sensorless pm brushless drives
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The University of SheffieldElectrical Machines & Drives Research Group
IEEE UK Chapter Seminar 15 December 2003
Sensorless PM Brushless Drives
Prof. D. Howe and Prof. Z. Q. Zhu
The University of SheffieldElectrical Machines & Drives Research Group
Outline
• Review of sensorless techniques
• Zero-crossing detection of back-emf waveform
• 3rd back-emf detection
• Flux observer
• Rotor saliency
• Extended Kalman filter
• Design of high-speed (>100krpm) BLDC motor for sensorless operation
• Vector control
• Flux weakening control
• Direct torque control
The University of SheffieldElectrical Machines & Drives Research Group
Sensorless Techniques
Why sensorless?• Reduced component count
• Improved reliability
• Eliminates mechanical/hysteresis problems of discrete sensors
Key consideration:• Simple algorithm
• Accurate rotor position estimates to dynamic load disturbances
• Robust to parameter variations
The University of SheffieldElectrical Machines & Drives Research Group
Sensorless Techniques
• Brushless DC:
- Back-emf zero crossing detection
- Third harmonic voltage detection
- Freewheel diode approach
- ...
• Brushless AC
- Flux/position observer
- Inductance variation
- Kalman filter
- …
The University of SheffieldElectrical Machines & Drives Research Group
Sensorless Techniques
• Sensitive to parameter variation• Poor performance at low speed• Initial position not identifiable• May not work at zero speed
Existing problems
Rotor saliency based approaches
• Operational at zero & low speed• Rotor saliency required
Key issues: - two zeros
• Zero crossing of back-emf waveform for BLDC• Zero speed for both BLDC & BLAC
The University of SheffieldElectrical Machines & Drives Research Group
Typical Sensored Brushless Drive System
Position sensors
Speed controller-
-Speed demand
Currentfeedback
3 ∅BLACmachine
DClink
3 ∅Inverter
DSP
Speed estimator
Switching logic
Currentcontroller
The University of SheffieldElectrical Machines & Drives Research Group
Typical Sensorless Brushless Drive System
3 ∅PM
machineNo sensor
Measurements from motorterminals
Switching logic
Currentcontroller
Speed controller-
-Speed demand
Currentfeedback
Rotor Position& Speed estimator
DSP
DClink
3 ∅Inverter
Sensorless controller
The University of SheffieldElectrical Machines & Drives Research Group
(1) Detection of Zero-Crossing of Back-EMF Waveform
0 30 60 90 120 150 180 210 240 270 300 330 360
phase voltageemf
detection point
Diode conduction angle
ideal current waveform
current waveform
• Most common technique for sensorless operation of brushless DC motors • Appropriate switching devices commutated 30oelec. after detection of zero-
crossing of back-emf waveform when phase is unexcited• Conduction angle of free-wheeling diodes must <30oelec.
- may be problem at high speed or high load condition- not suitable for flux-weakening operation
• Starting and low speed operation problems (due to absence of emf)
The University of SheffieldElectrical Machines & Drives Research Group
Detection of zero-crossing of back-emf waveform
Example:
Time (0.1ms/div)
Cur
rent
(0.5
A/d
iv)
Vol
tage
(50V
/div
)
Phase CurrentPhase Voltage
Measured @120krpm
Various commercial ICs, e.g.- Micro-linear 4425/4426/4428
Sensorless PWM motor controller
Mode of operation:• Initial alignment• Synchronous open-loop run-up• Sensorless close-loop
The University of SheffieldElectrical Machines & Drives Research Group
Design of 120krpm high-speed motor for sensorless operationOptimal design: within same space envelope, maximum efficiency,
diode conducting duration significantly <30oelec.
Motor A Motor B• Longer stator core• Fewer turns/coil• Shorter end winding• More iron, less copper• Relatively high unbalanced magnetic pull.
