IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 6, JUNE 2014 8101314

Modeling of a Permanent Magnet Synchronous Machine WithInternal Magnets Using Magnetic Equivalent Circuits

Wolfgang Kemmetmller, David Faustner, and Andreas Kugi

Automation and Control Institute, Vienna University of Technology, Vienna 1040, Austria

The design of control strategies for permanent magnet synchronous machines (PSM) is almost exclusively based on classicaldq0-models. These models are, however, not able to systematically describe saturation or nonhomogenous air gap geometries typicallyoccurring in PSM. This paper deals with a framework for the mathematical modeling of PSM based on magnetic equivalent circuits.Different to existing works, the model equations are derived by means of graph theory allowing for a systematic choice of a minimalset of state variables of the model and a systematic consideration of the electrical connection of the coils of the motor. The resultingmodel is calibrated and verified by means of measurement results. Finally, a magnetically linear and a dq0-model are derived andtheir performance is compared with the nonlinear model and measurement results.

Index Terms Electric motor, magnetic equivalent circuit, permanent magnet motors.

I. INTRODUCTION

PERMANENT magnet synchronous motors (PSM) arewidely used in many technical applications. Numerouspapers and books dealing with the design of PSM havebeen published in recent years [1][7]. The mathematicalmodels proposed in these papers range from finite element(FE) analysis over reluctance models to classical dq0-models.FE models exhibit a high accuracy for the calculated magneticfields and allow for an exact consideration even of complexgeometries of the motor [5][10]. Due to their high (numeric)complexity, these models are, however, hardly suitable fordynamical simulations and a controller design.

The design of control strategies for PSM is typically basedon classical dq0-models, which, in their original form, assumea homogenous air gap and unsaturated iron cores [11][18].Many modern designs of PSM (including, e.g., PSM withinternal magnets) exhibit considerable saturation and nonsinu-soidal fluxes in the coils. To cope with these effects, extensionsof dq0-models have been reported in literature, which are allbased on a heuristic approach and are limited to a very specificmotor design [16][18]. In most cases, these models are notable to accurately describe the motor behavior in all operatingconditions.

Magnetic equivalent circuits have become very popular forthe design and the (dynamical) simulation of PSM in therecent years [1], [2], [19][35]. This is due to the significantlyreduced complexity in comparison to FE models and theircapability to systematically describe saturation and nonho-mogenous air gap geometries. The accuracy and complexityof reluctance models can be easily controlled by means of thechoice of the reluctance network. While reluctance networkswith a rather high complexity are necessary to accuratelydescribe field profiles in the motor, models of significantlyreduced complexity already represent the behavior with respectto the torque, currents and voltages of the motor in sufficient

Manuscript received September 25, 2013; revised November 8, 2013 andDecember 18, 2013; accepted December 20, 2013. Date of publicationJanuary 9, 2014; date of current version June 6, 2014. Corresponding author:W. Kemmetmller (e-mail: kemmetmueller@acin.tuwien.ac.at).

Digital Object Identifier 10.1109/TMAG.2014.2299238

detail. Thus, models based on adequately chosen reluctancemodels promise to be a good basis for dynamical simulationsand the (nonlinear) controller design.

In this paper, a framework for the systematic derivationof a state-space model with a minimum number of nonlinearequations and state variables is presented. The main purposeof the derived model is to provide a state-space representationfor advanced model-based control strategies, and thus toreproduce the dynamic input-to-output behavior of the motorin an accurate manner. The framework developed here isuniversal, it is applied here to a specific internal magnetPSM that exhibits both large cogging torque and saturation.Section II presents the considered model and a completereluctance model of the motor. To obtain a minimal set of(nonlinear) equations that describe the reluctance network, amethod based on graph theory is proposed. This method, wellknown from electrical networks [36][38] is adjusted to theanalysis of magnetic networks. Subsequently, the descriptionof the electrical connection of the coils and the choice ofa suitable set of state variables for the dynamical system isoutlined. It should be noted that the framework presentedin this section can be applied to any PSM. Section IIIis concerned with a reduced model based on findings ofsimulation results of the complete model. Section IV showsthe systematic calibration of the reduced model and a com-parison with measurement results. Starting from the non-linear model, a magnetically linear model, and a classicaldq0-model are systematically derived in Section V. Finally, theresults of the nonlinear model, the magnetically linear modeland the dq0-model are compared with measurement results.

