Power Converter: Watts, kW, hp and Metric PS

Power Unit Converter

Power is how fast energy is used or produced. A power unit converter rewrites a power figure from one unit into another, such as kilowatts into horsepower. This page converts between watts, kilowatts, mechanical horsepower and metric horsepower, and shows all four values at the same time.

What the converter does

A visitor types one number and picks the unit that number is written in. The page returns the same power in all four units.

• Results are shown to eight significant figures. Only the display is rounded. The conversion runs on the number as typed.
• A result below 0.0001 in size, or of a billion or more, is shown in scientific notation. Zero is shown as a plain 0.
• Negative values are accepted. A motor that is generating rather than drawing power has a negative one.
• An empty box produces a prompt rather than a number. A link carrying a unit the page does not offer produces a message rather than a conversion.

The four units

The watt is the SI unit: one joule of energy every second. Each of the other three units is a fixed number of watts. None of those three numbers is a measurement, so none of them changes over time.

Power conversion formula

Every unit here is a fixed multiple of the watt, so a conversion is one multiplication followed by one division. The value is carried into watts, then back out into the unit wanted:

$P_2 = \frac{P_1 \times k_1}{k_2}$

where $P_1$ is the power typed in, $k_1$ is the number of watts in one unit of the chosen unit, $k_2$ is the number of watts in one unit of the wanted unit, and $P_2$ is the answer.

Nothing is added or subtracted along the way, because all four scales read zero at the same power. Only the size of the step changes. Temperature is the case where that is untrue: Celsius and Fahrenheit start counting from different points, so a converter that only multiplied would report 0 °C as 0 °F instead of 32 °F.

Where the two horsepower figures come from

Mechanical horsepower is defined as 550 feet-pounds-force per second. A foot-pound-force is the work done when a force of one pound-force moves something one foot. Both parts of that are fixed exactly. The International Yard and Pound Agreement of 1959 sets the foot at 0.3048 m and the pound at 0.45359237 kg. A pound-force is that pound pulled by standard gravity, which comes to 4.4482216152605 N.

$1\ \mathrm{hp} = \frac{550 \times 0.3048\ \mathrm{m} \times 4.4482216152605\ \mathrm{N}}{1\ \mathrm{s}} = 745.69987158227022\ \mathrm{W}$

Metric horsepower, written PS, is defined as 75 kilogram-force metres per second. A kilogram-force is one kilogram pulled by that same standard gravity, so it is 9.80665 N.

$1\ \mathrm{PS} = \frac{75 \times 9.80665\ \mathrm{N} \times 1\ \mathrm{m}}{1\ \mathrm{s}} = 735.49875\ \mathrm{W}$

Standard gravity was fixed at 9.80665 m/s² by the 3rd General Conference on Weights and Measures in 1901. It is an agreed value rather than a local measurement, which is what lets a mass unit stand in for a force unit. NIST Special Publication 811, Appendix B, lists mechanical horsepower rounded to 745.6999 W.

Example: a 100 kW car

A car rated at 100 kW produces 100,000 watts. Dividing that by the watts in one mechanical horsepower gives the figure used in the United States and the United Kingdom:

$P = \frac{100\,000}{745.69987158227022} \approx 134.10221\ \mathrm{hp}$

Dividing the same 100,000 watts by the watts in one metric horsepower gives the figure printed in most European brochures:

$P = \frac{100\,000}{735.49875} \approx 135.96216\ \mathrm{PS}$

Both answers describe the same car. They differ because the units differ, not because anything about the engine changed. For this input the page shows 100,000 W, 100 kW, 134.10221 hp and 135.96216 PS side by side.

Two units share the name horsepower

Mechanical horsepower and metric horsepower are close in size, so they are easy to mistake for each other, and they are not equal:

$1\ \mathrm{hp} = 1.013869665424\ \mathrm{PS}$

The gap is about 1.4%. A 100 hp engine is 101.38697 PS, and a 100 PS engine is 98.632007 hp. Mechanical horsepower is the figure normally quoted in the United States and the United Kingdom. PS is short for Pferdestärke, the German name for metric horsepower, and it is the figure normally quoted in continental Europe. A page that offered one unnamed "horsepower" would be wrong by 1.4% for half its visitors and would give no sign of it, so both are listed under their own names.

Common power conversions

Each row shows what one unit produces in the other three, at eight significant figures.

Which values are exact and which are rounded

All four units rest on definitions, but only some of those definitions can be written out in full as decimals.

• 1 kW is exactly 1,000 W. Kilo is the SI prefix for a thousand.
• 1 hp is exactly 745.69987158227022 W. The decimal stops there, because the foot and the pound-force are themselves exact decimals.
• 1 PS is exactly 735.49875 W, from 75 times standard gravity.
• 1 hp is exactly 1.013869665424 PS. That division happens to come out even.
• 1 PS is 0.98632007061953105… hp, with digits that never end, so any decimal written for it has been cut short.

The converter builds each horsepower figure from the defining constants rather than storing a shortened decimal, so the rounding a reader sees comes from the display and not from the numbers behind it. Eight significant figures is wide enough to show a defining value clearly and narrow enough to hide the last digits of the computer's arithmetic, which carry rounding noise rather than information.

Frequently asked questions

How many watts is one horsepower? That depends on which horsepower is meant. Mechanical horsepower is exactly 745.69987158227022 W, which the page shows as 745.69987 W. Metric horsepower is exactly 735.49875 W.

Is hp the same as PS? No. One mechanical horsepower is exactly 1.013869665424 metric horsepower, a difference of about 1.4%. A 100 hp engine is 101.38697 PS, and a 100 PS engine is 98.632007 hp.

How is kW converted to hp? Divide the kilowatts by 0.74569987158227022, or multiply by about 1.341 for a rough answer. One kilowatt is 1.3410221 mechanical horsepower, or 1.3596216 metric horsepower.

Why are European car figures given in kW or PS? The kilowatt is the SI unit, and PS is the horsepower unit that grew out of the metric system. Mechanical horsepower is the figure normally quoted in the United States and the United Kingdom, so the same car can appear with two different horsepower numbers.

Can a negative power be converted? Yes. Power is signed, so a value below zero describes energy flowing the other way, such as a motor generating instead of drawing. The page converts it rather than refusing it.

Why are so many digits shown? Eight significant figures keeps a defining value legible without showing the last digits of the arithmetic, which carry rounding noise. The stored constants are never shortened, so a shorter display never feeds back into the next conversion.