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

- 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 × Ia^{1.5±}

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 :