How to calculate the Torque Calculation for DC Motor With Explanation?

S.JAYAJAYA GANESH | 342 Views | digital electronics | 15 Aug 2016


  • 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.

Torque vs RPM :

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


"T" is torque, 
"V" is voltage,
"N" is rpm,
"Kt" is the torque constant, and
"Ke" is the induced voltage constant.
Theory (Torque & 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 windings 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 essence 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.
Runaway :
  • 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.