Power calculation from torque
Power calculation from torque
(OP)
Hi All,
I hope this is the best forum for this question - because it is not directly related to electrical motors, but I believe this would be the best place to find the answer.
The problem I have actually relates to the calculation of power based on varying torque and angular velocity (on a bicycle of all places).
Background:
A power meter on a bicycle measures a cyclists power output by measuring torque (sampling it at around 70Hz and averaging those 70 samples) then multiplies this average by angular velicity derived from the rpm to provide a power reading once per second.
It is common that a cyclist experiences difficulty in achieving the same power outputs when riding on an indoor trainer (which uses a flywheel and applies a resistive load to the rear wheel) as apposed to riding outdoors.
My hypothesis is that the torque pattern changes between these two environments (do to the removal of the mass damping effect that the riders body mass provides) and that using an arithmetic mean to calculate the average torque per rotation is in fact not correct, and that the torque for the rotation should be calculated using a root-mean-square.
It's been 20 years since my undergraduate days so I'm dusting out the cobwebs
I hope this is the best forum for this question - because it is not directly related to electrical motors, but I believe this would be the best place to find the answer.
The problem I have actually relates to the calculation of power based on varying torque and angular velocity (on a bicycle of all places).
Background:
A power meter on a bicycle measures a cyclists power output by measuring torque (sampling it at around 70Hz and averaging those 70 samples) then multiplies this average by angular velicity derived from the rpm to provide a power reading once per second.
It is common that a cyclist experiences difficulty in achieving the same power outputs when riding on an indoor trainer (which uses a flywheel and applies a resistive load to the rear wheel) as apposed to riding outdoors.
My hypothesis is that the torque pattern changes between these two environments (do to the removal of the mass damping effect that the riders body mass provides) and that using an arithmetic mean to calculate the average torque per rotation is in fact not correct, and that the torque for the rotation should be calculated using a root-mean-square.
It's been 20 years since my undergraduate days so I'm dusting out the cobwebs





RE: Power calculation from torque
If you want average power, just divide energy into time. And if you want instantaneous power, just use your torque times speed products.
No root mean square should be used.
Gunnar Englund
www.gke.org
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100 % recycled posting: Electrons, ideas, finger-tips have been used over and over again...
RE: Power calculation from torque
I don't know currently if the power meter is averaging torque first then multiplying by speed, or whether it is multiplying each torque sample by speed to get instantaneous power, then averaging instantaneous power.
We are just observing that the cyclist is not able to achieve the same power for the same duration when riding indoors - and as far as I can see, the only physical difference between the two systems is a change in mass damping.
RE: Power calculation from torque
Gunnar Englund
www.gke.org
--------------------------------------
100 % recycled posting: Electrons, ideas, finger-tips have been used over and over again...
RE: Power calculation from torque
Mechanical power delivered at any instant is torque times angular velocity. Both are signed quantities, so the direction depends on whether they have the same sign or different signs. In your case, you are only worried about the case where both are "positive". Energy transfered over a time interval is the power integrated over that time. Average power delivered in that time interval is the total energy divided by the time. Fundamentally, that's it -- period.
Let's look at an electrical analogy to see how you can get confused. You have a "black box" electrical load with two leads. You can measure the voltage across the leads and the current into a lead. Electrical power delivered at any instant is (signed) voltage times (signed) current. Energy transferred is power integrated over time, and average power transferred is this energy divided by the time. This is true for any pattern of voltage and current - AC or DC - and any electrical load (but not all three are independent, of course). You don't even have to know what the load is.
Now let's look at the case where you've got a fixed (frequency and voltage) AC input and a simple resistive load. Instantaneous current is simply instantaneous voltage divided by the resistance. In this case, voltage and current change sign at the same time, so power transfer is in the same direction all the time. If you have voltage and current measurements (and can process it fast enough), you can calcuate instantaneous power, net energy, and average power as explained above.
