Transformer Efficiency Equation, Derivation & Rearrangements

Core idea

Overview

Transformer efficiency measures how effectively a transformer transfers electrical power from the primary circuit to the secondary circuit. It is defined as the ratio of the output power delivered to the load compared to the total input power absorbed from the source, accounting for internal energy losses like heat and flux leakage.

When to use: Use this equation when evaluating the performance of electrical transformers under specific load conditions. It assumes a steady state where input and output power can be measured or calculated based on voltage and current, typically in a laboratory or industrial setting.

Why it matters: High efficiency is critical for minimizing energy waste in power distribution grids and preventing overheating in industrial equipment. Modern transformers often exceed 95% efficiency, which is essential for sustainable energy management and reducing the environmental footprint of electricity transmission.

Symbols

Variables

eta = Efficiency, W = Output Power, W = Input Power

e t a

Efficiency

V a r iab l e

W

Output Power

V a r iab l e

W

Input Power

V a r iab l e

Walkthrough

Derivation

Formula: Transformer Efficiency

Calculates how effectively a transformer transfers power from primary to secondary.

Power factor is 1 (purely resistive load) so P=IV.

1

Define Efficiency:

Standard efficiency definition.

Efficiency = \frac{Useful Power Out}{Total Power In} \times 100%

2

Use Electrical Power:

With P=IV, the ratio of output to input power gives the efficiency.

η = \frac{I _{s} V _{s}}{I _{p} V _{p}}

Result

η = \frac{I _{s} V _{s}}{I _{p} V _{p}}

Source: OCR A-Level Physics A — Electromagnetism

Free formulas

Rearrangements

Solve for $η$

Make eta the subject

η = \frac{P _{o u t}}{P _{in}}

Start from Transformer Efficiency. To make eta the subject, simply rewrite the division as a fraction.

Difficulty: 2/5

Solve for $P_{o u t}$

Make Output Power the subject

P_{o u t} = η P_{in}

Rearrange the transformer efficiency formula to solve for output power.

Difficulty: 2/5

Solve for $P_{in}$

Transformer Efficiency

P_{in} = \frac{P _{o u t}}{η}

Rearrange the transformer efficiency formula to solve for input power ( $P_{in}$ ).

Difficulty: 2/5

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

Visual intuition

Graph

The graph follows an inverse curve where efficiency decreases as input power increases, approaching the horizontal axis for positive values. For a physics student, this means that as input power grows significantly larger than the output power, the system becomes increasingly inefficient. The most important feature is that the curve never reaches zero, meaning that even at very high input power levels, the transformer maintains a theoretical efficiency greater than zero.

Graph type: inverse

Why it behaves this way

Intuition

Imagine electrical power flowing into a transformer as if through a funnel, where a portion of the power (Pin) enters, and a slightly smaller portion (Pout)

η

Transformer efficiency

Represents the fraction of input power that is successfully converted into useful output power; a higher value means less energy is wasted.

Pout

Output power delivered to the load

The useful electrical power that the transformer provides to the device connected to its secondary coil.

Input power absorbed from the source

The total electrical power drawn from the source by the transformer's primary coil.

Free study cues

Insight

Canonical usage

This equation calculates a dimensionless ratio representing the efficiency of a transformer, typically expressed as a decimal or percentage.

Common confusion

A common mistake is using inconsistent units for input and output power (e.g., kW for input and W for output) or forgetting to convert the decimal efficiency to a percentage when required for reporting.

Dimension note

Efficiency is a ratio of two quantities with identical dimensions (power), resulting in a dimensionless value. It indicates the fraction of input power converted to useful output power.

Unit systems

$P o u t$ W · Output power delivered by the transformer to the load.

$P in$ W · Input power supplied to the transformer from the source.

$η$ 1 · Transformer efficiency, a dimensionless ratio.

Ballpark figures

Quantity:

One free problem

Practice Problem

A power transformer receives 5000 Watts of input power and delivers 4850 Watts to a load. Calculate the efficiency as a decimal.

Input Power5000

Output Power4850

Solve for: $e t a$

Hint: Divide the output power by the total input power.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Estimating losses in a power adapter.

Study smarter

Tips

Ensure both power values are in the same units, such as Watts or kiloWatts.
Remember that efficiency is always less than 1.0 (or 100%) due to copper and iron losses.
Total input power can be thought of as the sum of output power and total internal losses.

Avoid these traps

Common Mistakes

Using percent without converting to decimal.
Swapping Pin and Pout.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

Calculates how effectively a transformer transfers power from primary to secondary.

Use this equation when evaluating the performance of electrical transformers under specific load conditions. It assumes a steady state where input and output power can be measured or calculated based on voltage and current, typically in a laboratory or industrial setting.

High efficiency is critical for minimizing energy waste in power distribution grids and preventing overheating in industrial equipment. Modern transformers often exceed 95% efficiency, which is essential for sustainable energy management and reducing the environmental footprint of electricity transmission.

Using percent without converting to decimal. Swapping Pin and Pout.

Estimating losses in a power adapter.

Ensure both power values are in the same units, such as Watts or kiloWatts. Remember that efficiency is always less than 1.0 (or 100%) due to copper and iron losses. Total input power can be thought of as the sum of output power and total internal losses.

References

Sources

Halliday, Resnick, Walker, Fundamentals of Physics
Wikipedia: Transformer
Halliday, Resnick, Walker Fundamentals of Physics, 10th ed.
NIST Chemistry WebBook
Chapman, S. J. (2012). Electric Machinery Fundamentals (5th ed.). McGraw-Hill Education.
Halliday, D., Resnick, R., & Walker, J. (2014). Fundamentals of Physics (10th ed.). John Wiley & Sons.
Wikipedia: Transformer (article title)
OCR A-Level Physics A — Electromagnetism

Transformer Efficiency

Overview

Variables

Derivation

Define Efficiency:

Use Electrical Power:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Transformer Turns Ratio

Frequently Asked Questions

Sources