TorqueSpeedAndPower

From Simreal

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The force you ultimately use from your motor is called torque, from the Latin torques via torquere or to twist.

Torque

Torque, τ, is a twisting force, defined by a force at a distance from the center of rotation. In SI units it is described in newton-meters and in English it is pound-feet.

SI	English
newton-meters {Nm}	pound-feet {lb/ft} pound-inches {lb/in} ounce-inches {oz/in}
1 Nm = 0.737561 lb/ft 1 Nm = 8.85075 lb/in 1 Nm = 141.6116 oz/in	1 lb/ft = 1.35582 Nm 1 lb/in = 0.1129848 Nm 1 oz/in = 7.06155E-03 Nm
	1 lb/ft = 12 lb/in 1 lb/in = 16 oz/in 1 lb/ft = 192 oz/in

Torque applied to a wheel, pulley, or sprocket creates a tangential force that is based on the torque applied at the axel and the radius of the wheel.

When no load (friction, resistance, or other force pushing back) is applied to the motor, it really isnít generating any torque, itís just spinning freely at a rate defined by its design and the applied voltage. As resistance is applied to the motor, it pushes back and slows down. Enough resistance will stop the motor entirely, or stall it. At this point, the motor is generating its maximum torque.

There are typically two torque ratings specified for a motor; stall torque τ_stall and continuous torque rating τ_cont. While motors may burn out if operated at τ_stall for very long, they can run indefinitely at the τ_cont load.

As the torque demanded from the motor increases, so will the motorís hunger for current. The current draw of a motor is directly related to its torque, as shown in the diagram to the right. If the current supplied to the motor is limited, the torque will also be limited.

Motors have a current rating and exceeding that rating for any length of time can melt something important. Most motors can carry a 400% current overload for a short time (perhaps a minute). Larger motors with higher current ratings will have a lower resistance to electrical flow, and hence less internal heating.

Speed

Speed in a motor (or any rotating object) is defined as a rotational speed or angular velocity, ω. ω is the number of revolutions the shaft makes per minute (RPM), revolutions per second (RPS), radians per second (rad/s), or degrees per second (deg/s).

Most motors specify ω in RPM, but the conversions between units are fairly straightforward.

ωrpm = 60ωrps
ωrps = ωrpm / 60

ωrps = 360ωdeg/s 
ωdeg/s = wrps / 360

ωrps = 2πωrad/s
ωrad/s = ωrps / 2π

Given the angular velocity you can find the tangential (rolling) velocity for any given radius. The calculation of the wheelís circumference is based in radians, so the natural units for ω for calculating the tangential velocity is radians per second. Specifying the wheelís radius r in inches gives:

v = r * ω

vin/s = rin * ωrad/s

Given ω in revolutions per second gives:

vin/s = 2πrin * ωrad/s

Of course, the feet per second and miles per hour are derived easily from this:

vft/s = vin/s / 12

vmph = 3,600vft/s / 5,280 = vft/s / 1.467

The no-load speed of a motor ω_nl is determined by both its design and the voltage applied to it. The higher the voltage the higher the speed, as shown in the diagram. The upper speed limit is defined by mechanical factors inside the motor such as increasing friction, vibration and imbalance, electrical arcing in the commutator, and other problems.

Increasing the applied voltage only increases the speed and not the current draw. Current draw is determined only by torque. |}

As torque is applied to the motor, it begins to slow down until it stalls (stops) at ω_s=0. The speed vs. torque relationship is very nearly linear for a motor with no (or very simple) gearing.

For a given speed ω (and a few motor parameters from the manufacturer), you can estimate the torque on the motor τ_mot as:

τmot = τstall * ( 1 - (ω / ωnl))

For a known load τ, you can estimate the speed of the motor ω_mot as:

ωmot = ωnl * (1 - (τ / τstall))

Power

Power is the work that the motor is able to perform, as defined by a force applied over time. As such power is a combination of both torque and angular velocity. Power is specified in watts (W), newton-meters per second (Nm/s), pound-feet per second (lb-ft/s), or horsepower (HP).

PW = PNm/s
PW = Plb-ft/s / 1.355818	Plb-ft/s = 1.355818 PW
PW = PHP / 745.6999	Plb-ft/s = PHP / 550

Since power is a combination of both torque and speed there is no power (that is, work being done) at zero speed and maximum torque, or at maximum speed but no torque. Maximum power lies in the middle of the torque and speed curves.

The power output of the motor at a given torque τ is given by:

Pτ = ωnl τ - ( ωnl / τstall) τ2

Likewise, at a given speed ω, the power is:

Pω = τstall ω - ( τstall / ωnl ) ω2

As these formulas show, the greatest power is found at the middle torque and speed.