# Finite Element Approach to Calculation of Parameters of an

Document Sample

```					          Finite Element Approach to Calculation
of Parameters of an Interior Permanent
Magnet Motor
Damir Žarko1, Drago Ban1, Ratko Klarić2
1
University of Zagreb
Faculty of Electrical and Engineering and Computing
Department of Electrical Machines, Drives and Automation
Unska 3, 10000 Zagreb, Croatia
2
Istarski vodovod d.o.o.
Sv. Ivan 8, 52420 Buzet, Croatia
1
E-mail: damir.zarko@fer.hr, Tel: +385 1 6129 706, Fax: +385 1 6129 705
1
E-mail: drago.ban@fer.hr, Tel: +385 1 6129 673, Fax: +385 1 6129 705
2
E-mail: ratko.klaric@ivb.hr, Tel: +385 52 602 226, Fax: +385 52 602 201

Abstract − The method for calculation of parameters of an
interior permanent magnet (IPM) motor at any operating                phase B
q
point using 2-D finite element method is presented. This                axis
approach is suited to be used in the design stage where it is                           IqsXq
necessary to determine the motor parameters, namely                                                           IRa
inductances, which are simultaneously the function of the               IdsXd                                  E
motor dimensions and the terminal voltage constraints. An
analytical technique based on a 3-D geometric model of the
end winding region in which each coil is modeled as a set of                      V             I
serially connected straight filaments has been used for                   Vqs                             γ
calculation of the end winding leakage inductance. The                                   Iqs ϕ
calculation of the mutual inductance of the end coils is based
on the multiple solutions of the Neumann integral. This                           Vds               Ids             Ψmd   d   phase A
axis
approach to calculation of motor parameters has been applied
in the design of a 1.65 kW IPM motor for which a prototype
has been built and tested.
Key words − Permanent magnet motors, modelling, simula-
tion, testing

I. INTRODUCTION
phase C
Interior PM motors are attractive for applications where                  axis
operation in a wide speed range is required (e.g. traction).       Fig. 1. Phasor diagram of an interior PM motor.
Unlike surface PM motors which have the same value of
inductance in d and q axes and where all the torque is           motors the maximum torque per amp operation occurs at
produced by the magnet flux, interior PM motors have             γ = 0.
different d and q inductances which results in an additional
torque component called reluctance torque. The funda-            There are two distinct regimes of operation of an IPM
mental torque equation can be derived from the phasor            motor: constant torque below corner speed and constant
diagram shown in Fig. 1 and is given by                          power above corner speed, as indicated in Fig. 2. The
corner speed is defined as the maximum speed at which
⎡                               ⎤                        rated torque can be developed with rated current flowing
3 ⎢                                ⎥                        without exceeding the maximum terminal voltage available
(         )
Tem = p ⎢ Ψ md I qs − Lq − Ld I qs I ds ⎥ , Lq > Ld
2 ⎢                                ⎥
(1)    from the inverter. Above that speed it is possible to
⎢ Electromagnetic Reluctance
⎥                        maintain constant power, but it is not possible to develop
⎣ torque          torque        ⎦                        rated torque without exceeding the voltage constraint
where p is the number of pole pairs, Ψmd is the magnet flux,     imposed by the power supply. Moreover, the ability to
Ld, Lq, Ids and Iqs are the d and q axis inductances and         maintain constant power is not universally attainable for all
currents respectively. The maximum torque per amp                interior PM motor designs. Only designs which satisfy the
operation in this case will occur when the current phasor is     optimal flux weakening condition first derived by Schiferl
shifted by an angle γ relative to the q axis. In surface PM      and Lipo [1] will give the maximum constant power output
in the field weakening regime above corner speed. This                     simple rectangular shape which are cheap to manufacture.
