J = 2.10-4 kg.m2
System moment of inertia
R = 2
Ω
Motor resistance
Ka = 2
Amp/Volt
Current amplifier gain
N = 1000
Counts/rev
Encoder line density
The DAC of the DMC-1600 o/-10V for a 14-bit command of +/-8192 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/Js2 = 1000/s2
Amp
Ka = 2 [Amp/V]
DAC
Kd = 10/32768 = .0003
Encoder
Kf = 4N/2
π
= 636
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).
L(s) = M(s) Ka Kd Kf H(s) =3.17
∗
106/[s2(s+2000)]
Then the open loop transfer function, A(s), is
A(s) = L(s) G(s)
Now, determine the magnitude and phase of L(s) at the frequency
ω
c = 500.
L(j500) = 3.17
∗
106/[(j500)2 (j500+2000)]
This function has a magnitude of
|L(j500)|
=
0.00625
and a phase
Arg[L(j500)] = -180
°
- tan-1(500/2000) = -194
°
G(s) is selected so that A(s) has a crossover frequency of 500 rad/s and a phase margin of 45
degrees. This requires that
|A(j500)|
=
1
Arg [A(j500)] = -135
°
However, since
DMC-1600
Chapter 10 Theory of Operation
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