• Shorter stator core• More turns/coil• Longer end winding• Less iron, more copper• Lower unbalanced magnetic pull
The University of SheffieldElectrical Machines & Drives Research Group
Design of 120krpm high-speed motor for sensorless operation
Motor A
• Sinusoidal back-emf waveform
Motor B
• High conduction angle, almost continuous current waveform• Low diode conduction angle
The University of SheffieldElectrical Machines & Drives Research Group
Design of 120krpm high-speed motor for sensorless operation
Motor A Motor B• Suitable for sensorless control • Unsuitable for sensorless control
Phase terminal voltage
Line terminal voltage
Phase terminal voltage
Line terminal voltage
emf
emfemf
emf
Zero-crossing
Zero-crossing
No zero-crossing
No zero-crossing
The University of SheffieldElectrical Machines & Drives Research Group
Sensorless high-speed PM brushless motors
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
0 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007
Time (s)
Cur
rent
(A)
MeasuredPredicted
Sensorless control board
Inverter connectionboard
Heat sink
Current waveforms on no-load, 125,000rpm
The University of SheffieldElectrical Machines & Drives Research Group
Pros & cons of back-emf zero-crossing detection
Simple, fast, commercial IC chips available
Cases in which zero-crossing of back emf is not detectable:
• BLDC - High speed operation (high reactance)
• BLDC - High load
• BLDC - Flux-weakening operation
• BLAC - Brushless ac operation
Current is continuous or almost continuous
Alternative sensorless technique is required
The University of SheffieldElectrical Machines & Drives Research Group
(2) 3rd Harmonic Back-EMF Detection
rmsn Eeu θ3sin33 −=≈
crmKreKre t∆⋅+= − ωθθ )1()(usn uxn
3 ways of detecting e3in literature:
usn, uhn, uhs
The University of SheffieldElectrical Machines & Drives Research Group
Features of 3rd harmonic back-EMF detection
• 3rd harmonic back-EMF can be extracted from voltage
• Only voltage usn is suitable for extracting 3rd harmonic back EMF in both BLDC and BLAC drives
• Independent of motor operation mode
• Applicable to both BLDC and BLAC operations
• Open-loop starting & close-loop operation as conventional back-emf detection
• Most suitable for high-speed application
• Example: 18 slots, 6 poles, surface-mounted magnet rotor, overlapping winding, 1 slot pitch skew
The University of SheffieldElectrical Machines & Drives Research Group
Detection of 3rd harmonic back-EMF - Voltage usn
The University of SheffieldElectrical Machines & Drives Research Group
BLDC operation with/without commutation advance
Low speed 320rpm, 4.62Nm; without advanced commutation, θad =0o
High speed 1950rpm, 0.25Nm; with advanced commutation, θad =45o
The University of SheffieldElectrical Machines & Drives Research Group
BLDC operation with/without commutation advance
0
1
2
3
4
5
0 500 1000 1500 2000Speed (rpm)
Toq
ue (N
m)
0
10
20
30
40
50
Opt
imal
ang
le (e
lec-
deg.
)
with optimal commutation advance optimal commutation
anglewithout commutation advance
The University of SheffieldElectrical Machines & Drives Research Group
BLAC operation with/without flux-weakening control
High speed 2010rpm, 0.27Nm; with flux-weakening control
Low speed 320rpm, 4.63Nm; without flux-weakening control
The University of SheffieldElectrical Machines & Drives Research Group
BLAC operation with/without flux-weakening control
0
1
2
3
4
5
0 500 1000 1500 2000Speed (rpm)
Toqu
e (N
m)
0
20
40
60
80
100
Opt
imal
ang
le (e
lec-
deg.
)
with optimal flux-weakening
optimal angle
without flux-weakening
The University of SheffieldElectrical Machines & Drives Research Group
Restriction of 3rd harmonics back-emf detection
)( 3333333 skdpwm kkkBkBE ⋅⋅⋅⋅=⋅⋅∝ ωωAbsented Em3
B3=0 - Sinusoidal shaped magnet, 120oelec. pole arc magnet, Halbach magnetised motor
kp3=0 - Conventional 3 slot / 2 pole BLDC (120oelec. coil pitch)
Reduced Em3
kd3 → 0 - Distributed winding
ksk3 → 0 - Skewed winding/magnet
Low speed, as conventional back-EMF based technique
The University of SheffieldElectrical Machines & Drives Research Group
(3) Based on rotor saliency
Method 1:
