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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°
As long as PM is positive, the system is stable. However, for a well damped system, PM should be between 30° and
45°. The phase margin of 70° given above indicated over-damped response.
System Design and Compensation
The closed-loop control system can be stabilized by a digital filter, which is preprogrammed in the DMC-21x5
controller. The filter parameters can be selected by the user for the best compensation. The following discussion
presents an analytical design method.
The Analytical Method
The analytical design method is aimed at closing the loop at a crossover frequency, ω
The system parameters are assumed known. The design procedure is best illustrated by a design example.
Consider a system with the following parameters:
K
= 0.2
t
J = 2 * 10
R = 2
K
= 2
a
N = 1000
The DAC of theDMC-21x5 outputs ±10V for a 16-bit command of ±32768 counts.
The design objective is to select the filter parameters in order to close a position loop with a crossover frequency
of ω c = 500 rad/s and a phase margin of 45 degrees.
The first step is to develop a mathematical model of the system, as discussed in the previous system.
Motor
M(s) = P/I = Kt/Js
Amp
K
= 2 [Amp/V]
a
DAC
K
= 10/32768 = .0003
d
Encoder
K
= 4N/2π = 636
f
ZOH
H(s) = 2000/(s+2000)
Compensation Filter
G(s) = P + sD
The next step is to combine all the system elements, with the exception of G(s), into one function, L(s).
Chapter 10 Theory of Operation ▫ 132
Nm/A
-4
2
kg.m
Ω
Amp/Volt
Counts/rev
2
2
= 1000/s
, with a phase margin PM.
c
Torque constant
System moment of inertia
Motor resistance
Current amplifier gain
Encoder line density
DMC-21x5 User Manual 1.0a1
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