condition is given by                                                      The motor has been designed using Differential Evolution
optimization algorithm [3] combined with the finite
Ψ md = Ld I R                                                        (2)   element method (FEM). The FEM was used to calculate the
where Ψmd is the flux of the magnets alone linked by the                   motor parameters for each design that emerged out of the
armature winding, Ld is the d axis inductance and IR is the                optimization scheme. The design objectives were to
rated armature current. The normalized characteristic                      maximize the electromagnetic torque at corner speed and
current is the quantity used to show the properties of the                 maximize the characteristic current of the motor. The
IPM motor in the flux weakening regime with respect to                     optimization has been formulated as a constrained
the optimal flux weakening condition. It is defined as                     multiobjective minimization problem where a population of
nondominant solutions is determined from which a single
Ψ md                                                                solution is selected as the best compromise [4]. The details
Ic =                                                                 (3)   of the optimization scheme used to design the prototype
Ld I R
IPM motor can be found in [5]. Here we only present the
The IPM motor design which satisfies (2) apparently has                    results related to calculation of the motor parameters.
Ic=1.
d axis
q axis
const. torque               const. power regime

Trated

Torque

Prated = Trated ω0

Power

ω0               Rotor speed                  ωmax
(corner speed)

Fig. 2 Capability curves of an interior PM motor.

The important design parameters of the IPM motor are its
inductances, and the magnet flux linkage, since they greatly
affect the amount of torque the motor can develop and its
power output in the flux weakening regime above corner
speed. The calculation of parameters presented in this                     Fig. 3. Stator and rotor laminations of the prototype IPM motor.
paper is based on finite element method with “permeance                    To simulate the operating point at the corner speed, a
freezing” which allows one to use superposition to extract                 magnitude of the current vector I and its position with
individual parameters of the motor under load and preserve                 respect to the q axis (angle γ) need to be determined first.
information about saturation. This principle has been                      These two parameters are at the same time constrained by
mentioned previously in literature [2].                                    the rated terminal voltage, which in this case is Vll=230 V
However, the main feature of the approach presented in this                (line-to-line value). The parameters which are known are
paper is its suitability for repetitive calculations in the                the armature current density J and the slot fill factor ffill.
design stage because it allows one to extract information                  They are used to calculate the available number of ampere-
about the motor inductances, the number of turns per coil                  turns per slot which follows from
and the rated current and torque from magnetostatic finite                 N c I = JSf fill                                                   (4)
element simulations while taking into account the terminal
voltage constraints. This principle of parameter calculation               where Nc is the number of turns per coil and S is the surface
has been integrated into an optimization scheme used to                    area of the slot. The slot area is calculated from the slot
design a 1.65 kW prototype IPM motor which has been                        dimensions which result from the stator outer and inner
built and tested.                                                          diameters and the yoke and tooth thicknesses.

II. CALCULATION OF IPM MOTOR PARAMETERS                                   The number of turns per coil and the current need to be
separated. The separation can be done in a straightforward
The proposed IPM motor topology analyzed in this paper is                  manner using a phasor diagram and considering the fact
shown in Fig. 3. The motor has two layers of cavities in the               that the sum of the motor back emf and the voltage drops
rotor which are shaped to accommodate magnets of a                         on the armature resistance and inductance have to be equal
to the terminal voltage V. The phasor diagram for the                         with
interior PM motor is shown in Fig. 1. The steady state
equations after resolving voltages and currents into d and q                               2⎛        1        1      ⎞
Ψ ds 0 =      ⎜ Ψ a0 − 2 Ψ b0 − 2 Ψ c0 ⎟
components can be written in the general form                                              3⎝                        ⎠
(10)
1
Vqs = Ra I qs + ωΨ ds                                                         Ψ qs 0     =    ( Ψ b0 − Ψ c0 )
3
Vds = Ra I ds − ωΨ qs                                                   (5)
The subscript 0 is used to emphasize the fact that flux
V = Vqs + Vds
2     2
linkages are calculated with Nc = 1. The currents Iqs and Ids
are also not known, but the total ampere turns per slot NcI
where
are known from (4). Using γ and NcI one can calculate
Ψ qs = Ψ mqd + Lq I qs + Lqd I ds
(6)    N c I qs = N c I cos γ
Ψ ds = Ψ md + Ld I ds + Ldq I qs                                                                                                                     (11)
N c I ds = N c I sin γ
Equations (6) also include the cross saturation terms which                   Considering the fact that Ψds = NcΨds0 and Ψqs = NcΨqs0, the
will be further discussed in the paper. The motor is                          torque equation (7) can be written in the form
designed for a rated operating point at corner speed
(1000 rpm). The parameters are initially calculated with an                               3
assumption that there is only one turn per coil with NcI as                   Tem =
2
(
p Ψ ds 0 N c I qs − Ψ qs 0 N c I ds   )                  (12)
the total current in the coil. The available number of
ampere turns per coil NcI is obtained from (4). The                           where NcIqs and NcIds are calculated from (11). With the
additional problem with the IPM motor is the lack of                          torque calculated for five values of γ, cubic spline is used
knowledge about the current control angle γTmax (see Fig. 4)                  to generate a smooth Tem vs. γ curve as shown in Fig. 4 and
at which maximum torque per amp is achieved. To                               find the value of Tmax where the maximum torque occurs.
