Saliency Modeling in Radial Flux Permanent Magnet Synchronous

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NORPIE 2004, Trondheim, Norway
Saliency Modeling in Radial Flux Permanent
Magnet Synchronous Machines
Sigurd Ovrebo ([email protected])
Smartmotor AS, Stiklestadveien 1
N-7041 Trondheim, Norway
Prof. Roy Nilsen, Prof. Robert Nilssen
Norwegian University of Science and
Technology, 7491 Trondheim, Norway
d − axis
Abstract—Sensorless
control
of
Permanent
Magnet
Synchronous Machines is popular for several reasons: cost saving
and system reliability. The basis for most low and zero speed
sensorless control is that there exist a difference in the direct and
quaderature inductance of the machines. This difference is also
reefed to as saliency. In this paper the different contributors to
the resulting saliency is modeled. This type of modeling is useful
for several reasons: increased understanding of the fundamentals
of sensorless control, incorporating the combined effects of
leakage and main inductance variations, incorporating the effects
of loading of the machine and finally it gives valuable insight
when the saliency ratio is attempted increased by machine
design.
Ld
q − axis
Lq
Figure 1 Salient pole synchronous machine
Index Terms—Sensorless control, saliency modeling, PMSM
I.
S
INTRODUCTION
ensorless control of electric machines has been a research
topic for more than two decades. The objective for this
work is to eliminate the need for a position sensor by
using the machine as the position sensor. The position sensor,
cabling and connectors have been a source of failure for motor
control applications. For small drives the sensor contributes
considerable to the overall cost. The basis for most low and
zero speed sensorless control [1-5] is the presents of a
difference in the d and q inductances. This difference is
referred to as saliency. There are several sources of saliencies
in PM machines: rotor inherent saliency, saturation based
saliency (yoke, teeth) [3], rotor stator teeth harmonics [1],
lamination direction based saliency [7], eddy current based
saliency [8], rotor eccentricity based saliency. In this paper a
single saliency model is developed from the rotor inherent
saliency and the saturation induced saliency. The term
saliency can be understood when considering a salient pole
synchronous machine structure (Figure 1). In this structure the
rotor has an inherent saliency due to the shape of the rotor. In
the d-axis the air-gap is small while there is a large air-gap in
the q-axis. The d-axis inductance will in this case be larger
then the inductance in the q-axis. The injection based
sensorless control schemes [1-5] extract rotor position
information from the difference in the inductance.
II. SALIENCY MODELING
In Sensorless Control of PMSM the rotor inherent saliency or
the saturation based saliency is usually used as the basis for
the saliency. The presented saliency model is simplified as
only these sources of saliency are included. In order to
illustrate the saturation based saliency a simplified model of a
30 kW IPMSM machine was simulated in FEMLab. The
relative permeability function (from supplier) was included in
the FEM simulation. In Figure 2 the machine is unloaded and
only the flux from the permanent magnets contributes to the
field. Three regions of the machine can be seen to have low
relative permeability values: the permanent magnet (in rotor),
the stator teeth (in the main flux path) and the stator yoke (in
the main flux path). The stator teeth are more saturated
compared to the portions of the stator yoke that is saturated.
d − axis
q − axis
Yoke saturation
Tooth saturation
Figure 2 Relative permeability, IPMSM (8-pole) FEM simulation
NORPIE 2004, Trondheim, Norway
The modeling of saliency is done by modeling the inductance
in a dq-rotor oriented reference frame. The inductance is
separated in two parts: the leakage inductance and the main
inductance. In this way the model can be used to incorporate
the saliency sensed by carrier signals that is concentrated in
the leakage flux paths or main flux path (or both). The
sensorless schemes [1-5] will in general sense the resulting
inductance but the high frequency carrier flux distribution
determines the components of main- or leakage flux. In the
first stage of the modeling the machine will be modeled at no
load (only flux from the permanent magnets). The load
dependency is included in the last section of the modeling.
Figure 3 illustrates why the main and leakage inductance
should be modeled separately: the main flux and the leakage
flux are 90 electrical degrees spatially shifted.
In order to simplify the modeling some simplifications are
made:
• Only fundamental components are considered
• The flux from the magnets and the windings are
assumed sinusoidal distributed over the air gap. The
distributed windings are simplified by representing
them as a two axis model.
• Saturation effects are assumed to be sinusoidal
distributed (superposition applied).
• Relative permeability assumed to wary linear with
flux density
These simplifications may seem crude. Some lamination
materials may have extremely non linear relation between
relative permeability and flux density. The machine in the test
setup has laminations from Congent (M235-35). The relative
permeability is derived from the BH curves from the supplier.
