Calculul Parametrilor Pm U [607598]

538 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 20, NO. 3, SEPTEMBER 2005
Calculation of Parameters of Single-Phase PM Motor
for Design Optimization
H. B ¨ulent Ertan, Member, IEEE ,B¨ulent Da ˇg, and G ´erard-Andr ´e Capolino, Fellow, IEEE
Abstract —This paper presents methods of calculation of param-
eters of single-phase permanent-magnet (SPPM) motor, in termsof motor dimensions and material properties, which are utilizedin the dynamic model of the motor. The intention of the study isto develop means of SPPM performance calculations, which lendthemselves to be employed within a mathematical design optimiza-tion approach. The calculated parameters are compared with mea-sured values and are shown to be accurate for the purpose of thestudy.
Index Terms —AC motors, design methodology, design optimiza-
tion, parameter estimation, permanent magnet motors.
NOMENCLA TURE
ε Phase angle of the supply voltage at starting.
θ Position of the rotor.
θ0 Initial position of the rotor.
θp Phase shift of the fundamental component of air gappermeance variation.
L Inductance of the stator winding.
R
cuResistance of the stator winding.
RfeResistance corresponding to core loses.
N Number of turns of stator winding.
e Back emf introduced by the rotor.
J Moment of inertia of rotor and load.
T1 Load torque.
Trp Peak reluctance torque.
ΦrpPeak rotor flux.
ΛFePermeance of the stator core.
ΛσPermeance of the leakage path outside the stator legs.
ΛrPermeance of the rotor.
ΛFPermeance of the stator core.
ΛgPermeance of air gap.
ΛsPermeance of the air gap between the stator legs.
µp Permeability of the permanent magnet rotor.
/IfracturrEquivalent mmf of the permanent magnet rotor.
/Ifractur With many different subscripts not mentioned either in
the nomenclature.
zg Axial length of the rotor.
r Radius of the rotor.
rh Radius of the hole.
r1 Radius of the small step.
Manuscript received February 1, 2005. Paper no. TEC-00033-2005.
H. B. Ertan and B. Da ˇg are with Electrical and Electronics Engineering
Department, Middle East Technical University, 06531 Ankara, Turkey (e-mail:[anonimizat]).
G.-A. Capolino is with the Department of Electrical Engineering, Uni-
versity of Picardie-Jules Verne, 80039 Amiens Cedex 1, France (e-mail:
[anonimizat]).
Digital Object Identifier 10.1109/TEC.2005.852962
Fig. 1. Cross section of a single-phase synchronous motor with a permanent-magnet rotor.
r2 Radius of the large step.
g1 Length of the small air gap between the rotor and stator.
g2 Length of the large air gap between the rotor and stator.
gr1Length of the outermost boundary of the flux tube 1.
gr2Length of the outermost boundary of the flux tube 2.
gr3Length of the outermost boundary of the flux tube 3.
yg Length of the air gap between the parallel sections of
the stator core.
xw Length of the section of the stator core housing statorwinding.
x
g Length of the section of the stator core not housing
stator winding.
zs Axial length of the stator core.
zsf Corrected axial length of the stator core taking into
account fringing.
xs Width of the stator core.
xsf Corrected width of the stator core taking into account
fringing.
σ Carter’s coefficient.
I. INTRODUCTION
SINGLE-PHASE permanent-magnet (SPPM) synchronous
motors of the type shown in Fig. 1 have found growing ap-
plications especially in low-power household appliances. This
is because this motor is well suited to mass production pur-poses and has a simple construction, and therefore has a cost
advantage. Also, they are known to be more efficient than rival
motors [1].
On the other hand, like all PM motors, if the motor is desired
to be self-starting, problems arise. Starting behavior of this typeof motor is dependent on the initial position of the rotor, on the
load inertia and the frictional load [2]. The starting behavior
is also shown to be dependent on the electrical parameters,especially on the phase of the stator voltage at the switching
0885-8969/$20.00 © 2005 IEEE

ERTAN et al. : CALCULATION OF PARAMETERS OF SINGLE-PHASE PM MOTOR FOR DESIGN OPTIMIZATION 539
Fig. 2. Electrical equivalent circuit of an SPPM motor.
instant [3]. As noted in [2], the motor oscillates at startup with
its Eigen frequency. If this frequency is close enough to the
mains frequency, the amplitude of the oscillations increase and
the motor begins to spin continuously. As the motor gets largerthe Eigen frequency becomes smaller and therefore starting with
50-Hz mains frequency is not possible.
At starting, the motor may rotate in either direction. There-
fore, the applications are limited to loads, which are not sensitive
to direction of rotation.
The problems mentioned above, however, can be overcome
with a simple electronic aid and the motor may remain attractive
for higher power levels. To be able to reach a sound decisionon this issue, one needs to understand the factors (parameters)
which control the performance of the motor. Based on this un-
derstanding, a design which satisfies the requirements may bereached.
Of course, the first thing one needs is an accurate enough
model to describe the behavior of the motor. The usual syn-chronous machine electrical circuit model may be used for anal-
ysis of the performance in the form given in Fig. 2, along with the
mechanical equations. In Fig. 2, the resistance R
feis employed
to take into account the core losses.