However, for steady-state linear AC circuits, RMS calculations simplify things enormously. If
V = A sin(wt)
then
I = (A/R) sin(wt)
and
P = (A^2/R) sin^2(wt)
With the time-averaged value of sin^2 being 0.5,
Pavg = 0.5 (A^2/R)
If the AC quantities are described by their RMS values, you can simplify these calculations. The RMS value of a sinusoid is the peak divided by sqrt(2). So
Vrms = A/sqrt(2)
and
Irms = A/R/sqrt(2)
so
Vrms*Irms = A^2/R/2 = Pavg
Remember that this is just a convenience to shorten the calculations in these cases.
Curt Wilson
Delta Tau Data Systems
RE: Power calculation from torque
I'd think you need to calculate power on a sample by sample basis and then average out the samples. Maybe some type of moving window to calculate the power for, say, the last minute or 5 minutes. You don't sum up these values over time or use a delta time multipier - the rpm already includes the time.
RMS is a statistical meausre of the magnitude of a varying quantity or to put it simply it just calculates the area under the curve.
RE: Power calculation from torque
As one rides a bike their entire moving mass 150lbs (more unfortunately in my case) is there to provide an inertial average(storage) of the full pedal rotations. Even though the rider is not accelerating, or even able to accelerate twice per rotation, there is still the smooth pedal motion and the sensory feel of moving smoothly.
Now on an exercise machine you end up with a load that is totally non cyclic. It is there always! No variation. And there is much less inertia available. This means those two zero points in a pedal stroke result in rapid variations in rotary motion/feel. That's why the "machines" feel so hard to pedal.
Can your measurements be missing the fact that there are actually "zero points"?
Keith Cress
Flamin Systems, Inc.- http://www.flaminsystems.com
RE: Power calculation from torque
This is NOT correct! RMS is the (square) Root of the Mean of the Square of the signal in question. It is not the "area under the curve".
The mathematical expression for the voltage signal out of a North American residential socket is:
V(t) = 170 * sin(2*pi*60*t)
The average voltage (area under the curve divided by time)
is zero.
The RMS voltage is 170/sqrt(2) = 120Vrms
The RMS value is useful for things like power calculations. If you put a 144-ohm load across this, you can calculate the average power as:
Pavg = Vrms^2 / R = 14,400 / 144 = 100W
Using RMS values makes power calculations for these (zero-centered) AC signals into linear loads as easy as for DC signals. That is, 120Vdc through a 144-ohm load also would have 100W of power transfer.
Bruce, I don't think your method of sampling torque 70 times per second and averaging, then multiplying by a single speed measurement for that second (which is probably some kind of effective average for the second anyway) will cause significant errors, because these quantities will not change that quickly.
As a side note, I've never felt that stationary cycles capture the "feel" of a real bike's loading on my legs, and that I'm not as "efficient" on a stationary cycle.
Curt Wilson
Delta Tau Data Systems
RE: Power calculation from torque
At this stage, the common theory to explain this phenomenon is rider motivation is not as good riding indoors as outdoors. However, our experiences show a fairly constant discrepency - even when this is done by professional cyclists who are focused athletes. The discrepencies can be as much as 10%.
Googling around for the term "RMS Torque" I see that it is used when rating motors that have varying torque cycles as the equivalent constant torque that causes the motor to heat up to the same temperature.
The only physical difference that I can see between the two environments is the difference in inertial load. We are actually using the same bicycle, with the same power meter, indoors and outdoors. The only impact I can see that a difference in inertial load can have, is on the shape of the torque wave during the pedalling rotation, hence my theory that we are actually measuring power incorrectly.
I put together an example to try to confirm my hypothesis, using two extremes.
When riding on the road, with the mass damping effect of the rider, lets say the torque samples look like this
0 1 2 3 4 3 2 1 0 1 2 3 4 3 2 1
average torque for this wave is 2, rms torque is 2.34
Now, when riding on the indoor trainer lets say the samples look like:
0 0 0 4 4 4 4 0 0 4 4 4 4 0 0 0
average torque for this wave is 2, rms torque is 2.82
This is an extreme example to illustrate a point.