Figure 4 shows the relative permeability for the laminations
(measurement errors in the 100 and 200 Hz curves). From the
50 Hz curve an approximate linear relation can be obtained in
the region 0.4 to 1.3 T. The non linearity in the lower region is
related to the first magnetization curve of the material. When
the machine model is considered this effect is neglected as the
laminations have been exposed to a
β
s
Is
Leakage
flux
regions
ξ
Figure 4 Relative permeability for 50,100,200 and 400 Hz excitation
frequencies (Congent M235-35 lamination-supplier data).
magnetic field and the time constant for the material to return
to its original sate is very large.
A. No load model
In the no load model only the flux from the permanent
magnets are considered. The inductance will be modeled in
dq-rotor reference frame and the main and leakage inductance
will be modeled separately.
1) Main inductance
In the first stage the machine is considered as a uniform
structure with no magnets or saturation.
Ld −main = L0
Lq−main = L0
(1)
(2)
The sources of saliency for a no load condition is shown in
Figure 5. These effects are included in the model as:
Ld −main = L0 − L0 rd − L0ts − L0 ys
(3)
L0
- main inductance (symmetrical machine)
L0rd - rotor design (low relative permeability in magnets
and induced currents in magnets)
L0ts - stator tooth saturation
ψs
L0 ys - yoke saturation
αs
The yoke saturation is assumed to have minimal impact on the
q-axis inductance. If high frequency carrier is used there may
be induced currents in the magnets due to the carrier flux. The
effect of induced currents in the magnets is reduced d-axis
inductance. This effect is included in the model by the term
L0rd (combined effect of low permeability and induced
currents). The induced current is highly dependent on the
carrier frequency [8].
Figure 3 Current distribution, current and flux space vectors
NORPIE 2004, Trondheim, Norway
βs
stator teeth
saturation
Stator
Leakage Flux
q
d
permanent
magnets
θ
αs
Rotor
stator yoke
saturation
Figure 6 Leakage flux path
3) Resulting no load model
In the resulting no load model the leakage and main
inductance are summed up in each axis:
Ld = Ld −main + Ld −leakage
Figure 5 Sources of saliency – no load
2) Leakage inductance
If we consider a flux vector (space vector) the leakage flux is
located 90 electrical degrees spatially shifted compared to the
main flux distribution. When the dq- inductances are modeled
the reference system is referred to the main flux. The effects
on the leakage inductance are there fore shifted 90 electrical
degrees. The basis for the leakage inductance model is also a
uniform structure with no saturation:
Ld −leakage = Lσ
(4)
Lq−leakage = Lσ
(5)
The leakage flux paths are illustrated in Figure 6. Three
regions may lead to modeling of the inductance: stator teeth,
stator yoke and rotor surface. Variations (or saturation) in the
rotor surface is neglected in this model as the surface of the
test machine is uniform and the saturation level is modest. If
we consider a stator flux vector oriented in the magnet d-axis
the leakage flux path for this main vector can be influenced by
yoke saturation (saturation due to flux from the permanent
magnets). This effect is included in the model as a reduction
of the d-axis leakage inductance by a factor Lσ ys . If a stator
current vector is oriented in the q-axis the leakage inductance
may be influenced by tooth saturation (saturation due to flux
from the permanent magnet). This effect is included in the
model as a reduction of the q-axis leakage inductance by a
factor Lσ ts :
Ld −leakage = Lσ − Lσ ys
(6)
Lq−leakage = Lσ − Lσ ts
(7)
Ld = L0 − L0 rd − L0ts − L0 ys + Lσ − Lσ ys
Lq = Lq−main + Lq −leakage
(9)
Lq = L0 + Lσ − Lσ ts
The difference between
(8)
Ld and Lq is the basis for saliency
in the machine. The position dependent parts of the inductance
can be illustrated by expressing the inductance on vector
form:
L = ΣL + ∆1Le j 2θ
(10)
where:
θ
- rotor position
ΣL =
ΣL =
Ld + Lq
2
2 L0 − L0 rd − L0 ts − Lys + 2 Lσ − Lσ ts − Lσ ys
∆1L =
∆1L =
(11)
2
Lq − Ld
2
− Lσ ts + Lσ ys + L0 rd + L0 ts + L0 ys
(12)
2
If the main inductance variation is used as the reference the
variation in the leakage inductance influence the magnitude of
position dependent inductance. If the leakage yoke saturation
term ( Lσ ys ) in (12) is larger than the leakage tooth saturation
Lσ
- leakage inductance (symmetrical machine)
Lσ ys - yoke saturation effect on leakage inductance
term ( Lσ ts ) the leakage terms adds to the main inductance
Lσ ts
oppose the position dependent terms from the main
inductance.