Considering the electrical equivalent circuit in Fig. 2, equa-
tions of the motor may be written as in [2] as follows:
iL=1
L/integraldisplay
(Vn+e)dt
Vn=Vs−isRcu
is=iL+ife
ife=Vn
Rfe. (1)
And the mechanical equation
J¨θ=NiΦrpsinθ−Trpsin[2(θ−θ0)]−Tl. (2)
In (1) and (2),Vsis the supply voltage with a frequency of w
and phase angleεat the switching instant. RcuandLare resis-
tance and inductance of the stator winding, respectively. Rfeis
the resistance corresponding to core loses. In the back emf ex-
pression(e=˙θNΦrpsin(θ−θ0))Nis the number of turns of
the stator winding and Φrpis the peak rotor flux linked by stator
winding.θis the position of the rotor (angle between rotor and
stator axis) andθ0is the initial position of the rotor. In the torque
balance equation, Jis the moment of inertia of the rotor and the
load,Trpis the peak reluctance torque, and Tlis the load torque.
Fig. 3. Magnetic equivalent circuit of the rotor of an SPPM motor.
Fig. 4. Typical B-H characteristics of a permanent magnet.
Note that in (1) and (2), rotor flux linked by stator winding and
reluctance torque are assumed to be sinusoidal function of rotorposition(θ). In practice, these variables do not show such pure
variations, but the approximations are acceptable.
For the application of optimum design techniques, one needs
to express the electrical and mechanical parameters of the motor
in terms of the dimensions of the motor. In this manner, the ob-
jective function (cost, size, etc.) can be expressed in terms ofdimensions as well as the constraints introduced by requirements
like rated torque, efficiency, and the starting performance, in ad-
dition to constraints due to materials and manufacturing process.
In this study, methods for the calculation of peak rotor flux and
reluctance torque, initial position of the rotor under no excitation
(rest position), and stator inductance will be introduced and
verified with experiments.
II. R
OTOR PARAMETERS
In this section, peak-reluctance torque and peak rotor flux
will be calculated. For these calculations, one needs to consider
the magnetic equivalent circuit of the motor with no stator exci-
tation. In this case, the equivalent circuit is as shown in Fig. 3.In Fig. 3, Λ
F(θ)is the stator magnetic circuit permeance and
will be neglected here for the sake of simplicity. Λrrepresents
the rotor permeance
Λr=µp·Am
/lscriptm(3)
whereµpis defined as in Fig. 4. Amis the cross section of the
rotor magnet and/lscriptmrepresents the mean flux path length in the
magnet.Fris the rotor MMF and is expressed as
Fr=Hc·/lscriptm (4)
whereHcis defined as in Fig. 4. Note that Amandlmare not
constant because of the cylindrical shape of the rotor, so all

540 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 20, NO. 3, SEPTEMBER 2005
Fig. 5. Differential analysis of rotor parameters.
the parameters will be defined differentially as explained in the
following sections.
Λg(θ)is the air gap permeance. In this paper, the rotor mmf
is taken to be constant and the variation of the air gap flux withθis modeled by taking Λ
gto be a function of θ. Assuming
that the variation of this permeance can be represented by a dc
component and a fundamental component
Λg(θ)=Λ g0+Λ g1cos2θ. (5)
However, since air gap is not uniform and the rotor length is
not constant, elements of the magnetic circuit will be defined
differentially.
A. Differential Rotor MMF and Permeance
Consider a differential portion of rotor represented by a hori-
zontal differential length dxas shown in Fig. 5(a). It is possible
to express the flux path /lscriptmfor this section as
/lscriptm=rcosθs (6)
whereθsis defined with respect to the rotor axis as in Fig. 5(a)
and is independent of rotor position, θ.ris the radius of the
rotor. Therefore, from (4), rotor MMF for the section of therotor excluding the shaft hole becomes
/Ifractur
r(θs)=Hcr·cosθs. (7)
For the section of the rotor including shaft hole rotor, MMF
must be treated differently. As the permeability of the magnet is
very close to that of air, it is acceptable to assume that flux linesfollow a straight line (Fig. 6) through the shaft hole and magnet.
With this assumption, for the section of the rotor including the
shaft hole, rotor MMF becomes
/Ifractur
rh(θs)=Hc(rcosθs−rhcosθsh) (8)
whererhis the radius of the hole [Fig. 5(b)] and
θsh=θsπ/2
θhfor−θh<θ s<θ h (9)
whereθhis defined as in Fig. 5(a).
To calculate the differential rotor permeance, consider again
section of the rotor marked by dxin Fig. 5(a)
x=rsinθs
dx=rcosθsdθs (10)
Fig. 6. Magnetic field solution results for several rotor positions.
dΛr(θs)=µpdAm
/lscriptm=µpzgdx
rcosθs=µpzgdθs (11)
wherezgis the axial length of the rotor. Expression given in (11)
is valid for the section of the rotor with shaft hole, too, because
the hole is assumed to be occupied by air and permeability of
the permanent magnet around the hole (µp)is very close to that
of air.
