# cbloom rants

## 5/08/2011

### 05-08-11 - Torque vs Horsepower

This sometimes confuses me, and certainly confuses a lot of other people, so let's go through it a bit.

I'm also motivated by this page : Torque and Horsepower - A Primer in which Bruce says some things that are slightly imprecise in a scientific sense but are in fact correct. Then this A-hole Thomas Barber responds with a dickish pedantic correction which adds nothing to our understanding.

We're going to talk about car engines, the goal is to develop sort of an intuition of what the numbers mean. If you look on Wikipedia or whatever there will be some frequently copy-pasted story about James Watt and horses pulling things and it's all totally irrelevant. We're not using our car engine to power a generator or grind corn or whatever. We want acceleration.

The horizontal acceleration of the car is proportional to the angular acceleration of the wheels (by the circumference of the wheels). The angular acceleration of the wheels is proportional to the angular acceleration of the flywheel, modulo the gear ratio in the transmission. The angular acceleration of the flywheel is proportional to the torque of the engine, modulo moment of inertia.

For a fixed gear ratio :

torque (at the engine) ~= vehicle acceleration

(where ~= means proportional)

So if we all had no transmission, then all we would care about is torque and horsepower could go stuff itself.

But we do have transmissions, so how does that come into play?

To maximize vehicle acceleration you want to maximize torque at the wheels, which means you want to maximize

vehicle acceleration ~= torque (at the engine) * gear ratio

where gear ratio is higher in lower gears, that is, gear ratio is the number of times the engine turns for one turn of the wheels :

gear ratio = (engine rpm) / (wheel rpm)

which means we can write :

vehicle acceleration ~= torque (at the engine) * (engine rpm) / (wheel rpm)

thus at any given vehicle speed (eg. wheel rpm held constant), you maximize acceleration by maximizing [ torque (at the engine) * (engine rpm) ] . But this is just "horsepower" (or more generally we should just say "power"). That is :

horsepower ~= torque (at the engine) * (engine rpm)

vehicle acceleration ~= horsepower / (wheel rpm)

Note that we don't have to say that the power is measured at the engine, because due to conservation of energy the power production must be the same no matter how you measure it (unlike torque which is different at the crank and at the wheels). Power is of course the energy production per unit time, or if you like it's the rate that work can be done. Work is force over distance, so Power is just ~= Force * velocity. So if you like :

horsepower ~= torque (at the engine) * (engine rpm)

horsepower ~= torque (at the wheels) * (wheel rpm)

horsepower ~= vehicle acceleration * vehicle speed

(note this is only true assuming no dissipative forces; in the real world the power at the engine is greater than the power at the wheels, and that is greater than the power measured from motion)

Now, let's go back to this statement : "any given vehicle speed (eg. wheel rpm held constant), you maximize acceleration by maximizing horsepower". The only degree of freedom you have at constant speed is changing gear. So this just says you want to change gear to maximize horsepower. On most real world engines this means you should be in as low a gear as possible at all times. That is, when drag racing, shift at the red line.

The key thing that some people miss is you are trying to maximize *wheel torque* and in almost every real world engine, the effect of the gear ratio is much more important that the effect of the engine's torque curve. That is, staying in as low a gear as possible (high ratio) is much more important than being at the engine's peak torque.

Let's consider some examples to build our intuition.

The modern lineup of 911's essentially all have the same torque. The Carrera, the GT3, and even the RSR all have around 300 lb-ft of torque. But they have different red lines, 7200, 8400 and 9400.

If we pretend for the moment that the masses are the same, then if you were all cruising along side by side in 2nd gear together and floored it - they would accelerate exactly the same.

The GT3 and RSR would only have an advantage when the Carrera is going to hit red line and has to shift to 3rd, and they can stay in 2nd - then their acceleration will be better by the factor of gear ratios (something like 1.34 X on most 2nd-3rd gears).

Note the *huge* difference in acceleration due to gearing. Even if the upshift got you slightly more torque by putting you in the power band of the engine, the 1.34 X from gearing is way too big to beat.

(I should note that in the real world, not only are the RSR/R/Cup (racing) versions of the GT3 lighter, but they also have a higher final drive ratio and some different gearing, so they are actually faster in all gears. A good mod to the GT3 is to get the Cup gears)

Another example :

Engine A has 200 torques (constant over the rpm range) and revs to 4000 rpm. Engine B has 100 torques and revs to 8000 rpm. They have the exact same peak horsepower (800 torques*krpm) at the top of their rev range. How do they compare ?

Well first of all, we could just gear down Engine B by 2X so that for every two turns it made the output shaft only made one turn. Then the two engines would be exactly identical. So in that sense we should see that horsepower is really the rating of the potential of the engine, whereas torque tells you how well the engine is optimized for the gearing. The higher torque car is essentially steeper geared at the engine.

How do they compare on the same transmission? In 1st gear Car A would pull away with twice the acceleration of Car B. It would continue up to 4000 rpm then have to change gears. Car B would keep running in 1st gear up to 8000 rpm, during which time it would have more acceleration than car A (by the ratio of 1st to 2nd gear).

So which is actually faster to 100 mph ?

You can't answer that without knowing about the transmission. If gear changes took zero time (and there was no problem with traction loss under high acceleration), the faster car would be the higher torque car. In fact if gear changes took zero time you would want an infinite number of gears so that you could keep the car at max rpm at the time, not because you are trying to stay in the "power band" but simply because max rpm means you can use higher gearing to the wheels.

