Horsepower v torque

There is a lot of confusion about the relationship between horsepower and torque, I was guilty until I decided to do some investigating. Here is my summary. Lets cut to the chase, in the context of this article it is torque and torque alone that accelerates a car. That's great you think, my big block sticks out 750lbf-ft of torque and that small high revving engine only puts out 250lbf-ft, I'll blow him away. Sadly its not quite as straight forward as that, there is a small caveat I did not mention, what I should of said was it's torque at the wheel and torque at the wheel alone that accelerates a car. All things being equal the car that has the most torque at the wheel will be accelerating harder. Well your first reaction may be 750lbf-ft is 3 times 250 so there is no contest, wrong. It's all about gearing, the best way to explain is an example, see diagram below.

 

Torque

OK, we have two engines both putting out 714hp, one makes 750lbf-ft at the engine and the other 250lbf-ft. The important factor is the speed at which the torque is made. The small engine is revving 3 times faster, so we can gear it down with a 3:1 ratio to achieve the same rpm at the output shaft (the wheels in real life). The consequences of this is the torque is multiplied 3 times, hey presto we have the same torque at the output shaft.

So after having said that torque alone accelerates a car, the torque figure is meaning less if we don't know at what rpm the torque is being made, this is where horsepower come in. Think of horsepower as a way of factoring in rpm to the torque equation.

You may be surprised, I know I was, but horsepower is simply a calculation based on rpm and torque. Dynos and rolling roads only ever measure torque. Horsepower is calculated after. Here's the formula:


Horsepower = Torque * RPM / 5252


So how is this formula arrived at, and where did 'horsepower' come from?

Well it's all down to the Scottish engineer James Watt (1736 to 1819), he established that a horse could pull a 550lb weight up from a coal mine at a rate of one foot every second, for an eight hour shift. This converts to 33,000 foot pounds per minute. He published these observations, stating this figure to be the equivalent of one horsepower.

We need to convert from the rotary motion of an engine to a linear motion, like the horse example. Remember we can only measure the torque of the engine, this is expressed in pound feet. A pound foot of torque is the twisting force necessary to support a one pound weight on a weightless horizontal bar, one foot from the centre of rotation. Lets rotate the one pound weight one revolution, the distance travel is 6.2832, that's Pi multiplied by the diameter, 2 feet. We have now done 6.2832 foot pound of work.

One horsepower equals 33,000 foot pounds per minute, so if we divide 6.2832 in to 33,000 we can conclude that we at 5252 rpm we are producing one horsepower, we are moving our one pound weight 33,000 feet every minute.
Therefore if we multiply the torque by rpm and then divide by 5252 we can calculate the horsepower.

Lets see what we now know:

  • A car will be accelerating hardest in any given gear when at its peak torque
  • Horsepower is a calculation based on torque.
  • Torque means nothing without knowing what rpm it occurs at.
  • Horsepower factors rpm in to the torque equation.
  • It's better to make torque at high rpm rather than low rpm, so we can take advantage of gearing.

So which is king?
Lets look at a constant rpm example, a really interesting one is a constantly variable transmission, CVT. These transmissions vary the gear ratio on the fly and effectively have an infinite number of gears. Do you think we should keep the engine at peak torque or peak horsepower for maximum acceleration?

From my previous statement: A car will be accelerating hardest in any given gear when at its peak torque, it's got to be peak torque, WRONG. Lets do some calculations to prove why.

Engine Rpm Engine Torque Engine hp Variable Gear Ratio Torque at Wheel Target Wheel rpm
1800 53 18 4.50 238.50 400
2100 123 49 5.25 645.75 400
2400 184 84 6.00 1104.00 400
2700 268 138 6.75 1809.00 400
3000 291 166 7.50 2182.50 400
3300 306 192 8.25 2524.50 400
3900 333 247 9.75 3246.75 400
4200 349 279 10.50 3664.50 400
4800 346 316 12.00 4152.00 400
5100 344 334 12.75 4386.00 400
5400 341 351 13.50 4603.50 400
5700 339 368 14.25 4830.75 400
6000 325 371 15.00 4875.00 400
6300 306 367 15.75 4819.50 400

In the above tabulation, you can see typical torque and horsepower values, the column on the right is a chosen nominal wheel speed, it could have been any speed, as long as it's constant. The idea is to see where peak torque at the wheels occurs, in other words the point at maximum acceleration. The Gear ratio column is Engine rpm divided by the Target wheel rpm. The Torque at wheel column is Gear ratio multiplied by Engine torque.

