating supply, never connect a demountable transformer to the mains, switch off before rearranging the coils or the core, and allow warm coils to cool before handling them.

Identifying the type of transformer

Count the turns on each coil in a diagram: the coil drawn with more loops is the one with the larger potential difference.
Compare the two potential differences instead when the turns are not given, since a larger output means a step-up transformer.
Use the ratio as a factor: a transformer with $NpNs=116\dfrac{N_{\mathrm{p}}}{N_{\mathrm{s}}}=\dfrac{1}{16}$ multiplies the potential difference by $16$ .
Read the labels carefully, because a transformer is only a step-up transformer with respect to the coil chosen as the primary.

Exam technique

Setting out a turns-ratio calculation

Write $VpVs=NpNs\dfrac{V_{\mathrm{p}}}{V_{\mathrm{s}}}=\dfrac{N_{\mathrm{p}}}{N_{\mathrm{s}}}$ before substituting, so the method earns credit even if the arithmetic slips.
List the given values with their subscripts, then rearrange for the unknown as a separate line.
Convert kilovolts and millivolts into volts in a visible step rather than doing it in your head.
Give the answer with a unit, in volts for a potential difference and with no unit for a number of turns.
Finish by stating whether the transformer is step-up or step-down, since many questions award a mark for that comparison.

Common Mistake

Do not swap the primary and secondary values, because the equation is symmetrical and the arithmetic gives no warning.
Do not leave a potential difference in kilovolts alongside one in volts when substituting.
Do not give a number of turns as a decimal, since turns of wire are counted in whole numbers.
Do not assume that a step-up transformer increases the power available, because raising the potential difference lowers the current.
Do not use the turns-ratio equation on a direct supply, since no potential difference is induced when the field is steady.

Self review

State the turns-ratio equation and identify each of its four quantities.
Explain why a coil with more turns has a larger potential difference across it.
Calculate the secondary potential difference of a transformer with $2000$ primary turns, $400$ secondary turns and a $V230\ \text{V}$ primary supply.
Calculate the number of primary turns needed to step $kV12\ \text{kV}$ up to $kV132\ \text{kV}$ using $150$ secondary turns.
Describe how to test the turns-ratio equation using a demountable transformer.

The turns-ratio equation

Definition

Turns ratio

The turns ratio of a transformer is the number of turns on its primary coil divided by the number of turns on its secondary coil.

Definition

Step-up transformer

A step-up transformer is a transformer with more turns on its secondary coil than on its primary coil, so the output potential difference is larger than the input potential difference.

Definition

Step-down transformer

A step-down transformer is a transformer with fewer turns on its secondary coil than on its primary coil, so the output potential difference is smaller than the input potential difference.

The potential differences and the numbers of turns of a transformer are linked by $VpVs=NpNs\dfrac{V_{\mathrm{p}}}{V_{\mathrm{s}}}=\dfrac{N_{\mathrm{p}}}{N_{\mathrm{s}}}$ .
In that equation $VpV_{\mathrm{p}}$ is the primary potential difference in volts, $VsV_{\mathrm{s}}$ is the secondary potential difference in volts, $NpN_{\mathrm{p}}$ is the number of turns on the primary coil and $NsN_{\mathrm{s}}$ is the number of turns on the secondary coil.
The ratio of the potential differences matches the ratio of the turns because the same changing magnetic field passes through every turn of both coils, so each turn has the same potential difference induced across it.
A coil with more turns therefore has more of those equal contributions added in series, which is why more turns means a larger potential difference.
A step-up transformer has $Ns>NpN_{\mathrm{s}}>N_{\mathrm{p}}$ , so $Vs>VpV_{\mathrm{s}}>V_{\mathrm{p}}$ and the potential difference is raised.
A step-down transformer has $Ns<NpN_{\mathrm{s}}<N_{\mathrm{p}}$ , so $Vs<VpV_{\mathrm{s}}<V_{\mathrm{p}}$ and the potential difference is lowered.
Equal numbers of turns give $Vs=VpV_{\mathrm{s}}=V_{\mathrm{p}}$ , which is useful when two circuits must be kept electrically separate without changing the potential difference.
The turns ratio $NpNs\dfrac{N_{\mathrm{p}}}{N_{\mathrm{s}}}$ is a pure number with no unit, so a transformer described as $10 : 1$ steps the potential difference down by a factor of ten.