II. CONSIDERED MOTOR AND COMPLETE MODELThe motor considered in this paper is a permanent mag-

net synchronous motor with internal magnets. It comprises12 stator coils, each wound around a single stator tooth,and eight NdFeB-magnets in the rotor, which are alternatelymagnetized. The setup of the motor is periodically repeatedevery 90 (number of pole pairs p = 4), such that only aquarter of the motor has to be considered. Fig. 1 shows asectional view of the PSM and the permeance network used

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8101314 IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 6, JUNE 2014

Fig. 1. Sectional view of the PSM with permeance network.

to model the stator and the rotor (air gap permeances are notincluded in this figure).

The motor is designed to exhibit a large cogging torqueby means of an inhomogeneous construction of the air gap,Fig. 1. This is due to the fact that the motor is used inan application where external torques beyond a certain limitshould not yield large changes in the rotor angle . Thelarge cogging torque, however, makes the design of high-performance control strategies more involved. Thus, tailoredmathematical models which accurately describe the coggingtorque and the saturation are required for the controllerdesign.

A. Permeance Network

As already outlined in Section I, a network of nonlinearpermeances is utilized for the derivation of a model of themotor. Fig. 2 shows the proposed permeance network of themotor. The permeances describing the core of the stator andthe rotor are approximated by cuboids of length l and area A.To account for saturation effects in the core, the relativepermeability r is defined as a function of the absolute valueof the magnetic field strength H = u/ l, i.e., r (|u| / l),where u denotes the magnetomotive force. Fig. 3 showsthe relative permeability r for the applied core materialM800-50A.

The nonlinear permeances of the stator teeth then read as

Gsj(usj

) =Ast0r

( |usj |lst

)

lstj = 1, 2, 3 (1)

with the area Ast , the length lst and the magnetomotive forceusj of a stator tooth, and the permeability 0 of free space.The permeances of the stator yoke can be found analogously

Fig. 2. Permeance network of the PSM.

in the form

Gsjk(usjk

) =Asy0r

( |usjk|lsy

)

lsyjk {12, 23, 31} (2)

where Asy is the area, lsy describes the length, and usjk isthe corresponding magnetomotive force. The center of therotor is divided into four elements, which are described bythe permeances

Gr jk(ur jk

) =Ar0r

( |ur jk |lr

)

lrjk {11, 12, 21, 22} . (3)

Here, Ar is the effective area, lr the effective length, and ur jkthe magnetomotive force of the rotor element. The permanentmagnets are placed inside the rotor of the motor. The resultingconstruction of the rotor exhibits parts, which have the formof very slender bars. The circumferential bars are described by

Gbjk(ubjk

) =Abc0r

( |ubjk|lbc

)

lbcjk {11, 12, 21, 22} (4)

with the area Abc, the length lbc, and the magnetomotive forceubjk. The permeances of the radial bars read as

Gbj(ubj

) =Abr0r

( |ubj |lbr

)

lbrj = 1, 2. (5)

KEMMETMLLER et al.: MODELING OF A PSM WITH INTERNAL MAGNETS 8101314

Fig. 3. Relative permeability r of the core material M800-50A.

Again, Abr denotes the area, lbr the length, and ubr j themagnetomotive force of the radial bar element.

The air gap of the motor is modeled by two types of per-meances: the permeances Gljk , jk {12, 23, 31}, describingthe leakage between adjacent stator teeth, and Gajk, jk {11, 12, 21, 22, 31, 32}, describing the coupling between statorand rotor. The leakage permeances are defined as

Gljk = Al0ll j k {12, 23, 31} (6)

with the effective area Al and length ll . The air gap permeancesGajk are, of course, functions of the relative rotation of therotor with respect to the stator. A geometric model of thesepermeances using an approximate air gap geometry is possiblebut yields inaccurate results due to stray fluxes not coveredby the approximate air gap geometry. Therefore, a heuristicapproach, as has been proposed in [2] and [19], is used toapproximate the coupling between the rotor and stator, i.e.,the air gap permeance Ga

Ga () =

0 4 Ga,max

2(1 + cos (

)) < 0 < 4 .