determine γTmax, five nonlinear magnetostatic FE
simulations with γ ranging from -200 to -600 are run first.                          17
Cubic spline
The electromagnetic torque is calculated for each value of γ                              Temmax                                      FE
using the well known equation                                                        16
3
2
(
Tem = p Ψ ds I qs − Ψ qs I ds            )                              (7)
15

where p is the number of pole pairs, Ψds, Ψqs, Ids and Iqs are
Tem (Nm)

the d and q components of the flux linkage and current                               14
respectively. The flux linkages Ψds and Ψqs are determined
from the flux linkages of phases A, B and C. For instance,
13
the phase A flux linkage is calculated as a sum of the flux
linkages of all phase A coils. The flux linkage of each
individual coil is equal to the line integral of the magnetic                        12
vector potential Az along the contour of the coil. In a 2-D
FE model this integral is proportional to the difference                                             γTmax
between the average potentials in the meshed geometric                               11
-60                -50               -40        -30            -20
regions occupied by the coil sides located in different pole                                                         γ (degrees)
regions. The phase A flux linkage is then                                     Fig. 4. Torque versus control angle curve used to find the control angle
for maximum torque. Cubic spline is used to generate the curve from five
1⎛                    ⎞
Qcoilp         Qcoilp
points obtained by FE method.
Ψ a = p ∑ Ψ coilk = p ∑                  ⎜ ∫ Az dS − ∫ Az dS ⎟ N c la   (8)
k =1              k =1       S ⎜ S1
⎝          S2
⎟
⎠                Another nonlinear simulation with γTmax as the control
angle is run to get the field solution for this operating point.
where Qcoilp is the number of coil sides per phase in one                     The flux lines and the distribution of permeabilities for the
pole region, p is the number of pole pairs, Nc is the number                  rated operating point at corner speed are shown in Figs. 5
of turns of each coil, la is the length of the stator core and S              and 6 respectively. The most effective approach to
is the cross-section of the coil region. Subscripts 1 and 2                   calculate saturated Ld, Lq and Ψmd at this operating point
denote coil sides located in different pole regions. Since the                and still preserve all the information about the saturation in
number of turns per coil is not known at this stage, the flux                 the motor is to “freeze” the permeabilities in the nodes of
linkages Ψa0, Ψb0 and Ψc0 are calculated assuming Nc = 1.                     the finite element mesh. Once the permeabilities are frozen
The flux vector can now be formed                                             the problem becomes linear and the parameters can be
2π         determined one at the time. Three linear magnetostatic
2
(                           )
j
Ψ 0 = Ψ ds 0 + j Ψ qs 0 =      Ψ a 0 + aΨ b 0 + a 2 Ψ c 0 , a = e 3 (9)       simulations are needed to determine Ld, Lq and Ψmd. In
3                                                addition, cross-saturation parameters Lqd, Ldq and Ψmqd can
be determined from the same simulations. Although small                       The d and q components of the current vector are then
in this particular case, the effect of cross-saturation has
been generally recognized as the phenomenon caused by                         ids = 1A, iqs = 0                                                    (14)
saturation which manifests itself as the flux linkage in the                  Fig. 7 shows the field solution for this case The flux
axis perpendicular to the axis where the excitation is
linkages of phases A, B and C are calculated according to
applied. At this stage the inductances will be calculated for                 (8). The flux vector is formed and its d and q components
Nc = 1 since the actual Nc is still unknown.                                  are calculated using (9) and (10). The inductances for one
turn per coil are then
Ψ ds 0
Ld 0 =
ids
(15)
Ψ qs 0
Lqd 0 =
ids

Inductance Ld0 only takes into account the flux linkage in
the core region. The 2-D FE simulation neglects the
leakage flux in the end region. Hence, the end winding
leakage inductance Lew0 is calculated separately using a 3-D
analytical approach described later in this chapter and
added to Ld0 to calculate the total Ld. For the inductance
Lqd0 2-D FE simulation is sufficient since there is no cross-
saturation effect in the end region because the end winding
is located in the air.