• Inject high frequency signal into motor terminals
• AC current component resulting from injected signal ii ∼ sin(2∆θ)sin(ωit)≈ 2∆θsin(ωit)
• ∆θ =instantaneous difference between estimated rotor position and actual rotor position
• ∆θ fed into observer that updates velocity and position to force error to zero
Method 2:
• Current variation from hysteresis current PWM controller
• Inductance ∼ 1/(∆I/∆t)
• ∆I=current variation over ∆t
• ∆t=current rise or decay time
• Rotor position obtained from variation of winding inductance
∆t
∆I
dtdiL
dtdiLV
/1
∝
=
• Applicable to PM motors with rotor saliency (interior and inset magnet rotors)
• Winding inductance is rotor position dependent
The University of SheffieldElectrical Machines & Drives Research Group
(4) Flux observer and rotor position estimation
• Suitable for brushless ac machines• Based on machine model• Influenced by parameter variations due to temperature & saturation• Speed obtained from differentiation of estimated rotor position• Filtering necessary
1. Voltage and current vectors - measured:
2. Stator flux-linkage vector - observed:
3. Excitation flux-linkage vector - observed:
4. Rotor position - calculated:
)0(0
)( st
s dtIRU Ψ+⋅−=Ψ ∫ &&&&
ILssf &&& −Ψ=Ψ
α
β
ψ
ψ=θ
f
festr arctan._
IU && ,
The University of SheffieldElectrical Machines & Drives Research Group
High pass filter to eliminate influence of DC offset on flux observer
)0(0
)( st
s dtIRU Ψ+⋅−=Ψ ∫ &&&&Stator flux-linkage vector
time
Flux locus without high pass filter Flux locus with high pass filter
The University of SheffieldElectrical Machines & Drives Research Group
Low pass filter on observed excitation flux-linkage vector
Rotor position
- calculated from observed excitation flux-linkage vector
α
β
ψ
ψ=θ
f
festr arctan._
With low pass filter:
Smooth locus of flux-linkage vector.Reduced ripple in estimated position.Time delay in position estimation.High frequency position error still exists, causes ripple in estimated speed.
The University of SheffieldElectrical Machines & Drives Research Group
Low pass filter on observed excitation flux-linkage vectorFlux locus Estimated and measured rotor position
0
1000
2000
3000
4000
0 0.01 0.02 0.03 0.04 0.05
Time (s)
Rot
or p
ositi
on (p
ulse
s)
-200
-100
0
100
200
Erro
r of e
stim
ated
pos
ition
(pul
ses)
actual positionestimated positionestimation error
0
1000
2000
3000
4000
0 0.01 0.02 0.03 0.04 0.05
Time (s)
Rot
or p
ositi
on (p
ulse
s)-200
-100
0
100
200
Erro
r of e
stim
ated
pos
ition
(pul
ses)
actual positionestimated positionestimation error
Without low pass flux-filter
α-axis (0.05Wb/div)
β-ax
is (0
.05W
b/di
v)
• High frequency ripple exists in flux locus
• Large position error exists
With low pass flux-filter
α-axis (0.05Wb/div)
β-ax
is (0
.05W
b/di
v)
• Smooth flux locus
• Significant phase shift exists
The University of SheffieldElectrical Machines & Drives Research Group
Speed estimation 1 - Differential of estimated rotor position
ttdtd errractrestrestr
p ∆
θ∆−θ∆=
∆
θ∆≈
θ=ω ._._._._
† _est.: estimated value; _act.: actual value; _err.: error.
∆t small, ∆θr_act. small comparable to error ∆θr_err.
0
1000
2000
3000
4000
0 1 2 3 4 5Time (s)
Spee
d (r
pm)
estimated speedactual speed
0
1000
2000
3000
4000
0 1 2 3 4 5Time (s)
Spee
d (r
pm)
estimated speedactual speed
Comparison of Estimated and Actual Speeds
Sensorless operationSensored operationError in estimated position causes ripple in estimated speed.System maybe unstable if estimated speed used as feed-back.
The University of SheffieldElectrical Machines & Drives Research Group
Speed estimation 2 - Average speed estimation
)()( ._t
LPFLPF estrpd ∆
θ∆=ω=ω † LPF: low pass filter.
0
1000
2000
3000
4000
0 1 2 3 4 5Time (s)
Spee
d (r
pm)
estimated speedactual speed
0
1000
2000
3000
4000
Time (0.5s/div)
Spee
d (r
pm)
estimated speedactual speedestimated speedactual speed
time delay
Sensorless operationSensored operation
Comparison of Estimated and Actual Speeds
Accurate estimation during steady-state operation.Time delay in estimated speed during transient operation.System maybe unstable if estimated speed used as feed-back.