B. Differential Air-Gap Permeance
Calculation of the variation of Λg(θ)is made particularly
difficult due to the existence of the fringing field.
At this stage, it is wise to consider the variation of the motor
magnetic field with angle θsand, hence, obtain an idea of how
the fringing field changes with position. For this purpose, field
solutions of the motor are obtained for different rotor positions,θand the result of this study is given in Fig. 6.

ERTAN et al. : CALCULATION OF PARAMETERS OF SINGLE-PHASE PM MOTOR FOR DESIGN OPTIMIZATION 541
Fig. 7. Analysis of air gap permeance density distribution.
By studying this figure, it is decided to represent the fringing
flux by flux tubes with constant cross section, defined in termsof the dimensions of the motor such as y
gandx1as displayed
in Fig. 7. In defining the tubes, it is assumed their boundary is
a fringing flux line defined by a circle. The permeance of theflux tubes on either side of the rotor can now be written in terms
of the cross section of the tube (A
i)and their mean flux path
lengthgri. Lengths of the outer flux lines seen in Fig. 7 can be
calculated from (12)–(14). Length of fringing flux lines, gris
assumed to vary uniformly between the inner and outer gaps ofthe tubes
g
r1=g2+πyg
4(12)
gr2=g1+πyg
4+/parenleftBig
r1+yg
2−r1cosθ1−x1/parenrightBigπ
2(13)
gr3=g1+x1(π/2+θ2/2−θ1) (14)
where
θ1=s i n−1(yg/2r1) (15)
θ2=2s i n−1(x1/2r1). (16)
It is now possible to draw the variation of the differential air
gap size with surface angle θs. Therefore, for a motor with axial
lengthZgdifferential air gap permeance can be expressed as
follows:
dΛg(θs)=µ0zgrg
grdθs (17)
wherergis defined as mean air gap radius and is given by
rg=r+(g1+g2)/4. (18)
The variation of differential air gap permeance can be visu-
alized in a simplified manner if it is normalized, for example,
with respect to the permeance of the small air-gap section. In
this case, the differential permeance is only a function of g1/gr.
The variation of normalized permeance (g1/gr)predicted in
this manner is plotted in Fig. 8(a). For the sake of simplic-ity, the variation of differential air gap permeance dΛ
g(θsθ)is
represented by a dc component and its fundamental harmonic
component. For this purpose, Fig. 8(b) can be used which isthe representation of (g
1/gr)by its dc and fundamental compo-
Fig. 8. Normalized air-gap permeance variation and its representation with
dc and fundamental components. (a) Normalized air gap permeance variation
and (b) representation of 100 g1/g1by 9th order polynomial and its dc and
fundamental components.
nents. In this case
g1
gr(θs)=0.75+0.32cos(θs−θp) (19)
whereθpis the phase shift of the fundamental component (found
from Fourier expansion; see Fig. 8(b) and is 0.1068 rad (6.12◦).
Hence, considering a mean air-gap radius rgand taking the
depth of the motor Zgdifferential air gap permeance becomes
dΛg(θs,θ)
=/parenleftbigg
0.75θ0zgrg
g1+0.32θ0zgrg
g1cos2(θs−θp+θ)/parenrightbigg
dθs.(20)
Note that in (20), the rotation of the rotor is introduced by a
phase shift of rotor position (θ)in the ac component.

542 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 20, NO. 3, SEPTEMBER 2005
Fig. 9. Measured and calculated reluctance torque against position.
C. Calculation of Reluctance Torque
The reluctance torque can be calculated in terms of magnetic
co-energyWco
Tre=dWco
dθ(21)
the differential co-energy is given by
dWco(θs,θ)=1
2F2
r(θs)dΛre(θs,θ) (22)
whereFr(θs)is the rotor MMF and dΛreis equivalent differ-
ential permeance seen by the rotor. Considering the magneticequivalent circuit in Fig. 3, dΛ
reis calculated as follows:
dΛre(θs,θ)=dΛr(θs)dΛg(θs,θ)
dΛr(θs)+dΛg(θs,θ). (23)
Then, from (11) and (20)–(23) and taking into account the
shaft hole, reluctance torque for any rotor position becomes
Tr(θ)=/integraldisplay−θh
−π/2/Ifractur2
rdΛ/prime
re+/integraldisplayθh
−θh/Ifractur2
rhdΛ/prime
re+/integraldisplayπ/2
θh/Ifractur2
rdΛ/prime
re
(24)
wheredΛ/prime
reis given by
dΛ/prime
re=−2c2bsin2(θs−θp+θ)
[c+a+bcos2(θs−θp+θ)]2dθs (25)
with
a=0.75µ0zgrg/g1 (26)
b=0.32µ0zgrg/g1 (27)
c=µpzg. (28)
The measured reluctance torque variation of the test motor
and the torque calculated from (24) is displayed in Fig. 9, which
shows that the computed peak torque is 2.57 Ncm, while the
measured value is 2.62 Ncm. This is a very acceptable level ofaccuracy. The place of the peak torque is however is not well
predicted. This is almost certainly due to the approximate repre-
sentation of Λ
g(θs,θ)in (20). However, this does not have any
significance as far as the performance calculation is concerned.