.
Fig. 5. Flux lines in the cross section of the prototype IPM motor for the
case of rated torque at corner speed of 1000 rpm.

Fig. 7. Flux lines in the cross section of the prototype IPM motor with
Fig. 6. Distribution of relative permeabilities in the cross section of the   frozen permeabilities used for calculation of saturated inductances Ld0 and
prototype IPM motor for the case of rated torque at corner speed of           Lqd0.
1000 pm.
Inductances Lq0 and Ldq0
Inductances Ld0 and Lqd0                                                      The inductances Lq0 and Ldq0 are calculated in a similar
The first linear simulation is used to calculate Ld0 and Lqd0.                manner, only in this case the current vector needs to be
The magnet flux is turned off by setting the magnet                           aligned with the q axis. The phase currents are then
remanence Br to zero. The current vector must be aligned
with the d axis. Since the problem has become linear after                                            3
ia = 0, ib = −ic =        A                                          (16)
freezing the permeabilites, the magnitude of the current                                             2
vector can be chosen arbitrarily. If the magnitude is chosen                  which in turns gives
to be 1 A, then to align the current vector with the d axis
the instantaneous phase currents have to be defined as                        ids = 0, iqs = 1A                                                    (17)
ia                                                   The field solution is shown in Fig. 8. After calculating the
ia = 1A, ib = ic = −        = −0.5 A                                 (13)
2                                                    flux components, the inductances for one turn per coil are
given by
Ψ qs 0                                                               End winding leakage inductance Lew0
Lq 0 =
iqs                                                                The end winding leakage inductance is a part of the total
(18)     leakage inductance of the phase winding. In the core region
Ψ ds 0
Ldq 0 =                                                                      the coils are located in the slots which, with the assumption
iqs                                                               that iron is infinitely permeable, makes it easier to predict
the distribution of the leakage flux and calculate the
leakage inductance. The end winding region is more
difficult to analyze because its magnetic circuit is entirely
in the air and its winding structure is often characterized by
complex three-dimensional geometry of the coils. An
additional difficulty is the effect that adjacent coils and
phases have on each other.
The most accurate approach to the end winding leakage
inductance calculation would be to use the 3-D finite
element method. The main problem with the 3-D FE
method is the fact that the drawing of a complex 3-D
geometric structure, mesh generation and solution of a very
large system of equations are extremely time consuming.
This is the main obstacle for practical utilization of the 3-D
FE method for the end winding leakage inductance
calculation in the design stage, especially if optimization is
involved. The analytical approach is the remaining
alternative. There are different closed form analytical
Fig. 8. Flux lines in the cross section of the prototype IPM motor with       solutions that can be found in literature [6]-[8]. One of the
frozen permeabilities used for calculation of saturated inductances Lq0 and
Ldq0.
major problems of these solutions is that they have been
derived for an assumed geometric shape of the end coil
Flux linkages Ψmd0 and Ψmqd                                                   which may not be applicable for all types of windings and
all types of machines with different power ratings.
