How Many Watts Never Reach the Shaft? Motor Losses in Numbers

10.39 kW goes in. 6 kW comes out of the shaft. The missing 4.392 kW leaves as heat. Those are the numbers for a three-phase motor pulling 15 A at 400 V, and they work out to an efficiency of 57.74%. Run it 4,000 hours a year and the losses alone come to 17,569 kWh, which at 20 cents a unit is 3,514 a year spent on warming up a motor housing.

Except that reading is wrong, and it is wrong in a way almost everyone gets wrong the first time. Hold that thought for two sections.

That is the answer. The rest of this article is where it comes from, because the same handful of equations that decides how much of your electricity reaches the shaft also decides how hard a magnet pushes on a wire and which single frequency a circuit will pick out of everything you feed it. Three questions, one piece of physics.

Why the supply changes the arithmetic before the motor does

Efficiency is output divided by input, and nobody argues about the output: it is the shaft power, either stated on the nameplate or worked out from torque and speed. The input is where the mistakes live, because the formula changes with what is feeding the machine.

On direct current the input is simply voltage times current. On a single-phase supply you have to multiply by the power factor, because voltage and current no longer peak together. On a three-phase supply you multiply by the power factor and by the square root of three. Skip the root three and a perfectly healthy motor will read as though it is producing more power than it consumes.

All three at 4,000 running hours and 18 cents per kWh. Work out your own on the electric motor efficiency calculator, or start from the supply you actually have: three-phase, single-phase or direct current.

The first two rows are the same motor, the same meter and the same afternoon. The only difference is the power factor: left at 1 in the first row, measured at 0.86 in the second. That single field moves the answer from 57.74% to 67.13%, drops the input from 10.39 kW to 8.937 kW, and cuts the losses from 4.392 kW to 2.937 kW, which is 32.87% of the input instead of 42.26%. In money over a year it is the difference between 3,514 and 2,350.

Nothing about the motor changed between those rows. The power factor sits inside the input formula, so a wrong value there walks straight into the answer. Leave it at 1 and the machine is charged for current it draws but does not turn into work, which makes a perfectly average motor look like a disaster.

Further down, a small single-phase motor running at 70.33% wastes 464 W, or 29.67% of everything it draws, while looking like a modest 1.1 kW machine. Its apparent power is 1.84 kVA, and apparent power is what the cable and the breaker have to carry regardless of how much of it does useful work. The three-phase machine draws 10.39 kVA for the same reason, while the direct-current motor at the bottom has no power factor at all and loses 16.67% of its input to nothing more complicated than resistance and friction.

The power factor moves the answer more than anything else you type

That nine point swing between the first two rows is not an edge case. The power factor is the one input people guess, because a clamp meter gives volts and amps without argument while power factor needs either a meter that reports true power or a nameplate you are willing to trust.

Guess it high and the motor looks worse than it is. Guess it low and it looks better. Take it from a nameplate rather than from a meter and you are not measuring efficiency at all, you are measuring the nameplate. If your instrument reports true power in watts directly, use that number and skip the multiplication, because then the power factor is already inside it.

That is also why an efficiency number on its own does not tell you the IE class. IEC 60034-30-1 sets a different limit for every combination of rated power, pole count and supply frequency, so the same percentage can clear the top class on a small motor and fall short on a large one. And a single measurement is a single working point: most induction motors read highest somewhere around three quarters of rated load and fall away on either side.

Where the turning actually comes from

Strip a motor down and what is left is a wire carrying current inside a magnetic field. The force on that wire is the whole reason the shaft turns, and it is a small enough number to be worth seeing written out.

Put half an amp through 10 cm of wire lying at 20 degrees to a 0.5 T field and you get 8.551 mN of force, or 85.51 mN/m along the wire. That is enough to hold up 871.9 mg, about the weight of a paperclip. A motor is not impressive because any one wire pushes hard; it is impressive because there are hundreds of turns and the pushes add.

The angle matters as much as the current, and 20 degrees is a bad angle. The sine of it is 0.342, so the wire is giving up two thirds of what it could. Square the same wire on to the field and the force climbs to 25 mN without changing a single component.

Same field, same current, same length, turned through the field. Note how flat the top of the range is: 60 degrees still gives 21.65 mN, which is 86.6% of the maximum, and even 45 degrees keeps 17.68 mN. Losing alignment costs almost nothing until the angle gets small, and then it costs everything, with 15 degrees leaving only 6.47 mN.

One law, three questions

The wire is one of three ways the same magnetism shows up, and the calculator handles each of them because they are different questions rather than different numbers.

Each case has its own page: force on a current-carrying wire, force on a moving charge, force between two parallel wires.