The University of SheffieldElectrical Machines & Drives Research Group
Speed estimation 3 - from induced EMF & excitation flux-linkage
0
1000
2000
3000
4000
0 1 2 3 4 5Time (s)
Spee
d (r
pm)
estimated speedactual speed
no time delay
0
1000
2000
3000
4000
0 1 2 3 4 5Time (s)
Spee
d (r
pm)
estimated speedactual speed
Sensored operation Sensorless operation
Comparison of Estimated and Actual Speeds
No time delay in estimated speed.System stable if estimated speed used as feed-back.Estimation error even during steady-state operation.
dsf
qsqqe iL
piLRiu+Ψ
⋅−−=ω( )
+ω+⋅+=
Ψω=
mdsqsqq
fm
EiLpiLRiu
E
f
qqe
RiuΨ
−≈ω
The University of SheffieldElectrical Machines & Drives Research Group
Speed estimation 4 - Improved estimation (combined 2 & 3)
.. 1
1)(
111
comddifd
dededh
TsTs
TsTs
TsTs
Ts
ω+ω=+
ω+ω=
+ω−ω+ω=
+ω+
+ω=ω
† ωdif.: difference of estimations 2. & 3.; ωcom: compensation.Accurate speed estimation during steady-state (s 0, ωh ωd).Fast dynamic response to speed changes (s ∞, ωh ωe).System stable if estimated speed used as feed-back.
The University of SheffieldElectrical Machines & Drives Research Group
Comparison of estimated and actual speedsSensored operation Sensorless operation
-500
0
500
1000
1500
2000
2500
3000
3500
0 1 2 3 4 5Time (s)
Spee
d (r
pm)
actual speed
estimated speed
speed differencespeed compensation
no time delay
0
1000
2000
3000
4000
0 1 2 3 4 5Time (s)
Spee
d (r
pm)
estimated speedactual speed
Accurate speed estimation during steady-state (s 0, ωh ωd).Fast dynamic response to speed changes (s ∞, ωh ωe).System stable if estimated speed used as feed-back.
The University of SheffieldElectrical Machines & Drives Research Group
(5) Based on Extended Kalman FilterEKF - An optimal recursive estimation algorithm for nonlinear systems
Applications - High-accuracy estimates of non-linear system
• State variables (current, speed from measured terminal variables and machine model)
• Model parameters (influence of temperature on resistance and back-emf, or saturation)
• Eliminating measurement noise (combined state observer & filtering functions)
Pros and cons
Pros
• Non-linear system
Cons:
• Computation requirement
• Parameter sensitivity
• Initial conditions (particularly noise behaviour)
The University of SheffieldElectrical Machines & Drives Research Group
Extended Kalman filter (only for illustration)
Non-linear discrete models with white noise )())(),(()1( kwkukxfkx +=+)())(()( kvkxhky +=
I. Prediction stage - calculates states at time (k+1) from those at time k
(a) State estimation neglecting noise
(b) Estimation of an error covariance matrix
II. Correction stage (filtering stage) - corrects estimation process in recursive manner based on deviation of estimated values from measured values
(c) Computation of a Kalman filter gain
(d) Update of an error covariance matrix
(e) State estimation
))(),/(ˆ()/1(ˆ kukkxfkkx =+
QkkkPkkkP T +ΓΓ=+ )()/()()/1(
[ ] 1)()/1()()()/1()1( −+∆+∆∆+=+ RkkkPkkkkPkK TT
[ ] )/1()()1()1/1( kkPkkKIkkP +∆+−=++
[ ]))/1(ˆ()1()1()/1(ˆ)1/1(ˆ kkxhkykKkkxkkx +−++++=++
The University of SheffieldElectrical Machines & Drives Research Group
Extended Kalman filter (only for illustration)Linearization is required at each sampling interval
=
))(),((......
))(),(())(),((
))(),(( 2
1
kukxf
kukxfkukxf
kukxf
N
If
Jacobian matrices are given by:
)/(ˆ)(21
2
2
2
1
2
1
2
1
1
1
...............
...
...
