Transformer Turns Ratio Equation, Derivation & Rearrangements

Q: What are common mistakes with the Transformer Turns Ratio formula?

Swapping primary and secondary. Forgetting ratio direction.

Q: What is a real-world example of the Transformer Turns Ratio formula?

Converting mains voltage to a lower device voltage.

Q: What are some study tips for the Transformer Turns Ratio formula?

Ensure that Vp and Vs are in the same units, typically Volts. A primary-to-secondary ratio greater than 1 indicates a step-down transformer. The equation only applies to AC; transformers do not work with steady state DC. Treat N as a dimensionless count of wire loops.

Core idea

Overview

The transformer turns ratio equation defines the proportional relationship between the voltages across the primary and secondary coils and the number of wire loops in each winding. It describes how magnetic induction allows for the modification of alternating current voltage levels while maintaining the frequency of the electrical signal.

When to use: Apply this formula when analyzing ideal transformers where energy loss through heat or flux leakage is assumed to be zero. It is used specifically for alternating current (AC) systems to determine how many turns of wire are needed to achieve a target output voltage.

Why it matters: This principle is fundamental to the global power grid, enabling high-voltage transmission over long distances to reduce energy loss before stepping the voltage down for safe consumer use. It allows electronic devices to operate at low, safe voltages even when plugged into high-voltage wall outlets.

Symbols

Variables

V = Primary Voltage, V = Secondary Voltage, Np = Primary Turns, Ns = Secondary Turns, Voltage Ratio = Voltage Ratio

V

Primary Voltage

V a r iab l e

V

Secondary Voltage

V a r iab l e

N p

Primary Turns

V a r iab l e

N s

Secondary Turns

V a r iab l e

V o l t a g e R a t i o

Voltage Ratio

V a r iab l e

Walkthrough

Derivation

Understanding the Transformer Equation

Relates turns ratio to voltage ratio for an ideal transformer.

Transformer is ideal (no losses).
All flux from the primary links the secondary (no flux leakage).

1

Apply Faraday's Law to Each Coil (Magnitude):

Both coils experience the same rate of change of flux linkage in an ideal transformer.

V_{p} = N_{p} \frac{ΔΦ}{Δ t} and V_{s} = N_{s} \frac{ΔΦ}{Δ t}

Note: Strictly, induced emf is $ε = - N d Φ/ d t$ ; the ratio uses magnitudes.

2

Take the Ratio:

Dividing cancels $ΔΦ/Δ t$ , leaving the turns ratio rule.

\frac{V _{s}}{V _{p}} = \frac{N _{s}}{N _{p}}

Result

\frac{V _{s}}{V _{p}} = \frac{N _{s}}{N _{p}}

Source: AQA A-Level Physics — Electromagnetic Induction

Free formulas

Rearrangements

Solve for $V$

Make Vp the subject

V = \frac{V N p}{N s}

Exact symbolic rearrangement generated deterministically for Vp.

Difficulty: 3/5

Solve for $V$

Make Vs the subject

V = \frac{V N s}{N p}

Exact symbolic rearrangement generated deterministically for Vs.

Difficulty: 3/5

Solve for $N p$

Make Np the subject

N p = \frac{V N s}{V}

Exact symbolic rearrangement generated deterministically for Np.

Difficulty: 3/5

Solve for $N s$

Make Ns the subject

N s = \frac{V N p}{V}

Exact symbolic rearrangement generated deterministically for Ns.

Difficulty: 3/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

The graph is a straight line passing through the origin because primary voltage is directly proportional to the voltage ratio, with the turns ratio acting as a constant slope. For a physics student, this linear relationship means that doubling the primary voltage will always result in a doubling of the voltage ratio. Large x-values represent high primary voltage inputs, while small x-values indicate low primary voltage inputs. The most important feature is that the constant slope represents the fixed turns ratio, s

Graph type: linear

Why it behaves this way

Intuition

Imagine a core material guiding a fluctuating magnetic field, generated by the primary coil's alternating current, which then cuts across the turns of the secondary coil, inducing a proportional voltage in each coil

Vp

The root-mean-square (RMS) voltage applied across the primary coil of the transformer.