(7)

Therein, is the relative rotation mapped to the interval(/4, /4) by means of a modulo operation. In addition, is a parameter which can be approximately determined by thegeometrical overlap between a permanent magnet and a statortooth, and Ga,max is the maximum value at = 0. Given Gaof (7), the air gap permeances between the individual statorteeth and permanent magnets are defined as

Gajk = Ga( ( j 1)

6 (k 1)

4

)(8)

with j = 1, 2, 3 and k = 1, 2.The NdFeB-magnets exhibit an almost linear behavior in

the operating range, which can be modeled in the form ofa constant magnetomotive force umsj , j = 1, 2 and a linear

permeance

Gmj(umj

) = Am0rmlm

j = 1, 2 (9)

with the constant relative permeability rm , the effective areaAm , and the length lm . Given the coercive field strength Hc ofthe magnets, their magnetomotive forces are described by

ums1 = ums2 = Hclm . (10)The stator coils with Nc turns are modeled by

ucs j = Ncicj j = 1, 2, 3 (11)with icj being the electric current through the coil j .

B. Balance EquationsTwo approaches for the derivation of the balance equa-

tions (Kirchhoffs node and branch equations) are typicallyused for magnetic reluctance networks: 1) mesh analysis[33][35] and 2) node potential analysis [2], [19][21], [23],[25][28]. While a proper choice of meshes, yielding a setof independent equations might be tricky, the node poten-tial analysis automatically guarantees the independence ofthe resulting equations. Therefore, node potential analysis istypically favored.

In this paper, an alternative approach for the systematicderivation of a minimal set of independent equations basedon graph theory is proposed. It uses a tree, which con-nects all nodes of the network without forming any meshes.This approach is well known from electric network analysis[36][38], and can be, as will be shown in this paper, directlyapplied to magnetic permeance networks, see also [29].

The chosen tree has to connect all nodes of the networkwithout forming any meshes. In addition, all magnetomotiveforce sources have to be included in the tree, which is alwayspossible for nondegenerated networks. It further turns out tobe advantageous to exclude as many air gap permeances fromthe tree as possible. One possible choice of a tree is givenin Fig. 2 by the components depicted in black. The cotree isthen composed of all components which are not part of thetree (depicted gray in Fig. 2). Adding one cotree element tothe tree yields a single mesh.

For the subsequent derivation, it is useful to subdivide theelements of the tree into magnetomotive force sources of thecoils (index tc), magnetomotive force sources of the permanentmagnets (index tm), and permeances (linear, nonlinear, angledependent, index tg). Then, the overall vector of the tree fluxest =

[Ttc,

Ttm,

Ttg

]Tis defined by

tc = [cs1, cs2, cs3]T (12a)tm = [ms1, ms2]T (12b)tg = [s1, s2, s3, s12, s23, b1, b2, r11, r12,

r21, m1, m2, a11]T . (12c)The vector of the corresponding tree magnetomotive forces utis defined in an analogous manner.

8101314 IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 6, JUNE 2014

The cotree only comprises permeances such that the vectorof the cotree fluxes is given by

c = [l12, l23, l31, s31, b11, b12, b21, b22,r22, a12, a21, a22, a31, a32]T (13)

and the vector of cotree magnetomotive forces uc is defined inthe same way. Now, the following relations between the treeand cotree fluxes and magnetomotive forces, respectively, canbe formulated:

t = Dc (14a)uc = DT ut . (14b)

The incidence matrix D describes the interconnection of theindividual elements of the permeance network and its entriesare either 1, 0, or 1. It can be decomposed into a partDc linking the cotree fluxes with the tree coil fluxes, a partDm linking the cotree fluxes with the tree permanent magnetfluxes, and a part Dg , which connects the cotree fluxes withthe fluxes of the tree permeances, i.e., DT =

[DTc , DTm , DTg

].

