Transformer Turns Ratio

In one sentence: A transformer changes an AC voltage up or down by the same ratio as the number of wire loops on its two coils.

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

Read it out loud: “V sub s divided by V sub p equals N sub s divided by N sub p. The secondary voltage, V sub s, divided by the primary voltage, V sub p, equals the number of secondary turns, N sub s, divided by the number of primary turns, N sub p.”

Before you start

The ideas and signs this page uses, explained from zero. For powers of ten, Greek letters and other notation in general, see Math basics.

Coils and turns, $N$: A coil is a length of wire wound into many loops. Each loop is called a turn. The letter $N$ counts turns, so it is a plain count with no unit. A coil with 500 turns has 500 loops of wire wound one after another around the same core. More turns means more loops for the magnetic field to pass through.
Primary, secondary and subscripts $p$ and $s$: The primary coil is the input coil, connected to the supply. The secondary coil is the output coil, which feeds the load. The small letters after a symbol are subscripts, labels that say which coil a quantity belongs to. $V_p$ is the primary voltage and $N_s$ is the number of secondary turns. A subscript is a label, not a multiplication.
Magnetic field and induction: A magnetic field is the invisible influence around a magnet or a current, which pushes on other magnets and currents. A current that keeps changing makes a field that keeps changing. A changing field passing through a nearby coil pushes a voltage into that coil. This push is called induction, and it is how a transformer moves energy from one coil to the other.
Alternating current (AC) and direct current (DC): Direct current (DC) flows steadily in one direction, like the current from a battery. Alternating current (AC) reverses direction again and again, like the wall supply, which repeats 60 times each second in US homes. A transformer needs a changing current, so AC works. A steady DC current does not.
Iron core and the ideal transformer: The two coils sit on a shared core made of thin sheets of iron. Iron is a magnetic material, so it guides the magnetic field strongly along a path that passes through both coils. The ideal transformer is a simple model that loses no power at all. Real transformers lose a little, mostly as heat, but the ideal model is close to real behavior for well-built transformers.
Ratios, like $N_s / N_p$: A ratio compares two amounts by division. $N_s / N_p = 50 / 500 = 0.1$ means the secondary has one tenth as many turns as the primary. A ratio has no unit when both amounts have the same unit, or when both are simple counts like turns. The ratio tells you how one quantity scales when the other changes.

What each symbol means

Symbol	Meaning
$V_s$	Secondary voltage, the voltage across the output coil, measured in volts (V). One volt is one joule of energy for each coulomb of charge. In the example the secondary voltage is 12 V, a low voltage that is much safer to touch than mains voltage.
$V_p$	Primary voltage, the voltage across the input coil, measured in volts (V). In the example the input is the 120 V wall supply found in many US homes, so the primary voltage is 120 V.
$N_s$	Secondary turns, the number of wire loops on the output coil. Turns are counted, not measured, so the symbol has no unit. The example’s output coil has 50 turns, a count you could make by hand on a short coil.
$N_p$	Primary turns, the number of wire loops on the input coil. Turns are counted, so there is no unit. The example’s input coil has 500 turns, which is ten times as many loops as the output coil has.
$I_s$	Secondary current, the current the output coil delivers to the load, measured in amperes (A). One ampere is one coulomb of charge passing a point each second. The example’s output carries 5 A, which sets how thick the output wire must be.
$I_p$	Primary current, the current drawn from the supply by the input coil, measured in amperes (A). The example’s input draws only 0.5 A, which is a small current compared with the 5 A on the output side.

The formula piece by piece

$N_p$: Start with the number of loops on the primary coil. It is a count with no unit. It is the number the turns ratio divides by, so it sets the scale of the output relative to the input.
$N_s$: The number of loops on the secondary coil. Every loop of the output coil sees the same share of the changing field as every loop of the input coil, so the output voltage depends on how many loops it has.
$\frac{N_s}{N_p}$: The turns ratio: how many output loops there are for each input loop. It is a pure number with no unit. A ratio of 0.1 means the output coil has one tenth the turns of the input coil.
$\frac{V_s}{V_p}$: The voltage ratio: the output voltage divided by the input voltage. The rule says this equals the turns ratio, so a turns ratio of 0.1 gives a voltage ratio of 0.1 too, and the output voltage is one tenth of the input.
$\frac{V_s}{V_p} = \frac{N_s}{N_p}$: The whole formula in one sentence: the secondary voltage divided by the primary voltage equals the secondary turns divided by the primary turns.
$V_s = V_p\,\frac{N_s}{N_p}$: The same rule solved for the output voltage. Multiply the input voltage by the turns ratio. A step-down transformer has a turns ratio less than one, so its output voltage is smaller than its input voltage.
$\frac{I_s}{I_p} = \frac{N_p}{N_s}$: The current rule for an ideal transformer. The currents are in the opposite ratio to the turns, so you flip the turns ratio upside down to get the current ratio. More output turns means less output current, and the reverse.

What the formula is doing, and why

A transformer has two coils of wire wound on a shared iron core. The primary coil is connected to the input, and the secondary coil feeds the output. The formula links the voltage on each coil to the number of turns on it: $V_s / V_p = N_s / N_p$ . Voltage is measured in volts (V), and the turns are counted loops, so the ratio has no unit.

Why the voltage follows the turns. The changing current in the primary makes a changing magnetic field in the core, and that field passes through every turn of both coils. Each turn sees the same change in field, so each turn picks up the same small voltage. The voltage on a whole coil is that small voltage times the number of turns. A secondary with ten times as many turns gives ten times the voltage. This is Faraday’s law of induction at work.

