A hybrid two-dimensional (2-D) finite-element/analytical technique is described for predicting the d-axis and q-axis inductances of permanent magnet (PM) brushless ac machines, with due account for the influence of skew. Predicted inductances are compared with measured values for two machines having identical stators, which are skewed by one slot-pitch, but which have different rotor topologies, one having surface-mounted magnets and the other having interior magnets

Chen, Y.S.

Howe, D.

Zhu, Z.Q.

English

Crossref

Calculation of DQ-axis inductances of PM brushless ac machines accounting for skew

White Rose Research Online

This is a repository copy of Calculation of d- and q-axis inductances of PM brushless ac machines accounting for Skew .White Rose Research Online URL for this paper:http://eprints.whiterose.ac.uk/860/Article:Chen, Y.S., Zhu, Z.Q. and Howe, D. (2005) Calculation of d- and q-axis inductances of PM brushless ac machines accounting for Skew. IEEE Transactions on Magentics, 41 (10). pp.3940-3942. ISSN 0018-9464 https://doi.org/10.1109/TMAG.2005.854976eprints@whiterose.ac.ukhttps://eprints.whiterose.ac.uk/Reuse Unless indicated otherwise, fulltext items are protected by copyright with all rights reserved. The copyright exception in section 29 of the Copyright, Designs and Patents Act 1988 allows the making of a single copy solely for the purpose of non-commercial research or private study within the limits of fair dealing. The publisher or other rights-holder may allow further reproduction and re-use of this version - refer to the White Rose Research Online record for this item. Where records identify the publisher as the copyright holder, users can verify any specific terms of use on the publisher’s website. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing eprints@whiterose.ac.uk including the URL of the record and the reason for the withdrawal request. 3940 IEEE TRANSACTIONS ON MAGNETICS, VOL. 41, NO. 10, OCTOBER 2005Calculation of d- and q-Axis Inductances of PMBrushless ac Machines Accounting for SkewY. S. Chen1;2, Z. Q. Zhu2, Senior Member, IEEE, and D. Howe2College of Electrical Engineering, Zhejiang University, Hangzhou, P.R. China 310027Department of Electronic and Electrical Engineering, University of Sheffield, Sheffield S1 3JD, U.K.A hybrid two-dimensional (2-D) finite-element/analytical technique is described for predicting the -axis and -axis inductances ofpermanent magnet (PM) brushless ac machines, with due account for the influence of skew. Predicted inductances are compared withmeasured values for two machines having identical stators, which are skewed by one slot-pitch, but which have different rotor topologies,one having surface-mounted magnets and the other having interior magnets.Index Terms—Brushless machine, permanent magnet (PM), permanent magnet (PM) machine, winding inductance.I. INTRODUCTIONTHE -axis and -axis inductances have an important influ-ence on both the steady-state and dynamic performance ofpermanent magnet (PM) brushless ac machines [1]. Thus, theiraccurate prediction is essential, not only for predicting perfor-mance aspects such as the torque and flux-weakening capabili-ties, but also for designing control systems, in order to maximizethe efficiency, power factor, etc. The influence of magnetic sat-uration is usually accounted for by determining current-depen-dent inductances [1], [2], i.e., and .However, since high local saturation may occur due to the com-bined influence of the PMs and the armature reaction field, espe-cially in interior-magnet rotor machines, analytical methods fordetermining the winding inductances are often inadequate, anda numerical method must be employed. When the stator slotsor rotor magnets are skewed, however, three-dimensional (3-D)calculations are required, which remain time consuming.In this paper, the influence of skew on the -axis and -axisinductances is investigated by employing a hybrid two-dimen-sional (2-D) finite-element/analytical technique, with due ac-count for skew and end effects. Predictions are validated bymeasurements, using a dc inductance bridge technique [2], ontwo brushless ac machines having identical stators, which areskewed by one slot-pitch but different rotor topologies.II. CALCULATION OF -AXIS AND -AXIS INDUCTANCESVarious techniques have been employed to predict thewinding inductances using the finite-element method [3]–[5].When the magnetic circuit can be assumed to be linear, the