36
1 UPC Field Oriented Control in Permanent Magnet Synchronous Motors m r d q JCEE 06. Novembre 2006 Dr. Antoni Arias. Universitat Politècnica de Catalunya. Catalonia
1
UPC
Field Oriented Control in Permanent Magnet Synchronous Motors
m
r
dq
JCEE 06. Novembre 2006
Dr. Antoni Arias. Universitat Politècnica de Catalunya.
Catalonia
2
UPCINDEX
• Electrical Motors Classification• Permanent Magnet Synchronous Machine Model• Field Oriented Control
– Torque, speed and position control
• Sensorless Field Oriented Control
3
UPCClassification
• Types of electrical machines– DC – Universal – AC
• Single-phase AC Induction Motors• Single-phase AC Synchronous Motors• Three-phase AC Induction Motors• Three-phase AC Synchronous Motors
– Stepper– Permanent Magnet
• PMDC – Brushless • PMAC – SMPM, IPM
– Linear– Nano
Most important and used
4
UPCPermanent Magnet Machines
• Magnetic material to establish the rotor flux.• Most common magnetic material are samarium-cobalt (SmCo) and
neodymium-iron-boron (NdFeB) introduced in 1983 having superior magnetic characteristics at room temperature.
• Advantages:• No rotor currents => no rotor losses. • Higher efficiency => energy saving capability.• Smaller rotor diameters, higher power density and lower rotor
inertia.• Higher torque per ampere constant.• Weight and volume less than other type of machine for the same
power. Attractive for aerospace applications such as aircraft actuators.
• Other applications: machine tools, position servomotors (replacing the DC motors).
• Inconvenience: • synchronous machines => need for rotor position.• Price
5
UPCPM Motor Types
• Considering the shape of the back EMF• PMDC – brushless - trapezoidal• PMAC – sinusoidal
– Surface Mount PM– Interior Mount PM
SMPM IPM
6
UPCPMSM Dynamic Equations
• Space vector transformation– combines the individual phase quantities in to a single
vector in the complex plane
– Similar transformation are applied to• Stator voltages • Stator flux linkage
3
43
2
)()()(32
j
c
j
ba etietitii
( ))t(i)t(i33
i;)t(ii cbβaα ==
svs
ψ
a
b
c
i
ib
ic
b
c
a
i
ava +-vc
+
-
-
+vb
7
UPCPMSM Dynamic Equations
• Basic equation for phase windings voltages
• Total flux linkage
flux produced by the rotor magnet
Leakage inductance
Magnetising inductance
Self inductance
Mutual inductance
c
b
a
c
b
a
s
c
b
a
dtd
iii
rvvv
)cos()cos(
)cos(
343
2
r
r
r
m
c
b
a
ccbca
bcbba
acaba
c
b
a
ψiii
LMMMLMMML
ψψψ
mψ
lLmL
mlcba LLLLL
2m
cabcabLMMM baab MM; = cbbc MM; = acca MM; =
a
b
c
i
ib
ic
b
c
a
ava +-
vc+
-
-
+vb
m
r
8
UPCPMSM Dynamic Equations
• voltage vector equation in the stationary - frame • Replacing the inductances values and applying the space vector transformation
)cos(
)cos(23
2
r
rmmls dt
dψii
dtdLL
ii
rvv
i
v +-
m
r
i
v
+
-
a
b
c
i
ib
ic
b
c
a
ava +-
vc+
-
-
+vb
m
r
Clarketransformation
9
UPCPMSM Dynamic Equations
• Saliency– Variation of the stator phase inductance as function of the rotor position.
– For example
)2cos(
)2cos(
)2cos(
32
34
rmmla
rmmlb
rmmla
LLLL
LLLL
LLLL
)2cos(2
)2cos(2
)2cos(2
34
32
rmm
ca
rmm
bc
rmm
ab
LLM
LLM
LLM
m
d
q
mml
rmmla
LLL
)θ2cos(LLLL
Δ-+=
=Δ-+=
m
r
dq
ml
rmmla
LL
)θ2cos(LLLL
+=
=Δ-+=
m
r
d
q
mml
rmmla
LLL
)θ2cos(LLLL
Δ++=
=Δ-+=
10
UPCPMSM Dynamic Equations
• voltage vector equation in the stationary - frame considering saliency
• Where
• If there is no saliency and it is obtained the previous equation.
