Efficiency Calculator

What Is the Efficiency Calculator?

How to use the efficiency calculator

1. Choose the unknown. The Solve for menu opens on Efficiency, and the other three choices are the useful output, the total input and the Overall efficiency of a chain. The boxes a mode does not read are hidden while that mode is showing.
2. Type the two figures. They open on 9 and 12, and they must be in the same unit as each other. Which unit that is never reaches the arithmetic — joules over joules and watts over watts give the identical number.
3. Read the headline. Four significant figures with trailing zeros dropped by the engine, and a per cent sign beside it. At the opening boxes it reads 75 %, which the working prints as 75.00 per cent.
4. Read the fraction, not the percentage, if you are carrying it onwards. The second reading is 0.7500. A percentage the page has already printed has been rounded, and a rounded figure multiplied into the next step moves the answer.
5. Read the waste. The last two readings are 3.000 and 25.00 % at the opening boxes. That is the figure that changes a decision; the percentage on its own rarely is.
6. Switch per cent to fraction if you prefer. Every efficiency box carries a two-entry unit menu and the engine converts for you, so 75 in per cent and 0.75 as a fraction are the same entry.
7. Switch the menu to the chain. Choose Overall efficiency and four stage boxes appear instead, opening on 38, 96, 92 and 88. A stage at 100 per cent is a stage that is not there.
8. Read the chain against its worst stage. The worst single stage is 38.00 % and the overall figure is 29.53 %. The chain is below its worst stage, which is the whole reason the mode exists.
9. Open Show working. The division or the multiplication appears line by line, then the waste, then the ceiling. The chain mode also prints the average and the sum, both labelled as the wrong answers they are.

This page is the arithmetic, and the efficiency formula guide is where the method is set out with eight worked problems, the energy form against the power form, and the chain diagram. The two figures you type usually come from somewhere else: the guide to power in physics is where a power is worked out from work and time, and the work and power calculator is the tool for that step.

Three mistakes account for most wrong answers here. The first is mixing the two units: the ratio cancels one unit, not two different ones, so kilojoules over joules reads a thousand times out. The second is deciding the wrong figure is "useful" — a heater's waste heat is its useful output, and the guide to energy conversion examples is the page that hands this question over to this one.

The third is combining a chain by adding or averaging. Both wrong answers are printed beside the right one in the chain mode so that the gap is visible rather than described. Neither is close: two stages at 80 and 90 per cent give 72.00 %, where the average says 85.00 % and the sum says 1.700, which is above 1 and so cannot be an efficiency at all.

Worked example: change one thing at a time

The first table starts at the opening boxes, runs the three stated-given cases the page quotes, then changes one thing at a time: the same lamp over an interval, both sides doubled, the menu moved to each of the other two single-device directions, and finally the entries the calculator declines. Every answer and reading cell was read out of the running widget rather than worked out by hand, so where a cell and the tool ever part company, believe the tool.

Rows 1 to 4 are four stated givens and nothing more. The lamp, the kettle and the motor produce 75.00 %, 88.00 % and 72.67 % because those are the figures that were put in, not because any device was measured. Row 4 is the boundary the page allows: a useful output exactly equal to the input returns 100.00 %, the ideal lossless device.

Rows 5 and 6 change one thing each and neither moves the answer. Row 5 runs the lamp over 60 seconds instead of per second, so 540 joules useful out of 720 joules in — the interval appears top and bottom of the same fraction and cancels exactly. Row 6 doubles both figures and gets 75.00 % again, which is the scale-free property written out.

Rows 7 to 11 are the same relation rearranged. Row 7 asks for the useful output at a stated 35 % and gets 140000; row 8 asks what input a 500 useful output needs at 62.5 % and gets 800. Row 9 is the device that wastes everything, which is a real device and is answered rather than refused.

Rows 10 and 11 close the round trip on row 7's own state. Feeding 140000 and 400000 back in returns 35 %, and feeding 140000 at 35 % returns 400000. That is a tautology and it is worth saying so: one division rearranged three ways can only disagree with itself in the last floating-point digit.

Rows 12 to 18 sit outside the domain and are declined there rather than answered. Row 12 is the one that matters most: 13 useful out of 12 in would be an efficiency above 100 per cent, which would mean energy appearing from nowhere, so the page says so instead of printing a number. The conservation of energy guide is where that principle is set out properly.

The last three are the conditional ones. An efficiency above 100 per cent is refused as an input in row 16 for the same reason row 12 is refused as an output. An efficiency of zero is fine in row 9 and refused in row 17, because only one of those two directions divides by it — and a useful output of zero is refused in row 18 only, because there the answer would be a total input of zero.

