Why Exercise Bike Watts Go Up as Cadence Squared

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On a magnetic bike, a watt on the console is not a measurement. There is no force sensor anywhere in the machine. The number is a calculation from your cadence and an assumed property of the brake, and because of how a magnetic brake works, the calculation squares your cadence.

That has a consequence you can check on your own machine in thirty seconds. Leave the knob exactly where it is and pedal twice as fast. The resistance you feel doubles, and the power doubles twice, to four times what it was. Nothing on the bike moved.

What the knob actually sets

A magnetic brake is a magnet held near the rim of a spinning metal flywheel. The metal moving through the field has a voltage induced in it, and that voltage is proportional to how fast it is moving. The voltage drives currents in the flywheel, proportional to the voltage. Those currents sit in the magnet's field and feel a retarding force, proportional to the current.

Follow the chain and the braking torque comes out proportional to speed. Not constant with speed, the way a friction pad is. Proportional to it.

Power is torque multiplied by speed. So if the torque is itself proportional to speed, the power is proportional to speed squared. The thing the knob changes is the constant in front, and that constant has units of newton metres per radian per second. It is not a force. It is not a torque. It is not a power. It is a damping coefficient, and there is no everyday word for it, which is exactly why the dial has numbers on it instead of units.

Our own page on why a drill clutch number is not a torque establishes the general version of this and this page is a harder case of it. A clutch number at least maps onto one torque, even if the maker will not tell you which. A magnetic resistance level maps onto a coefficient, and the torque it produces depends on how fast you are going, so two riders on the same level are not experiencing the same anything.

The ladder, at one fixed knob position

Anchor it somewhere readable. Suppose a given knob position gives 100 watts at 60 revolutions per minute, which fixes the coefficient at 2.533 newton metre seconds and the brake torque at 15.92 newton metres. Now change nothing but your legs.

At 70 revolutions per minute the power is 136.1 watts. At 80 it is 177.8. At 90, 225. At 100, 277.8. At 110, 336.1. At 120, exactly 400. The torque has doubled from 15.92 to 31.83 newton metres over that range and the power has quadrupled, because both the torque and the speed went up together.

Run it backwards and it is more useful still. If you want to hold 100 watts, you must hold 60 revolutions per minute, and a drift of 10 revolutions costs you 36 watts, which is 36 percent. Cadence discipline on a magnetic machine is not a style preference. It is the only thing holding the power steady.

A JLL magnetic indoor exercise bike is the machine this question is really about, and its record contains the most interesting thing on this page by not containing something. The console, it says, "shows time, speed, distance, RPM, odometer and pulse (30 to 240)". Six metrics. Power is not one of them.

That is correct, and the reason is in the next line of the same record: "Resistance is stepless magnetic, controlled by a tension knob for instant intensity changes." A stepless knob has no position number. With no position number there is nothing for a console to look the brake coefficient up from, and without the coefficient there is no power to display. So the bike measures the one thing it genuinely can measure, which is revolutions per minute, and declines to multiply it by an assumption. Nothing in the listing explains that this is what is happening, and it is the single most honest design decision in the category.

Why the error doubles

The square law has a corollary worth having. If the power is proportional to the square of the cadence, then a small proportional error in the cadence becomes twice that proportional error in the power. Three percent out on cadence is six percent out on watts. Five percent is ten.

And the brake coefficient is worse, because it is not measured at all. Whatever the console's table assumes for a given level, any discrepancy between that assumption and the actual magnet position, flywheel temperature, belt tension and age of the machine passes straight into the displayed power at full strength.

Flywheel temperature is not a throwaway. An eddy current brake dissipates every watt you produce as heat in the flywheel itself, and the electrical resistivity of the flywheel metal rises with temperature. Higher resistivity means smaller induced currents, which means less torque for the same speed, which means the coefficient drifts downward as you warm the machine up. A console reading from a cold table will therefore read high after twenty minutes, and the size of that drift is not something I can put a number on, because no listing publishes the flywheel material, let alone its temperature coefficient.

Sixteen levels, and where the sixteen levels actually are

A YCCHENG 16 level magnetic stair stepper takes the other approach and numbers its dial. Its record says "Sixteen levels of magnetic resistance let you dial intensity from a gentle warm-up to a hard climb, and progress as you get fitter", which is a reasonable description and leaves open the question everybody assumes the answer to: are the levels the same size?

They cannot be. The braking coefficient goes as the square of the magnetic field at the flywheel, and the field falls steeply as the magnet is drawn away from it. So the dial is converting a roughly linear mechanical movement into a strongly nonlinear resistance, and then squaring the nonlinearity.

Put representative numbers on it, clearly labelled as representative because no listing states any of them. Suppose the magnet gap runs from ten millimetres at level one down to two millimetres at level sixteen, in fifteen equal steps, with the field inversely proportional to the gap. Then the resistance at level sixteen is twenty five times the resistance at level one. The step from level fifteen to level sixteen is 81.5 times the size of the step from level one to level two. Levels one through eight, exactly half the dial, cover six percent of the machine's range. And half of the total range lies above level fourteen and a half, which is the top nine percent of the numbers printed on the knob.

Change the field law and the magnitudes change: if the field falls only as the square root of the gap, the range compresses from twentyfive to five and half of it sits above level eleven and a half instead. What does not change is the shape, because the squaring is not an assumption. Whatever the gap does to the field, the coefficient does it twice. The dial is compressed at the bottom and explosive at the top, and that is why every machine of this type feels like nothing is happening for the first third of the dial and then suddenly happens all at once.