)(
kkxkxN
NNN
N
N
xf
xf
xf
xf
xf
xf
xf
xf
xf
k
=
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
=Γ)/(ˆ)(
)())(),(()(
kkxkxi kxkukxfk
=∂
∂=Γ
)/1(ˆ)()()(()(
kkxkxi kxkxhk
+=∂∂
=∆
The University of SheffieldElectrical Machines & Drives Research Group
Surface-mounted PM motor (only for illustration)
−−−
+−−
++−
=
ω
ωθθλ
θωλ
θωλ
θω
αβ
ββ
αα
β
α
JTn
JDii
Jn
Lu
Li
LR
Lu
Li
LR
ii
dtd
Lpmp
ss
m
s
s
ss
m
s
s
)sincos(23
cos
sin
2
I&
β
αa,
d
sΨ&
fΨ&
αψ f
βψ f
rθ
qILs &
V
+−−
++−
=
ω
θωλ
θωλ
θω
ββ
αα
β
α
0
cos
sin
ss
m
s
s
ss
m
s
s
Lu
Li
LR
Lu
Li
LR
ii
dtd
Decoupled electrical and mechanical equations
The University of SheffieldElectrical Machines & Drives Research Group
Salient-pole PM motor (only for illustration)
+
−=
β
α
β
α
θωθωθωθω
ii
LRLLLR
uu
s
s
2sin22cos22cos22sin2
22
22
−
++
β
α
θθθθ
ii
LLLLLL
&
&
2cos2sin2sin2cos
202
220
−+
θθ
ωλcossin
m
++++−+−−−−
−=
β
α
β
α
θωθθωωθθωωθθωθ
ii
LLLRLRLLLLRLLLLRLLLRLR
LLii
sss
sss
2sin22cos2cos222sin2cos222sin2sin22cos1
022002222
022220220
22
20
&
&
−
−+
+−−−
−−
θθωλ
θθθθ
β
α
cossin
2cos2sin2sin2cos1
20202
22022
20 LLu
uLLL
LLLLL
m
where
2
2
2
0
qd
qd
LLL
LLL
−=
+=
θ
θθ
αβ
β
α
2sin
2cos2cos
2
20
20
LL
LLLLLL
=
−=+= where, Ld = d axis inductance
Lq = q axis inductance
The University of SheffieldElectrical Machines & Drives Research Group
Comments on application to PM brushless machines
In addition to the state observer, such as position and speed
It can be used to estimate:
• Stator resistance and/or emf, for high temperature applications
• Winding inductances, for better modeling of magnetic saturation
• Load torque and/or rotor inertia, to improve dynamic speed control
• It is still far too complicated to implement the full-order EKF observer
• Hence, the reduced-order EKF is most desirable
The University of SheffieldElectrical Machines & Drives Research Group
(6) Example - Sensorles DTC based on simplified EKF
dtdiRu s
sssψ
+=Stator voltage equation:
Stator flux linkage vector obtained from measured stator voltages and currents:
dtiRu ssss )(∫ −=ψ
This equation can be expressed in stationary reference frame:dtRiu sss )(∫ −= αααψ
)(23
αββ ψψ ssssa iipT −=
22βα ψψψ ss +=
dtRiu sss )(∫ −= βββψ
Magnitude of stator flux linkage:
Electromagnetic toque equation:
The University of SheffieldElectrical Machines & Drives Research Group
Control strategy of DTC
Block diagram of DTC for PM BLAC drive
The University of SheffieldElectrical Machines & Drives Research Group
Sensorless DTC
• Conventional approach:
s
ksks
Tdtd )1()( −−
≈=θθθω
β
α
ψψ
θs
ss arctan=
dtdiRu s
sssψ
+=
dtiRu ssss )(∫ −=ψ
dtRiu sss )(∫ −= αααψ
dtRiu sss )(∫ −= βββψ
This equation can be expressed in stationary reference frame:
Stator voltage equation:
Stator flux linkage vector obtained from measured stator voltages and currents:
From DTC
Estimated stator flux position
Estimated speed Need filters
The University of SheffieldElectrical Machines & Drives Research Group
Sensorless DTC based on simplified EKF
u(k)=0Output variables
=
α
β
ψψ
s
s
kyky
)()(
2
1 Input variables
+
=
)()(
)(sin)(cos
)()(
2
1
2
1
kvkv
kk
kyky
θθFor brushless ac drives,
fundamentals of fluxes are sinusoidal
)()()1( kwkFxkx +=+)())(()( kvkxhky +=
=
10011001 sT
F
=
)(sin)(cos
))((kk
kxhθθ
Tr wx ]',,[ ωθ=
−⋅
=
θθθθˆˆˆˆ
000
3
2
1
cossinsincos
kkk
K
where k1, k2, and k3 are tuning parameters, and can be pre-computed from simulations, by using, for example, the Matlab DLQE command for Kalman estimator design of discrete-time systems
State variables
State-space model
Kalman filter gain can now be significantly simplified and is given by
The University of SheffieldElectrical Machines & Drives Research Group
Sensorless DTC based on simplified EKF
• Simplified extended Kalman filter (EKF) based sensorless DTC:
=
α
β
ψψ
s
s
kyky
)()(
2
1Recursive Algorithm:
)(ˆsin)()(ˆcos)()( 12 kkykkyk θθε −=
)]()(ˆ)(ˆ[)1(ˆ1 kkkTkk rs εωθθ ++=+
)()(')(ˆ)1(ˆ 2 kkkwkk rr εωω ++=+
)()(')1(' 3 kkkwkw ε+=+
The University of SheffieldElectrical Machines & Drives Research Group
Phase current and stator flux linkage
-3.75
-2.5
-1.25
0
1.25
2.5
3.75
0 0.02 0.04 0.06 0.08Time (ms)
Cur
rent
(A)
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
-0.15 -0.1 -0.05 0 0.05 0.1 0.15alfa (Wb)
beta
(Wb)
(a) Phase current (simulation)Time (20ms/div)
Cur
rnet
(1.2
5A/d
iv)
(c) Phase current (experiment)
alfa (0.05wb/div)
beta
(0.