The second table is the chain mode, which is the half of this page the single formula cannot do. Each row types four stage efficiencies and reads back the overall figure, the worst single stage, how many stages are actually doing anything, and the average of those stages — which the widget labels as not the answer, because it is the commonest wrong way to combine a chain.

The first three rows are the whole lesson. Two stages at 80 and 90 per cent give 72.00 %, below both of them. Three stages give 31.28 %, below the 40.00 % one that was already the worst.

Four stages give 29.53 %, where the average of those same four stages reads 78.50 %. The chain is below its worst stage and nowhere near their average, which is the difference between multiplying and averaging.

Row 4 types the same four stages backwards and gets the same 29.53 %, because multiplication does not care about order. Row 5 puts three perfect stages in front of one bad one and still lands on 50.00 %: the overall figure is set by the size of the loss, not by where it sits or by how many stages surround it.

Rows 6 and 7 are the two ends. A chain with every stage at 100 per cent has no chain in it at all, so the reading says so in words rather than printing an average of nothing. One dead stage takes the whole chain to 0.00 % however good the other three are, which is why that reading counts the stages below 100 per cent rather than assuming four.

The last two rows are refused. A stage above 100 per cent or below zero is not an efficiency, so the chain is declined rather than multiplied out — the same ceiling as row 12 of the first table, applied one link at a time.

Formula and symbol reference

One division does nearly all the work. Efficiency is the useful output over the total input, eta = E_useful / E_input, and multiplying by 100 turns that pure number into a percentage. Rearranged for each of the other two unknowns it is E_useful = E_input × eta and E_input = E_useful / eta, which are the second and third positions of the Solve for menu.

The same two boxes take powers as happily as energies, and that is not a convenience but the reason the boxes carry no unit. If both figures are measured over the same interval then that interval sits top and bottom of one fraction and cancels exactly: a lamp at 9 watts useful out of 12 watts in is 75.00 %, and over 60 seconds it is 540 joules out of 720, which is 75.00 % again. The guide to what energy is in physics is where the joule itself is set out, and the power calculator turns work and time into the watts this page then divides.

The fourth position of the menu is a different equation. Stages in series multiply, eta_total = eta1 × eta2 × eta3 × eta4, because every stage takes as its input whatever the one before it passed on. The waste follows from the efficiency rather than needing its own measurement: E_waste = E_input - E_useful, which is the same as the input times one minus the efficiency, and it is the figure worth quoting.

Two neighbouring quantities are called efficiency on this site and are not this one. The ratio of two mechanical advantages is the mechanical advantage guide's quantity and has its own calculator; the temperature-driven ceiling on a heat engine is the Carnot efficiency guide's, and neither is computed anywhere on this page.

The physics: why efficiency can never pass 100 per cent

The ceiling is not a property of machines but of the arithmetic the question is made of. The useful output is one part of the input, and the waste is the rest; a part cannot be bigger than the whole it was taken from. That is conservation of energy and nothing more, which is why this page refuses the entry rather than reporting a figure above 100.

It is worth being precise about what the ceiling is not. It is not the second law, and it is not a statement about heat. A heat engine does carry a further ceiling, set by the two temperatures it works between and lower than 100 per cent, but that is an extra restriction stacked on top of this general one — and it is computed by the Carnot efficiency calculator rather than here.

The other apparent counterexample is a heat pump quoted above 100 per cent, and it is a different ratio rather than a violation. What is being quoted there is heat moved per unit of work supplied, which compares two different quantities and has its own name and its own page. Nothing on this page computes one.

Which brings the weight back onto the one judgement this page cannot make for you: which figure is useful. A lamp's useful output is light and its heat is waste; an electric heater's useful output is the heat, and the same joules that were waste in one case are the whole point in the other. The arithmetic is identical and the answer is not, so it is the device in front of you and not the formula that settles the numerator — and where that numerator is a quantity of work, the guide to work done in physics is where the figure is worked out before this page ever sees it.

Where the efficiency calculator breaks down

The arithmetic here is exact to the last digit a double can hold, and there is no physical constant in it to be uncertain about. What goes wrong is everything around the division: which figure you called useful, where the two numbers came from, and whether a single percentage was ever the thing worth knowing.

The calculator cannot tell you which figure is useful

This is the one judgement the page cannot make and the one that decides the answer. Light from a lamp, lifting from a hoist, heat from a heater — the same joules can be the useful output in one problem and the waste in the next, and only what the device is for settles it. Give the page the wrong numerator and it will divide it perfectly.

A percentage on its own rarely changes a decision

Two devices at 90 per cent waste wildly different amounts if one draws ten times the power of the other, which is why the readings put the wasted figure next to the percentage. The signature of this method is the waste, not the ratio. A reader who only takes the percentage away has learnt something they could mostly have guessed.

