142 • Chapter 10 Theory of Operation DMC-1500
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 degrees and 45 degrees. The phase margin of 70 degrees given above indicated
overdamped response.
Next, we discuss the design of control systems.
System Design and Compensation
The closed-loop control system can be stabilized by a digital filter, which is preprogrammed in the
DMC-1500 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
c
, with a phase
margin PM. The system parameters are assumed known. The design procedure is best illustrated by a
design example.
Consider a system with the following parameters:
K
t
= 0.83 Nm/A Torque constant
J = 2.10
-4
kg.m
2
System moment of inertia
R = 2
Ω
Motor resistance
K
a
= 2 Amp/Volt Current amplifier gain
N = 1000 Counts/rev Encoder line density
The DAC of the DMC-1500 outputs +/-10V for a 16-bit command of +/- 32,768 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 = K
t
/Js
2
= 4150/s
2
Amp
K
a
= 2 [Amp/V]
DAC
K
d
= 20/65536 = .0003
Encoder
K
f
= 4N/2π = 636
ZOH
H(s) = 2000/(s+2000)
Compensation Filter
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