A 25% difference. Does this explain the difference that the athlete experiences between the two environments?
RE: Power calculation from torque
Your goals and definitions need stiffening up a bit. Are you interested in calculating power or are you interested in studying the effect that a variable power has on the human body?
The former case is clear-cut and there are no doubts as to what methods to use. The latter case is something entirely different.
If losses or fatigue in the human body are proportional to developed power squared, then your example is probably valid. Using RMS will then be an acceptable and well established method. But, if losses follows some other function of develpoed power, then you have to find and use some other way of calculating "specific load" or whatever the name for that may be.
Gunnar Englund
www.gke.org
--------------------------------------
100 % recycled posting: Electrons, ideas, finger-tips have been used over and over again...
RE: Power calculation from torque
When a rider is on a bicycle his body tends to stay verticle and the bicycle oscilates from side to side. In some cases, when the rider is striving for maximum acceleration, the side to side movement of the bicycle is extreme.
The vector force of the arms on the handle bars will have some effect, but basically, when you put all your weight on one foot, If your body's center of gravity is not over your foot print, you will fall down. On a bicycle, if your center of gravity is not over the center of balance (the center line when going straight ahead) you and the bike will fall.
A bicycle rider has four ways to compensate for the transfer of pressure from one side to another;
1> By moving the bicycle from side to side. (Low mass involved)
2> By moving his body from side to side. (High mas involved)
3> By swerving slightly from side to side. (Almost no mass involved)
4> Varying the force on the handle bars from one side to the other.
On the machine, the rider may only compensate and get maximum pedal pressure by moving his body from side to side, and by varying pressure on the handle bars.
Much of the effort exerted on the handle bars may be to move the body from side to side.
When a rider is pumping hard, and swinging the bicycle from side to side, and swerving slightly, virtually all of the handle bar pressure is in line with the balance points and is making the maximum contribution to increased pedal pressure.
I humbly suggest that these effects may a significant part of the explanation as to why a cyclist is able to preform better on the road. His range of options to transfer energy efficiently to the pedals are limited on the machine.
Just the energy wasted moving the body from side to side on the machine may be a large part of what you are trying to resolve as a measuring error.
A question? Do you get a close correlation between the road bicycle and the machine at low levels of exertion when the rider is seated comfotably on the seat, and progresively greater discrepancies as the exertion level increases?
If so, I rest my case.
respectfully.
Ps; re; rms torque. For a constant speed, horsepower is related to torque, and rms horsepower is used to determine an electric motor's suitability to handle a cyclic load that at times may exceed the horsepower rating of the motor.
Edward H. Cowern P.E. of Baldor Motors has prepared an excellent series of papers on electric motors. He describes the RMS horsepower method of predicting motor heating under cyclic loads and overloads.
Available at
http://www.baldor.com/pdf/literature/PR2525.pdf
"COWERN PAPERS"
On the machine, the only
RE: Power calculation from torque
The discrepency is happening at what we call the Functional Threshold Power - which is the power that is sustainable for 60 minutes. Which is porbably about 20% - 30% of peak power - i.e. power produced in a sprint. At this power, the upper body is very still, and the arms are relaxed. In fact the same power can be achieved on a Time Trial bike - in which position the elbows are actually resting on the handlebars and there is very little if not zero side-to-side movement.
skogsgurra: I posted a lengthy discussion yesterday regarding the relevance of power etc - trying to answer your questions, but I see it is not on the forum? Not sure why, but I will post it again today!
RE: Power calculation from torque
Thank you for your reply.
Can you tell us a little about your instrumentation? How and where does your power meter measure torque? Brand and model?
Have you tried varying the mass of the flywheel?
respectfully
RE: Power calculation from torque
Essentially we are measuring the training effect that riding at a certain power level has on the athletes body (i.e. fitness).