A handed-down percentage may already be rounded

The figures on a label or in a question have usually been rounded before they reached you, and multiplying a rounded figure carries that rounding forward. Two thirds times 0.850 is 56.67 % from the fractions and 56.70 % from a stage printed as 66.7 per cent first. The gap is 0.0283 percentage points — invisible in the first decimal, wrong in the second — which is why the chain mode asks for stages rather than for a percentage you typed back in.

Four stages is the limit, and a stage at 100 per cent is a stage that is not there

The chain mode multiplies exactly four numbers. A longer chain has to be done in two passes, which is arithmetically identical because the product does not care how it is grouped, but the page will not do it in one go. Leaving a stage at 100 per cent is how you tell it a stage is absent, and the stages below 100 per cent reading is there so that an absent stage never quietly joins the average.

Nothing here is a measurement, and no device is given a real efficiency

Every figure on this page is an arithmetic exercise with stated givens. The preset names are shorthand for the numbers behind them and name no manufacturer, model or standard, and none of the percentages is a specification of anything. If you need the real efficiency of a real device, verify it against that device before you use it.

The heat-engine ceiling and the heat-pump ratio are not computed here

A heat engine cannot reach 100 per cent even with perfect engineering, because its two working temperatures set a lower ceiling on top of the general one. That calculation is not performed anywhere on this page; it belongs to the Carnot efficiency calculator. The figure above 100 per cent that heat pumps are quoted with is a different ratio again, and is also not computed here.

Precision and uncertainty are not modelled

The page prints percentages to two decimal places and fractions to four significant figures because that is the house ceiling, not because your inputs justify it. Two figures carrying two digits each cannot support a four-figure answer, however many the headline shows, and nothing here propagates an uncertainty or offers an error bar. Where to round in a chain is a different question from how many figures an answer deserves, and only the first is treated here.

A refusal is an answer, not a fault

A useful output larger than the total input, a total input of zero or less, a negative useful output, an efficiency above 100 per cent as an input, an efficiency of zero when the input is the unknown, and a stage outside 0 to 100 per cent are all declined in words rather than answered with an infinity or a NaN. Two of those are conditional and worth reading twice: an efficiency of zero is a perfectly good input in one direction, and so is a useful output of zero.

Where efficiency is actually used

Turning two measurements into one number

This is the forward question and the reason the menu opens where it does. You have an output you wanted and a supply you paid for, both in the same unit, and the division tells you what fraction of the second became the first. Everything else on the page hangs off that one step.

Working out what a stated efficiency will actually deliver

Point Solve for at the useful output, type the efficiency you were given and the input you have, and the answer is what you can expect out. This is the direction a specification is usually read in, and it is also where the page refuses an efficiency above 100 per cent rather than multiplying it out.

Sizing the supply for an output you have already fixed

Point the menu at the total input instead and the question reverses: how much has to go in for the output you need? At 62.5 per cent a 500 joule useful output needs 800 joules supplied, and the 300.0 difference is the cost of the inefficiency in the unit you care about.

Finding the stage that is setting the scale

In the chain mode the worst single stage reading sits next to the overall figure, and the gap between them is how much the rest of the chain is costing you. Four stages at 38, 96, 92 and 88 per cent give 29.53 % against a worst stage of 38.00 %, so the three good stages are still taking a bite out of a chain that stage had already capped. Improving the worst link is nearly always worth more than improving any of the others.

Costing the waste rather than quoting the ratio

The wasted reading is the one that turns into money, heat or a cooling requirement. A kettle at 88.00 % leaves 22800 joules that went somewhere other than the water, and that figure is the one worth putting in a report. The percentage tells you how good something is; the waste tells you what it costs.

Checking somebody else's number

Checking a quoted figure is the easiest use of all, and it needs no theory. Type the two numbers behind the claim and see whether the percentage comes back. If the division does not reproduce the quoted figure, one of the two numbers is not measuring what you assumed — and if it comes back above 100 per cent, the widget declines the combination, and the reason for that is set out above rather than in the refusal itself.

Where to go next

For the method rather than the arithmetic — eight worked problems, the energy form against the power form, the chain diagram and the two wrong ways to combine one — read the efficiency formula guide, which this tool is the calculating half of. The energy conversion examples guide is the page that handed this question over, and the conservation of energy guide is where the ceiling comes from.

Four more guides pick up what this page leaves out. The mechanical advantage guide and the Carnot efficiency guide own the two neighbouring quantities that are also called efficiency, the guide to power in physics is where the watts come from, and the specific heat capacity guide is where the kettle problem gets its stated given.

Four tools sit next to this one. The work and power calculator and the power calculator produce the two figures this page divides; the mechanical advantage calculator and the Carnot efficiency calculator compute the two efficiencies this one deliberately does not.

The full physics lab library and calculator index are open too.