This simulation is used to calculate magnet flux linked by
Therefore a more flexible method has been used in this
the armature winding. In this case the magnets are turned
paper which models the end coil as a set of serially
on by setting the magnet remanence Br to the actual value
connected straight filaments. This allows one to define the
while phase currents are equal to zero. The field solution is
end coil geometry of an arbitrary shape and still calculate
shown in Fig. 9. The flux linkages of the phase windings
the inductance in a unique manner.
are calculated again using the previously described
procedure. Equation (10) is then used to calculate the flux                   The mutual inductance between any two coils in the end
linkages Ψmd0 and Ψmqd0. Note that although the magnets                       region is calculated by adding the contributions of all
act only in the direction of the d axis, there will also be a                 possible pairs of filaments in both coils. The method has
small cross saturation flux linkage in the q axis.                            been described in detail in [9], [10] for the case of a
turbogenerator with double-layer involute winding. It has
been modified and adapted in this paper to be used for a
single-layer end winding structure typical for small
permanent magnet motors. The method also takes into
account the influence of the iron core by applying the
method of images. Some assumptions on the magnetic
properties of the core and the geometry of the end region
are necessary to simplify the problem. The following
• The permeability of the iron core is constant,
• The iron core surface extends infinitely and fills the
entire half-space,
• The influences of slots, air gap and rotor shaft are
neglected,
• The coils are represented by infinitely thin conductors.
The three-dimensional contour of the end coil is replaced
by an arbitrary number of straight filaments depending on
Fig. 9. Flux lines in the cross section of the prototype IPM motor with       the desired accuracy of the geometric model of the coil.
frozen permeabilities used for calculation of the flux linkages Ψmd0 and      Such model for a single layer overlapping winding of a
Ψmqd0.                                                                        prototype IPM motor is shown in Fig. 10. The complete
model of the end winding containing all the coils is shown      to find the actual value of the end winding inductance.
in Fig. 11.                                                     Table I contains the values of the end winding leakage
inductance calculated for different values of relative
Talking about mutual inductance of the end coils is
permeability of iron. When µr=0, it is assumed that the iron
incorrect, because an end coil does not represent a closed
circuit. For the lack of a better term, we shall refer to the   is infinitely conductive and impermeable. When µr=1, the
mutual inductance of the end coils by which we shall imply      influence of iron is neglected. Finally, when µr=1, it is
the contribution to the mutual inductance of the end parts      assumed that the iron is infinitely permeable and infinitely
of the coils. Generally, the mutual inductance between two      resistive. For calculation purposes, infinity is replaced by
closed infinitely thin contours l1 and l2 in space can be       the value of 109 for the relative permeability.
calculated as                                                   In an actual motor, the core, the shaft and the frame would
act more as a conductive screen than as a permeable, highly
µ0         dl1dl2
M =
4π I   ∫
l1
r
(19)    resistive surface. Therefore, it is reasonable to take the
value of the end winding leakage inductance calculated
with µr=0 as an actual motor parameter. At this stage the
where r is the distance between segments dl1 and dl2 . If       leakage inductance is calculated assuming that Nc=1.
the contour of the end coil is replaced by a sequence of
straight filaments, then (19) is replaced by a sum of
integrals of the form (Neumann integrals)
B      b                                                                 15
dl2
N = cos ϕ ∫ dl1 ∫                                       (20)

z (mm)
10
A      a
r
5

which are solved for all possible combinations of straight                             0
50
filaments in both coils. This integral is based on the
40
integration of the magnetic vector potential due to the
current in one coil along the contour of the other coil and                                                   30

represents the flux linkage. The angle ϕ in (20) is the angle                                                         20                                                     60
10                                    50
between the directional vectors of two straight lines in 3-D                                                      y (mm)                            40
0
space while A, B, a and b denote starting and ending points                                                                                                    x (mm)

of each filament. The general solution of the Neumann           Fig. 10. 3-D model of the end coil of a single-layer overlapping winding
integral for two straight filaments in an arbitrary position    comprised of 20 straight filaments.
relative to each other in space can be found in [10].