The Funtional Threshold Power (FTP) - or power that is sustainable for 60 minutes is the baseline that is used to calculate all other power levels. Each level has specific physiological adaptions that are targeted in that level, so the athlete would target a certain level when the goal is to train that particular adaption. The levels have been defined as:
Level 1: Active Recovery <55% FTP
Level 2: Endurance 56% - 75% of FTP
Level 3: Tempo 76% - 90% of FTP
Level 4: Lactate Threshold 91% - 105% of FTP
Level 5: VO2Max 106% - 120% of FTP
Level 6: Anearobic Capacity > 121% of FTP
Level 7: Neuromuscular Power Maximal Effort
See http:
Most cyclists wish to switch between indoor and outdoor environments, it is a lot easier to control external influences such as wind, traffic, gradient etc when training indoors, whereas it is more enjoyeable training outdoors. A loot of cyclists would use the indoor trainer to do interval training targeting the higher levels.
Now, if the load exerted on the athlete is different between indoors and outdoors, power based training is compromised and will be less effective.
RE: Power calculation from torque
There are five types of power meatures currently available:
1. Powertap - uses strain gauges and is effectively a hub fitted to the rear wheel of the bicycle. Torque is sampled at a rate of 70Hz, and velocity once per revolution.
2. SRM - also uses strain gauges, and replaces the front crank and attaches to the front sprockets. I don't know the sampling rate etc, but this device is considered to best one on the market.
3. Ergomo - uses two perforated discs on either side of the axle connecting the left and right crack. An LED and photosensor are used to create a square wave on each disc - as the axle twists, the phase difference between the two waves is calculated, from which the torque is derived. Note that this device can only measure the left leg and doubles it to calculate overall power.
4. Polar - measures the vibration and speed of the chain and calculates the torque being applied. This device is difficult to calibrate correctly and is not very popular.
5. iBike - uses a wind speed sensor, an accelerometer, a tilt sensor, rider aerodynamic coefficient etc to calculate the velocity vectors etc. Using the principle that the power output of the rider is equal and opposite to the forces of gravity and wind resistance, the power is calculated. This device obviously cannot be used indoors.
RE: Power calculation from torque
If you draw the torque curve, and the speed curve, can you agree that the speed curve lags by (less than*) 90 degrees compared to the torque curve? If I replace the system with an equivalent resistor and inductor - the resistor is the friction, and the inductor is the flywheel of the indoor equipment - the angle between torque and speed gets larger if the inductance is increased.
* When the torque reaches a peak and falls slightly, the bike is still accelerating. the bike will stop accelerating when the reducing torque matches the torque required to keep the bike going at that slightly higher speed.
In effect, I think, you should try to get the 'power factors' for outside and inside, the same. Which means you must match friction and inertia.
RE: Power calculation from torque
RE: Power calculation from torque
RE: Power calculation from torque
A test is a good idea. Try a coast down test.
RE: Power calculation from torque
RE: Power calculation from torque
RE: Power calculation from torque
Going back to your post of 28 Oct at 2:51, where you wonder about simple averages versus RMS:
You are always correct, regardless of signal pattern, if you can sample torque and speed at high rates, multiply the products together, then take the simple average (mean) of these values over the time span of interest. (In certain situations, RMS calculations permit you to shortcut these lengthy computations.)
You say that you are sampling torque at 70 Hz, and that torque is cyclic at a few Hz. This seems to be to be a high enough sampling rate. You are effectively sampling velocity at 1 Hz. Now obviously the velocity varies far less than the torque, but your calculations essentially assume that the velocity is constant over this second. If it were constant, then (simple) averaging of the torque and multiplying by this velocity value is equivalent to multiplying each torque sample by the velocity and then averaging, and so would yield a correct average power value for the interval.
The question is: How good is this assumption of constant velocity over a second? Do you access to any better (higher frequency) measurements for the purposes of characterization? Also, it would be a good math exercise (I would just use Excel) to compare your averaging with exact calculations for various X% oscillations in velocity. You should make sure that the velocity curve lags the torque curve (a purely inertial load would lag 90 degrees).
Curt Wilson
Delta Tau Data Systems
RE: Power calculation from torque
Unfortunately I don't have access to the data samples. The sampling is performed using electronics embedded in the device and the results are transmitted.