The middle row hides something neat. An electron and a proton crossing that field at the same speed feel exactly the same force, 8.011 x 10^-17 N, because they carry the same amount of charge. What differs is what the force does to them. The electron curves on a radius of 11.37 nm and goes round at 14 GHz; the proton, about 1,836 times heavier, sweeps a radius of 20.88 µm and takes 131.2 ns per turn, which is 7.623 MHz. Same push, wildly different paths.

Worth remembering alongside that: a magnetic field does no work on a moving charge. The force is always at right angles to the velocity, so it bends the path and never changes the speed. Every kilowatt in the table above is lost to resistance and to the iron, never to the magnetism itself.

The bottom row is the one that used to define the ampere. Two wires carrying 10 A each, 5 cm apart, pull on one another with 400 µN/m, which over a 10 cm run is 40 µN, about the weight of 4.08 mg. Halve the separation and the force doubles, because it falls as one over the distance rather than the distance squared.

• At 1.25 cm apart: 1.6 mN/m, or 160 µN over 10 cm.
• At 2.5 cm apart: 800 µN/m, or 80 µN over 10 cm.
• At 5 cm apart: 400 µN/m, or 40 µN over 10 cm.

Currents in the same direction attract, opposite directions repel. That is why the conductors inside a transformer or a busbar need mechanical bracing: under a short circuit those currents are not 10 A, and a force that scales with the product of two currents gets serious very quickly.

The circuit that picks out one frequency

The third question is the one a motor drive, a radio front end and a crossover all ask: out of everything arriving, which frequency do we keep? Put a resistor, an inductor and a capacitor together and the answer is set by the inductor and the capacitor alone.

Take 100 ohms with 10 mH and 100 nF. Resonance lands at 5.033 kHz, and it lands there whether the parts sit in series or in parallel, because the same L and C set it. Everything else about the two circuits is opposite.

Run your own parts through the RLC circuit calculator, or go straight to the arrangement you are building: series or parallel.

The quality factor is what separates them here. In series the circuit reaches 3.16, which gives a bandwidth of 1.592 kHz and half-power points at 4.3 kHz and 5.891 kHz. In parallel, with the same parts, Q falls to 0.32 and the bandwidth opens out to 15.92 kHz, running from 1.458 kHz to 17.37 kHz. One is a filter, the other barely selects anything.

Step off resonance and the circuit stops being simple. Drive those same parts at 5 kHz, a whisker below the 5.033 kHz peak, and the impedance is 100.1 Ω with a phase angle of -2.4°, so the circuit reads as slightly capacitive: the reactances are 314.2 Ω and 318.3 Ω and no longer quite cancel. That is 0.7% off the resonant frequency showing up as a measurable phase shift.

There is a practical warning buried in that first column. At resonance the reactances are each 316.2 ohms and they cancel, but they only cancel together. The voltage across the coil and across the capacitor individually reaches Q times the driving voltage, so a 10 V source puts 31.6 V across parts that a quick glance at the supply rail says will never see more than 10 V. Capacitors have been chosen on that quick glance and have not survived it.

Measuring your own motor without guessing

1. Read the voltage at the motor terminals, not at the panel and not from the nameplate. Volt drop along a long run is real and it lowers the input power you should be using.
2. Clamp the current while the machine is doing its normal work. A reading taken at no load describes a working point you do not care about.
3. Measure the power factor rather than assuming it. This is the twelve point swing from earlier. If your meter gives true power directly, use that and skip the multiplication entirely.
4. Get the shaft power honestly. Either the rated output at that load, or torque times speed, which is what the torque calculator and the mechanical power calculator are for.
5. Put the losses in currency before deciding anything. A motor at 57.74% wasting 3,514 a year is a different conversation from one wasting 160.

Tools discussed in this article

• Electric motor efficiency calculator, with a page each for three-phase, single-phase and direct current supplies.
• Magnetic force calculator, with a page each for a current-carrying wire, a moving charge and two parallel wires.
• RLC circuit calculator, with a page each for series and parallel arrangements.

More physics tools

Momentum · Force · Kinetic energy · Potential energy · Work · Mechanical power · Torque · Kinetic friction · Pendulum period · Thermal expansion · Specific heat · Thermal conductivity · Hooke's law · Simple harmonic motion · Buoyancy · Acceleration · Density · Temperature converter · Thermal efficiency · Gravitational force · Snell's law · Refractive index · Thin lens · Magnifying glass · Wavelength to color · Wave frequency · Photon energy · Speed of sound · Echo distance · Ohm's law · Electrical power · Amps · Parallel plate capacitor · Capacitor energy · Transformer · Battery life · Battery charging time · Power bank capacity · Inductance