Fig. 10. Representation of the fringing field at the rest position.
Therefore, more attention is not paid to this issue at this mo-
ment. It may be worthwhile to investigate whether an accurate
representation of Λg(θs,θ)leads to a more accurate prediction
of peak torque position.
An important issue is determination of the rest position. The
rest position corresponds to the position of the rotor where rotor
flux is maximum. Fig. 8(b) indicates that this occurs at θpwhich
is0.1068rad = 6.12◦. The measured rest position of the rotor is
6◦. This corresponds to a remarkably good accuracy. However,
this could be a coincidence. Therefore, a more accurate method
for cross checking is given in Section II-D.
When the peak-pole flux value is calculated from the equa-
tions derived in this section, it is found that the value found does
not compare well with the measured value. This is an expectedresult, as the reluctance variation of the air gap is approximated
by its fundamental and dc components only. For this reason,
a more accurate approach for predicting the peak rotor flux isdeveloped and is presented in the following section.
D. Calculation of Peak Rotor Flux
When the approximate representation of air gap permeance in
(20) is used for the calculation of peak rotor flux, it is observed
that the result deviates considerably from the measurement.
Therefore, for this calculation the actual air-gap permeance vari-
ation needs to be considered. Since peak rotor flux is of interest
here, analysis is to be performed just in the vicinity of the restposition of the rotor where rotor flux is maximum. The rotor
flux linked by the stator winding is assumed to be sinusoidal
function of rotor position (θ)as in (1) and (2).
When the field lines of the test motor for the rest position
in Fig. 6(a) is investigated it is observed that the fringing field
is important only in the region adjacent to the smaller air gap.It is essential to take into account the effect of this fringing
field to determine the peak flux with acceptable accuracy. For
this purpose the field distribution in the fringing field regionis studied and it is found that, for all practical purposes, it is
sufficient to consider a flux tube with the dimensions shown
in Fig. 10. The angle θ
fin this figure can be represented as
follows:
θf=2s i n−1/parenleftbiggx1
r1/parenrightbigg
. (29)

ERTAN et al. : CALCULATION OF PARAMETERS OF SINGLE-PHASE PM MOTOR FOR DESIGN OPTIMIZATION 543
Fig. 11. Equivalent rotor shape with uniform airgaps.
The length of the outermost flux line gfcan be calculated
from
gf=g1+2×1(π/2+θf/2−θ2)+x1π/2 (30)
whereg1is the length of small air gap.
To simplify the handling of the problem, the magnetic circuit
around the rotor can be represented with an equivalent air gapregion composed of two uniform air gaps with lengths g
1and
g2. Such a representation is shown in Fig. 11. In this figure,
the transition region, represented with the angle θtin Fig. 10,
is equally included in both uniform air gap regions of (g1)and
(g2), and the fringing field effect is represented by extending
the constant air gap region (g1), by an angleθfe.θfeis calculated
from (31), which is determined so as to give the same perme-
ance as the actual fringing region (a linear variation is assumed
betweeng1andgf)
θfe=θf
2/parenleftbigg
1+g1
gf/parenrightbigg
. (31)
With the assumptions considered above, the air gap now has
two sections (see Fig. 11) where rg1andrg2are the mean
radiuses of the respective sections. The differential permeance
of the sections are then calculated as follows, for Section 1:
dΛg1(θs)=µ0rg1zg
g1dθs. (32)
Similarly, for Section 2
dΛg2(θs)=µ0rg2zg
g2dθs. (33)
The next issue is determining the flux crossing the air gap.
Considering the magnetic equivalent circuit given in Fig. 3, forthe section of the rotor facing air-gap Section 1, the equivalentTABLE I
VARIA TION OF ROTOR FLUXΦRWITHRESPECT
TOROTOR POSITION
differential permeance becomes
dΛre1(θs)=dΛ0(θs)dΛg1(θs)
dΛ0(θs)+dΛg1(θs)(34)
and for Section 2
dΛre2(θs)=dΛ0(θs)dΛg2(θs)
dΛ0(θs)+dΛg2(θs). (35)
Total flux linking the stator coil is calculated from
Φ=/integraldisplay
/IfracturdΛ. (36)
Therefore, considering Fig. 11, rotor flux linked by stator
winding is calculated as follows:
Φr(θ)=/integraldisplay(θg1−θt/2−θ)
θh/Ifractur0(θs)dΛre1(θs)
+/integraldisplayθh
−(θ+θt/2)/Ifractur0h(θs)dΛre1(θs)
+/integraldisplay−(θ+θt/2)
−θh/Ifractur0h(θs)dΛre2(θs)
+/integraldisplay−θh
−(θg2+θt/2+θ)/Ifractur0(θs)dΛre2(θs) (37)
whereθis the position of the rotor and Φrpshould be cal-
culated forθ=θ0. In (37), the second and third components
correspond to the contribution of the section of the rotor with
shaft hole. In Table I. some values of Φrpcalculated from (37)
for several values of rotor position (θ)are shown. As seen, it
reaches a peak value of 1.28 10−4Wb at the vicinity of measured
rest position (θ0=6◦)while it was measured experimentally as
1.2510−4Wb. This certainly is an acceptable prediction accu-
racy.