I agree, the sampling rate is more than adequate, even with the harmonics that would be present, the torque wave fundamental frequency is generally 1 - 2 Hz.
Thinking about what you are saying though makes sense - the torque curve does not change between the two environments as this is a function of the pedaling dynamics of a particular rider, but the velocity curve may change significantly due to the change in intertial loading.
I know on an indoor trainer - the coast down when pedaling stops is a lot quicker than on the road.
I guess one way to test this would be to place an LED and a sensor on the wheel (through the spokes) and take a look at the pattern on a scope.
RE: Power calculation from torque
Thinking about where the discrepancy between inside/outside could lie, I can only imagine that the frictional force is not constant with speed, and 'outside' the speed variation is skewed by the power of three wind resistance, and is not sinusoidal any longer (effectively, the top of the speed curve is flattened), whereas if the inertia is larger, the speed does not vary much.
Another thought is that the torque input is not sinusoidal, either.
The flattened speed curve would give the same average. From my spreadsheet this shows a reduction in power for the high inertia load.
This could seem to conflict with your statement "coast down takes longer outside", which could mean that inertia is higher for outside. But I cling to the hope that coast down takes longer outside because the wind resistance is much less at lower speeds, and perhaps your indoor trainer has higher resistance at lower speeds?
I hope I haven't totally confused you, but it is quite an interesting problem.
RE: Power calculation from torque
That's confusing: it should say
The flattened speed curve would give the same average speed. From my spreadsheet this shows an increase in power required for the low inertia load, with same average speed.
RE: Power calculation from torque
You are saying though that there is an increase in power required for low inertial load - this is modelling our problem if I undertand you correctly?
RE: Power calculation from torque
Keith Cress
Flamin Systems, Inc.- http://www.flaminsystems.com
RE: Power calculation from torque
I hope so. The effect, I think, comes from the higher speed 'pulses' which require ^3 more power. So to keep the same average speed, more power is required. If the trainer has more inertia, then speed will vary less. If the trainer doesn't have the same friction variation with speed, then it's probably even more different.
I checked the influence of phase shift torque/speed - v. little effect.
I checked the influence of calculation of speed more often - v. little effect. By the way, the speed sample every rotation seems to be OK. The torque samples must be made more often, since a once-per-rotation sample would not represent the rest of the rotation whereas the speed sample does represent the average of the rest of the rotation.
Magnetic braking: this may be more of a constant retarding torque, rather than a varying torque with speed i.e completely different compared to real world conditions.
RE: Power calculation from torque
These speed pulses could well be fatiguing the muscles more, the question is, how do we measure this? Given the limitations of the equipment that we are using.
In my mind a practical way to solve this is to develop weights that can be attached to the rear wheel of the bike. This would allow a rider to adjust the weights based on his body weight.
RE: Power calculation from torque
Keith Cress
Flamin Systems, Inc.- http://www.flaminsystems.com
RE: Power calculation from torque
Gunnar Englund
www.gke.org
--------------------------------------
100 % recycled posting: Electrons, ideas, finger-tips have been used over and over again...
RE: Power calculation from torque
From Bruce's initial statement and in his follow up posts it seems that the athelete can produce more power on the road as opposed to what he can achieve on an indoor trainer.
I am assuming that the levels of power are measured on the same bike using the same set of instruments in both cases.
Let me generalize for a while before getting back to specifics.
first up, power is defined as the the rate of doing work, and Work is defined as a force operating over a distance. (I know we all know this, but i like to define terms)
Now the way I see things is that the total work done is no different if the job is done fast or slow. At the end of the alocated time, the net work done is the same.
Lets think of two guys loading a pallet of bricks each, up onto a truck from the ground by hand.
Assume an impatient guy did it all in 20 minutes and then rested for 40, and compare him to a Plodder who paced himself and took exactly 60 minutes to do the job.
