Actuators - Introduction to Biomechatronics - page 102

For a coil with N turns, the maximum force generated by N parallel but insulated
conductors running along the segment AB is

F = NBIl (3.17)

The peak torque generated in the coil as a result of this force, F (N), is the product of
the force and the distance from AB to the rotational axis O - O'. If this distance is R (m),
then the peak torque, τ p (Nm), is

τ p = RNBIl (3.18)

Because the same amount of torque is developed by conductors running along segment
CD, the peak torque will be doubled

τ p = 2 RNBIl (3.19)

and l are fixed, leaving the user control of the
current I (A) only. Therefore, it is convenient to specify a motor in terms of the relationship
between the current and the torque. This is known as the torque constant, K m (Nm/A),
K m = I = 2 RNBl

For any manufactured motor, R , N , B ,

(3.20)

One further characteristic of DC motors is important. It was shown earlier that when
a conductor of length l (m) cuts through a magnetic field with strength B (weber/m 2

) at a

velocity v (m/s), an EMF, ε (V), is developed across the conductor.

ε = Bl v (3.21)

If the same assumptions are adopted as were for the development of the motor, it can
be shown that an EMF will develop across the brushes if an external torque is applied
to rotate the coils physically. The magnitude of this voltage is dependent on the motor
configuration and the angular velocity. This allows a constant of proportionality that
describes the relationship between the angular velocity and the voltage to be determined.
It is generally known as the back EMF constant, K e (V per rad/s).
When a direct current (DC) motor that is not driving a load is supplied with a voltage
V i (V), its speed will increase until the back EMF just equals the applied voltage. This is
known as the no-load speed, ω n (rad/s). As the load increases and the motor is required

to provide an increasing amount of torque, the motor speed will decrease in a reasonably
linear manner until the motor stalls and the torque is at a maximum. This is known as the
stall torque, τ s (Nm). In other words, there is a trade-off between how much torque a DC
motor can generate and how fast it spins, as shown in Figure 3-12.
It is important to realize that the graphs shown in Figure 3-12 are for a specific applied
voltage. If the applied voltage is altered, then the slope of the line remains the same but is
just displaced vertically, as shown in Figure 3-13.
Given the two points at the extremes of the graph, an equation can be written describing
the torque in terms of the rotation speed, or the motor speed in terms of the torque
τ mot = τ s − ωτ s
ω n

(3.22)

and

= (τ s − τ ) ω n
τ s

ω (3.23)

mot