XIX International Conference on Electrical Machines - ICEM 2010, Rome
Review of Position Estimation Methods for
IPMSM Drives Without a Position Sensor
Part I: Nonadaptive Methods
O. Benjak, D. Gerling
Φ
Abstract --This paper presents a review in state of the art
techniques for position sensorless permanent magnet
synchronous motor drives . In particular nonadaptive methods
like estimators using monitored stator voltages or currents, flux
estimators and back-EMF based estimators are described.
Furthermore the authors give an overview for current methods
to the integrator drift problem.
Index Terms—permanent magnet (PM) machines, positions
estimation, sensorless drives, review, flux estimator, back-EMF,
extended back-EMF, integrator drift.
I
I.
INTRODUCTION
n the past few years great efforts have been made in the
field of speed – and/ or shaft position sensorless
controlled drives. These drives are usually referred to as
“sensorless” drives. This expression refers to the speed and
shaft sensors, but there are still other sensors, e.g. current
sensors [1].
Permanent Magnet Synchronous Motors provide an
excellent power density (compactness), a high energy
efficiency and a high torque to inertia ratio. In the last years
the price of rare-earth magnet material decreased
significantly. For this reason PM-Machines are available for
standard drives up to 300 kW [40].
The main drawback of a PM-Machine is the position sensor,
which is vulnerable for electromagnetic noise in hostile
environments and has a limited temperature range. For PM
motors rated up to 10kW the cost of an encoder is below
10% of the motor manufacturing cost, but e.g. for
applications in the automotive industry with the high number
of produced units the elimination of the position sensor is
preferable. Thus the elimination of the electromechanical
sensors reduces the hardware costs, reduces the installation
complexity of the system (because of associated cabling),
decreases the system inertia, increases the robustness and the
reliability and reduces obviously the noise sensitivity of the
electrical drive [2],[3].
PM-Machines can be divided in two categories which are
based on the assembly of the permanent magnets (PM’s).
The PM’s can be mounted on the surface of the rotor
(surface permanent magnet synchronous motor - SPMSM)
or inside of the rotor (interior permanent magnet
synchronous motor - IPMSM). These two configurations
have an influence on the shape of the back electromotive
force (back-EMF) and on the inductance variation.
Subsequent only the IPMSM machine with a sinusoidal
excitation is examined.
The aim of this paper is to provide a review of a position
estimation for IPMSM drives without a position sensor
O. Benjak is with the Institute of Electrical Drives, University of Federal
Defense Munich, 85577 Neubiberg Germany
(e-mail: oliver.benjak@unibw.de).
D. Gerling is the head of the the Institute of Electrical Drives, University of
Federal Defense Munich, 85577 Neubiberg, Germany
(e-mail: dieter.gerling@unibw.de).
978-1-4244-4175-4/10/$25.00 ©2010 IEEE
which is based on nonadaptive methods and presented in
recent publications. The author renounces a regard of special
kinds of rotor construction for the IPMSM.
This paper is organized as follows:
• First, a classification for the strategies of position
estimation for interior PM synchronous machines is
given.
• Second, the motor equations for the PMSM in
different coordinates are illustrated.
• Third, the nonadaptive methods are described.
II.
STRATEGIES FOR POSTION ESTIMATION OF IPMSM
In general there are three strategies for position estimation of
IPMSM [2], [4]:
1. Fundamental Excitations
a. Nonadaptive Methods,
b. Adaptive Methods,
2. Saliency and Signal Injection,
3. Artificial Intelligence.
a) Nonadaptive Methods: will described in Section III,
are divided in the following three categories [2], [4] :
1. Techniques using the measured DC-Link,
2. Estimators using monitored stator voltages, or
currents,
3. Flux based position estimators
4. Position estimators based on back-EMF.
b) Adaptive Methods: are divided in four categories [4] :
1. Estimator based on Model Reference Adaptive
System (MRAS).
2. Observer-based estimators.
3. Kalman estimator.
4. Estimator which use the minimum error square.
c) Saliency and Signal Injection: In many ac machines,
the position dependence is a feature of the rotor. In the
case of the interior PMSM there is a measurably spatial
variation of inductances or resistances (saliencies) in the
dq direction due to geometrical and saturation effects,
which can be used for the estimation of rotor position [2],
[25], [27].