The constitutive equations of the permeances can be sum-marized in the form

tg = Gt utg (15a)c = Gcuc (15b)

with the permeance matrices Gt and Gc of the tree andcotree, respectively. Note that in general these matrices arefunctions of the corresponding magnetomotive forces (due tosaturation) and the displacement of the rotor, i.e., Gt (utg, )and Gc(uc, ). For the permeance network of Fig. 2 thesematrices read as

Gt = diag [Gs1, Gs2, Gs3, Gs12, Gs23, Gb1, Gb2,Gr11, Gr12, Gr21, Gm1, Gm2, Ga11] (16a)

Gc = diag [Gl12, Gl23, Gl31, Gs31, Gb11, Gb12, Gb21,Gb22, Gr22, Ga12, Ga21, Ga22, Ga31, Ga32] . (16b)

Inserting (15) into (14), we find the following set ofequations:

tctm

Gt utg

= DGc[DTc , DTm , DTg

]

utcutmutg

. (17)

If it is assumed that the coil currents ic = [ic1, ic2, ic3]T andthus the magnetomotive forces utc are given, the unknownvariables of (17) are tc, tm and utg . A simple reformulationof (17) yields

I 0 DcGcDTg0 I DmGcDTg0 0 Gt + DgGcDTg

tctmutg

= DGc(

DTc utc + DTmutm)

(18)with the identity matrix I. It can be easily seen that a setof dim

(utg

) = n = 13 nonlinear algebraic equations hasto be solved for utg . All other quantities of the networkcan be calculated from simple linear equations. A proof ofthe existence and uniqueness of a solution of the nonlinearalgebraic equations (18) is given in the Appendix.

C. Torque EquationStarting from the magnetic coenergy of the permeance

network, the electromagnetic torque of the motor is definedas

= 12

p(

uTtgGt

utg + uTcGc

uc

)(19)

with the number p of pole pairs [2]. With the help of (14b)this equation can be reformulated in the form

= 12

p(

uTtgGt

utg + uTt DGc

DT ut)

(20)

with uTt =[uTtc, u

Ttm, u

Ttg

].

D. Voltage EquationThe mathematical model (18) and (20) allows for a calcu-

lation of the magnetomotive forces, fluxes and the torque forgiven currents ic. This model is useful for a static analysisof the motor. In a dynamical analysis, however, the coilvoltages vc must be used as inputs. This relation is providedby Faradays law

dcdt

= Rcic vc (21)with the flux linkage c = Nctc, the winding matrix Nc =diag[Nc, Nc, Nc], the electric resistance matrix Rc =diag[Rc, Rc, Rc], and the electric voltages vc = [vc1, vc2,vc3]T . Here, Nc is the number of turns, Rc the electricresistance, and vcj the voltage of the respective coil j =1, 2, 3. Equation (21) links the fluxes tc of the coils with theircurrents ic. Thus, either tc has to be defined as a function ofic or vice versa. For nonlinear permeance networks, it provesto be advantageous to express the coil currents ic as functionsof the fluxes by reformulating (18) in the form

DcGcDTc 0 DcGcDTgDmGcDTc I DmGcDTgDgGcDTc 0 Gt + DgGcDTg

K1

utctmutg

x

=

tc00

M1

DGcDTmutm M2

. (22)

This means that the dynamical model of the motor is givenby a set of nonlinear differential-algebraic equations (DAE),i.e., (21) and (22). Now, the following questions arise.

1) Do the state variables of (21) represent the minimumnumber of states or is it possible to reduce the numberof states?

2) Does the nonlinear set of equations (22) have a uniquesolution?

3) How can the electric interconnection of the coils (e.g.,delta or wye connection) be systematically considered?

To answer the first two questions consider the matrix K1of (22). Using the results of the Appendix, it turns out thatK1 is singular if the rows of Dc are linearly dependent.

KEMMETMLLER et al.: MODELING OF A PSM WITH INTERNAL MAGNETS 8101314

Let us assume that Dc Rmn , m < n has m lineardependent rows. Then, the column space DIc = span (Dc) hasdimension m m and the orthogonal complement Dc =span

(a Rm |aT b = 0, b DIc

)has dimension m. Let DIc

be a matrix composed of m m independent vectors ofDIc (i.e., the image of Dc) and Dc be composed of mindependent vectors of Dc (i.e., the kernel of DTc ). Then,(Dc

)T Dc = 0 holds and the nonsingular matrix

T1 =

T1c 0 00 I 00 0 I

(23)

with

T1c =[(

Dc)T

(DIc

)T

]