Using and rearranging the equation

The rearranged forms are $Vs=VpNsNpV_{\mathrm{s}}=\dfrac{V_{\mathrm{p}}N_{\mathrm{s}}}{N_{\mathrm{p}}}$ , $Vp=VsNpNsV_{\mathrm{p}}=\dfrac{V_{\mathrm{s}}N_{\mathrm{p}}}{N_{\mathrm{s}}}$ , $Ns=NpVsVpN_{\mathrm{s}}=\dfrac{N_{\mathrm{p}}V_{\mathrm{s}}}{V_{\mathrm{p}}}$ and $Np=NsVpVsN_{\mathrm{p}}=\dfrac{N_{\mathrm{s}}V_{\mathrm{p}}}{V_{\mathrm{s}}}$ .
List the four quantities and label each one as primary or secondary before substituting, because swapping a pair turns a step-up answer into a step-down answer.
Convert every potential difference into the same unit first, so $kV132\ \text{kV}$ becomes $V132\,000\ \text{V}$ and $mV25\ \text{mV}$ becomes $V0.025\ \text{V}$ .
Numbers of turns are counted, so they never carry a unit and must never be rounded to a fraction of a turn.
A useful shortcut is to work out the ratio first: if the secondary has $120\dfrac{1}{20}$ of the turns, the secondary potential difference is $120\dfrac{1}{20}$ of the primary potential difference.
Check the finished answer against the turns: the coil with more turns must always be the one with the larger potential difference.

Example

Finding an output potential difference

A transformer has $1150$ turns on its primary coil and $50$ turns on its secondary coil, and its primary is connected to a $V230\ \text{V}$ alternating supply.
Rearranging the turns-ratio equation gives $Vs=VpNsNpV_{\mathrm{s}}=\dfrac{V_{\mathrm{p}}N_{\mathrm{s}}}{N_{\mathrm{p}}}$ .
Substituting gives $Vs=230×501150V_{\mathrm{s}}=\dfrac{230\times50}{1150}$ .
The secondary potential difference is $VV_{\mathrm{s}}=10\ \text{V}$ .
The secondary has fewer turns than the primary, so this is a step-down transformer and the smaller output is the expected result.

Example

Finding a number of turns

A power station transformer raises an alternating potential difference from $kV25\ \text{kV}$ to $kV400\ \text{kV}$ , and its primary coil has $500$ turns.
Both potential differences are converted to volts, giving $VV_{\mathrm{p}}=25\,000\ \text{V}$ and $VV_{\mathrm{s}}=400\,000\ \text{V}$ .
Rearranging the turns-ratio equation gives $Ns=NpVsVpN_{\mathrm{s}}=\dfrac{N_{\mathrm{p}}V_{\mathrm{s}}}{V_{\mathrm{p}}}$ .
Substituting gives $000N_{\mathrm{s}}=\dfrac{500\times400\,000}{25\,000}$ .
The secondary coil needs $Ns=8000N_{\mathrm{s}}=8000$ turns.
The potential difference has been raised by a factor of $16$ , and the number of turns has risen by the same factor of $16$ , which confirms the arithmetic.

Practical

Measuring the turns ratio of a demountable transformer

Aim: to test whether the ratio of the potential differences across two coils equals the ratio of their numbers of turns.
Apparatus: demountable transformer kit with a laminated C-core and clamping yoke, a set of coils of known turns such as $300$ , $600$ and $1200$ turns, low-voltage alternating supply, two multimeters or a data logger with two voltage sensors, and connecting leads.
Variables: the number of turns on the secondary coil is the independent variable, the secondary potential difference is the dependent variable, and the primary coil, the primary potential difference, the core, the yoke tightness and the supply frequency are controlled.
Method, setting up:
- Slide the $600$ turn coil onto one arm of the C-core to act as the primary, and a coil of known turns onto the other arm to act as the secondary.
- Clamp the yoke firmly across the open ends of the core so the magnetic circuit is closed, and use the same tightness for every run.
- Connect the primary to the low-voltage alternating supply set to about $V4\ \text{V}$ , and connect a meter across each coil so both potential differences are read at the same moment.
- Record the number of turns marked on each coil before switching on.
Method, taking the readings:
- Switch on, read the primary and secondary potential differences together, then switch off promptly so the coils do not warm up.
- Repeat the reading three times, switching off between runs, and calculate a mean for each potential difference.
- Swap the secondary for a coil with a different number of turns without moving the primary or altering the supply, and repeat the readings.
- Collect results for at least four different secondary coils, including one with fewer turns than the primary and one with more.
- Reverse the roles of the two coils for one pair, so the step-down arrangement becomes a step-up arrangement, and check that the ratio inverts.
Measurements and processing: tabulate $NpN_{\mathrm{p}}$ , $NsN_{\mathrm{s}}$ , $VpV_{\mathrm{p}}$ and $VsV_{\mathrm{s}}$ with the mean values, then calculate $NpNs\dfrac{N_{\mathrm{p}}}{N_{\mathrm{s}}}$ and $VpVs\dfrac{V_{\mathrm{p}}}{V_{\mathrm{s}}}$ for each pair and compare the two columns.
Graph: plot $VsV_{\mathrm{s}}$ against $NsN_{\mathrm{s}}$ for a fixed primary, which should give a straight line through the origin.
Expected pattern: the two ratios agree closely, and the measured secondary potential difference is usually a little lower than the predicted value because the transformer is not perfectly efficient.
Sources of uncertainty: an air gap left by a loose yoke reduces the field reaching the secondary, the coils warm up and their resistance rises during a run, meters have a limited resolution on low readings, and the supply potential difference may drift.
Improvements: clamp the yoke to a fixed tightness every time, monitor the primary potential difference throughout so any drift is spotted, keep each run short, and use a data logger so both readings are captured at the same instant.
Safety: use only the low-voltage alternating supply, never connect a demountable transformer to the mains, switch off before rearranging the coils or the core, and allow warm coils to cool before handling them.