80
The mutual inductances are calculated between the first
coil and all other coils obtained by rotating every                                             60

subsequent coil by an angle that corresponds to two slot                                        40
pitches. Thus, the mutual inductance between any two coils
amounts to calculation of the mutual inductance of the first                                    20

coil and another coil shifted by a certain number of slot
y (mm)

0
pitches. The mutual inductance of the coils of one phase is
-20
calculated by adding the mutual inductance of the first coil
and all the others that belong to the same phase. The                                           -40

obtained result is multiplied by two to take into
-60
consideration both sides of the machine. The mutual
inductance of two phases contains the sum of mutual                                             -80

inductances of every coil from the first phase and every                                              -80     -60     -40   -20      0
x (mm)
20      40       60        80

coil from the second phase. Due to symmetry, the mutual
a)
inductances of any two phases are equal. The total
inductance per phase is equal to the sum of the self
inductance and the mutual inductance with other two
z (mm)

10
phases. With the existence of the parallel paths in the                        0

winding, the obtained result is divided by the squared
50
number of paths ( a 2 ) because the inductance per phase is
p
0
proportional to the squared number of turns connected in                                                                                                                     50
series.                                                                                                     -50                                           0
y (mm)                                       -50
The mutual inductances between the first coil and all other                                                                                       x (mm)

coils for different values of iron permeability are shown in                                                                      b)
Fig. 12. The curves obtained for µr=0 and µr=1 represent        Fig. 11. Full 3-D model of the single-layer overlapping end winding of
the lower and upper bounds within which one would expect        the six pole prototype IPM motor. a) XY plane, b) Perspective view.
0.03                                                                                                                                    V
µr=0                                    Nc =
2
0.025                                                  µr=1                                              ⎡ Ra 0 N c I ds − ω Lq 0 N c I qs − ω Lqd 0 N c I ds − ωΨ mqd 0 ⎤ +
⎣                                                               ⎦
µr=∞
(23)
0.02                                                                                                                                                                    2
⎡ Ra 0 N c I qs + ω Ld 0 N c I ds + ω Ldq 0 N c I qs + ωΨ md 0 ⎤
⎣                                                              ⎦
Mutual inductance (µH)

0.015
Thus the number of turns per coil Nc is uniquely
0.01                                                                                          determined from the terminal voltage V and the motor
parameters calculated assuming that Nc = 1 for the case of
0.005
rated torque at corner speed. Once the number of turns per
0
coil is known the rated current of the motor is simply
Nc I
-0.005                                                                                         IR =                                                                                 (24)
Nc
-0.01
0             50            100        150         200       250      300       350
Angular position (degrees)                          where NcI is calculated from (4). The inductances, the flux
linkages and the number of turns per coil of the prototype
Fig. 12. Mutual inductance between two end coils as a function of relative
coil positions for the single-layer winding of the prototype IPM motor                                                   IPM motor have been calculated in the proposed manner
assuming one turn per coil.                                                                                              and are shown in Table II together with some other
TABLE I END WINDING LEAKAGE INDUCTANCE FOR DIFFERENT VALUES OF
important data of the motor.
RELATIVE IRON PERMEABILITY AND ONE TURN PER COIL                                           TABLE II PARAMETERS OF THE PROTOTYPE IPM MOTOR
Mutual                                 Parameter                            Symbol Value
Relative                                                                                            Total leakage
Self inductance               inductance                               Rated power [W]                         P     1651
permeability of                                                                                         inductance
of one phase                between two                               Rated line voltage [V]                  V     230
iron core                                                                                          Lew0=Lsew0+Mew0
Lsew0 [µH]                    phases                                 Rated current [A]                        I    7.63
µr                                                                             [µH]
Mew0 [µH]                               Rated corner speed [rpm]                nr    1000
0                              0.9625                  0.1188                 1.0813
Maximum speed [rpm]                   nmax    6000
1                              1.0082                  0.1444                 1.1527
∞                              1.0539                  0.1701                 1.2240          Rated electromagnetic torque [Nm]                                Tem             16.09
Rated shaft torque [Nm]                                           T              15.70
Power factor                                                    cos ϕ            0.628
Armature winding resistance Ra0                                                                                           Efficiency                                                        η              0.887