Another method to estimate the rotor position is to add a
high frequency stator voltage (current) component and
evaluate the effects of the machine anisotropy on the
amplitude of the correspondent stator voltage (current)
component [20].
d) Artificial Intelligence: Artificial intelligence describe
neural network, fuzzy logic based systems and fuzzy
neural networks. These kind of methods do not require a
mathematical model of the drive, exhibit good noise
rejection properties, can easily be extended and modified,
can be robust to parameter variations and are
dq-voltages can be rewritten as:
computationally less intensive [2]
III. MATHEMATICAL MODEL OF IPMSM
Coordinates, used in this paper, are defined in Fig. 1, which
shows the model of a IPMSM.
⎡ud⎤ ⎡Rs +Ldhp−ωrLqd −ωrLq +Ldqhp ⎤ ⎡id⎤ ⎡ 0 ⎤
⎥⎢ ⎥+⎢
⎢ ⎥=⎢
⎥ . (7)
Rs +Lqhp−ωrLdq ⎦⎥ ⎣⎢iq⎦⎥ ⎣ωrψm ⎦
⎣⎢uq⎦⎥ ⎣⎢ ωrLd +Lqdhp
Subsequently only (4) is examined. In a position sensorless
drive system the d-q-model can not be utilized because the
rotor position is not detected and the real rotating coordinate
is unknown. A d-q-coordinate is the rotating coordinate. A γδ-coordinate is the estimating rotating coordinate having the
Δθ difference from the d-q-coordinate.
Because of the saliency the transformed equation of a
IPMSM based on the mathematical motor model becomes
very complicated. The transformation matrix is defined by:
⎡cos(Δθ) −sin(Δθ)⎤ .
Tdq−γδ =
(8)
⎢ sin(Δθ) cos(Δθ) ⎥
⎣
Fig. 1. Model of an IPMSM with definition of the used coordinates [41]
The 3-phase voltages of an IPMSM are given by [50]:
⎡ dψu ⎤
⎢
⎥
⎡ uu ⎤ ⎡Rs 0 0 ⎤ ⎡ iu ⎤ ⎢ dt ⎥
⎢ ⎥ ⎢
⎥ ⎢ ⎥ ⎢ dψv ⎥
⎢ uv ⎥ = ⎢ 0 Rs 0 ⎥ ⎢ iv ⎥ + ⎢
⎥ .
⎢u ⎥ ⎢ 0 0 R ⎥ ⎢i ⎥ ⎢ dt ⎥
s⎦ ⎣ w⎦ ⎢ dψ ⎥
⎣ w⎦ ⎣
w
⎢ dt ⎥
⎣
⎦
⎡uγ ⎤ ⎡Rs +pLd −ωrLq ⎤ ⎡iγ ⎤
⎡−sin Δθ⎤
⎥⎢ ⎥ +ω ψ ⎢
⎢ ⎥=⎢
⎥ +"
r
m
⎢⎣uδ⎥⎦ ⎢⎣ ωrLd Rs +pLq ⎥⎦ ⎣⎢iδ⎦⎥
⎣ cos Δθ ⎦
(1)
Where:
⎡
⎤
−(Ld −Lq )sin2 (Δθ) ⎥
⎢−(Ld −Lq )sin(Δθ)cos(Δθ)
Lb = ⎢
⎥,
2
(Ld −Lq )sin(Δθ)cos(Δθ)⎥⎦
⎢⎣ −(Ld −Lq )sin (Δθ)
⎡ dψd
⎤
⎡ud⎤ ⎡Rs 0 ⎤ ⎡id⎤ ⎢ dt −ωrψq⎥
⎥.
⎢ ⎥=⎢
⎥⎢ ⎥+⎢
⎢
⎥
⎣⎢uq⎦⎥ ⎣⎢ 0 Rs⎦⎥ ⎣⎢iq⎦⎥ ⎢ dψq +
ωrψd⎥
⎣ dt
⎦
(2)
Where ud, uq, id, iq, ψd and ψq are the dq-axis-voltages,
currents and flux linkages, respectively. ωr is the rotor
electrical angular velocity. ψd and ψq can be written as [4],
[50]:
(3)
Where Ld, Lq, are the self-inductance and ψm the flux linkage
due to permanent-magnets. Apply (3) into (2) the d-q
voltages can be rewritten as:
⎡ud⎤ ⎡Rs +pLd −ωrLq ⎤ ⎡id⎤ ⎡ 0 ⎤
⎥⎢ ⎥+⎢
⎢ ⎥=⎢
⎥.