self-and mutual-winding inductances can be evaluated by energizingthe appropriate stator windings, with the PMs replaced by air[5], and determining the stored energy or winding flux linkage,such that the -axis and -axis inductances can subsequentlybe deduced. However, if magnetic saturation is significant,both the self- and mutual inductances, and the -axis and-axis inductances, are current dependent. The incremental andDigital Object Identifier 10.1109/TMAG.2005.854976apparent self- and mutual inductances should then be calcu-lated by either the energy perturbation method or the currentperturbation method, and the influence of the PMs should beconsidered [3], [4]. However, although generally applicable, themethod described in [4] requires multiple finite-element fieldsolutions, for a range of current perturbations in each winding,and, therefore, demands significant computational effort.The -axis and -axis inductances, and , respectively,which are used in the flux-weakening control of PM brushlessac machines [6], can be calculated from [2], as follows:(1)(2)where , and are the d-axis and -axis currents andflux-linkages, respectively; and is the flux-linkage due to thePM rotor.However, while 2-D finite-element analysis can account forthe airgap, slot leakage, tooth-tip leakage and magnetomotiveforce (MMF) harmonic leakage components of inductance to ahigh accuracy, it cannot account for the end-winding componentof leakage inductance or the influence of skew.While skew is effective in reducing the harmonic contentin the flux linkage and back-electromotive force (EMF) wave-forms, as well as in reducing the cogging torque, skewing ei-ther the stator teeth or the rotor magnets results in a 3-D mag-netic-field distribution in the active region of a machine. How-ever, since the axial variation of saturation due to skew is gen-erally small in most PM machines, the relationship between the-axis and -axis inductances, and , which are calcu-lated by 2-D finite-element analysis of the active region of amachine without skew and the -axis and -axis inductances,and , for the same machine with skew can be derivedanalytically, as will be described.The self-inductance and the mutual inductance can beexpressed as functions of the rotor electrical angle [7], asfollows:(3)120 (4)0018-9464/$20.00 © 2005 IEEECHEN et al.: CALCULATION OF - AND -AXIS INDUCTANCES OF PM BRUSHLESS ac MACHINES ACCOUNTING FOR SKEW 3941where and are the components of the self- and mutualinductances due to the space-fundamental airgap flux, is theadditional leakage flux component, and the second harmoniccomponents and result from saliency of the rotor.Skew changes the relative position between the stator androtor along the axial length of a machine. Hence, the magnetic-field distribution also varies. However, the self-inductanceand the mutual inductance of a machine with skew canbe deduced from the inductances of the same machine withoutskew by subdividing the active length of the machine into axialslices, with each slice having a skew angle of . Hence,(5)120 (6)where is the electrical skew angle, and(7)(8)where the skew factor is given by(9)Thus, the constant components of the winding inductances re-main unchanged when skew is employed, while the second har-monic components are reduced by the skew factor , whichis , i.e., the inductance variation due to saliency is reducedby skew.In general, from the Park transformation, the relationship be-tween the phase self- and mutual inductances, and , and the-axis and -axis inductances, and , are given by [7](10)(11)Thus, from (5)–(9), the -axis and -axis inductances, ac-counting for the effect of skew, can be obtained as(12)(13)Combining (11)–(13), the relationship between the induc-tances and , which are deduced from 2-D finite-ele-ment analyses of the active region of a machine neglecting skew,and the inductances and , which account for skew, canbe derived as(14)(15)Fig. 1. Magnetic field distributions. (i) Surface-mounted magnet rotor:(a) Open circuit; (b) rated d-axis current; and (c) rated q-axis current.