)cos()cos(
)2cos()2sin()2sin()2cos(
2
r
rm
rssrs
rsrsss
dtdψ
ii
LLLLLL
dtd
ii
rvv
ms
mls
LL
LLL
23
23
0mL
11
UPCPMSM Dynamic Equations
• voltage vector equation in the synchronous reference d/q frame fixed on the rotor – Angle chosen equal to the PM position
differential operator ; direct axis and quadrature axis inductances pssd LLL Δ= - ssq LLL Δ+=
m
r
d
id+
-vd
vq
+
-
iq
q
Parktransformation
i
v +-
m
r
i
v
+
-
10
rmq
d
qrd
rqd
q
ds
q
d ψii
pLLLpL
ii
rvv
12
UPCPMSM Dynamic Equations
• expression for the instantaneous torque for the PM synchronous machine
• first term, usually called as magnet torque, is directly proportional to and independent of .
• second term, or reluctance torque, is only present in salient machines where and is proportional to the current product .
– Motion equation: • J rotor inertia. • D friction
)(2
3qdqdqme LLiiiψ PT
qidi
0 qd LL qdii
13
UPC
Field Oriented Control of PMSM
14
UPCFOC in PMSM
• instantaneous torque for the PM synchronous machine
• Id will be kept to zero, for not demagnetizing the PM machine. Therefore, the reluctance torque will be zero.
• Electromagnetic torque will be regulated with Iq.
)(2
3qdqdqme LLiiiψ PT
m
r
dq
Iq>0
m
r
dq
Iq<0
Te > 0 Te < 0
15
UPCFOC Scheme
+- PI
SMPMSMSVM PowerConverter
dq
dq
PI
id*=0
PI +-
+-
iq*r
*
id
iq
32
i
i
e
vd*
vq*
pddt
r
r
v
v
• 3 PI control loops– 2 identical inner current loops, d and q axis.– 1 outer speed loop.
16
UPCCurrent PI Control
• Inner faster loop.• D and Q current loops are closed by identical PI.• From
• Eliminating d-q coupling terms
sdqdq
dq
rsL1
)s(v)s(i
+=
10
rmq
d
qrd
rqd
q
ds
q
d ψii
pLLLpL
ii
rvv
17
UPCCurrent PI Control
45.9 (Nm)Peak:
15.3 (Nm)Continuous stall:
8.3 (mH)L (ph-ph):
0.94 (Ohms)R (ph-ph):
20.5 (kgcm²)Inertia:
98.0 (Vrms/krpm):Ke:
1.6 (Nm/Arms)Kt:
3.82 (kW)Rated power
12.2 (Nm)Rated torque:
3000 (rpm)Rated speed:
6Number of poles:
142UMC30Model:
Manufacturer’s data for the PM machine from Control Techniques under the commercial name
UNIMOTOR.
47.01015.41
)()(
3 ssv
si
dq
dq
sdqdq
dq
rsL1
)s(v)s(i
+=
18
UPCCurrent PI Control
• The plant can be simplified as follows• First order with one pole
• PI transfer function:
+- PI
i dq*
vdq*
sdqdq
dq
rsL1
)s(v)s(i
+=
dqs
Lr-s =
sKsK
sKK)s(PI IPI
P
+=+=
19
UPC
• Root locus and step response with a P controller• E0. Position Error• Slow dynamics.