⎣⎢uq⎦⎥ ⎣⎢ ωrLd Rs +pLq ⎦⎥ ⎣⎢iq⎦⎥ ⎣ωrψm⎦
(4)
Where p is the differential operator d/dt.
[33], [44], [45] and [46] proposed a modified two axis
model, which contains the non-linearity ψd, ψq, dψd/dt and
dψq/dt due to the influence of magnetic saturation. With this
assumption (3) can be written as:
ψd=Ldid+Ldqiq+ψm ,
ψq=Lqiq+Lqdid .
(5)
Where Ldq and Lqd are the mutual-inductances. The
differentiation of ψd and ψq can be described as [50]:
dψd ∂ψd did ∂ψd diq
=
+
=L ⋅pi +L ⋅pi ,
dt ∂id dt ∂iq dt dh d dqh q
dψq ∂ψq diq ∂ψq di
d =L ⋅pi +L ⋅pi .
=
+
dt ∂iq dt ∂id dt qh q qdh d
(6)
Where Ldh, Lqh, Ldqh and Lqhd the incremental self- and
mutual-inductance. Apply (5) and (6) into (2) the modified
(9)
⎡iγ ⎤
" + ⎣⎡Lap+ωr Lb +( Δωr −ωr )Lc ⎦⎤ ⎢ ⎥ .
⎢⎣iδ⎥⎦
⎡
⎤
2
(Ld −Lq )sin(Δθ)cos(Δθ)⎥
⎢ −(Ld −Lq )sin (Δθ)
La = ⎢
⎥,
(Ld −Lq )sin2 (Δθ) ⎥⎦
⎢⎣(Ld −Lq )sin(Δθ)cos(Δθ)
Where uu, uv, uw, iu, iv, iw, ψu, ψv and ψw are the phase
voltages, currents and flux linkages respectively and Rs the
phase resistance. (1) can be transformed into the rotor
reference dq-frame [50]:
ψd=Ldid+ψm ,
ψq=Lqiq .
⎦
Transforming (4) into the γ-δ frame the model is obtained as
follows [35]:
⎡
⎤
2
2
⎢ (Ld −Lq )sin(Δθ)cos(Δθ) −Ld sin (Δθ)−Lq cos (Δθ)⎥
Lc = ⎢
⎥.
2
2
⎢⎣Ld sin (Δθ)+Lq cos (Δθ) −(Ld −Lq )sin(Δθ)cos(Δθ) ⎥⎦
Some authors prefer to transform the d-q- coordinates in α-β
coordinates directly, which gives the rotor angle θ instantly.
The transform matrix is with Δθ=θ similar to (8). The
equation (4) can be written in α-β frame as [29] ,[32]:
⎡uα⎤ ⎡Rs +pLα −ωrLαβ ⎤ ⎡iα⎤
⎡−sin θ⎤
⎥⎢ ⎥ +ω ψ ⎢
⎢ ⎥=⎢
r m cos θ ⎥ .
⎢⎣ uβ⎥⎦ ⎢⎣ ωrLαβ Rs +pLβ⎥⎦ ⎢⎣iβ⎥⎦
⎣
⎦
Where,
(
)
L0 = Ld + Lq / 2,
(
(10)
)
L1 = Ld − Lq / 2,
Lα = L0 + L1 cos2θ, Lβ = L0 − L1 cos2θ,
Lαβ = L1 sin 2θ .
IV. NONADAPTIVE METHODS
In this section common nonadaptive methods, which use
measured currents and voltages as well as fundamental
machine equations of the IPMSM, are presented.
A. Techniques using the measured DC-Link
The Method of measuring the DC-Link is one of the oldest
in order to get adequate signal for feedback. It is an easy and
low-cost process.
In [10] the authors presented two energy optimizing
techniques, which are based on measuring the DC-link
current and a voltage/ frequency (V/f) control method. The
assumption of these methods is that the machine is working
in a stationary operating point, hence the load torque and the
rotor speed is constant at each frequency.