(ii) Interior-magnet rotor: (a) Open circuit; (b) rated d-axis current; and(c) rated q-axis current.The end-winding component of inductance may be esti-mated using a well-proven method that has been applied exten-sively to induction motors [8]. Thus, the resultant -axis and-axis inductances are(16)(17)III. COMPARISON AND EXPERIMENTAL VALIDATIONPredicted -axis and -axis inductances have been comparedwith measured values for two 1.3-kW, three-phase PM brushlessac motors having identical stators, with six poles, 18 slots, anoverlapping winding, and a skew of one slot pitch, but differentrotors, viz. one with surface-mounted magnets and another withinterior magnets. The main design parameters are as follows:stator outside diameter 106.6 mm, active length 32 mm,airgap length 0.75 mm, rotor diameter 60.5 mm, andwidth of stator slot openings 2.5 mm. Both rotors have sin-tered NdFeB magnets with a remanence of 1.17 T and a relativerecoil permeability of 1.07. It will be noted that the rotor back-iron of the machine with the surface-mounted magnet rotor hasair spaces to reduce the inertia and minimize the influence of-axis magnetic saturation, and thereby maximize the overloadtorque capability.Fig. 1 shows field distributions when the two motors are onopen circuit and on load when rated current (4 A at peak) isapplied to the and axes. Fig. 2 compares measured and pre-dicted - and -axis inductances, from which the influence ofmagnetic saturation can be clearly observed. The predictionsinclude the end-winding component of inductance, viz. 6.9 mH3942 IEEE TRANSACTIONS ON MAGNETICS, VOL. 41, NO. 10, OCTOBER 2005Fig. 2. Measured and predicted d-axis and q-axis inductances: (i) Surface-mounted magnet rotor motor and (ii) interior-magnet rotor motor.[8]. The -axis inductance was measured by locking the rotoralong the axis, which was determined by applying a -axiscurrent and and allowingthe rotor to rotate to the position at which the torque was zero,and using a dc inductance bridge [2]. The -axis inductancewas similarly measured. In general, very good agreement is ob-tained. As will be seen, due to the different rotor topologies, thevariation of and with and , respectively, is signifi-cantly different for the two motors.Fig. 3 shows the influence of skew on the predicted -axis and-axis inductances for both motors. It will be noted, both fromFig. 3 and the foregoing equations, that irrespective of and, the smallest inductance component is increased as a resultof skew, while the other component is reduced. Hence, the useof skew reduces the influence of any rotor saliency, and bothand will approach the value as the skew angleis increased. For a machine without saliency, i.e., , theinfluence of skew will generally be negligible.Fig. 3. Predicted d-axis and q-axis inductances with and without skew:(i) surface-mounted magnet rotor motor and (ii) interior-magnet rotor motor.IV. CONCLUSIONA hybrid two-dimensional (2-D) finite-element/analyticaltechnique has been proposed for predicting the -axis and-axis inductances of permanent magnet (PM) brushless acmachines with skew. Predicted -axis and -axis inductanceshave been compared with measurements in two machineshaving identical stators, which are skewed by one slot pitch,but different rotor topologies, and good agreement has beenachieved.REFERENCES[1] B. J. Chalmers, S. A. Hamed, and G. D. Baines, “Parameters andperformance of a high-field permanent magnet synchronous motor forvariable-frequency operation,” Proc. Inst. Elect. Eng.—B, vol. 132, pp.117–124, 1985.[2] P. H. Mellor, F. B. Chaaban, and K. J. Binns, “Estimation of parametersand performance of rare-earth permanent-magnet motors avoiding mea-surement of load angle,” in Proc. Inst. Elect. Eng.—B, vol. 138, 1991,pp. 322–330.[3] L. Chang, “An improved FE inductance calculation for electrical ma-chines,” IEEE Trans. Magn., vol. 32, no. 4, pp. 3237–3245, Jul. 1996.[4] T. W. Nehl, F. A. Fouad, and N. A. Demerdash, “Determination of satu-rated values of rotating machinery incremental and apparent inductancesby an energy perturbation method,” IEEE Trans. Power App. Syst., vol.101, no. 12, pp. 4441–4451, Dec. 1982.[5] D. Pavlik, V. K. Garg, J. R. Repp, and J. Weiss, “A finite elementaltechnique for calculating the magnet sizes and inductances of perma-nent magnet machines,” IEEE Trans. Energy Convers., vol. 3, no. 1, pp.116–122, Mar. 1988.[6] S. Morimoto, M. Sanada, and Y. Takeda, “Wide-speed operation of in-terior permanent magnet synchronous motors with high-performancecurrent regulator,” IEEE Trans. Ind. Appl., vol. 30, no. 4, pp. 920–926,Jul.–Aug. 1994.[7] A. E. Fitzgerald, C. Kingsley, and S. D. Umans, Electric Ma-chinery. New York: McGraw-Hill, 1989, pp. 272–276.[8] M. Liwschitz-Garik and C. Whipple, Alternating-Current Ma-chines. New York: Van Nostrand, 1961, pp. 510–512.Manuscript received February 7, 2005.