• Solution: add a PI controller• Add a zero and a pole
-400 -350 -300 -250 -200 -150 -100 -50 0-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1Root Locus Editor (C)
Real Axis
Imag
Axi
s
Step Response
Time (sec)
Ampl
itude
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.0160
0.1
0.2
0.3
0.4
0.5
0.6
0.7
dqs
Lr-s =
0K;1K IP ==
PI
KK
IP
-s:zero0s:pole
;s
KsK)s(PI
==
+
Current PI Control
20
UPCCurrent PI Control
-1600 -1400 -1200 -1000 -800 -600 -400 -200 0-800
-600
-400
-200
0
200
400
600
800Root Locus Editor (C)
Real Axis
Imag
Axi
s
Step Response
Time (sec)
Ampl
itude
0 1 2 3 4 5 6 7 8
x 10-3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
• Root locus and step response with a PI controller • E0=0• 2nd order system and response• Damping factor equal to 0,707
800-KK5,5;pK
p
I ==
21
UPCCurrent PI Control
• Implementing PI in a DSP– From S to Z domain
– Ts=100us
Step Response
Time (sec)
Ampl
itude
0 1 2 3 4 5 6 7 8
x 10-3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1-z)Ts-1(-z
K)z(PIs
sK)s(PI P
IP
IK
K
P
KK
P =→+
=
1-z,920-z
5,5)z(PIs800s
5,5)s(PI =→+
=
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1Root Locus Editor (C)
Real Axis
Imag
Axi
s
0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2Root Locus Editor (C)
Real Axis
Imag
Axi
s
22
UPCCurrent PI Control
// start iq PI controlleriq_error = iq_ref-iq;
vq_ref = vq_ref_last+c_Kp*(iq_error-c_Ki_Kp*iq_error_last);iq_error_last = iq_error;
if (vq_ref > VPI_MAX) vq_ref = VPI_MAX;if (vq_ref < -VPI_MAX) vq_ref = -VPI_MAX;vq_ref_last = vq_ref;// end iq PI controller
• Implementing a PI Controller in a DSP– From Z to discrete time domain
– C code for the TI DSP 6711
1-z)Ts-1(-z
K)z(error_iq
)z(ref_vq)z(PI P
IK
K
P==
))]Ts-1(-z(K)[z(error_iq)1-z)(z(ref_vq PI
KK
P=
))]K_K_c(z-1(K)[z(error_iq)z-1)(z(ref_vq pi-1
P-1 =
))K_K_c(last_error_iq-error_iq(Klast_ref_vqref_vq piP+=
23
UPCSpeed PI Control
• The plant can be simplified as follows• First order with one pole • Mechanical time constant might be 50 times slower than the
electrical one. Current loop is neglected.• Typical sampling time 5ms.
JD-s =
+- PI
T e*
DsJ1+
r*
24
UPC
Sensorless FOC by High Frequency Injection
25
UPCGeneral Scheme
HF Injection Vector
)tcos()tsin(
V̂vv
i
ii
i
i
kHz 1V 20V̂
i
i
3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4
-0.1
-0.05
0
0.05
0.1
0.15
Compensated Position Signals
Time [s]
I' al
pha,
I' b
eta
[A]
26
UPCDynamic HF PMSM model
• An AC machine is said to be salient if Ld Lq
• In SMPMSM, the geometric saliency is very small. Therefore, it is track the saturation saliency.
)sin()cos(
)2cos()2sin()2sin()2cos(
00
00
r
rm
rssrs
rsrss
s
s
ii
LLLLLL
ss
ii
rr
vv
α
β
dqa
a'
b
b'
c
c' θ r
PM
Saturation
Where: ; 2dq
sLL
L
2LL
LΔ dqs
27
UPC
α
β
dq
a
a'
b
b'
c
c' θr
PM
Saturation
HF
1kHz
• Injecting a rotating HF voltage vector:
)cos()sin(ˆ
tt
Vvv
i
ii
i
iiv
)2sin()sin()2cos()cos(
10
10
tItItItI
ii
iri
iri
i
iii
• The following HF current is obtained:
• The amplitude of the negative sequence is proportional to the saliency L = (Ld - Lq)/2:
iqd
i
LLLVI
ˆ
0iqd
i
LLΔLVI
ˆ1 ;
• Frequency domain representation: fe 1kHz-2fe 1kHz
I1
I0
Dynamic HF PMSM model
28
UPC
i
BPF
j(-w i t)e
j(2wi t)e
2fe
I1
2kHz-2fe
I1
1kHz-2fe 1kHz
I1
I0
fe 1kHz-2fe 1kHz
I1
I0
2kHz-2feDC
I1
I0
Homodyne Signal Processing
• Frequency domain
29
UPC
( )tω2θ2j
γδi
sidqif
ieeLLωLV̂
i~ -Δ=
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5-10
-5
05
10
i alp
ha, i
bet
a [A