Identifying the type of transformer

Count the turns on each coil in a diagram: the coil drawn with more loops is the one with the larger potential difference.
Compare the two potential differences instead when the turns are not given, since a larger output means a step-up transformer.
Use the ratio as a factor: a transformer with $NpNs=116\dfrac{N_{\mathrm{p}}}{N_{\mathrm{s}}}=\dfrac{1}{16}$ multiplies the potential difference by $16$ .
Read the labels carefully, because a transformer is only a step-up transformer with respect to the coil chosen as the primary.

Exam technique

Setting out a turns-ratio calculation

Write $VpVs=NpNs\dfrac{V_{\mathrm{p}}}{V_{\mathrm{s}}}=\dfrac{N_{\mathrm{p}}}{N_{\mathrm{s}}}$ before substituting, so the method earns credit even if the arithmetic slips.
List the given values with their subscripts, then rearrange for the unknown as a separate line.
Convert kilovolts and millivolts into volts in a visible step rather than doing it in your head.
Give the answer with a unit, in volts for a potential difference and with no unit for a number of turns.
Finish by stating whether the transformer is step-up or step-down, since many questions award a mark for that comparison.

Common Mistake

Do not swap the primary and secondary values, because the equation is symmetrical and the arithmetic gives no warning.
Do not leave a potential difference in kilovolts alongside one in volts when substituting.
Do not give a number of turns as a decimal, since turns of wire are counted in whole numbers.
Do not assume that a step-up transformer increases the power available, because raising the potential difference lowers the current.
Do not use the turns-ratio equation on a direct supply, since no potential difference is induced when the field is steady.

Self review

State the turns-ratio equation and identify each of its four quantities.
Explain why a coil with more turns has a larger potential difference across it.
Calculate the secondary potential difference of a transformer with $2000$ primary turns, $400$ secondary turns and a $V230\ \text{V}$ primary supply.
Calculate the number of primary turns needed to step $kV12\ \text{kV}$ up to $kV132\ \text{kV}$ using $150$ secondary turns.
Describe how to test the turns-ratio equation using a demountable transformer.

13.2.2 The transformer turns-ratio equation

The turns-ratio equation

Using and rearranging the equation

Finding an output potential difference

Finding a number of turns

Measuring the turns ratio of a demountable transformer

Identifying the type of transformer

Setting out a turns-ratio calculation

Review 13.2.2 The transformer turns-ratio equation by teaching Genie

13.2.2 The transformer turns-ratio equation

The turns-ratio equation

Using and rearranging the equation

Finding an output potential difference

Finding a number of turns

Measuring the turns ratio of a demountable transformer

Identifying the type of transformer

Setting out a turns-ratio calculation

Review 13.2.2 The transformer turns-ratio equation by teaching Genie

13.2.2 The transformer turns-ratio equation

The turns-ratio equation

Using and rearranging the equation

Finding an output potential difference

Finding a number of turns

Measuring the turns ratio of a demountable transformer

Identifying the type of transformer

Setting out a turns-ratio calculation

Review 13.2.2 The transformer turns-ratio equation by teaching Genie

13.2.2 The transformer turns-ratio equation Revision Guide

13.2.2 The transformer turns-ratio equation

The turns-ratio equation

Using and rearranging the equation

Finding an output potential difference

Finding a number of turns

Measuring the turns ratio of a demountable transformer

Identifying the type of transformer

Setting out a turns-ratio calculation

Review 13.2.2 The transformer turns-ratio equation by teaching Genie

13.2.2 The transformer turns-ratio equation Revision Guide