Page 569 - Mechanical Engineers' Handbook (Volume 2)

560 Servoactuators for Closed-Loop Control

X 2 X (n /n ) 2
1
2
2
X inductive reactance due to magnetic field
m
Z stator impedance R jX 1
1
1
Z reflected impedance of rotor at S (R 2 /S ) jX 2
R
R
2
Z 2 reflected impedance of rotor at (2 S ) R 2 /(2 S ) jX 2
R
R
Z impedance due to magnetic field generation in the stator [(1/R ) (1/jX )] 1
m
m
m
phase angle between fixed and control voltages
motor speed (rad/s)
m
synchronous speed (rad/s)
s
Figure 13 shows typical torque–speed characteristics for a two-phase ac motor. The
characteristics are clearly nonlinear. However, about the origin they may be treated as linear.
Dynamic Model
Combining the equation of motion, the electrical lag (due to stator inductance and resistance),
and the torque–speed characteristic of Eq. (20) yields a nonlinear dynamic model. For an
approximate static analysis, Eq. (20) may be linearized about an operating point. For many
servosystem applications the most interesting and useful operating point is k 0 and S
R
1.0. This linearization yields

Figure 13 Torque–speed characteristics of a two-phase ac servomotor. (From J. E. Gibson and F. B.
Tueter, Control System Components, McGraw-Hill, New York, 1958, p. 288.)