III. R ESTPOSITION OF THE ROTOR(θ0)
When there is no stator excitation, the rotor settles in a posi-
tion where the permeance of the magnetic circuit is maximum.
Knowing this position is important for predicting the starting
behavior of the motor with acceptable accuracy. In calculating
the initial (rest) position of the rotor, the position of the rotor atwhich there is minimum (zero) reluctance torque or maximum
flux in the air gap could be considered by using the related cal-
culations in previous sections. Indeed the rest position predictedin Section II-D is found to match the measurement well for the

544 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 20, NO. 3, SEPTEMBER 2005
Fig. 12. Equivalent rotor shape with uniform airgaps for the rest position
calculation.
test motor. However, in view of the approximations made in the
calculations of previous sections, it is felt that a cross-checkwould be appropriate.
Although several approaches with varying degree of accuracy
has been experimented with, one of these will be presented here.
In this approach, the rotor MMF along the rotor surface has been
assumed to be constant with an average value. In other words, itis assumed that the permeance of the air gap alone determines
the rest position.
Consider the equivalent rotor shape with two uniform air
gaps given in Fig. 12, whose permeance expression derivation
is explained in the previous section. The MMF on the surface of
the rotor may be viewed as to be proportional to the mean fluxpath in the rotor magnet. Then, neglecting the effect of shaft
hole, for example, for the section of the rotor facing the small
air gap region (g
1)the mean flux path ( lr1in Fig. 12) in the
rotor can be calculated as
lr1=/integraltext(θg1−θt/2−θ)
−(θt/2+θ)rcosθ
θg1
=rsin(θg1−θt/2−θ)+sin(θt/2+θ)
θg1. (38)
Then, the equivalent MMF on the rotor surface facing Section
1 becomes
/Ifracturs1=Hk·/lscriptr1 (39)
whereHkis the average operating magnetic flux intensity of
the rotor.
The permeance of the air gap region (Section 1) in Fig. 11,
for a unit axial length of the rotor is
Λg1=µ0rg1θg1
g1. (40)
Fig. 13. Field distribution just for stator excitation.
Then, the flux crossing the air gap in the considered section
is given by the following equation:
Φg1=/Ifracturs1Λg1
=Hkrµ0rg1
g1[sin(θg1−θt/2−θ)+sin(θt/2+θ)].
(41)
In the same way, with changing parameters, the air-gap flux
of the uniform gap region (g2)is obtained as follows:
Φg2=Hkrµ0rg2
g2[sin(θg2+θt/2+θ)−sin(θt/2+θ)].
(42)
The total air-gap flux is
φg=φg1+φg2. (43)
The position at which the air-gap flux has its maximum value
is the rest position of the rotor and can be found by solving (44)
forθ
dφg
dθ=0. (44)
From (42), this rest position of the rotor is calculated from
sin/parenleftBig
θg1−θt
2−θ0/parenrightBig
sin/parenleftBig
θg2+θt
2+θ0/parenrightBig=Λ2
Λ1=g1rg2sin(θg2/2)
g2rg1sin(θg1/2). (45)
By substituting the dimensions of the test motor into (45),
the rest position is found as 6.8◦. The measured rest position
is 6◦. It is quite clear that the approach presented here is quite
accurate, especially when the error in measuring the rest position
is considered.
IV . C ALCULA TION OF STATOR INDUCTANCE
For the calculation of the inductance of the stator coil it is
essential to consider the three-dimensional (3-D) field distribu-
tion. This is because the air gap, as seen by the stator coil, is very
large (permanent magnet rotor has very low permeability) andthe leakage and fringing flux becomes important. This makes the

ERTAN et al. : CALCULATION OF PARAMETERS OF SINGLE-PHASE PM MOTOR FOR DESIGN OPTIMIZATION 545
Fig. 14. Magnetic equivalent circuit for stator excitation.
Fig. 15. Magnetic circuit of the stator with a stepped-air gap.
calculation of stator inductance particularly difficult. To obtain
an idea of what dimensional parameters are important for this
calculation, 2-D and 3-D field solutions of the test motor aredone at rated current. The result of one of the 2-D field solutions
is shown in Fig. 13. This figure illustrates the importance of the
flux leaking out of the magnetic circuit. Fig. 14 is the magnetic
equivalent circuit of the motor when only the stator is excited.
In Fig. 14, Λ
Feis the permeance of the stator core and is
neglected since the flux densities in the core are very low. Λl
parallel core section embracing the rotor, is the permeance of
the air gap between the stator legs including the fringing fluxbetween the stator legs. Λ
σis the permeance of the leakage path
outside the stator legs, flux bulging out between stator legs and
will be neglected. Ni is the MMF of the stator excitation whereiis the stator winding current.
In attempting to derive the equation of the stator inductance
here, a number of simplifications are made. This is becausethe method developed here will be used within an optimization
algorithm and is likely to be used hundreds of times within the
procedure. Therefore, it is essential to avoid including details,which improve the calculation accuracy little, at the expense of
much computational burden.
A. Simplification of Stator Magnetic Circuit
Actual magnetic circuit of a SPPM motor is shown in Fig. 13.