This is the input alternating current (AC) voltage that drives the magnetic flux in the primary winding.

Vs

The root-mean-square (RMS) voltage induced across the secondary coil of the transformer.

This is the output AC voltage provided by the transformer to a connected load.

Np

The number of turns (loops) in the primary winding of the transformer.

Represents how many times the wire is coiled on the input side, influencing the strength of the magnetic field generated for a given current.

Ns

The number of turns (loops) in the secondary winding of the transformer.

Represents how many times the wire is coiled on the output side, determining how much voltage is induced from the changing magnetic flux.

Signs and relationships

Vp/Vs: This ratio quantifies the voltage transformation. If Vp > Vs (ratio > 1), it signifies a step-down transformer; if Vp < Vs (ratio < 1), it signifies a step-up transformer.
Np/Ns: This ratio directly dictates the voltage transformation. If Np > Ns, the primary has more turns, resulting in a step-down in voltage (Vs < Vp).

Free study cues

Insight

Canonical usage

This equation is used to relate the ratio of voltages across the primary and secondary coils to the ratio of the number of turns in those coils. Both sides of the equation are dimensionless ratios.

Common confusion

A common mistake is to use different units for Vp and Vs (e.g., kilovolts for Vp and volts for Vs) without converting them to a consistent unit before calculating the ratio, leading to incorrect results.

Dimension note

The transformer turns ratio equation expresses a relationship between two dimensionless quantities: the ratio of primary to secondary voltage (Vp/Vs) and the ratio of primary to secondary turns (Np/Ns).

Unit systems

$V p$ Volt (V) · Must be in the same unit as Vs for the ratio Vp/Vs to be dimensionless.

$V s$ Volt (V) · Must be in the same unit as Vp for the ratio Vp/Vs to be dimensionless.

$N p$ dimensionless · Represents a count of wire turns, which is dimensionless.

$N s$ dimensionless · Represents a count of wire turns, which is dimensionless.

One free problem

Practice Problem

A step-down transformer is used to reduce a household voltage of 120V to power a small device at 6V. If the primary coil has 400 turns, how many turns are in the secondary coil?

Primary Voltage120

Secondary Voltage6

Primary Turns400

Solve for: $N s$

Hint: The ratio of primary to secondary voltage must equal the ratio of primary to secondary turns.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Converting mains voltage to a lower device voltage.

Study smarter

Tips

Ensure that Vp and Vs are in the same units, typically Volts.
A primary-to-secondary ratio greater than 1 indicates a step-down transformer.
The equation only applies to AC; transformers do not work with steady state DC.
Treat N as a dimensionless count of wire loops.

Avoid these traps

Common Mistakes

Swapping primary and secondary.
Forgetting ratio direction.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

Relates turns ratio to voltage ratio for an ideal transformer.

Apply this formula when analyzing ideal transformers where energy loss through heat or flux leakage is assumed to be zero. It is used specifically for alternating current (AC) systems to determine how many turns of wire are needed to achieve a target output voltage.

This principle is fundamental to the global power grid, enabling high-voltage transmission over long distances to reduce energy loss before stepping the voltage down for safe consumer use. It allows electronic devices to operate at low, safe voltages even when plugged into high-voltage wall outlets.

Swapping primary and secondary. Forgetting ratio direction.

Converting mains voltage to a lower device voltage.

Ensure that Vp and Vs are in the same units, typically Volts. A primary-to-secondary ratio greater than 1 indicates a step-down transformer. The equation only applies to AC; transformers do not work with steady state DC. Treat N as a dimensionless count of wire loops.

References

Sources

Halliday, Resnick, and Walker, Fundamentals of Physics
Wikipedia: Transformer
Halliday, Resnick, Walker, Fundamentals of Physics
Halliday, Resnick, Walker - Fundamentals of Physics, 10th Edition
Wikipedia: Transformer (article title)
AQA A-Level Physics — Electromagnetic Induction

Transformer Turns Ratio

Overview

Variables

Derivation

Apply Faraday's Law to Each Coil (Magnitude):

Take the Ratio:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Transformer Efficiency

Frequently Asked Questions

Sources