- tooth saturation effect on leakage inductance
saliency. On the other hand if
Lσ ts > Lσ ys the leakage terms
NORPIE 2004, Trondheim, Norway
B. Load dependency
Several factors may change from an unloaded to a loaded
machine: flux and saturation levels, flux and saturation
orientation and saturation regions. In this model the
orientation of the resulting stator flux is considered in the first
stage. In the next stage saturation from the load current is
included. In order to simplify the model the control optimal
control of the PMSM is not considered. The control strategy
of the machine is assumed to be
Stator
Load
dependent
Leakage Flux
Saturated
Regions
id = 0 and iq = is* .
Ψs
I q Lq
∆θ
Rotor
Ψm
Figure 8 Saturation due to leakage flux from load current
Figure 7 Load dependency on stator flux
The stator flux increases when the machine is loaded (Figure
7). The resulting flux vector is shifted by a term ∆θ due to
the loading of the machine.
⎡
⎤
iq Lq
⎢
⎥
∆θ ≈ a sin
⎢ ψ M 2 + (iq Lq ) 2 ⎥
⎣
⎦
L = ΣL' + ∆ 4 Le jξ
(13)
The shift in the stator flux results in a shift of the stator yoke
and teeth saturation regions. In addition to the main flux
effects the leakage effects must also be included. Figure 8
illustrates typical saturation regions due to the leakage flux
from the load current. This type of saturation will result in a
load dependent term in the q-axis leakage inductance ( Llds ):
Lq−leakage = Lσ − Lσ ts − Llds
(14)
Lq = L0 + Lσ − Lσ ts − Llds
(15)
The spatial shift in the stator flux is incorporated in the model
by shifting the saturation based saliency by the angle ∆θ :
L = ΣL' + ∆ 2 Le − j 2θ + ∆ 3 Le − j 2(θ +∆θ )
(16)
Where the upper index ‘ indicates that there may be a change
in the saturation level due to the loading (as the stator flux
increase) and the terms in (16) can be expressed as:
'
ΣL =
L'd + L'q
2
'
'
1 ⎛ 2 L0 − L0 rd − L0 ts − Lys + ⎞
'
ΣL = ⎜
⎟
2 ⎜⎝ 2 Lσ − L'σ ts − L'σ ys − Llds ⎟⎠
L
∆ 2 L = 0 rd
2
− L'σ ts + L'σ ys + L'0 ts + L'0 ys − Llds
∆3L =
2
In this model the saturation is assumed sinusoidal distributed
and linear. The different terms may be combined to form a
resulting single saliency:
(17)
(18)
(19)
ξ = a tan(
∆4L =
(20)
sin( ∆θ ) ∆ 3 L
)
∆ 2 L + cos( ∆θ ) ∆ 4 L
( ∆ 2 L + cos( ∆θ )∆ 3L )
2
(21)
+ (sin( ∆θ ) ∆ 3 L) 2 (22)
The resulting model in (20), (21) and (22) describes the
relation between the different sources of saliency in the
IPMSM and load current in the q-axis.
III. EXPERIMENTAL WORK
A 30 kW radial flux IPMSM (motor data in Table 1) was used
in the experimental work. The machine was designed for high
fundamental flux (500-600 Hz) operation during field
weakening.
Table 1 IPMSM parameters
PN
[kW]
UN
[V]
IN
[A]
nN
[rpm]
Rs [Ω]
Ld[mH]
Lq[mH]
p
30
110
181
2500
0.011
0.123
0.381
4
The self and mutual inductance was estimated by applying a
transient excitation [4] to the machine. The excitation
sequence (t1-t5) is illustrated in Figure 9. The time interval
between t1/t2 and t4/t5 was 12.5 us and t2/t3 and t3/t4 was 25
us.
NORPIE 2004, Trondheim, Norway
011
100
011
a
A
inverter
Control signals to inverter leg A,B,C
IPMSM
Ia
100
A
V
Ua
n
B
V
Uc
V
Ub
b
c
C
Figure 10 Measurements of self and mutual inductance
t
Figure 9 Transient excitation in the zero vector
Figure 10 shows the connection between the inverter and the
IPMSM. The pulses were applied across the a-phase and
neutral point while the b and c phase was open. The current in
the a-phase, the applied voltage and the induced voltage in the
b and c-phase was measured. Current and voltage
measurements where sampled at time instances t2, t3 and t4.
