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This site:(https://www.motioncontroltips.com/faq-what-are-dc-shunt-motors-and-where-are-they-used/ ) explains the speed regulation of DC shunt motor as follows: If the load on the motor is increased, the armature rotation slows and back EMF is reduced, since back EMF is proportional to speed. With less back EMF voltage and a constant supply voltage the net voltage increases. The increase in net voltage results in an increase in armature current. Since torque is proportional to armature current, torque also increases. Finally, this increased torque allows the motor to increase its speed and compensate for the slowdown due to loading. Hence, a DC shunt motor is able to self-regulate its speed, and can be referred to as a constant speed motor.

Considering that torque is inversly related to speed for a DC shunt motor, wouldn't this entire explanation be wrong as the last line mentions increase in speed due to increase in torque ?

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Considering that torque is inversely related to speed for a DC shunt motor, wouldn't this entire explanation be wrong as the last line mentions increase in speed due to increase in torque ?


torque is inversely related to speed

This means

The maximum Torque is applied at Zero speed, because torque is inversely related to speed.

However

as speed increases, Torque reduces

until

The required design speed and it's associated value of Torque are reached.

If additional load is applied speed reduces

as speed reduces, Torque increases

Which then takes us back to where we started. The increased Torque will accelerate the speed, similar to what it did on starting, only in this case we are not increasing from zero or stall speed. But some point slower than the designed speed of the D.C. Shunt motor.

as speed increases, Torque reduces

until

The required speed and it's New associated value of Torque are reached.

Therefore if you relate this to the explanation. And view them as steps in a process not simultaneous events,

The explanation is correct.

this increased torque

allows the motor to increase its speed and compensate for the slowdown due to loading

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