Let's look down the Torque at wheel column to find the highest value: 4875, move to the left and you will see that we are at peak horsepower!

Oh boy this is getting really confusing, does this then mean there must be a point when accelerating through the gears that maximum acceleration occurs at peak horsepower and not peak torque?

No, acceleration still occurs at peak torque. This is because we are in a fixed gear situation. We are using the increase in engine rpm to move the car forward. The CVT uses constant rpm and changes the gear ratio to move the car forward.

Lets add a few more columns to the chart just to clarify things a bit more.

Constantly variable transmission Fixed gear transmission
Engine Rpm Engine Torque Engine hp Variable Gear Ratio Torque at Wheel Target Wheel rpm Fixed Gear Ratio Torque at Wheel Wheel Speed
1800 53 18 4.50 238.50 400 15 795 120
2100 123 49 5.25 645.75 400 15 1845 140
2400 184 84 6.00 1104.00 400 15 2760 160
2700 268 138 6.75 1809.00 400 15 4020 180
3000 291 166 7.50 2182.50 400 15 4365 200
3300 306 192 8.25 2524.50 400 15 4590 220
3900 333 247 9.75 3246.75 400 15 4995 260
4200 349 279 10.50 3664.50 400 15 5235 280
4800 346 316 12.00 4152.00 400 15 5190 320
5100 344 334 12.75 4386.00 400 15 5160 340
5400 341 351 13.50 4603.50 400 15 5115 360
5700 339 368 14.25 4830.75 400 15 5085 380
6000 325 371 15.00 4875.00 400 15 4875 400
6300 306 367 15.75 4819.50 400 15 4590 420


In the Fixed gear ratio column there is a value of 15, this is the ratio at which we made the most torque at the wheel with the CVT. The wheel speed is obviously changing but at 6000 rpm the CVT and fixed gear data match. Where does maximum acceleration occur? At maximum torque, but of course at a different wheel speed.

The fixed gear system makes 5235 lbf-ft torque at 280 wheel speed, out of interest lets see what torque would be made with a CVT:

Constantly variable transmission
Engine Rpm Engine Torque Engine hp Variable Gear Ratio Torque at Wheel Target Wheel rpm
1800 53 18 6.43 340.71 280
2100 123 49 7.50 922.50 280
2400 184 84 8.57 1577.14 280
2700 268 138 9.64 2584.29 280
3000 291 166 10.71 3117.86 280
3300 306 192 11.79 3606.43 280
3900 333 247 13.93 4638.21 280
4200 349 279 15.00 5235.00 280
4800 346 316 17.14 5931.43 280
5100 344 334 18.21 6265.71 280
5400 341 351 19.29 6576.43 280
5700 339 368 20.36 6901.07 280
6000 325 371 21.43 6964.29 280
6300 306 367 22.50 6885.00 280

6964 lbf-ft, the sooner they perfect this technology the better!

Well I've got side tracked a little, back to question in hand 'which is king' Torque or Horsepower? Well the question is in fact a non-starter, as stated, torque means absolutely nothing without rpm, so you cannot compare the two.
It's like saying which is faster 100miles or 100mph! Dumb question hey?
An excellent analogy I've seen on the web is the Waterwheel that produces 2600 lbf-ft of torque, that's impressive, if we connected the water wheel up the wheels of a car then we could shoot up to 60mph in a flash, without the Waterwheel noticing. Wrong! Back to gearing again, the ratio required to spin the wheels at 60mph would leave us with 43 lbf-ft of torque at the wheels, which equates to 6hp!

Answer: Horsepower is King! (the more horsepower the more torque at the wheel)

Back to the large and small engine example at the beginning, which will accelerate a car down the quarter mile the quickest? They both put out the same torque at the wheel, so I guess as the small engine is considerably lighter, it's the winner. Again its not quite as straight forward as that. What we are really interest in is the average torque at the wheels through out the complete run. High revving engines by there very nature have a small power band, in other word there torque curve is not very flat compared to large displacement engines. To keep the average torque at the wheel as high as possible we need loads of gears, and changing gear takes time. So to counter its weight disadvantage the large displacement engine has high torque through out the working rev range, therefore needing less gears. Never straight forward is it!

There you go, Torque v Horsepower according to my world!

If you have any comments please let me know, good or bad, this is a working document.