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V
Σ
Figure 10.7 - Mathematical model of the control system
The open loop transfer function, A(s), is the product of all the elements in the loop.
A = 390,000 (s+51)/[s 2 (s+2000)]
To analyze the system stability, determine the crossover frequency, ω c at which A(j ω c ) equals one.
This can be done by the Bode plot of A(j ω c ), as shown in Fig. 10.8.
Magnitude
4
1
0.1
Figure 10.8 - Bode plot of the open loop transfer function
For the given example, the crossover frequency was computed numerically resulting in 200 rad/s.
Next, we determine the phase of A(s) at the crossover frequency.
A(j200) = 390,000 (j200+51)/[(j200) 2 . (j200 + 2000)]
α = Arg[A(j200)] = tan -1 (200/51)-180° -tan -1 (200/2000)
α = 76° - 180° - 6° = -110°
Finally, the phase margin, PM, equals
PM = 180° + α = 70°
140 • Chapter 10 Theory of Operation
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FILTER
50+0.980s
ENCODER
318
50
200
ZOH
DAC
2000
0.0003
S+2000
2000
W (rad/s)
AMP
MOTOR
500
4
2
S
DMC-1000
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