The first technique is the “minimum input power method”,
which minimize the average DC-link current at a considered
fixed DC-link voltage. For the method the current is filtered
and put in a control algorithm, which adjust the reference
voltage. Problematical is the low frequency instability in the
class of synchronous motor drives. The damping of the local
oscillation can be improved by an dc-link current feedback
for the frequency modulation, which made it nearly
independent of the applied frequency.
The second technique is the “power factor method”, which is
based on the contour of the non-filtered DC-link current. The
power factor (PF) is for sinusoidal stator currents and
voltages is defined as PF = cos(θ) , where θ is defined as the
angle from the stator current phasor to the stator voltage
phasor. To get the optimum PF, the DC-link current contour
should has the form of Fig. 2.
⎛ u v −u w −R s ( i v −i w )− Ld p( i v −i w )− 3ωr ( L q −L d )i u ⎞
⎟⎟
⎜
3( u u − R s i u − L d pi u )+ωr ( Lq − Ld )( i v −i w )
⎝
⎠
(13)
The required speed ωr can be calculated as follows:
θ = arctan ⎜
ωr =
1
2
(u u R s pi u ) 2 + ⎡⎣ u v − u w − R s ( i v −i w )− Ld p( i v −i w ) ⎤⎦
3
ψm
(14)
The initial position of the rotor at t=0 can be determined by
substitution ωr=0 in (14).
[12] proposes a voltage-model and current-model based
control which works in the γ-δ reference frame with the
assumption that Ld=Lq=L. Furthermore it is supposed that
Δθ=0 and ωM=ωr. (4) can be transformed and rearranged in:
⎡uγ ⎤ ⎡R +pL −ω L ⎤ ⎡iγ ⎤ ⎡ 0 ⎤
r ⎢ ⎥+
⎢ ⎥=⎢ s
⎥
⎢
⎥
⎢⎣uδ⎥⎦ ⎣ ωrL Rs +pL⎦ ⎣⎢iδ⎦⎥ ⎣ωrψm ⎦
(15)
The estimated speed ωe can be obtain as follows:
u −(R s + pL)iδ
ωe = δ
Li γ +ψ m
Fig. 2. DC-link current when the stator current is in phase with the stator
voltage [10]
The PF is calculated from variation of the current, which is
controlled by a feedback system. Voltage is controlled by
using a PI-controller [13].
Rotor position oscillations in currents arise due to small load
change. To minimize this stability problem, the authors in
[10] applied a control input, which uses the measurement of
the DC-link current. This stability control increase the
reference angular velocity when the filtered DC-link current
decreases and vice versa. After Stabilization the current
pulsation stopped and stable current waveforms were
obtained.
The methods, which are using the DC-link as input gives
nearly the same efficiencies at different loads and speeds
(appr. 10-100% of the rated speed in without and appr. 5100% with current feedback). Disadvantageous is the low
dynamic performance and accuracy, compared with torquecontrolled PM drives [5].
B. Estimator using monitored stator voltages or currents
For estimating the rotor angle θ using monitored stator
voltages and currents different authors follow different
approaches regarding to the used reference frame. This
section shows examples for the perspective in three-axis and
γ-δ coordinates.
[8], [17] and [24] propose the approach with the three axis
model. Firstly transformation from the d-q coordinates to the
α-β coordinates have to be done. The transformation matrix
is based on (8):
⎡cos( θ) − sin(θ) ⎤
Tdq −αβ = ⎢
⎥
⎣ sin(θ) cos(θ) ⎦
(11)
The transformation matrix from the α-β coordinates to the
three-axis model is illustrate in (12).
⎡
⎢1
⎢
2⎢
T23 = ⎢ 0
3
⎢
⎢1
⎢2
⎣
1
2
3
−
2
1
2
−
1⎤
− ⎥
2
⎥
3⎥
2 ⎥
⎥
1 ⎥
2 ⎥⎦
The currents are detected by a current sensor and the
voltages are obtained by calculation which using information
on PWM-pattern, dc-voltage and dead time.
The speed estimation for the current-model is sample based
and can be obtained as:
n en +1
θn +1 −θM
K
ωnM+1 = M
= M T + E Δinγ
ψM
T
T
The final equation for the rotor position angle can be found
as:
(17)
With T and KE as sampling period and estimation gain
respectively. The current model-based sensorless control
algorithm does not need voltage information, that means the
control is free from inaccuracy of the voltage caused by the
inverter.