Now we look at the rate of doing work (or power). Over a 60 minute period, exactly the same amount of work was done in both instances, so by definition the average power is the same. (i am deliberately ignoring parasitic losses to keep it kind of simple
If you look at the instaneous power, though, you get a different story, Obviously the power profile will be nice and even for the guy that paced himself, where as for the impatient Fellow, he had a huge peak followed by a lull for 40 minutes.
Now in reality any two people doing such a task would each work at their own pace sometimes fast and sometimes slow and at different points in the cycle, but if they both did the same work in the same time frame they have both maintained the same FTP as Bruce Calls it.
Now I would assume that for elite athletes, Used to regular training, the variations in effort throughout the training session weather it be indoors on the trainer or outdoors would be minimal so I am fairly confident in saying that the minor variations mentioned by various people here will not really impact the total FTP in either situation. If the measuring equipment is doing it's job correctly then they should both read the correct power level for the duration.
I am a Motor & drives Kind of guy (28 years of it) and I have frequently come across statements along the lines that "one motor is delivering more power then another". The usual result is that the loads are different. Now one trueism about motors is
"A motor can only deliver power, that a load requires from it" The motor cannot "Push" excess power into a load if there is no need for it.
Now I see the athelete as being my Motor. He needs to output so much power to move his cycle along a road at given speed. If he goes faster, He will need more power, or if he starts to go up hill, he will need more power to maintain the constant speed.
So if he has a task to do in a set time (moving himself at a constant speed) he outputs whatever power required of the task. If it is easier (going down wind) then the power required will be less, because the losses will be less. If is fighting a head wind then the power to go a set speed will be higher.
I do not think Bruce says anything about different speeds, just different power output, so it could well be different environmental conditions requiring different levels of torque to maintain a given speed.
I keep mentioning constant speed because from my cycling days, i know that I felf more comfortable at a partivular speed, not faster, not slower, and i found myself tiring faster if i tried a different speed.
If you run a motor in constant torque mode, and the load drops, the motor will speed up until the torque of the load matches the motor torque, I do not think humans work this way.
Ok, So looking at the two different situations, What is different that could result in different torque requirements?
1
I am guessing that the outdoor track is level as would be the trainer, otherwise we wouldn't be having this discussion.
2
Obviously the outside ride will be influenced by the prevailing wind. whereas inside, the air resistance would be negligable.
3
What is the configuration of the trainer?. Do both of the cycle wheels turn or just the rear wheel? Does the rear wheel sit between two rollers or does it sit on top of a single roller?. What is the diameter of the roller?
I mention this because of what I call "Rolling Resistance". This is the extra forward force required to deform the cycle tyre as it flattens out due to the weight of the rider & bike on the road. A flat road will deform the tyre in one way, a small diameter roller will deform it in another way, and a large diameter roller even different again. Two contact points on the rear wheel will again be differnt as they are not on the bottom, but either side of the bottom. All of these things change the effort requred to move the cycle bearing a specific weight.
If the front wheel is not turning then it will offer no resistance to the forward motion meaning less torque is required to move at a given speed.
So my conclusion is that I do not think that your power differences are coming from the power measurement method but rather from the physical differences between the two activities. Two find the answer, You will need to define just how different the two situations are.
It is a bit long winded but I had nothing else to think about while sitting idle in a plane for a couple of hours this morning.
Tom
RE: Power calculation from torque
Tom
RE: Power calculation from torque
For your background, the trainers we are using have two types of speed vs. resistance curves - trainers based on magnetic braking have a linear curve i.e. resistance is proportional to speed, and fluid trainer which have an exponential resistance vs. speed curve.
We are measuring power either at the cranks or the rear hub of the bike. The source of the resistance downstream of the measuring point should theoretically be irrelevant.
Speed is not really a factor due to the gearing of the bike, so all the athlete has to do is generate the watts for a period of time (the watts he can generate for 60 minutes being his functional threshold power (ftp)). At the end of the day the athlete changes gears in order to achieve the cadence they desire at the speed that produces the appropriate resistance (based on the speed vs. resistance curve of the trainer).
Interesting problem this one - I'd be interested to know why car companies test power the way they do?