The armature winding resistance is initially also calculated                                                              RMS linear current density [A/m]                                 K1s             20538
assuming one turn per coil. The cross area of the conductor                                                               Magnet remanence [T]                                              Br              0.20
is calculated from the slot area S and the slot fill factor ffill                                                         Magnet relative permeability                                      µr              1.15
The resistance for a single-layer lap winding with one turn                                                               Number of turns per coil                                          Nc               27
per coil is                                                                                                               Armature resistance at 75oC [Ω]                                   Ra              1.01
Saturated q axis inductance at corner speed [mH]                  Lq              76.1
1          lturn Qs
Ra 0 =                                                                                                          (21)      Saturated d axis inductance at corner speed [mH]                  Ld              32.4
κ 75          3a 2 f fill S
p                                                                           End winding leakage inductance [mH]                              Lew              0.788
Saturated saliency ratio                                          ξ                2.35
where lturn is the the total length of one coil turn, Qs is the                                                           Current control angle [deg.]                                      γ              -46.83
number of slots, ap is the number of parallel branches and                                                                Cross saturation inductance [mH]                                 Ldq              1.039
κ75 is the conductivity of copper at 750 C.                                                                               Cross saturation inductance [mH]                                 Lqd             1.039
Number of turns per coil and the actual parameters                                                                        Magnet flux (rms value) [Vs]                                    Ψmd              0.1012
Magnet cross saturation flux (rms value) [Vs]                   Ψmqd             -0.0017
The relationships between the actual parameters and the
RMS line-to-line back emf at 6000 rpm [V]                       Emax              160
ones calculated for one turn per coil are given by
The correctness of the IPM motor model based on
Ra = N c2 Ra 0
parameters thus calculated can be verified by comparing
Ld = N c2 ( Ld 0 + Lew0 )                                                                                                the electromagnetic torque calculated directly from the FE
Lqd = N c2 Lqd 0                                                                                                        simulation and calculated from the expression

Lq = N c2 Lq 0 + Lew0                  (                   )                                                    (22)                  (
Tem = 3 p Ψ ds I qs − Ψ qs I ds      )                                               (25)

Ldq = N c2 Ldq 0                                                                                                         with d and q flux and current components given as rms
ψ md = N cψ md 0                                                                                                         values. After taking into account that
ψ mqd = N cψ mqd 0                                                                                                       Ψ qs = Ψ mqd + Lq I qs + Lqd I ds
Ψ ds = Ψ md + Ld I ds + Ldq I qs                                                     (26)
The number of turns per coil can now be calculated by
combining (5), (6) and (22) in the following manner:                                                                     Ldq = Lqd , I qs = I cos γ , I ds = I sin γ
and substituting into (25), the torque equation takes the                                                                    Calculated
0.08
form                                                                                                                         Experimental

(         )
0.07
Tem = 3 p ⎡ Ψ md I qs − Ψ mqd I ds + Ld − Lq I ds I qs +
⎣
0.06

(         )
Ldq I qs − I ds ⎤
2       2
⎦                                                       0.05
(27)

L (H)
1
(
= 3 p [ Ψ md I cos γ − Ψ mqd I sin γ + Ld − Lq I 2)                        0.04

q
2
sin(2γ ) + Ldq I 2 cos(2γ ) ⎤ ⎦
0.03

0.02
The torque calculated using (27) with parameters from
Table III is                                                                   0.01

Tem = 3 ⋅ 3 ⋅ ⎡ 0.1012 ⋅ 7.63cos( −46.830 ) − ( −0.0017) ⋅ 7.63 ⋅
⎣
0
0    2     4       6              8   10      12          14
i q (A)
1
sin(−46.830 ) +   ( 0.0761 − 0.0324 ) 7.632 ⋅         (28)
a)
2
0.04
sin(−2 ⋅ 46.83 ) + 0.001039 ⋅ 7.63 cos(−2 ⋅ 46.83 ) ⎤
0                   2              2
⎦
Calculated
Experimental
0.035
= 16.13 Nm
0.03
which is very close to 16.09 Nm in Table II calculated
directly from the FE solution.                                                 0.025
L (H)

0.02
III. COMPARISON OF CALCULATE AND
d

MEASURED INDUCTANCES                                             0.015
The 1.65 kW prototype IPM motor has been designed using
the multiobjective optimization algorithm previously                            0.01

mentioned. The IPM motor coupled to a 25 kW induction
0.005
machine which is used as a load is shown in Fig. 13. The
purpose of designing and building the prototype has been                          0
0   2      4       6             8   10       12         14
to investigate if the physical properties of the motor                                                     -i (A)
d
predicted in the design stage can be actually achieved in
b)
practice. The armature winding inductances Ld and Lq have
Fig 14. Comparison of measured and calculated inductances of the
been measured using static tests with the locked rotor The             prototype IPM motor. a) q axis inductance, b) d axis inductance.