Calculation of d- and q-axis inductances of PM brushless ac machines accounting for Skew

This is a repository copy of Calculation of d- and q-axis inductances of PM brushless ac 
machines accounting for Skew .
White Rose Research Online URL for this paper:
http://eprints.whiterose.ac.uk/860/
Article:
Chen, Y.S., Zhu, Z.Q. and Howe, D. (2005) Calculation of d- and q-axis inductances of PM 
brushless ac machines accounting for Skew. IEEE Transactions on Magentics, 41 (10). pp.
3940-3942. ISSN 0018-9464 
https://doi.org/10.1109/TMAG.2005.854976
eprints@whiterose.ac.uk
https://eprints.whiterose.ac.uk/
Reuse 
Unless indicated otherwise, fulltext items are protected by copyright with all rights reserved. The copyright 
exception in section 29 of the Copyright, Designs and Patents Act 1988 allows the making of a single copy 
solely for the purpose of non-commercial research or private study within the limits of fair dealing. The 
publisher or other rights-holder may allow further reproduction and re-use of this version - refer to the White 
Rose Research Online record for this item. Where records identify the publisher as the copyright holder, 
users can verify any specific terms of use on the publisher’s website. 
Takedown 
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emailing eprints@whiterose.ac.uk including the URL of the record and the reason for the withdrawal request. 
3940 IEEE TRANSACTIONS ON MAGNETICS, VOL. 41, NO. 10, OCTOBER 2005
Calculation of d- and q-Axis Inductances of PM
Brushless ac Machines Accounting for Skew
Y. S. Chen1;2, Z. Q. Zhu2, Senior Member, IEEE, and D. Howe2
College of Electrical Engineering, Zhejiang University, Hangzhou, P.R. China 310027
Department of Electronic and Electrical Engineering, University of Sheffield, Sheffield S1 3JD, U.K.
A hybrid two-dimensional (2-D) finite-element/analytical technique is described for predicting the -axis and -axis inductances of
permanent magnet (PM) brushless ac machines, with due account for the influence of skew. Predicted inductances are compared with
measured values for two machines having identical stators, which are skewed by one slot-pitch, but which have different rotor topologies,
one having surface-mounted magnets and the other having interior magnets.
Index Terms—Brushless machine, permanent magnet (PM), permanent magnet (PM) machine, winding inductance.
I. INTRODUCTION
T
HE -axis and -axis inductances have an important influ-
ence on both the steady-state and dynamic performance of
permanent magnet (PM) brushless ac machines [1]. Thus, their
accurate prediction is essential, not only for predicting perfor-
mance aspects such as the torque and flux-weakening capabili-
ties, but also for designing control systems, in order to maximize
the efficiency, power factor, etc. The influence of magnetic sat-
uration is usually accounted for by determining current-depen-
dent inductances [1], [2], i.e., and .
However, since high local saturation may occur due to the com-
bined influence of the PMs and the armature reaction field, espe-
cially in interior-magnet rotor machines, analytical methods for
determining the winding inductances are often inadequate, and
a numerical method must be employed. When the stator slots
or rotor magnets are skewed, however, three-dimensional (3-D)
calculations are required, which remain time consuming.
In this paper, the influence of skew on the -axis and -axis
inductances is investigated by employing a hybrid two-dimen-
sional (2-D) finite-element/analytical technique, with due ac-
count for skew and end effects. Predictions are validated by
measurements, using a dc inductance bridge technique [2], on
two brushless ac machines having identical stators, which are
skewed by one slot-pitch but different rotor topologies.
II. CALCULATION OF -AXIS AND -AXIS INDUCTANCES
Various techniques have been employed to predict the
winding inductances using the finite-element method [3]–[5].
When the magnetic circuit can be assumed to be linear, the self-
and mutual-winding inductances can be evaluated by energizing
the appropriate stator windings, with the PMs replaced by air
[5], and determining the stored energy or winding flux linkage,
such that the -axis and -axis inductances can subsequently
be deduced. However, if magnetic saturation is significant,
both the self- and mutual inductances, and the -axis and
-axis inductances, are current dependent. The incremental and
