Notes on Motors

DC Motors

DC motors consist of one set of coils, called an armature, inside another set of coils or a set of permanent magnets, called the stator. Applying a voltage to the coils produces a torque in the armature, resulting in motion.

Small permanent magnet motors are cheap, but as size increases, the price advantage shifts to wound motors.

DC Motor Types


Torque vs RPM

For permanet magnet DC motors, there is a linear relationship between torque and rpm for a given voltage.

Torque versus RPM, showing linear relationship.

The maximum torque occurs at 0 rpm, and is called stall torque. The minimum torque (zero) occurs at maximum rpm, reached when the motor is not under a load, and is thus called free rpm. The formula for torque at any given rpm is:

T = Ts - (N Ts ÷ Nf)

where T is the torque at the given rpm N, Ts is the stall torque, and Nf is the free rpm.

Power, being the product of torque and speed, peaks exactly half way between zero and peak speed, and zero and peak torque. For the above graph, peak power occurs at 1500 rpm and 5 ft-lbs of torque; 1.4 hp. However, you do not generally want to run a motor at this speed, as it will draw much too much current and overheat. The above motor might be rated for only 0.5 hp (1 ft-lbs of torque at 2700 rpm).

Knowing the stall torque and the free rpm, we can derive two important constants for the motor in question. First is the induced voltage constant, which relates the back-voltage induced in the armature to the speed of the armature.

Ke = V ÷ Nf

where Ke is the induced voltage constant, Nf is the free rpm, and V is the voltage.

The second important constant is the torque constant which relates the torque to the armature current.

Kt = Ts ÷ V

where Kt is the torque constant, Ts is the stall torque, and V is the voltage.

Using these two constants, we can write the motor equation (these are all the same equation, solved for different variables):

T = Kt × (V - (Ke × N)

V = (T ÷ Kt) + (Ke × N)

N = (V - (T ÷ Kt)) ÷ Ke

where T is torque, V is voltage, N is rpm, Kt is the torque constant, and Ke is the induced voltage constant. The units don't matter, as long as they're the same units you used to calculate the constants.

Stall Torque ft-lbs
Free RPM rpm
Reference Voltage volts
Torque constant
Induced Voltage constant
RPM rpm
Torque ft-lbs
Voltage volts

Note that these formulas only apply to shunt motors and permanent magnet motors. Series motors behave differently.


Torque and Current

Torque is proportional to the product of armature current and the resultant flux density per pole.

T = K × f × Ia

where T is torque, K is some constant, f is the flux density, and Ia is the armature current.

In series wound motors, flux density approximates the square root of current, so torque becomes approximately proportional to the 1.5 power of torque.

T = K × Ia1.5±

where T is torque, K is some constant, and Ia is the armature current.

Speed, Voltage, and Induced Voltage

Resistance of the armature widings has only a minor effect on armature current. Current is mostly determined by the voltage induced in the windings by their movement through the field. This induced voltage, also called "back-emf" is opposite in polarity to the applied voltage, and serves to decrease the effective value of that voltage, and thereby decreases the current in the armature.

An increase in voltage will result in an increase in armature current, producing an increase in torque, and acceleration. As speed increases, induced voltage will increase, causing current and torque to decrease, until torque again equals the load or induced voltage equals the applied voltage.

A decrease in voltage will result in a decrease of armature current, and a decrease in torque, causing the motor to slow down. Induced voltage may momentarily be higher than the applied voltage, causing the motor to act as a generator. This is the essense of regenerative breaking.

Induced voltage is proportional to speed and field strength.

Eb = K × N × f

where Eb is induced voltage, K is some constant particular to that motor, N is the speed of the motor, and f is the field strength.

This can be solved for speed to get the "Speed Equation" for a motor:

N = K × Eb ÷ f

where N is rpm, K is some constant (the inverse of the K above), Eb is the induced voltage of the motor, and f is the flux density.

Note that speed is inversely proportional to field strength. That is to say, as field strength decreases, speed increases.


In a shunt-wound motor, decreasing the strength of the field decreases the induced voltage, increasing the effective voltage applied to the armature windings. This increases armature current, resulting in greater torque and acceleration. Shunt-wound motors run away when the field fails because the spinning armature field induces enough current in the field coils to keep the field "live".

In a series-wound motor, the field current is always equal to the armature current. Under no load, the torque produced by the motor results in acceleration. As speed increases, induced voltage would normally increase until at some speed it equalled the applied voltage, resulting in no effective voltage, no armature current, and no further acceleration; in this case, however, increasing speed decreases field current and strength, stabilizing induced voltage. Torque never drops to zero, so the motor continues to accelerate until it self-destructs.

Runaway does not occur in permanent magnet motors. Starter motors, electric car motors, and some golf cart motors are series wound, however, and can run away.


© 2003 W. E. Johns