Control of AC servo motors - University of...
Transcript of Control of AC servo motors - University of...
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436-459 Advanced Control and Automation
Control of AC servo motors
• 3-phase permanent magnet synchronous motors– “brushless DC” motor
• trapezoidal back-EMF profile• rectangular pulse current profile• requires only 3 Hall-effect position sensors for electronic
commutation– AC servo motor
• sinusoidal back-EMF profile• balanced sinusoidal current profile• requires precise motor position measurement (resolver or
encoder)
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Recall DC servo motor
http://www.servomag.com/flash/motor_types/brush_motor.swf
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Permanent magnet AC servo motor
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Control of brushless DC motor
Source: P. Krause, O. Wasynczuk, S. Sudhoff, Analysis of Electric Machinery and Drive Systems, Wiley (2002)
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Electrical torque production
• For balanced windings and 3-phase supply, current and back-EMF waveform shapes are identical, but displaced by 120ºE (electrical degrees)
• Hence, motor torque is
• Back-EMFm m a a b b c cT e i e i e iω = + +
( ) , ( ) , ( ) ,a a e m b b e m c c e me k e k e kθ ω θ ω θ ω= = =
2 2 2 23 3 3 3( ) ( ) ( ) ( ) ( ) ( ) ( )m e a e a e a e a e a e a eT k i k i k iπ π π πθ θ θ θ θ θ θ= + − − + + +
• Two ‘standard’ ways of producing constant torque:– Trapezoidal back-EMF and square wave current– Sinusoidal back-EMF and sinusoidal current
• Shape of back-EMF profile depends on geometry of magnets and windings
L1:6Brushless DC waveforms and torque production
Source: D. Hanselman, Brushless Permanent Magnet Motor Design, 2nd ed, 2003
• Three Hall-effect sensors provide commutation switching points– ‘six-step’ drive
KpIp
( ) 2e p pT K Iθ =m a a b b c cT e i e i e iω = + +
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http://www.servomag.com/flash/4-pole/smi-motor007.htm
L1:8AC servomotor waveforms and torque production
• Sinusoidal back-EMF:
2 2 2 23 3 3 3( ) ( ) ( ) ( ) ( ) ( ) ( )m e a e a e a e a e a e a eT k i k i k iπ π π πθ θ θ θ θ θ θ= + − − + + +
( ) cos( )a e p ek Kθ θ=
( ) cos( )a e p ei Iθ θ=• Sinusoidal phase current:
• Hence 2 2 22 23 3( ) cos cos ( ) cos ( )m e p p e e eT K I π πθ θ θ θ⎡ ⎤= + − + +⎣ ⎦
• Need precise measurement of rotor position to generate phase current with correct phase; i.e., encoder or resolver
3( )2m e p pT K Iθ =i.e.,
L1:9Mechanical and electrical position
• Balanced sinusoidally-distributed phase windings supplied with balanced 3-phase currents generate a spatially-sinusoidal MMF which rotates at angular speed ωm = (2/P)ωe radM/s(mechanical radians), where ωe is frequency of phase currents, P is number of motor poles:
Coil 1 ofphase a
Coil 2 ofphase a
3MMF cos2 2
ss p e s
N PI tP
ω φ⎛ ⎞ ⎛ ⎞= −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
4 pole motor
( ) cos( )a e p ei I tθ ω=
where φs = stator angular coordinate (radM)
o60 Eetω =
0etω =
o120 Eetω =
2
4
6
8
10
30
210
60
240
90
270
120
300
150
330
180 0
P = 4
o60 Eetω =
0etω =
o120 Eetω =
2
4
6
8
10
30
210
60
240
90
270
120
300
150
330
180 0
P = 4
2
4
6
8
10
30
210
60
240
90
270
120
300
150
330
180 0
P = 4
Ref: P. Krause, O. Wasynczuk, S. Sudhoff, Analysis of Electric Machinery and Drive Systems, Wiley (2002)
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Dynamics of AC servo motor• KVL for phases:
abcabc s abc s abc
ddt
= + +iv R i L e
[ ] [ ],T Tabc a b c abc a b cv v v i i i= =v i
3 3s phR ×=R I1 12 2
1 12 21 12 2
l ph ph ph
s ph l ph ph
ph ph l ph
L L L LL L L LL L L L
⎡ ⎤+ − −⎢ ⎥= − + −⎢ ⎥⎢ ⎥− − +⎣ ⎦
L
Ll , Lph = leakage and magnetising inductances of coils
23
23
cos2 cos( )
cos( )
r
abc r p r
r
KP
π
π
θω θ
θ
⎡ ⎤⎢ ⎥= −⎢ ⎥
+⎢ ⎥⎣ ⎦
e (sinusoidalback-EMF)
All this results in a very complex, nonlinear expression for the motor torque:
Tm = Tm(ia, ib, ic, θr)
L1:11Park transformation to qd0 variables• The equations are simplified by a transformation of variables from
the machine frame abc to a quadrature-direct-zero qd0 reference frame rotating with the rotor, at speed ωr = (P/2)ωm radE/s
• Transformation for voltage, current, flux linkage or charge variables:
( ) ( )( ) ( )
2 23 3
2 23 3
1 1 10 2 2 2
cos cos cos2 sin sin sin3
q r r r a
d r r r b
c
v vv vv v
π π
π π
θ θ θθ θ θ
⎡ ⎤− +⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥= − +⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦
( ) ( )( ) ( )
2 23 3
2 203 3
cos sin 1cos sin 1cos sin 1
a r r q
b r r d
c r r
v vv vv v
π π
π π
θ θθ θθ θ
⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥= − −⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥+ +⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦
• Then, total power:
( )0 0 03 22
abc a a b b c c
qd q q d d
P v i v i v i
P v i v i v i
= + +
= = + +
( )0qd s r abcθ=v K v
( )10abc s r qdθ−=v K v
L1:12Equations of motion in transformed variables• KVL
( ) ( )
( ) ( )0
0 0
2q
q ph q l ph r l ph d p r
dd ph d l ph r l ph q
ph l
di Pv R i L L L L i Kdtdiv R i L L L L idt
div R i Ldt
ω ω
ω
= + + + + +
= + + − +
= +
• Motor torque 32m p qT K i=
• Mechanical dynamics m m m lJ T B Tω ω= − −
i.e., 2 2r m r lJ T B T
P Pω ω⎛ ⎞ ⎛ ⎞= − −⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
iq is the ‘torque producing’component of the stator currents
2p m p rPK Kω ω⎛ ⎞= ⎜ ⎟
⎝ ⎠
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Balanced 3-phase set• Phase voltages are controlled to have a frequency
equal to the rotor speed (in radE/s): ωe = ωr
23
23
cos( )cos( )cos( )
a r v
abc b s r v
c r v
v tv V tv t
π
π
ω φω φω φ
+⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥= = − +⎢ ⎥⎢ ⎥
+ +⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦
v
• Quadrature and direct voltages and currents
( )0
0
cossin0
q v
qd s r abc d s v
vv Vv
φθ φ
⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥= = = −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦
v K v
• To maximise torque production and minimise losses, φv = 0
, 0q s dv V v= =
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Electromechanical model of AC servo motor
Source: S. Lyshevski, Electromechanica Systems, Electirc Machines, and Applied Mechatronics, CRC Press (2000)
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Control based on quadrature and direct currentsSo
urce
: SIE
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cum
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