The estimated speed is compared to the speed reference and
the speed error is processed in the PI-speed control.
C.
Flux estimators
A PMSM drive normally uses a Y connection line and
has no neutral one. The advantage of the flux calculation
method is, that the line voltages can be used [24].
The phase-voltage equation of the stator can be written in the
following form [2], [4]:
G
d G
G
(18)
u s = R s i s + Ψs
dt
G
There is uG s the input voltage, is the current, R s the
resistance and
G
ψ
s
the flux linkage of the stator. With
knowing the initial position, the machine parameters and the
relationship of the flux linkage to the rotor position, the
estimated stator flux is used to calculate the 3-phase stator
currents [24], [39]. The performance depends on the quality
and accuracy of the estimated flux linkage and measured
values of voltages and currents [39].
Change (18) to the stator flux space vector ψG s :
G
G
G
G
G
Ψs = ∫ es = ∫ (us−Rs i s)dt + Ψs0
where is e the induced back-EMF and
(12)
(16)
G
ψ
s0
(19)
the initial
position for the stator flux [2], [4], [34], [39], [59], [60]. (19)
written in α-β-coordinates depends on the terminal voltage
and the stator current. Equation (3) only needs the stator
current of the two phase d-q-reference frame [39], [59].
For determining the rotor position many ways are proposed,
which depend on the used reference frame. Using the α-βframe the equation for the rotor angle can be written as
follows [60]:
⎛ Ψ −Li ⎞
α ⎟.
θ=arctan ⎜ α
⎜ Ψβ − Lβ ⎟
⎝
⎠
(20)
Where L is the winding inductance.
The actual rotor angle using the d-q frame can be calculated
with [2], [4], [9], [39], [59], [60]:
⎛Ψ ⎞
θ=arctan ⎜ d ⎟.
⎜ Ψq ⎟
⎝
⎠
(21)
The estimation of the rotor position from the flux has the
difficulty of the pure integrator in (19). In [18], [22], [34],
[57] solutions for this major problem are presented.
Because of the noise, in the last decade a pure investigation
of the rotor position has gained less attention. Solutions with
adaptive or observer methods are more common [20], [31],
[42], [43].
D.
Position estimation based on back EMF
In PM machines , the movement of magnets relative to
the armature winding causes a motional EMF. The EMF is a
function of rotor position relative to winding, information
about position is contained in the EMF-waveform [43].
Back-EMF based sensorless methods were first developed to
estimate the rotor position of PMSM having a surface
mounted PM-rotor without rotor magnetic saliency [50].
[16] proposes to detect the rotor angle with a back-EMF
measurement which is preformed by adding MCDI
(Maximum Current Decaying Interval) test cycle to the
current control algorithm (cf. fig. 3).
system and improves its reliability. Instead of the measured
voltages reference voltages are used. The back-EMF is not
calculated by the integration of the total flux linkage of the
stator phase circuits because of the integrator drift problem.
The estimation of the rotor position is given by the
difference of the arguments of the back-EMF in the α-β
reference frame and the arguments of the same one in the
rotating d-q frame:
⎛ψ
⎛ eβ ⎞
M
θ=arctan ⎜
⎟ −arctan ⎜⎜
e
⎝ α⎠
⎝ Lq i q
⎞
⎟
⎟
⎠
(25)
(25) shows, that there is only a quadrature current
dependency of the rotor position. Furthermore the back-EMF
in subject to the reference voltage in α-β- coordinates can be
described as following function:
⎛ uβ − Riβ ⎞
ƒ (u α , u β ) = arctan ⎜⎜
⎟⎟
⎝ u α − Riα ⎠
(26)
The relationship between the actual motor voltages and the
reference ones is:
*
u α = u α + δu α
*
u β = u β + δu β
(27)
After substituting (27) in (26) we get:
(
) (
)
∂ƒ
∂ƒ
ƒ u*α ,u*β = ƒ u α ,uβ +
δu +
δu
∂u α α ∂uβ β
⎛ u* −Ri ⎞ V
β
β⎟
= arctan ⎜
− ωT
⎜ u* −Ri ⎟ E r
α⎠
⎝ α
(28)
The second term in (28) is the dependent of the back EMF
on the rotor speed. V, E and T are the RMS voltages, the
back-EMF and the lag time introduced by the inverter
respectively. Thus we get for the “estimated” position θ*:
⎛ u* −Ri ⎞
⎛ψ ⎞
β
β ⎟ V
θ*=arctan ⎜
− ω T −arctan ⎜ M ⎟.