(24)

can be defined. Applying the transformation matrix T1 in theform

T1K1T11 K2

T1

utctmutg

= T1M1 T1M2 (25)

results in a matrix K2 with the structure

K2 =[

0 00 K2r

](26)

where the number of zero rows and columns is m. To provethis statement, K2 is formulated as

K2 =

T1cDcGcDTc T11c 0 T1cDcGcDTgDmGcDTc T11c I DmGcDTgDgGcDTc T11c 0 Gt + DgGcDTg

. (27)

It can be seen that the product

T1cDc =[ 0(DIc

)T Dc

](28)

gives m zero rows. Of course, the right-hand side multiplica-tion with an arbitrary matrix does not change the zero rows. Toprove the zero columns in K2, the product DTc T

11c is analyzed.

The matrix T1c can be written in the form

T1c =

aT1...

aTm

bT1...

bTmm

(29)

where a j Dc and b j DIc . The inverse T11c =[v1, . . . , vm, w1, . . . , wmm

]has to meet T1cT11c = I and

therefore

aTj vk = j k aTj wk = 0 (30a)bTj wk = j k bTj vk = 0 (30b)

with the Kronecker symbol j k , holds. Obviously, this meansthat v j Dc and w j DIc . Based on this discussion

DTc T11c = [0, ] (31)

holds, where the number of zero columns is equal to m and is a matrix with m m nonzero columns. Thus, K2 hasm zero columns and rows.

The application of T1 to the vector of unknowns gives

T1

utctmutg

=

T1cutctmutg

(32)

the multiplication of M1 with the transformation matrix resultsin

T1

tc00

=

T1ctc

00

(33)

and T1 used in combination with M2 yields

T1DGcDTg utm =

T1cDc

DmDg

GcDTg utm (34)

which again has m zero rows.This discussion shows two important results: 1) from (25)

with (32) and (24) it can be seen that the part (Dc)T

utc of thecoil currents cannot be calculated from the permeance networkbut has to be defined by the electrical connection of the coils.Only the part

(DIc

)Tutc is determined by the (reduced) set of

nonlinear equations

K2r

(DIc

)Tutc

tmutg

=

(DIc

)Ttc

00

(DIc

)T DcDmDg

GcDTg utm . (35)

2) Not the entire part of tc is independent but the componentsare restricted to fulfill

(Dc

)Ttc = 0. (36)

This implies that the set of differential equations (21) for theflux linkage can be reduced to m m differential equations,where

(DIc

)Ttc is a possible choice of independent states.

Remark 1: To systematically obtain the reduced set ofnonlinear equations from the transformed set of equations (25),the reduction matrix H1

H1 =

H1r 0 0

0 I 00 0 I

(37)

H1HT1 = I with H1r = [0, I] R(mm)m

, is introduced.Multiplying (25) with H1 from the left side directly yields thereduced equations

H1K2HT1 K2r

H1rT1cutc

tmutg

=

H1r T1ctc

00

H1T1M2 (38)

8101314 IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 6, JUNE 2014

with

K2r =

H1rT1cDcGcDTc T11c HT1r 0 H1r T1cDcGcDTgDmGcDTc T11c HT1r I DmGcDTgDgGcDTc T

11c H

T1r 0 Gt + DgGcDTg

(39)and the new vector of unknowns

H1T1

utctmutg

=

H1r T1cutc

tmutg

. (40)

In a further step, the electrical connection of the coils will beconsidered by means of the interconnection matrix Vc, that is

utc = Vcutc. (41)Here, utc corresponds to the independent currents of the coils.Using, e.g., a wye connection of the three coils, the constraintreads as ic1 + ic2 + ic3 = 0, which can be accounted for bythe interconnection matrix

ic1ic2ic3

=

1 00 1

1 1

[

ic1ic2

]. (42)

Thus, utc = Nc [ic1, ic2]T has been chosen as the vector ofindependent currents. Replacing utc by (41) in the reducedvector of unknowns (40) results in

H1r T1cutc

tmutg

=

H1r T1cVcutc

tmutg

. (43)

If the matrix H1r T1cVc is nonsingular, utc can be used asthe new vector of independent unknown coil currents and nofurther action is necessary. In cases where the matrix is notsquare, the resulting nonlinear set of equations is overdeter-mined, i.e., there are more equations than unknowns. This candirectly be seen by calculating the left-hand side of the reducedset of equations (38) in the form K3