Why the current goes the other way. For an ideal transformer, the power going in equals the power coming out, so $V_p I_p = V_s I_s$ . Rearranging gives $I_s / I_p = V_p / V_s = N_p / N_s$ . A step-up transformer raises the voltage and lowers the current by the same factor. The power stays the same, because the transformer creates no energy of its own.

Why it needs AC. The secondary voltage exists only while the magnetic field through it is changing. AC keeps the field changing all the time, so a voltage is induced all the time. A steady DC current makes a steady field. A steady field induces nothing, so the secondary voltage drops to zero, and the primary coil acts as a plain resistor.

Step-up, step-down and real transformers. When $N_s$ is bigger than $N_p$ , the transformer steps the voltage up. When $N_s$ is smaller, it steps the voltage down. Real transformers lose a little power, mostly as heat in the wire and the iron core, so the output power is slightly less than the input power. The ideal rules are a close guide for well-built transformers.

An everyday picture

Think of the gears on a bicycle. A large front gear turning a small back gear makes the back wheel spin fast, but the rider has to push harder on the pedals. A small front gear turning a large back gear makes the wheel turn slowly, with less effort on the pedals. A transformer trades voltage against current in the same way that the gears trade speed against turning force. The power on the two sides stays about the same.

Worked example: Stepping 120 V down to 12 V

The primary coil of a transformer has $N_p = 500$ turns and is connected to a 120 V AC supply, the wall outlet found in many US homes. The primary draws $I_p = 0.5$ A. The secondary coil has $N_s = 50$ turns. Find the secondary voltage and current, then check that the power is the same on both sides.

Find the turns ratio. Divide the secondary turns by the primary turns: $N_s / N_p = 50 / 500 = 0.1$ . The output coil has one tenth as many loops as the input coil, so this is a step-down transformer, and the ratio tells us how the voltage will scale.
Find the secondary voltage. Multiply the primary voltage by the turns ratio: $V_s = V_p \times \frac{N_s}{N_p} = 120\ \text{V} \times 0.1 = 12\ \text{V}$ . The output voltage is one tenth of the input, because the voltage follows the number of turns.
Find the secondary current. The current goes the other way, so we use the flipped ratio, $N_p / N_s = 500 / 50 = 10$ . Then $I_s = I_p \times \frac{N_p}{N_s} = 0.5\ \text{A} \times 10 = 5\ \text{A}$ . The current is ten times bigger on the output side, because the power has to stay the same.
Check the power on the input side. Power is volts times amperes, and volts times amperes gives watts, which is energy used each second. $P_{\text{in}} = V_p I_p = 120\ \text{V} \times 0.5\ \text{A} = 60\ \text{W}$ . The supply sends 60 joules of energy every second into the primary coil.
Check the power on the output side. Multiply the output voltage and current the same way: $P_{\text{out}} = V_s I_s = 12\ \text{V} \times 5\ \text{A} = 60\ \text{W}$ . The two sides match, which is what an ideal transformer must do, because it creates no energy.
Compare the size of the change. The voltage fell by a factor of 10, since the turns fell by a factor of 10. The current rose by the same factor of 10, so the product of voltage and current stayed at 60 W. Any change that trades voltage this way must also trade current.

Answer: The secondary delivers 12 V and 5 A. The voltage went down by a factor of 10 and the current went up by the same factor. Both sides carry 60 W, which is what an ideal transformer should do.

The numbers in this example are recomputed by script every time the site is built.

Common mistakes

Using the voltage rule for the current. The voltage follows $N_s / N_p$ directly, but the current follows the flipped ratio $N_p / N_s$ . Using the wrong one gives a current that is off by the square of the turns ratio. In the example, that gives 0.05 A instead of 5 A, which is 100 times too small.
Expecting a transformer to create energy. The power out can never be bigger than the power in. A step-up transformer raises the voltage only by lowering the current, so the power stays the same.
Using a transformer with DC. A steady current makes a steady field, which induces no voltage in the secondary, so the secondary output is zero.
Mixing up the coils. The primary is the input coil, the one connected to the supply. Name the coils by how they are connected, not by how big they are.

Where you will meet it

Power grids, where transformers step the voltage up for long-distance transmission and back down for homes.
Many phone and laptop chargers, which use a small transformer to step the mains voltage down before it is turned into a low voltage for the device.
Doorbells and other low-voltage devices, which use a small step-down transformer on the mains.
Welding equipment, where a transformer supplies the high current that a weld needs at a low voltage.

Questions people ask

Why does a transformer need AC?

The secondary coil gets a voltage only when the magnetic field through it changes. AC keeps the field changing, while DC gives a steady field that induces nothing.

If the voltage goes up, does the current go up too?

No. For an ideal transformer the current goes the other way. A step-up transformer raises the voltage and lowers the current, so the power stays the same.

Is a transformer the same as a generator?

No. A generator turns mechanical energy into electrical energy. A transformer takes AC electrical energy that already exists and changes its voltage and current.

What does ideal mean here?

It means the transformer loses no power, so the power going in equals the power coming out. Real transformers lose a little energy as heat in the wire and the iron core, so their output power is slightly smaller than their input power.

Faraday's Law of Induction: A changing magnetic field through a loop of wire pushes a voltage into the wire, and that is how generators make electricity.
Electric Power: Electric power tells you how fast a device uses energy. You get it by multiplying the voltage, the push, by the current, the flow, and the answer is measured in watts.
RMS Voltage: RMS voltage is the steady voltage that heats a resistor as much as an alternating voltage does, and for a sine wave you get it by dividing the peak voltage by the square root of 2.
Ohm's Law: Ohm’s law says the push on a wire, called voltage, equals the flow through it, called current, times how hard the wire resists that flow, called resistance.