Modern Control System Theory and Design, 2nd Edition

3.8.  ILLUSTRATIVE PROBLEMS AND SOLUTIONS

This section provides a set of illustrative problems and their solutions to supplement the material in Chapter 3.

I3.1. The torque–speed curve of a two-phase ac servomotor is illustrated where it is assumed that this characteristic can be represented by a straight line:

Figure I3.1

(a)  Derive an equation for this torque–speed relationship, relating T and ω.

(b)  The motor-shaft output power is the product of the motor’s derived torque, T, and its speed, ω:

P = Tω.

Derive the speed, ω_max, where the motor output power is a maximum.

(c)  Derive an expression for the corresponding maximum motor-shaft output power, P_max.

(d)  Determine the maximum motor-shaft output power, P_max, if the no-load motor speed, ω_NL, is 5000 rev/min and the stall torque, T₀, is 4 in-oz.

SOLUTION: (a)                      

(b)  Substituting the equation for T derived in part (a) into the equation for the motor-torque output power

P = Tω

we obtain the following:

To find ω_max, we differentiate P with respect to ω as follows:

Therefore,

To check that this is a maximum, we ...