The variable dimensions of the air gap make the calculation
of the permeance of the space between stator legs unnecessar-
ily complicated. However, since the distance between the legs
is quite large, a simplification in the details is possible with-out losing accuracy. For this purpose, first the transition region
between the sections with air gaps g
1andg2is divided into
two equal parts and the stepped air-gap geometry in Fig. 15 isobtained.
Fig. 16. Stator magnetic circuit with simplified rotor gap.
Fig. 17. Simplified magnetic circuit of the stator.
A further simplification is made by replacing the stepped air
gap geometry with a equivalent geometry as in Fig. 17 so as
to give the same permeability. For this purpose, the following
actions are taken. First, the air gap is replaced with an equivalentunstepped geometry in Fig. 16 whose radius, R
s
Rs=θs1R1+θs2R2
θs1+θs2(46)
whereR1andR2are defined as in Fig. 15. Later, the unstepped
geometry is replaced with an equivalent rectangular geometrywith air-gap length, y
r(see Fig. 16) where
yr=Rssinθs
θs(47)
and
θs=c o s−1/parenleftbiggyg
Rs/parenrightbigg
. (48)
Finally, this rectangular geometry is replaced with another
rectangular geometry in Fig. 17 with air gap length of yg,t h e
actual length between the parallel sections of the stator legs (seeFig. 15). Now, the resulting width, x
/prime
gof the part of the stator

546 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 20, NO. 3, SEPTEMBER 2005
Fig. 18. (a) Top view of the stator core with fringing regions. (b) Approximate
model of stator core for fringing calculations.
core not bearing the excitation coil seen in Fig. 17 becomes
x/prime
g=xg−xr/parenleftbigg
1−yg
yr/parenrightbigg
. (49)
B. Fringing Flux Calculations
Because of the very large distance between the stator legs,
flux completing its path outside the magnetic core becomes
important. Fig. 18(a) is the top view of the stator core showing
the fringing regions. Flux paths on the side of the coils and the
fringing flux (Fig. 13) adjacent to the end of the core is takeninto account.
In Fig. 18(a)z
sis the actual axial length of the stator core
andzsfis the corrected axial length of the core including the
effect of the side fringing region. xsis the width of the stator
legs of the simplified geometry and xsfis the extension of the
stator core due to the front fringing region.
In this study,zsfandxsfare determined by assuming similar
imaginary structures as seen in Fig. 18(b) at a distance 2wffor
front fringing and 2wsfor side fringing. In this approximate
model, determination of the distances wfandwsare important.
Fringing flux regions of the test motor are analyzed by 3-D field
solutions. It is decided to take the values of wfandwsas the
distances between the points where the fringing flux lines start
and the point at which their density is reduced to 1% of the
starting value. In this way, it is guaranteed that the imaginarystructures has no any magnetic effect on the actual geometry.
As a result of this study, the distances are determined in terms
of motor dimensions as follows:
w
s=2zs (50)
wf=2xs. (51)
Fringing flux calculation is completed with the application of
Carter’s method to determine the effective core lengths zsfand
xsffollows:
zsf=zs+ws(1−σs) (52)
xsf=wf(1−σf) (53)where,σis Carter’s coefficient and is defined as
σ=2
π/parenleftBigg
arctan/parenleftbiggw
yg/parenrightbigg
−yg
2wln/bracketleftBigg
1+/parenleftbiggw
yg/parenrightbigg2/bracketrightBigg/parenrightBigg
.(54)
C. Stator-Inductance Expression
After the calculations above, the total permeance of the airgap
of stator core is evaluated as follows:
Λs=µ0xwzsf
3yg+µ0x/prime
gzsf
yg+µ0xsfzf
yg. (55)
In (55), the first component corresponds to the perme-
ance of the stator core bearing the excitation coil, including
side fringing, the second component corresponds to the per-
meance of the stator core beyond the excitation coil including
the side fringing, and the third one covers the permeance of
the front fringing region. In the first component, division by 3comes from the fact that in the corresponding region, flux lines
do not link the excitation coil completely [8].
Then, the inductance is simply calculated from
L=N
2Λs. (56)
Stator inductance is calculated as 2.27 H in this way while it
is measured as 2.38 H.
V. M EASUREMENT OF THE PARAMETERS
To determine the accuracy of the expressions obtained, ex-
periments are performed on a test motor with the following
dimensions:
1) Rotor diameter: 9.5 mm2) Shaft hole diameter: 3 mm
3) zs=25 mm, g1 =2 mm, g2 =2.5 mm
4) xw
txg=54 mm, yg =13 mm.
A. Measurement of Reluctance Torque and Peak Rotor Flux
Reluctance torque can be measured by suspending some suit-
able weights to the shaft of the rotor and by this way peakreluctance torque as well as the reluctance torque for any rotor
position can be determined.