The flywheel is a ripple number, not a resistance number

The bike publishes a 22 pound flywheel and makes a lot of it, and the flywheel mass is the one figure in the whole machine that cannot affect average power at all.

Here is what it does do. Twenty two pounds is 9.979 kilograms. Treated as a ring at a radius of 0.17 metres, the moment of inertia is 0.2884 kilogram metres squared. At a representative belt ratio of four and a half to one and 80 revolutions per minute at the cranks, the flywheel is turning 360 revolutions per minute, and the kinetic energy stored in it is 204.9 joules.

Compare that with the energy in one pedal revolution. At 150 watts and 80 revolutions per minute, each revolution takes 0.75 seconds and carries 112.5 joules. So the flywheel is holding 1.82 pedal revolutions' worth of energy. That is the buffer, and the buffer is what "smooth" means.

Work the ripple. Your legs deliver almost nothing through the top and bottom of the stroke, call it 20 percent of each revolution, or 0.15 seconds, during which the brake is still taking 150 watts and so withdraws 22.5 joules. Since energy goes as speed squared, the flywheel slows by the square root of one minus 22.5 over 204.9, which is a 5.65 percent dip. Halve the flywheel to 11 pounds and the same dead spot costs 11.66 percent. So the flywheel mass roughly inversely sets the speed ripple, which you feel directly, and leaves the average power exactly where it was.

One more figure makes the stored energy tangible. The record says "you press the knob to stop the flywheel". From a hard 110 revolutions per minute at the cranks there are 387.4 joules in there, which is the same 9.979 kilograms dropped from 3.96 metres. That is what the press has to absorb, and it is why these machines have a brake rather than just a lower setting.

Three ratings, one of which means something

It helps to put the magnetic level next to the two other ways home equipment states resistance, because the three are not the same kind of statement.

A FIAMEARMOR twister arm exerciser rated 22 to 440 pounds is the spring case. Its record quotes a "Wide 22 to 440 lb adjustable range" and, in its own drawbacks, concedes "No weight or dimensions stated". A spring's force depends on how far you have bent it, so a pound figure without a deflection is a number without a question attached to it, and a twentyfold range multiplies the ambiguity rather than resolving it. Our page on whether a 50 pound resistance band is really 50 pounds works that case through properly and this page takes its conclusion rather than repeating the arithmetic.

A pair of ATIVAFIT adjustable dumbbells dialling 5 to 25 pounds each is the honest case, and it is worth saying so because it is easy to take for granted. Its record says it "Dials from 5 to 25 lb each" in "Five clear steps for progression". Those steps are 5, 10, 15, 20 and 25, which are exactly equal, exactly a force, independent of how fast you move, and directly comparable with every other dumbbell ever made. Its drawbacks note "Dimensions not published", which for a mass genuinely does not matter: a 15 pound dumbbell is a 15 pound dumbbell whatever shape it is.

That is the comparison the original question needs. Twenty five pounds transfers. Level 12 does not transfer, between two machines or between two riders on the same machine, and 180 watts transfers only if you also know the cadence it was produced at and trust the table it was computed from.

What a console watt is and is not

Where a console does show watts, the number it is trying to show is mechanical power delivered to the brake. There are two things it is not.

It is not your power at the pedals, because a belt and two sets of bearings sit in between. At a typical belt efficiency the difference is a few percent, and it runs in the direction of the console under-reading your effort, which is the pleasant direction for once.

And it is not the energy you are spending. The chemical energy a person consumes to deliver a mechanical watt is several times that watt, and the multiplier is a figure about human physiology rather than a figure about this bike, so this page does not compute it and does not turn it into anything else. Both of our machines' records carry their own sensible notes about new exercisers and health conditions, and those stand. The point here is narrower and purely mechanical: the watt is at the brake, and the brake is the only place in the system where anything is known.

Our page on blender horsepower versus watts makes the related point about appliances, which is that a headline power figure describes an input rather than an outcome. On an exercise bike the problem inverts: the figure describes an output, and the input, which is the coefficient the knob sets, is the part nobody states.

The honest limit, and what to do with the dial

Three things on this page are exact and the rest are not. The square law is exact, because power is torque times speed and eddy current torque is proportional to speed. The error doubling is exact, because it follows from the exponent. And the dumbbell steps are exact, because they are published.

Everything else is representative and stated as such. The 100 watts at 60 revolutions per minute is an anchor to make the ladder readable, not a specification. The flywheel radius of 0.17 metres, the four and a half to one belt ratio, the 20 percent dead spot and the 150 watt working level are all assumptions, because the bike publishes a flywheel mass and nothing else mechanical. The ten to two millimetre magnet gap and the inverse field law behind the sixteen bars are the most assumed things here, and I have given the result for a gentler law so you can see which part survives.

What survives is the ratio structure. Doubling cadence always quadruples power. Halving the flywheel always roughly doubles the ripple and never changes the average. The top of a magnetic dial always holds a disproportionate share of its range. And a level is never comparable to a level on another machine, because the quantity it sets has no unit anyone prints.

Which suggests the one practical change worth making, and it costs nothing. Stop using the dial as the variable. Pick a knob position and leave it there, then use cadence as your intensity control and record the cadence, because cadence is the one number on a magnetic console that is genuinely measured. Two sessions at level 8 are not comparable. Two sessions at level 8 and 85 revolutions per minute are, on that machine, to within the drift of a warming flywheel. And if you ever want a figure that transfers to a different machine or a different year, the honest answer is that a magnetic bike cannot give you one, which is a limitation of the brake and not of the rider.