⎜ Lq iq ⎟
⎜ u* −Ri ⎟ E r
⎝
⎠
α⎠
⎝ α
Fig. 3. MCDI test cycle [16]
From the measured back-EMF, the actual rotor angle can be
achieved as follows [16], [39]:
− Ed K eωr sin( θ) sin(θ)
=
=
= tan ( θ )
Eq K eωr cos( θ) cos( θ)
⎛ −E
θ = arctan ⎜ d
⎜ Eq
⎝
⎞
⎟
⎟
⎠
(22)
(23)
The experimental results shows an acceptable accuracy in
the low speed range (10~100rpm). The rotor speed quantities
is fluctuating but can be decreased by a higher bandwidth
for the speed controller.
[33] proposes to estimate the rotor position with an approach
measure the current ripple under a conventional PWM
modulation to derive the back-EMF. Under a PWM
modulation the phase currents always present large ripple. If
this current ripple is measured, the rotor position dependent
inductance and back-EMF can be solved using the PMSM
machine model with the known voltage vectors applied.
The machine model in the α-β reference frame will be
discretized. Thus the rotor position can be expressed as :
⎛ e (n) ⎞ π
θ=arctan ⎜ α
⎟ −
⎜ eβ (n) ⎟ 2
⎝
⎠
(24)
[61] propagates the determination of the back-EMF without
the aid of voltage probes which reduces the cost of the
(29)
The proposed estimation in [61] appears to be very robust
against parameter variation. Furthermore the electrical drive
has a good dynamic performance. Disadvantageous is that
after torque application the error increase but stay never
exceeded 35 r/min during transient operation.
Because of the saliency the transformed equation of an
IPMSM based on the mathematical motor model (10)
becomes very complicated [32], [35], [40]. There are two
trigonometric functions of 2θ, which result from changing
stator inductance. A reason why in (10) 2θ terms appear is
that the impedance matrix is asymmetrical. The desired
matrix should have the following form:
G ⎡ R s + pLd
A=⎢
⎢⎣ ωr Lq
−ωr Lq ⎤
⎥
R s + pLd ⎥⎦
(30)
To eliminate this problem the following easy mathematical
method have to be applied for the desired voltage uq on (4):
uq = A21 + Rsiq + A22 +ωrψm +ωrLdiq + pLqiq − A21 − A22 (31)
with
A 21 = ωr Lq i d , A 22 = pL d i q
Substitute (30) in (4) we get:
⎡ud⎤ ⎡Rs +pLd −ωrLq ⎤ ⎡id⎤ ⎡
⎤
0
⎥⎢ ⎥+⎢
⎢ ⎥=⎢
⎥
⎢⎣uq⎥⎦ ⎢⎣ ωrLd Rs +pLq ⎥⎦ ⎣⎢iq⎦⎥ ⎢⎣(Lq −Ld )(ωrid −piq )+ωrψm ⎥⎦
Transforming (32) in the α-β frame can be written as:
(32)
Books:
⎡uα⎤ ⎡ Rs +pLd
ωr (Ld −Lq )⎤ ⎡iα⎤
⎥ ⎢ ⎥ +"
⎢ ⎥=⎢
⎣⎢ uβ ⎦⎥ ⎣⎢−ωr (Ld −Lq ) Rs +pLd ⎦⎥ ⎢⎣ iβ ⎥⎦
{
(33)
}
⎡−sin θ⎤
" + (Lq − Ld )(ωrid − piq ) +ωr ψm ⎢
⎥
⎣ cos θ ⎦
{
}
⎡−sin θ⎤
(Lq − Ld )(ωrid − piq ) +ωrψm ⎢
⎥
⎣ cos θ ⎦
⎡ u −(R +pL )i −ω (L −L )i ⎤
⎡−e ⎤
θ = arctan ⎢ e α ⎥ = arctan ⎢− α s d α r d q β ⎥
⎢⎣ uβ −(Rs +pLd )iβ +ωr (Ld −Lq )iα ⎥⎦
⎣⎢ β ⎦⎥
[4]
Papers from Conference Proceedings (Published):
[5]
[6]
[7]
[8]
(35)
[9]
[10]
(36)
For the calculation of the necessary speed a PI-controller is
used in [31], [35].