]0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
-1
-0.50
0.51
i alp
ha
i, i be
tai [
A]
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50.2
0.4
0.6
0.8i d
i, i qi
[A]
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5-0.2
-0.10
0.1
0.2
i dif,
iq
if [A
]
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5-0.2
-0.10
0.10.2
i alph
ap
os, i
bet
apo
s [A
]
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50
24
6
8
Time [s]
thet
ae [r
ad]
1 1.002 1.004 1.006 1.008 1.01-10
-50
510
i alp
ha, i
be
ta [A
]
1 1.002 1.004 1.006 1.008 1.01-1
-0.50
0.51
i alp
hai
, ib
eta
i [A]
1 1.002 1.004 1.006 1.008 1.010.2
0.4
0.6
0.8
i di, i
qi [A
]
1 1.002 1.004 1.006 1.008 1.01-0.2
-0.10
0.10.2
i dif, i
qif [A
]
1 1.002 1.004 1.006 1.008 1.01-0.2
-0.10
0.10.2
i alp
ha
pos
, ib
eta
po
s [A
]
1 1.002 1.004 1.006 1.008 1.010
2
4
68
Time [s]
thet
ae [r
ad]
Processing Signals ZoomProcessing SignalsProcessing
Steps
e
BPF
( ){ }tω2θ2jss
γδi
idqi
ieeLLLLω
V̂i~ -Δ+=
tωj ie -
αβi~
( ){ }tωθ2js
tωjs
γδi
iiαβ
iδi eLLLLω
V̂i~ -Δ+=
HPF
tω2j ie
atan/2
eθ2j
γδi
sipos_αβ e
LLωLV̂
i~Δ
=
• Time domain
Homodyne Signal Processing
30
UPCImproving the position signals
• Harmonics exist on resolver signals– Non-sinusoidal distribution of saturation– Inverter effects – dead time & device voltage drop
0 3 6 9 12 15 180
0.025
0.05
0.075
0.1
0.125
Sampled Alpha Position
Am
plitu
de
[A
]
0 3 6 9 12 150
0.025
0.05
0.075
0.1
0.125Sampled Beta Position
Frequency [Hz]
Am
plitu
de
[A
]
3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
I alp
ha,
I b
eta
[A]
Sampled Position Signals
Time [s]
0 3 6 9 12 15 180
0.025
0.05
0.075
0.1
0.125
Compensated Alpha Position
Am
plitu
de
[A]
0 3 6 9 12 15 180
0.025
0.05
0.075
0.1
0.125Compensated Beta Position
Frequency [Hz]
Am
plitu
de
[A
]
3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
Compensated Position Signals
Time [s]I'
alph
a, I
' bet
a [A
]
Cleaning process
SMP Tables
31
UPCExperimental results
4kW SMPM machine sensorless Position Control – 0% load
• Response to 180 position demand
32
UPCExperimental results
4kW SMPM machine sensorless Position Control – 100% load
• Response to 180 position demand
- no integrator in control loop (incremental position only)
- isq (torque current) limited to 1.3 x rated
33
UPC
Experimental Set up
● Matrix Converter 7.5kW
● Surface Mount Permanent Magnet Motor 4kW
Sensorless Control with Matrix Converter
- Basics for DSP implementation C:\MATLAB7\work\foc File: foc.mdl
34
UPCConclusions
• Permanent Magnet Synchronous Motors– Features / Advantages
• No rotor currents => no rotor losses. • Higher efficiency => energy saving capability.• Smaller rotor diameters, higher power density and lower rotor
inertia.• Weight and volume less than other type of machine for the same
power. Attractive for aerospace applications such as aircraft actuators.
• Other applications: machine tools, position servomotors (replacing the DC motors).
– Inconvenience: • synchronous machines => need for rotor position.• Price
35
UPCConclusions
• PMSM motor model has been obtained• Field Orineted Control for PMSM has been presented
– PI control loops
• HF injection Sensorless vector control has been introduced. – Scheme and principles.– Saturation saliency has been tracked to estimate the rotor position.– 4 step homodyne demodulation. – Position signal improvements:
• Dead time compensation.• SMP.
36
UPC
Questions / Debate
• JCEE 06. Novembre 2006
Moltes gràcies
