L1:1

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)

L1:2

Recall DC servo motor

http://www.servomag.com/flash/motor_types/brush_motor.swf

L1:3

Permanent magnet AC servo motor

L1:4

Control of brushless DC motor

Source: P. Krause, O. Wasynczuk, S. Sudhoff, Analysis of Electric Machinery and Drive Systems, Wiley (2002)

L1:5

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 = + +

L1:7

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)

L1:10

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⎛ ⎞= ⎜ ⎟

⎝ ⎠

L1:13

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= =

L1:14

Electromechanical model of AC servo motor

Source: S. Lyshevski, Electromechanica Systems, Electirc Machines, and Applied Mechatronics, CRC Press (2000)

L1:15

Control based on quadrature and direct currentsSo

urce

: SIE

MEN

S du

cum

enta

tion

for S

IMO

DR

IVE

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