05w
b/di
v)
(d) Locus of stator flux linkage (experiment)(b) Locus of stator flux linkage (simulation)
The University of SheffieldElectrical Machines & Drives Research Group
Comparison of measured and estimated speed
Time (1s/div)
Spee
d (1
000r
pm/d
iv)
E stimatedM easured
(a) Using encoder for feedback, estimated speed derived from stator flux-linkage
without speed filter
Time (1s/div)
Spee
d (1
000r
pm/d
iv)
E stim atedM easured
(b) Using estimated speed for feedback, speed derived from stator flux-linkage with speed filter
With delay
Tim e (1s/div)Sp
eed
(100
0rpm
/div
)
E stim atedM easured
(c) Using estimated speed for feedback, speed derived from stator flux-linkage by simplified EKF
No delay
The University of SheffieldElectrical Machines & Drives Research Group
Comparison of measured and estimated rotor position
Time (10ms/div)
Posi
tion
(100
0 pl
uses
/div
)
Erro
r (10
0 pl
uses
/div
)
MeasuredEstimatedError
Time (10ms/div)
Posi
tion
(100
0 pl
uses
/div
)
Erro
r (10
0 pl
uses
/div
)
MeasuredEstimatedError
(a) Estimated directly from stator flux-linkage
(b) Estimated by using simplified EKF
The University of SheffieldElectrical Machines & Drives Research Group
Acknowledgment
Dr Jason EDE
Dr Jian Xin SHEN
Mr Yan Feng SHI
Mr Yong LIU
The University of SheffieldElectrical Machines & Drives Research Group
Speed and electromagnetic torque
0
500
1000
1500
2000
2500
3000
3500
4000
0 1 2 3 4 5 6 7 8Time (s )
Measured speedSpeed reference
-0 .6
-0 .4
-0 .2
0
0 .2
0 .4
0 .6
0 .8
1
1.2
1.4
0 1 2 3 4 5 6 7 8Time (s )
Est imated to rq ueTorq ue reference
(a) Speed (simulation)
(b) Electromagnetic torque (simulation)
Time (1s /d iv)
Measured sp eedSp eed reference
(c) Speed (experiment)
Time (1s /d iv)
Es t imated To rq ueTo rq ue reference
(d) Electromagnetic torque (experiment)
The University of SheffieldElectrical Machines & Drives Research Group
Comparison of measured and estimated rotor position
Tim e (10m s/div)
Posi
tion
(100
0 pl
uses
/div
)
Erro
r (10
0 pl
uses
/div
)
M easuredEstim atedError
Tim e (10m s/div)
Posi
tion
(100
0 pl
uses
/div
)
Erro
r (10
0 pl
uses
/div
)
M easuredEstim atedError
(a) Estimated directly from stator flux-linkage
)3
2(
rs
s
pTL
arctgψψ
δ =
δθθ −= sr
)(23
αββ ψψ ssssa iipT −=
)](2sin)()sin(2[43
δψδψψ
dqsqrqd
s LLLLL
pT −−=
for a surface-mounted permanent magnet BLAC motor, Ld=Lq=Ls
The stator flux-linkage position is converted to the rotor position by subtracting the load angle, δ, that is
)sin(2
3δψ
ψr
s
s
Lp
=
Thus,
Since the electromagnetic torque can be estimated as:
• Position estimation:
(b) Estimated by using simplified EKF
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