Torque is the rotational version of force. The more torque an engine produces, the more force it can exert at the rim of a flywheel of a given radius. Power

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The Tradeoffs between Power and Torque in Engines and Motors
So why do the power and torque figures that are quoted in the technical specifications for engines and motors matter so much in vehicle design?
Torque is the rotational version of force. The more torque an engine produces, the more force it can exert at the rim of a flywheel of a given radius.
Power is force multiplied by speed. The more power an engine generates, the more work it can do in a given time.
A typical automobile engine will run evenly from somewhere around idle speed (about 800 revolutions per minute or rpm) to its "red line" which might be anywhere from 4000 rpm for an older engine to 12000 rpm for a Formula One engine. The power and torque will vary through this range.
Torque increases as rotational speed increase from idle to a certain figure and then falls as the rotational speed increase above this figure. Acceleration is proportional to the amount of force pushing the vehicle forward; so maximum acceleration in a given gear is obtained when maximum torque is obtained.
Power is force (~torque) multiplied by speed (~rotational speed) so power increases with rotational speed up to and past the point of maximum torque. However, at still higher rotational speed the engine starts to be limited by the amount of air that it can draw in (4 valves per cylinder help) and torque then decreases more rapidly than the rotational speed increase, and therefore power also decreases. (Electric motors, though their torque and horsepower behavior differ considerably from that of internal combustion engines, experience a similar loss of torque and power at high rotational speeds due to electromagnetic effects.)
Power is force multiplied by speed, and maximum acceleration is obtained by having maximum propulsive force at the wheels. Use of a low gear ratio multiplies the engine torque at the wheel (at the price of having the engine rotate more quickly). Maximum acceleration at a given speed is obtained by having the engine operate at maximum power.
Driving a car is easier and more relaxed if it has a flexible engine. Flexibility even becomes a safety issue in four-wheel drive vehicles. Maximum torque is obtained at a certain rotational speed, and maximum power at a higher rotational speed. An engine is flexible if these maximums occur at widely different rotational speeds, say in the ratio 1:2 or more.
A vehicle engine operating at a rotational speed above its maximum-torque point is in a "stable" speed regime. If it slows down by a small amount (due to the vehicle encountering an incline, head-wind, etc.) engine torque will tend to increase and resist the slowing. Conversely, if it speeds up by a small amount, torque will tend to decrease and discourage a further increase in speed.
A vehicle engine operating at a rotational speed below its maximum-torque point is in an unstable speed regime. If it slows down by a small amount, the torque decreases and its speed will fall further. The driver can compensate by opening the throttle. Conversely, if the speed increases then the torque increases and the speed increases even more. The driver can compensate by closing the throttle (or risk a speeding ticket). The driver has to actively compensate for these variations (or has to rely on automatic cruise control), so the car is considerably less fun and relaxing to drive.
The driver cannot correct for a fall in engine rotational speed and loss of torque if the throttle is already wide open, except by changing into a lower gear. This can be a safety matter on a steep road. If you start up a steep hill in too high a gear, or have to slow down due to obstacles, for example, the engine may fall below its maximum-torque point and be unable to recover from a downward speed spiral. Changing gears causes a loss of more speed and, additionally, traction may be lost as the clutch engages. Inaction on the part of the driver may lead to a stalled engine and a forced restart on a dangerous slope. These risks are minimized if maximum engine torque is designed to occur at low engine rotational speed.
So, if you are into racing, particularly Formula One racing, you want an engine with the highest possible power output and this is best achieved by generating maximum power at very high rpm. Such engines also tend to generate maximum torque at very high rpm and are inflexible and difficult to drive -- but that is what the driving aces are paid big bucks for!
If you are into four wheel driving or drag racing, you want a very flexible engine that generates maximum torque, and lots of it, at low rotational speed and generates maximum power at a higher rotational speed - a ratio of 1:2 is good. A ratio of 1:3 (e.g. 1500rpm and 4500rpm) is excellent. As a side benefit, which is especially valuable for passenger vehicles, engine wear is largely determined by piston speeds and producing high torque at low rpm allows you to use a high "over-drive" top gear for quiet highway cruising and for long engine life.
[This information was adapted from an article on Power and Torque appearing on the web site.]
Graphing Torque and Horsepower for an Internal Combustion Engine
If you plot the torque and the horsepower versus the rpm values for an engine, what you end up with are torque and horsepower curves for that engine. This is what a machine called a dynamometer does. Typical torque and horsepower curves for a high-performance engine might look like those in the graph below (these happen to be the curves for the 300-horsepower engine in the Mitsubishi 3000 twin-turbo). The horizontal axis is rotational speed (in rpm) and the vertical axis is horsepower (for the horsepower curve) or torque (in ft-lbs for the torque curve).
Notice the very “flat” torque curve, with a maximum at 2500 rpm and only a gradual drop off in torque at the high end. Notice that the horsepower increases almost linearly until it peaks at about 6500 rpm. The ratio here is 2.6/1. This makes for a very “sweet” ride!
[This information is adapted from an article on the web site.]

The torque and horsepower curves for a typical small dc motor differ a bit from the internal combustion engine. The torque curve is very linear with maximum torque developed at zero rotational speed, and the horsepower curve peaks near the middle of the range. The following MATLAB plot for the small VEX dc motor illustrates these characteristics.

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