From these measurements the self and mutual inductance can
be derived as:
t4 − t2
I a (t2 ) − 2 I a (t3 ) + I a (t4 )
t 4 − t2
Lab = U b
I a ( t 2 ) − 2 I a ( t3 ) + I a ( t 4 )
t4 − t 2
Lac = U c
I a (t2 ) − 2 I a (t3 ) + I a (t4 )
Laa = U a
(23)
(24)
(25)
The rotor was turned by hand and the inductance as a function
of the position was estimated by current and voltage
measurements. In Figure 11 there is a large difference in the
position dependent part of the self and mutual inductance. In
classical machine modeling [9] the self and mutual inductance
are modeled as:
La (θ ) = Laσ + La 0 − Lg cos(2θ )
L
Lab (θ ) = − a 0 − Lg cos(2θ − 120)
2
Where:
Laσ - leakage inductance phase a
La 0 -average inductance (minus leakage) phase a
Lg -position dependent inductance phase a
(26)
Laσ (θ ) = Laσ 0 − ( Laσ ts − Laσ ys ) cos(2θ )
(28)
La (θ ) = Laσ 0 + La 0 + ( Laσ ts − Laσ ys − Lg ) cos(2θ ) (29)
Where:
Laσ 0 -average leakage inductance phase a
Laσ ts -tooth saturation dependent term phase a
Laσ ys -yoke saturation dependent term phase a
The b and c-phase did not conduct any current and the
leakage inductance in these phases where not measured. The
expression in (27) is therefore assumed to be correct for the
mutual inductance term.
The classical modeling uses the average inductance as the
basis for the model. In this paper a maximum inductance was
used. In order to relate the two models (29) was used in the
stationary inductance matrix and transformed to the dqrepresentation. By comparing the dq-inductance the relation
between the models where found (Table2).
la [mH]
t4 t5
0.4
0.3
0.2
0.1
0
(27)
200
300
400
500
600
700
800
900
200
300
400
500
600
700
800
900
200
300
400
500
600
700
800
900
0.1
lab [mH]
t3
0
-0.1
-0.2
-0.3
-0.4
0.1
lac [mH]
t1 t2
In the saliency modeling in the previous section variations in
the leakage inductance where assumed. If these variations are
included in the model the self inductance can be expressed as:
0
-0.1
-0.2
-0.3
-0.4
rotor position[el.deg]
Figure 11 Self and mutual inductance measured with transient excitation
Table 2 Relation between Theoretical model and measurements
NORPIE 2004, Trondheim, Norway
Theoretical model
Modified
model(measurements)
L0 + Lσ
3La 0
+ Laσ
2
L0 rd + L0ts + L0 ys
Lg
Lσ ts
Laσ ts
2
Lσ ys
Laσ ys
[1]
[2]
[3]
[4]
[5]
2
From the measurements done in this work it is difficult to
estimate all the components in the saliency model. Table 3
shows the terms that can be derived from the measurement.
Table 3 Measured inductance terms
La 0
Laσ
Laσ st − Laσ yt
[mH]
[mH]
[mH]
0.125
0.025
0.0375
V. REFERENCES
standard
Lg
Ld
Lq
[mH]
[mH]
[mH]
0.0875
0.1
0.325
The magnitude of the position dependent terms in the mutual
and self inductance indicates that there is a position
dependency in the leakage inductance. These variations can
be incorporated in the machine model by using saliency
modeling as presented in this paper. Table 3 shows the
estimated inductance terms for the test machine used in this
work obtained by transient excitation (Figure 9). The term
( Laσ st − Laσ yt ) describes the position dependent term in the
leakage inductance. In classical machine modeling this term
will equal zero.
IV. CONCLUSION
A new saliency model is presented in this paper. The model
incorporates the combined saturation effect in the stator
teeth/yoke, induced currents in the magnets and load
dependency. The model suggests that the leakage inductance
should be modeled with a position dependent part.
Measurements on an IPMSM show that there can be a
difference in the position dependent term in the self and
mutual inductance when transient excitation is used to
estimate the inductance. This difference can be incorporated in
the machine model by adding position dependent part to the
leakage inductance.
The new saliency model describes: the different sources of
saliency, the combined effect of saliency in main and leakage
inductance, saliency due to induced currents in magnets and
the load dependency in the machine.
[6]
[7]
[8]
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