[40], [54] propose a discrete calculation of the EEMF, which
is based on the approximation of the current in γ-δ-frame.
It has to be note, that the arctan calculation is sensitive to
the signal-noise-ratio. In the direct calculation it has no lag,
but suffers from a large position estimate error due to the
noise. This problem can be mitigate by using a state filter,
but the estimate then has lagging properties. The estimation
accuracy of the EEMF based state filter is determined by the
bandwidth of the state filter and the errors in parameter
estimation an reference voltage [34].
Further positon estimation techniques of EMF in conjunction
with flux estimation are [34], [58].
Just in section “flux estimators” the problem of noise is also
presented for back-EMF-methods. To overcome this
difficulty several authors uses observer and adaptive
methods [17], [29], [32], [35], [38], [41], [42], [48], [55],
[56], [62].
V.
[3]
(34)
In conventional estimation methods a phase-locked loop
(PLL) control for making the position error Δθ=0 is applied
[31], [32], [41],[51].
⎡ u −(R +pL )iδ−ωrLqiγ ⎤
⎥
Δθ = arctan ⎢− γ s d
⎢⎣ uδ −(Rs +pLd )iγ+ωrLqiδ ⎥⎦
K. Rajashekara, A. Kawamura, K. Matsuse, Sensorless control of AC
motor drives: speed and position sensorless operation, IEEE Press,
1996.
Vas, P., Sensorless Vector and Direct Torque Control, Oxford
University Press (UK), 1998.
Jacek F. Gieras, Mitchell Wing, Permanent Magnet Motor
Technology, 2nd ed., Marcel Dekker, Inc, 2002.
D. Schröder, Elektrische Antriebe - Regelung von Antriebssystemen.
3th edn., Berlin Heidelberg New York: Springer-Verlag, 2008 (in
German)
[2]
The second term on the right side of (33) is defined as the
extended EMF (EEMF). The physical meaning can be
explained as follows. The first term represent the induced
voltage by the rotating flux excited by the d-axis current.
The second term means an induced voltage by changing qaxis current. This differential term of iq is responsible that,
even when the motor’s velocity is near zero, the EEMF is
not zero when the q-axis current iq is changing. This property
can be use for standstill and low speed-drives. The last term
depict the EMF induced by the rotating permanent magnet.
Generally the position estimation calculated from the EEMF
[28], [32], [47]:
⎡eα⎤
⎢ ⎥=
⎢⎣ eβ ⎥⎦
[1]
CONCLUSION
IPMSM drives without mechanical sensors for motor
position or speed have the attraction of lower cost and higher
reliability. Algorithms which can be implemented in
standard microcontroller hardware, are of increasing interest
for industrial application.
A review of prior work in the field of non-adaptive methods,
especially techniques using the DC-link, stator voltages or
currents and flux- or back-EMF based position estimators
has been presented in this paper.
A review of adaptive methods will be given in a subsequent
publication.
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
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VI. BIOGRAPHIES
Oliver Benjak was born in Hennigsdorf in German Democratic
Republic, on April 28th, 1978. He graduated from the German Polytechnic
School, Hennigsdorf, Germany and studied Electrical Engineering at the
Technical University of Berlin, Germany where he received his diploma
degree in 2009. He joined the Institute for Electrical Drives of the
University of Federal Defense Munich in 2009 as research assistance.
Dieter Gerling, born 1961, got his diploma and Ph.D. degrees in
Electrical Engineering from the Technical University of Aachen, Germany
in 1986, 1992 respectively. From 1986 to 1999 he was with Phillips
Research Laborites in Aachen, Germany as Research Scientist and later as
Senior Scientist. In 1999 Dr. Gerling joined Robert Bosch GmbH in Bühl,
Germany as Director. Since 2001 he is a Full Professor and Head of the
Institute of Electrical Drives at the University of Federal Defense Munich,
Germany.