I wrote a little simulator. Using the real transmission ratios from a Porsche 993 :

```
Transmission Gear Ratios: 3.154, 2.150, 1.560, 1.242, 1.024, 0.820
Rear Differential Gear Ratio: 3.444
Rear Tire Size: 255/40/17  (78.64 inch cirumference)
Weight : 3000 pounds

```
and 1/3 of a second to shift, I get :
```
200 torque, 4000 rpm redline :

time_to_100 = 15.937804

100 torque, 8000 rpm redline :

time_to_100 = 17.853252

```
higher torque is faster. But what if we can tweak our transmission for our engine? In particular I will make only the final drive ratio free and optimize that with the gear ratios left the same :
```
200 torque, 4000 rpm redline :

c_differential_ratio = 3.631966
time_to_100 = 15.734542

100 torque, 8000 rpm redline :

c_differential_ratio = 7.263932
time_to_100 = 15.734542

```
exact same times, as they should be, since the power output is the same, with double the gear ratio.

In the real world, almost every OEM transmission is geared too low for an enthusiast driver. OEMs offer transmission that minimize the number of shifts, offer over-drive gears for quiet and economy, etc. If you have a choice you almost always want to gear up. This is one reason why in the real world torque is king ; low-torque high-power engines could be good if you had sufficiently high gearing, but that high gearing just doesn't exist (*), so the alternative is to boost your torque.

(* = drag racers build custom gear boxes to optimize their gearing ; there are also various practical reasons why the gear ratios in cars are limitted to the typical range they are in ; you can't have too many teeth, because you want the gears to be reasonably small in size but also have a minimum thickness of teeth for strength, high gear ratios tend to produce a lot of whine that people don't like, etc. etc.)

One practical issue with this these days is that more and more sports cars use "transaxles". Older cars usually had the transmission up front and then a rear differential. It was easy to change the final drive ratio in the rear differential so all the old American muscle cars talk about running a 4.33 or whatever different ratios. Nowadays lots of cars have the transmission and rear differential together in the back to balance weight (from the Porsche 944 design). While that is mostly a cool thing, it makes changing the final drive much more expensive and much harder to find gears for. But it is still one of the best mods you can do for any serious driver.

(another reason that car gear ratios suck so bad is the emphasis on 0-60 times means that you absolutely have to be able to reach 60 in 2nd gear. That means 1st and 2nd can't be too high ratio. Without that constraint you might actually want 2nd to max out at 50 mph or something. There are other stupid goals that muck up gearings, like trying to acheive a high top speed).

Let's look at a final interesting case. Drag racers often use a formula like :

```
speed at end of 1/4 mile :

MPH = 234 * (Horsepower / Pounds) ^ .3333

```
and it is amazingly accurate. And yet it doesn't contain anything about torque or gear ratios. (they of course also use much more complex calculators that take everything into account). How does this work ?

A properly set up drag car is essentially running at power peak the whole time. They start off the line at high revs, and then the transmission is custom geared to keep the engine in power band, so it's a reasonable approximation to assume constant power the entire time.

So if you have constant power, then :

```
d/dt E = P

d/dt ( 1/2 mv^2 ) = P

integrate :

1/2 mv^2 = P * t

v^2 = 2 * (P/m) * t

distance covered is :

x = 1/2 a t^2

and P = m a v

a = (P/m) / v

so

t = sqrt( 2*x*v / (P/m) )

sqrt( 2*x*v / (P/m) ) = v^2 / ( 2 * (P/m) )

simplify :

v = 2 * ( x * (P/m) ) ^(1/3)

```
which is the drag racer's formula. Speed is proportional to distance covered times power-to-weight to the one third power.

If you're looking at "what is the time to reach X" (X being some distance or some mph), the only thing that matters is power-to-weight *assuming* the transmission has been optimized for the engine.

Currently the two figures that we get to describe a car's engine are Horsepower (at peak rpm) and Torque (at peak rpm) (we also get 0-60 and top speed which are super useless).

I propose that the two figures that we'd really like are : Horsepower/weight (at peak rpm) and Horsepower/weight (at min during 10-100 run).

Let me explain why :

(Power/weight) is the only way that power ever actually shows up in the equations of dynamics (in a frictionless world). 220 HP in a 2000 pound car is better than 300 HP in a 3000 pound car. So just show me power to weight. Now, in the real world, the equations of dynamics are somewhat more complicated, so let's address that. One issue is air drag. For fighting air, power (ignoring mass) is needed, so for top speed you would prefer a car with more power than just power to weight. However, for braking and turning, weight is more important. So I propose that it roughly evens out and in the end just showing power to weight is fine.

Now, what about this "Horsepower/weight (at min during 10-100 run)" ? Well let's back up a second. The two numbers that we currently get (Power and Torque both at their peak) give us some rough idea of how broad the power band of an engine is, because Power is almost always at peak near the max rpm, and Torque is usually at peak somewhere around the middle, so a higher torque number (power being equal) indicates a broader power band. But good gearing (or bad gearing) can either hide or exagerate that problem. For example a tuned Honda VTEC might have a narrow power band that's all in the 7k - 10k RPM range, but with a "crossed" transmission you might be perfectly happy never dropping out of that rev range. Another car might have a wide power band, but really huge gear steps so that you do get a big power drop on shifts. So what I propose is you run the cars from 10mph-100 , shifting at red line, and measure the *min* horsepower the engine puts out. This will tell you what you really want to know, which is when doing normal upshifts do you drop out of the power band, and how bad is it? eg. what is the lowest power you will experience.

Of all the numbers that we actually get, quarter mile time is probably the best.