Peak rotor flux can be experimentally estimated by rotating
the rotor by a prime mover at a speed w
m. Rotor flux linked by
the stator winding is equal to
λr=NΦrpcosθ. (57)
Then, the induced voltage on the stator winding is
Vm=dλr
dt=NΦrpwmsinθ. (58)
The rms value of the induced voltage is
Vmrms=1√
2NΦrpwm. (59)
Therefore, by measuring Vrmsthe peak rotor flux can be
obtained from (60)
Φrp=√
2Vmrms
Nw m. (60)

ERTAN et al. : CALCULATION OF PARAMETERS OF SINGLE-PHASE PM MOTOR FOR DESIGN OPTIMIZATION 547
Fig. 19. Rotor position while measuring θ0.
Fig. 20. Electrical equivalent circuit without rotor excitation.
B. Measurement of the Rest Position of Rotor
In this paper, rest position (θ0)of the rotor is determined by
applying a dc current to the stator winding at rated value. Due to
the electrodynamic torque, the rotor is pulled towards the statoraxis with a deviation θ
d(see Fig. 19). The rotor, however, does
not align with the stator axis but remains at rest at angle θsdue
to reluctance torque. At the equilibrium position, by equating
the reluctance torque to the electrodynamic torque
Trpsin(2θd)=Φ rpNisinθs. (61)
Therefore, by measuring θdand calculatingθsfrom (61), the
rest position of the rotor can be estimated as
θ0=θd+θs. (62)
Note that in (61) it is assumed that the two flux paths (rotor
flux and stator winding flux) exist without causing saturation.
C. Measurement of Stator Inductance
Stator inductance can be obtained indirectly by a simple ex-
periment, in which the rotor is locked at θ=90◦so that the
effect of the permanent magnet rotor is minimized. In this case,
electrical equivalent circuit of the motor becomes as shown in
Fig. 20.
In Fig. 20,Rcuis resistance of the stator coil and RFeis the
resistance corresponding to the core losses. To determine the
stator inductance, L, 50 Hz, sinusoidal-excitation is applied tothe terminals of the stator coil to obtain rated current through
the winding. Then, the stator voltage, current and input power,
are measured, as well as R
cu. Following this, the calculation
of the stator inductance from the equivalent circuit in Fig. 20
is a simple matter. Below are the experimental results of theconsidered parameters;
1)V
s= 161 V(rms)
2)is=0.21A(rms)
3)Pin=8W4)Rcu= 135Ω .
By performing a circuit analysis with above measurements
Rfeand L are determined as
1)Rfe=1 2 kΩ.
2)L=2.38H.
VI. C ONCLUSION
The single-phase permanent-magnet motor presented here is
an efficient and inexpensive alternative for many household ap-pliance applications. However, its application so far has been in
small power range (tens of watts) for various reasons. In an at-
tempt to investigate whether larger size (hundreds of watts) mo-
tors, with the desired performance, can be achieved, expressions
are obtained for the equivalent circuit parameters of this type ofmotor in terms of the motor dimensions and the materials, used.
The expressions derived are in analytical form and can be
used in a mathematical optimization approach. In this manner,it is believed that the process of synthesizing a motor with
desired performance can be done quickly and better designs can
be reached in comparison to trial and error approach.
Comparison of the values of computed parameters are shown
to be quite accurate when compared with measured parameters
of a test motor. Although not presented here, when the perfor-mance of the test motor is calculated from computed parameters
the results are also found to be acceptably accurate.
The design optimization study for larger motors is continuing
and will be the subject of a future report.
A
CKNOWLEDGMENT
The authors would like to thank B. Avenoglu and L. B.
Yalc¸ıner for their contributions to FE solutions and harmonic
calculations, respectively. Thanks also go to Arc ¸elik Company
for providing the test motors.
REFERENCES
[1] E. C. Protas, “Energy saving by means of a new drive conception,” in Proc.
Int. Aegean Conf. Electrical Machines and Power Electronics (ACEMP
95),K u s¸adasi, Turkey, Jun. 1995, pp. 303–308.
[2] W. Teppan and E. Protas, “Simulation, finite element calculations and
measurements on a single phase permanent magnet synchronous motor,”
inProc. Int. Aegean Conf. Electrical Machines and Power Electronics
(ACEMP 95) ,K u s¸adasi, Turkey, Jun. 1995, pp. 609–704.
[3] G. Altenbernd and J. Mayer, “Starting of fractional horse-power single-
phase synchronous motors with permanent magnetic rotor,” in Proc. Elec-
trical Drives Symp. , Capri, Italy, Sep. 1990, pp. 131–137.
[4] G. Altenbernd, “Actual aspects of the development of fractional horse-
power single phase synchronous motors with permanentmagnetic rotor,”inProc. Int. Conf. on the Evolution and Modern Aspects of Synchronous
Machines , Zurich, Switzerland, Aug. 1991, pp. 1083–1089.
[5] H. Schemmann, “Stability of small single-phase synchronous motors,”
Philips Tech. Rev. , vol. 33, no. 8/9, pp. 235–243, 1973.
[6] G. Landolfi, M. Pasquali, and E. Santini, “Single-phase PM synchronous
machine: Evaluation of parameters and dynamic analysis,” in Proc. 3rd
Int. Symp. Advanced Electromechanical Motion Systems , Patras, Greece,
1999, pp. 107–112.