Digital Object Identifier 10.1109/TMAG.2005.854976
apparent self- and mutual inductances should then be calcu-
lated by either the energy perturbation method or the current
perturbation method, and the influence of the PMs should be
considered [3], [4]. However, although generally applicable, the
method described in [4] requires multiple finite-element field
solutions, for a range of current perturbations in each winding,
and, therefore, demands significant computational effort.
The -axis and -axis inductances, and , respectively,
which are used in the flux-weakening control of PM brushless
ac machines [6], can be calculated from [2], as follows:
(1)
(2)
where , and are the d-axis and -axis currents and
flux-linkages, respectively; and is the flux-linkage due to the
PM rotor.
However, while 2-D finite-element analysis can account for
the airgap, slot leakage, tooth-tip leakage and magnetomotive
force (MMF) harmonic leakage components of inductance to a
high accuracy, it cannot account for the end-winding component
of leakage inductance or the influence of skew.
While skew is effective in reducing the harmonic content
in the flux linkage and back-electromotive force (EMF) wave-
forms, as well as in reducing the cogging torque, skewing ei-
ther the stator teeth or the rotor magnets results in a 3-D mag-
netic-field distribution in the active region of a machine. How-
ever, since the axial variation of saturation due to skew is gen-
erally small in most PM machines, the relationship between the
-axis and -axis inductances, and , which are calcu-
lated by 2-D finite-element analysis of the active region of a
machine without skew and the -axis and -axis inductances,
and , for the same machine with skew can be derived
analytically, as will be described.
The self-inductance and the mutual inductance can be
expressed as functions of the rotor electrical angle [7], as
follows:
(3)
120 (4)
0018-9464/$20.00 © 2005 IEEE
CHEN et al.: CALCULATION OF - AND -AXIS INDUCTANCES OF PM BRUSHLESS ac MACHINES ACCOUNTING FOR SKEW 3941
where and are the components of the self- and mutual
inductances due to the space-fundamental airgap flux, is the
additional leakage flux component, and the second harmonic
components and result from saliency of the rotor.
Skew changes the relative position between the stator and
rotor along the axial length of a machine. Hence, the magnetic-
field distribution also varies. However, the self-inductance
and the mutual inductance of a machine with skew can
be deduced from the inductances of the same machine without
skew by subdividing the active length of the machine into axial
slices, with each slice having a skew angle of . Hence,
(5)
120 (6)
where is the electrical skew angle, and
(7)
(8)
where the skew factor is given by
(9)
Thus, the constant components of the winding inductances re-
main unchanged when skew is employed, while the second har-
monic components are reduced by the skew factor , which
is , i.e., the inductance variation due to saliency is reduced
by skew.
In general, from the Park transformation, the relationship be-
tween the phase self- and mutual inductances, and , and the
-axis and -axis inductances, and , are given by [7]
(10)
(11)
Thus, from (5)–(9), the -axis and -axis inductances, ac-
counting for the effect of skew, can be obtained as
(12)
(13)
Combining (11)–(13), the relationship between the induc-
tances and , which are deduced from 2-D finite-ele-
ment analyses of the active region of a machine neglecting skew,
and the inductances and , which account for skew, can
be derived as
(14)
(15)
Fig. 1. Magnetic field distributions. (i) Surface-mounted magnet rotor:
(a) Open circuit; (b) rated d-axis current; and (c) rated q-axis current.
(ii) Interior-magnet rotor: (a) Open circuit; (b) rated d-axis current; and
(c) rated q-axis current.
The end-winding component of inductance may be esti-
mated using a well-proven method that has been applied exten-
sively to induction motors [8]. Thus, the resultant -axis and
-axis inductances are
(16)
(17)
III. COMPARISON AND EXPERIMENTAL VALIDATION
Predicted -axis and -axis inductances have been compared
with measured values for two 1.3-kW, three-phase PM brushless
ac motors having identical stators, with six poles, 18 slots, an
overlapping winding, and a skew of one slot pitch, but different
rotors, viz. one with surface-mounted magnets and another with
interior magnets. The main design parameters are as follows:
stator outside diameter 106.6 mm, active length 32 mm,
airgap length 0.75 mm, rotor diameter 60.5 mm, and
width of stator slot openings 2.5 mm. Both rotors have sin-
tered NdFeB magnets with a remanence of 1.17 T and a relative
recoil permeability of 1.07. It will be noted that the rotor back-
iron of the machine with the surface-mounted magnet rotor has
air spaces to reduce the inertia and minimize the influence of
-axis magnetic saturation, and thereby maximize the overload
torque capability.
Fig. 1 shows field distributions when the two motors are on
open circuit and on load when rated current (4 A at peak) is
applied to the and axes. Fig. 2 compares measured and pre-
dicted - and -axis inductances, from which the influence of
magnetic saturation can be clearly observed. The predictions
include the end-winding component of inductance, viz. 6.9 mH
3942 IEEE TRANSACTIONS ON MAGNETICS, VOL. 41, NO. 10, OCTOBER 2005
Fig. 2. Measured and predicted d-axis and q-axis inductances: (i) Surface-
mounted magnet rotor motor and (ii) interior-magnet rotor motor.
[8]. The -axis inductance was measured by locking the rotor
along the axis, which was determined by applying a -axis
current and and allowing
the rotor to rotate to the position at which the torque was zero,
and using a dc inductance bridge [2]. The -axis inductance
was similarly measured. In general, very good agreement is ob-
tained. As will be seen, due to the different rotor topologies, the
variation of and with and , respectively, is signifi-
cantly different for the two motors.
Fig. 3 shows the influence of skew on the predicted -axis and
-axis inductances for both motors. It will be noted, both from
Fig. 3 and the foregoing equations, that irrespective of and
, the smallest inductance component is increased as a result
of skew, while the other component is reduced. Hence, the use
of skew reduces the influence of any rotor saliency, and both
and will approach the value as the skew angle
is increased. For a machine without saliency, i.e., , the
influence of skew will generally be negligible.
Fig. 3. Predicted d-axis and q-axis inductances with and without skew:
(i) surface-mounted magnet rotor motor and (ii) interior-magnet rotor motor.
IV. CONCLUSION
A hybrid two-dimensional (2-D) finite-element/analytical
technique has been proposed for predicting the -axis and
-axis inductances of permanent magnet (PM) brushless ac
machines with skew. Predicted -axis and -axis inductances
have been compared with measurements in two machines
having identical stators, which are skewed by one slot pitch,
but different rotor topologies, and good agreement has been
achieved.
REFERENCES
[1] B. J. Chalmers, S. A. Hamed, and G. D. Baines, “Parameters and
performance of a high-field permanent magnet synchronous motor for
variable-frequency operation,” Proc. Inst. Elect. Eng.—B, vol. 132, pp.
117–124, 1985.
[2] P. H. Mellor, F. B. Chaaban, and K. J. Binns, “Estimation of parameters
and performance of rare-earth permanent-magnet motors avoiding mea-
surement of load angle,” in Proc. Inst. Elect. Eng.—B, vol. 138, 1991,
pp. 322–330.
[3] L. Chang, “An improved FE inductance calculation for electrical ma-
chines,” IEEE Trans. Magn., vol. 32, no. 4, pp. 3237–3245, Jul. 1996.
[4] T. W. Nehl, F. A. Fouad, and N. A. Demerdash, “Determination of satu-
rated values of rotating machinery incremental and apparent inductances
by an energy perturbation method,” IEEE Trans. Power App. Syst., vol.
101, no. 12, pp. 4441–4451, Dec. 1982.
[5] D. Pavlik, V. K. Garg, J. R. Repp, and J. Weiss, “A finite elemental
technique for calculating the magnet sizes and inductances of perma-
nent magnet machines,” IEEE Trans. Energy Convers., vol. 3, no. 1, pp.
116–122, Mar. 1988.
[6] S. Morimoto, M. Sanada, and Y. Takeda, “Wide-speed operation of in-
terior permanent magnet synchronous motors with high-performance
current regulator,” IEEE Trans. Ind. Appl., vol. 30, no. 4, pp. 920–926,
Jul.–Aug. 1994.
[7] A. E. Fitzgerald, C. Kingsley, and S. D. Umans, Electric Ma-
chinery. New York: McGraw-Hill, 1989, pp. 272–276.
[8] M. Liwschitz-Garik and C. Whipple, Alternating-Current Ma-
chines. New York: Van Nostrand, 1961, pp. 510–512.
Manuscript received February 7, 2005.


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Calculation of d- and q-axis inductances of PM brushless ac machines accounting for Skew

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