Motor NotesRelations between Load Torque, Rotation Rate, and Motor Current of Brushed DC Motors

This article summarizes the relationships between load torque, rotation rate, and motor current, taking brushed DC motors as examples.

Relations between Load Torque, Rotation Rate, and Motor Current of Brushed DC Motors

The relations between load torque, rotation rate, and motor current for brushed DC motors are such that when the load torque is zero, the motor rotates at the maximum rotation rate, but when a load torque is applied, the rotation rate falls in proportion to the magnitude of the load torque. The motor current is smallest when the load torque is zero, and increases in proportion to the load torque, reach its maximum when the rotation rate is zero.

These relations are represented by equations below, based on the equivalent circuit for a brushed DC motor connected to a power supply.

The DC relationship in this equivalent circuit is:

Ea＝Ia×R＋Ec ……（1）

Because the motor generated voltage Ec is proportional to the rotation rate, it can be expressed as

Ec＝Ke×N ……（2）　where N is the rotation rate (rpm) and Ke is a generation constant (V/rpm)

Moreover, because the motor torque T (N･m) is proportional to the motor current, it can be expressed as

T＝Kt×Ia ……（3）　where Kt is a torque constant (N･m/A)

Substituting among equations (1) to (3) and rearranging to obtain the relation between the rotation rate N and the torque T, we obtain

N＝Ea/Ke－R/(Kt×Ke)×T ……（4）

The relationship of equation (4) of the rotation rate N to the torque T (T-N characteristic), and the relationship of equation (3) of the current I to the torque T (T-I characteristic), may be graphed as follows.

When the load torque on the motor is zero, the rotation rate is maximum, and as the load torque increases the rotation rate declines in proportion to the load torque, reaching zero at a certain load torque. The torque at this point is the maximum load torque.

The motor current increases in proportion to the load torque, and is maximum when the rotation rate becomes zero.

In actual motors, even when in the no-load state, friction in the motor itself and other factors cause some amount of load torque, so that even at the maximum rotation rate the current is not zero, and the motor characteristics are like those within the solid-line rectangle. If, in an ideal state, the load torque really were zero, the characteristics would be like those within the dotted-line rectangle.

As the power supply voltage is raised, the T-N characteristic is parallel-shifted upward and to the right, and the maximum rotation rate and maximum load torque are increased. As for the T-I characteristic, the maximum current value is larger.