results of experiments conducted on the motor showed that
the measured q axis inductance was approximately 10-20%                                     IV. CONCLUSION
lower than the calculated one, while the measured d axis
inductance was about 10% higher than calculated (Fig. 14a              A finite element based approach to calculation of
and 14b). The lower value of the inductance Lq could be                inductances of an interiror PM motor has been presented.
attributed to a larger effective air gap than assumed by the           This particular approach is suitable to be used in the design
design. The larger air gap can occur due to manufacturing              stage since it allows one to calculate the motor parameters
tolerances, but can also be created by changed properties of           while simultaneously taking into account the terminal
the core material due to punching and laser cutting of the             voltage constraint at the corner speed. Thus it is possible to
rotor laminations.                                                     determine the number of turns per coil and the rated
armature current from a single magneotstatic FE
simulation. This principle of parameter calculation has
been integrated into an optimization scheme used to design
the 1.65 kW prototype IPM motor which has been built and
tested. The difference between calculated and measured
inductances of the prototype motor is between 10% and
20%. Such difference could be attributed to manufacturing
imperfections or changed properties of the core material
due to punching of the stator laminations and laser cutting
of the rotor laminations.

ACKNOWLEDGMENT
The authors wish to thank the company KONČAR–
MES d.d., Zagreb, Croatia for building the prototype IPM
Fig. 13. Prototype IPM motor coupled to the induction machine via
torque transducer.                                                     motor.
REFERENCES.
[1]  R. F. Schiferl and T. A. Lipo, “Power Capability of Salient Pole
Permanent Magnet Synchronous Motors in Variable Speed Drive
Applications,” IEEE Transactions on Industry Applications, vol. 26,
pp. 115-123, Jan./Feb. 1990.
[2] N. Bianchi, S. Bolognani, “Magnetic Models of Saturated Interior
Permanent Magnet Motors Based on Finite Element Analysis,”
Thirty-Third IAS Annual Meeting, St. Louis, Missouri, Vol. 1,
pp. 27-34, 12-15 Oct 1998.
[3] R. Storn, K. Price, “Differential Evolution - a Simple and Efficient
Adaptive Scheme for Global Optimization over Continuous Spaces,”
Technical Report TR-95-012, ICSI, March 1995.
[4] C.A.C. Coello, R.L. Becerra, “Evolutionary Multiobjective Optimi-
zation Using a Cultural Algorithm,” Proceedings of the 2003 IEEE
Swarm Intelligence Symposium, pp. 6-13, April 2003.
[5] D. Žarko, A Systematic Approach to Optimized Design of Permanent
Magnet Motors With Reduced Torque Pulsations, Ph.D. thesis,
University of Wisconsin-Madison, USA, 2004.
[6] T. A. Lipo, Introduction to AC Machine Design. Wisconsin Power
Electronics Research Center, University of Wisconsin, 2004.
[7] M. Liwschitz-Garik and C. C. Whiple, Electric Machinery, Vol. 2
AC Machines. Van Nostrand Publishers, 1961.
[8] P. L. Alger, Induction Machines - Their Behavior and Uses. Gordon
and Breach Science Publishers, 1965.
[9] I. Mandić, Computation of Magnetic Fields in Saturated Iron
Structures with Special Reference to the Computation of Short
Circuit Performance of Induction Motors with Wound Rotors, Ph.D.
thesis, University of Southampton, Faculty of Engineering and
Applied science, Department of Electrical Engineering,
Southampton, United Kingdom, 1974.
[10] D. Ban, D. Žarko, and I. Mandić, “Turbogenerator End Winding
Leakage Inductance Calculation Using a 3-D Analytical Approach
Based on the Solution of Neumann Integrals,” IEEE Transactions on
Energy Conversion, Vol. 20, No 1, pp. 98-105, March 2005.

```
DOCUMENT INFO
Shared By:
Categories:
Stats:
 views: 9 posted: 5/11/2010 language: Croatian pages: 9
How are you planning on using Docstoc?