[uTtc,

Ttm, u

Ttg

]T, with K3

given by

K3 =

S11 0 H1r T1cDcGcDTgS21 I DmGcDTgS31 0 Gt + DgGcDTg

(44)

where

S11 = H1r T1cDcGcDTc T11c HT1r H1r T1cVc (45a)S21 = DmGcDTc T11c HT1rH1r T1cVc (45b)S31 = DgGcDTc T11c HT1rH1r T1cVc. (45c)

Under the previous assumption that H1r T1cVc is not square,the matrix K3 has more rows than columns, which impliesthat not all components of the reduced flux vector H1r T1ctcin (38) can be arbitrarily assigned and therefore used as statevariables in (21). Thus, a part of the reduced flux vector hasto be added to the vector of unknowns. Let us assume thatthe upper-left entry S11 of K3 has n dependent rows. Thetransformation T2 =

[S11, SI11

], where SI11 is the column space

of S11 and S11 is the orthogonal complement to SI11, is usedto introduce a transformed vector tc in the form

T2tc =

S11 [I, 0]

H3r

+SI11 [0, I] H4r

tc = H1r T1ctc. (46)

It can be seen that adding the first n elements of tc to thevector of unknowns results in a set of nonlinear equations witha unique solution. To do so, (46) is inserted into (38) with (41),(44), and (45) resulting in

[S2 K3

]

H3r tcutctmutg

=

SI11H4r tc

00

H1T1M2 (47)

with S2 =[(

S11)T

, 0, 0]T

. Obviously, H3r tc is obtained asa solution of (47) and H4r tc has to be used as independentstate in the dynamical equation [see (21), (38), and (41)]

ddt

H4r tc = H4r T12 H1r T1cN1c(

RcN1c Vcutc vc)

. (48)As a result of this modeling framework we get the DAE system(47), (48) which is of minimal dimension and systematicallyaccounts for the electric interconnection of the coils.

E. Simulation ResultsTo evaluate the behavior of the PSM, simulations of the

mathematical model were performed. In a first step, thetorque and the magnetomotive forces for fixed currents wereinvestigated using (18) and (20). Fig. 4(a) shows the coggingtorque, i.e., the torque for zero currents icj = 0, j = 1, 2, 3.It can be seen that a pronounced cogging torque with aperiodicity of 15 is present in the motor. The results given inFig. 2(b)(d) were obtained for ic2 = ic3 = 2.5 A,ic1 = 0 A, which approximately corresponds to the nominalvalue. A closer look at the torque in Fig. 4(b) shows that thecharacteristics of the torque is far from being sinusoidal, whichwould be expected for an ideal PSM. The magnetomotiveforces in the stator teeth and yoke depicted in Fig. 4(c) and (d)further reveal that the magnetomotive forces in the yoke aremuch smaller than for the teeth.

This fact gives rise to the development of a simplifiedpermeance network, which covers the essential effects of thecomplete model. A simplified model of reduced dimension andcomplexity is especially desirable for a prospective controllerdesign. Thus, the following simplifications will be made: 1) thepermeances Gs12, Gs23, and Gs31 of the stator yoke areneglected, i.e., set to . 2) Simulations show that the fluxes inthe radial rotor bars are very small compared to the fluxes inthe rest of the motor. Thus, the simplification Gb1 = Gb2 = 0is used. 3) With the last simplification, the circumferentialrotor bars and the center of the rotor can be modeled by asingle equivalent permeance Gb and Gr , respectively.

In the subsequent section, the simplified model will bepresented in more detail. A comparison of simulation resultsof the complete with the reduced model will justify thesimplifying assumptions being made.

KEMMETMLLER et al.: MODELING OF A PSM WITH INTERNAL MAGNETS 8101314

Fig. 4. Simulation results of the complete model (a) for zero currents and(b)(d) for ic2 = ic3 = 2.5 A, ic1 = 0 A.

