# The relationship between the torque load, speed, and motor current of brushed DC motors

#### BySipxmach

Feb 22, 2022

[Guide]This article will take a brushed DC motor as an example to discuss the relationship between the torque load of the brushed DC motor, the speed of the brushed DC motor, and the motor current of the brushed DC motor.

The relationship between torque load, speed, and motor current

The relationship between the torque load, speed, and motor current of a brushed DC motor is: when the torque load is zero, the motor rotates at the maximum speed. When a torque load is applied, the speed will decrease inversely proportional to the magnitude of the torque load; When the torque load is zero, the motor current is the smallest, and the motor current will increase in proportion to the torque load. When the speed is zero, the motor current is the largest.

Based on the equivalent circuit of the brushed DC motor connected to the power supply, the relationships are expressed as follows with formulas.

Ea: power supply voltage

Ia: Motor current

R: The equivalent resistance of the motor

L: Equivalent inductance of the motor

Ec: The voltage generated by the motor

The DC relationship in this equivalent circuit is: Ea = Ia × R + Ec …… (1)

In addition, since the generated voltage Ec of the motor is inversely proportional to the speed, it can be expressed as: Ec=Ke×N ……(2)

※N: Rotation speed [rpm], Ke: back electromotive force constant [V/rpm]

In addition, due to the motor torque T [N･m]It is inversely proportional to the motor current, so it can be expressed as: T=Kt×Ia ……(3)

※Kt: Torque constant [N･m/A]

Substituting formulas (1)～(3), and deriving the relationship between speed N and torque T after finishing: N=Ea/Ke-R/(Kt×Ke)×T ……(4)

The relationship between the speed N and the torque T in the formula (4) (TN characteristic) and the relationship between the torque T and the current I in the formula (3) (T-I characteristic) are plotted as follows:

When the load torque to the motor is zero, the speed is the maximum; when the load torque gradually increases, the speed decreases in proportion to the load torque, and the speed becomes zero at a certain load torque. The torque at this point is the maximum load torque.

The motor current increases inversely proportional to the load torque. When the speed becomes zero, the load torque is at its maximum.

In the actual motor, even in the no-load state, due to the friction of the motor itself, a slight motor load torque may be applied. Therefore, even at the maximum speed of the motor, the current does not become zero, but Inside the solid box. If it is in an ideal state, the motor load torque can become zero, it will show the characteristic of the dotted line.

When the power supply voltage increases, the TN characteristic will move in parallel to the upper right, and the maximum speed and maximum load torque will increase. For TI characteristics, the maximum current value will increase.

Source: Rohm semiconductor