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142 • Chapter 10 Theory of Operation DMC-2X00
Σ
50+0.980s
318
V
ENCODER
500
S
2
FILTER
2000
S+2000
0.0003 4
ZOH DAC
AMP
MOTOR
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.
1
4
0.1
50 200 2000 W (rad/s)
Magnitude
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°