[ 7 ] H . B .E r t a n , B .D a ˇg, and G. Capolino, “Calculation of some parameters
of single phase PM motor for design optimization,” in Proc. OPTIM 2002 ,
Bras¸ov, Romania, May 2002, pp. 351–356.
[8] B. Singh, Electrical Machine Design , Vikas, 1982.

548 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 20, NO. 3, SEPTEMBER 2005
H. B ¨ulent Ertan (M’05) received the B.S. and M.S.
degrees in electrical and electronics engineering in
1971 and 1973, respectively from the Middle East
Technical University (METU), Ankara, Turkey, andthe Ph.D. degree from the University of Leeds, Leeds,U.K., in 1977.
He has led many industry supported projects since
1977. He is currently a Professor in the Electrical andElectronic Engineering Department of METU and isalso heading the Intelligent Energy Conversion Group
at Turkish Scientific and Technical Research Coun-
cil (TUBITAK) Information Technologies and Electronics Research Institute(BILTEN), Ankara. He has authored numerous research papers and is co-editorof the book Modern Electrical Derives (Norwell, MA: Kluwer, 2000). His re-
search interests are on electrical machine design and drive systems.
Dr. Ertan is International Secretary of the steering committee of the In-
ternational Conference on Electrical Machines (ICEM) and founder of the
Aegean International Conference on Electrical Machines and Power Electronics
(ACEMP). He is a member of Turkish Chamber of Electrical Engineers and amember of the Institution of Electrical Engineers (IEE-U.K.) and IEE represen-tative in Turkey.
B¨ulent Da ˇgreceived the B.S. and M.S. degrees in
electrical and electronics engineering from MiddleEast Technical University (METU), Ankara, Turkey,in 1998 and 2002, respectively.
His research interests are electrical machines, ma-
chine design, and FE modeling.
G´erard-Andr ´e Capolino (A’77–M’82–SM’89–
F’02) received the B.Sc. degree in electrical engi-
neering from Ecole Sup ´erieure d’Ing ´enieurs de Mar-
seille, Marseille, France, in 1974, the M.Sc. degreefrom Ecole Sup ´erieure d’Electricit ´e, Paris, France,
in 1975, the Ph.D. degree from the University Aix-
Marseille I, Marseille, in 1978, and the D.Sc. degree
from the Institut National Polytechnique de Greno-ble, Grenoble, France, in 1987.
In 1978, he joined the University of Yaound ´e,
Cameroon, as an Associate Professor and Head of theDepartment of Electrical Engineering. From 1981 to 1994, he was Associate
Professor with the University of Dijon, Dijon, France, and the Mediterranean
Institute of Technology, Marseille, where he was Founder and Director of the
Modelling and Control Systems Laboratory. From 1983 to 1985, he was VisitingProfessor at the University of Tunis, Tunisia. From 1987 to 1989, he was theScientific Advisor of Technicatome SA, Aix-en-Provence, France. In 1994, he
joined the University of Picardie “Jules Verne” in Amiens, France, as a Full
Professor, Head of the Department of Electrical Engineering (1995–1998) andDirector of the Energy Conversion and Intelligent Systems Laboratory (1996–2000). He is now director of the Graduate School in Electrical Engineering at the
University of Picardie “Jules Verne” and Coordinator of the European Master
on Advanced Power Electrical Engineering (MAPEE). In 1995, he was a FellowEuropean Union Distinguished Professor of electrical engineering at Polytech-nic University of Catalunya, Spain. Since 1999, he has been the Director of
the Open European Laboratory on Electrical Machines (OELEM), a network of
excellence in between 50 partners from the European Union. In 2003, he waselected Emeritus Member of the French Institute of Electrical and Electronics
Engineers (SEE). He has published more than 250 papers in scientific journals
and conference proceedings since 1975. He has been the advisor of 15 Ph.D. andnumerous M.Sc. students. In 1990, he founded the European Community Groupfor teaching electromagnetic transients and he has coauthored the book Simula-
tion & CAD for Electrical Machines, Power Electronics and Drives , ERASMUS
Program Edition. His research interests are electrical machines, electrical drivespower electronics and control systems related to power electrical engineering.
Dr. Capolino is the Chairman of the France Chapter of the IEEE Power
Electronics, Industrial Electronics and Industry Applications Societies and the
Chairman of the IEEE France Section (2005–2008). He is also member ofthe AdCom of the IEEE Industrial Electronics Society and chairman of theTechnical Committee on Diagnostics for Power Electronics of the IEEE Power
Electronics Society. He is the co-founder of the IEEE International Symposium
for Diagnostics of Electrical Machines Power Electronics and Drives (IEEE-SDEMPED) that was held for the first time in 1997. He is a member of steeringcommittees for several high reputation international conferences. He is an As-
sociate Editor of the IEEE T
RANSACTIONS ON INDUSTRIAL ELECTRONICS and
the IEEE T RANSACTIONS ON POWER ELECTRONICS . He is also Vice Chairman
of the international steering committee of the International Conference on Elec-
trical Machines (ICEM), chairman of the technical committee on diagnostics
of the IEEE Power Electronics Society, and chairman of the electrical machinecommittee of the IEEE Industrial Electronics Society.