III. REDUCED MODELA. Permeance Network

Fig. 5 shows the reduced permeance network. Therein, theeffective permeances of the circumferential bars and the center

Fig. 5. Reduced permeance network of the PSM.

of the rotor are given by

Gb =Abc0r

( |ub|2lbc

)

lbc(49a)

Gr =Ar0r

( |ur |2lr

)

lr(49b)

while all other components remain the same as for the com-plete model.

Given the tree in Fig. 5, the flux vector of the tree perme-ances tg reads as

tg = [s1, s2, s3, b, m1, m2, a11]T (50)

and the vector of the cotree fluxes is given by

c = [r , l12, l23, l31, a12, a21, a22, a31, a32]T. (51)

The magnetomotive forces are defined accordingly and theremaining fluxes and magnetomotive forces are equal to thecomplete model. The permeance matrices of the tree andcotree reduce to

Gt = diag[Gs1, Gs2, Gs3, Gb, Gm1, Gm2, Ga11] (52a)Gc = diag[Gr , Gl12, Gl23, Gl31,

Ga12, Ga21, Ga22, Ga31, Ga32] (52b)

8101314 IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 6, JUNE 2014

and the components of the incidence matrix read as

Dc =

0 1 0 1 0 1 1 1 10 1 1 0 0 1 1 0 00 0 1 1 0 0 0 1 1

(53a)

Dm =[1 0 0 0 0 0 0 0 0

1 0 0 0 0 0 0 0 0

](53b)

Dg =

0 1 0 1 0 1 1 1 10 1 1 0 0 1 1 0 00 0 1 1 0 0 0 1 1

1 0 0 0 1 0 1 0 11 0 0 0 0 0 0 0 0

1 0 0 0 0 0 0 0 00 0 0 0 1 1 1 1 1

. (53c)

The balance and the torque equations are defined equally tothe complete model and are, therefore, not repeated here. Inthe subsequent section, however, the derivation of the voltageequations according to Section II-D is carried out for thereduced model.

B. Voltage EquationFollowing (22) the vector of unknowns x and the right-hand

side M1 for the reduced permeance network of Fig. 5 aregiven by

x = [ucs1, ucs2, ucs3, ms1, ms2, us1, us2, us3, (54a)ub, um1, um2, ua11]T

M1 = [cs1, cs2, cs3, 0, 0, 0, 0, 0, 0, 0, 0, 0]T. (54b)The column space DIc of Dc from (53a) reads as

DIc =

1 00 1

1 1

(55)

with the orthogonal complement Dc = [1, 1, 1]T . Thus, thetransformation matrix T1c according to (24) is given by

T1c =

1 1 11 0 10 1 1

(56)

and the matrix H1r , see (37) reads as

H1r =[

0 1 00 0 1

]. (57)

The linear combination of coil currents which can becalculated from the set of equations are defined by

H1r T1cutc =[

ucs1 ucs3ucs2 ucs3

](58)

and the sum of the currents(Dc

)Tutc = ucs1 + ucs2 + ucs3,

see (36), cannot be deduced from the permeance network.This is immediately clear, since applying the same current toall three coils does not change the fluxes in the machine.

The vector of independent coil fluxes is then given by

H1r T1ctc =[cs1 cs3cs2 cs3

](59)

Fig. 6. Electrical connection of the motor coils (delta).

and the constraint(Dc

)Ttc = cs1 + cs2 + cs3 = 0 has to

be met.The coils of the motor are connected in delta connection,

Fig. 6, which does not directly imply an additional constrainton the currents. The electric voltages, however, have to meetvc1 + vc2 + vc3 = 0. Using this constraint in the ode

Ncddt

(cs1 + cs2 + cs3) = Rc (ic1 + ic2 + ic3)+ vc1 + vc2 + vc3 (60)

ic1 + ic2 + ic3 = 0 can be directly deduced. Finally, the set ofindependent differential equations is given by

Ncddt

(cs1 cs3) = Rc (ic1 ic3) + vc1 vc3 (61a)

Ncddt

(cs2 cs3) = Rc (ic2 ic3) + vc2 vc3. (61b)Remark 2: Note that the electrical interconnection of the

coils does not have to be considered since H1r T1cVc = I with

Vc = 13

2 1

1 21 1

(62)

and utc = Nc [ic1 ic3, ic2 ic3]T .

