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

S.JAYAJAYA GANESH | 284 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`

Where

`"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.