C. Comparison With Complete Model and MeasurementsTo prove that the reduced model captures the essential

behavior of the complete model with sufficient accuracy, acomparison of the torques for zero current [Fig. 7(a)] and foric2 = ic3 = 2.5 A, ic1 = 0 A [Fig. 7(b)] is given. It can beseen that almost perfect agreement between the two modelscan be achieved. The comparison of the magnetomotive forceus3 in Fig. 7(c) shows some minor differences between thecomplete and reduced model, which, however, do not signif-icantly influence the torque. Therefore, the simplifications ofthe reduced model can be considered feasible.

For the evaluation of the model quality in comparison withthe behavior of the real motor, measurements at a test benchwere performed. The test bench given in Fig. 8 is composedof: 1) the PSM; 2) a torque measurement shaft; 3) a highlyaccurate resolver; 4) an inertia disk; and 5) a harmonic drive.The PSM is connected to a voltage source, where the terminalvoltage vc is adjusted to obtain a desired current ic while

KEMMETMLLER et al.: MODELING OF A PSM WITH INTERNAL MAGNETS 8101314

Fig. 7. Comparison of the complete with the reduced model (a) for zerocurrents and (b) and (c) for ic2 = ic3 = 2.5 A and ic1 = 0 A.

Fig. 8. Test bench for the PSM.

the terminal voltages va and vb are set to zero, Fig. 6. Tomeasure the torque as a function of the angle , the PSMis driven by a harmonic drive motor at a constant rotationalspeed of n = 2 rpm.

Fig. 9. Comparison of the reduced model with measurements (a) for zerocurrents, (b) for ic2 = ic3 = 2.5 A, ic1 = 0 A, and (c) for ic2 = ic3 =7.5 A, ic1 = 0 A.

Fig. 9 depicts a comparison of the measured and simulatedtorque of the PSM. It can be seen that a rather good agreementbetween measurement and reduced model is given, which isremarkable, since the model has only been parameterized bymeans of geometrical and nominal material parameters.

The reduced model, however, is not accurate enough for ahigh precision control strategy. Therefore, the next section isconcerned with the calibration of certain model parameters tofurther improve the model accuracy.

IV. MODEL CALIBRATIONThe main reason for the model errors are the inaccuracies

in the air gap permeances Gajk. Thus, the following strategyis introduced for the identification of the air gap permeances.

1) The torque is measured for fixed currents ics1 = 0,ics2 = ics3 = ics and a fixed step size in the angle ,resulting in a measurement vector km , k = 1, . . . , N with the

8101314 IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 6, JUNE 2014

corresponding angles k = k, the number of measurementsN and N = /2.

2) It is assumed that Ga = Ga,nom+Ga , with the nominalvalue Ga,nom and the corrective term Ga to be identified. Ofcourse, the corrective term has to meet the symmetry condi-tion (8), Galm = Ga ( (l 1)/6 (m 1)/4), l =1, 2, 3 and m = 1, 2. For fixed angles k the correspondingvalues are given by Glmalm , where the index lm is defined as

lm = mod(

k (l 1) N3

(m 1) N2

1, N)

+ 1. (63)

3) For each angle k , the relation(

Gt + DgGcDTg)

uktg = DgGc(

DTc utc + DTmutm)

(64)

with Gt (k, uktg) and Gc(k, uktg) has to be fulfilled, see (18).4) The derivation Ga/, needed for the calculation of

the torque, see (20), is approximated byGa

k+2

= Gk+1a Gka

. (65)

The corresponding magnetomotive forces of the air gap alsohave to be evaluated at k + /2. Since they are calculatedfrom (64) at the angles k , these values are obtained byaveraging the magnetomotive forces at the angles k and k+1,that is

ualm |k+2 =uk+1alm + ukalm

2(66)

with l = 1, 2, 3 and m = 1, 2.With these prerequisites, N torque equations in the form

k = km and 7N nonlinear equations defined by (64), aregiven. The N unknown values of Gka and the 7N unknownvectors uktg are given as the solution of this set of equations.This solution is found numerically using, e.g., MATLAB andresults in the desired corrective term Gka as a function of .

Fig. 10 shows a comparison of the nominal air gap perme-ance Ga () model adopted from [2] with the identified values,where measurements with fixed currents ics1 = 0, ics2 =ics3 = ics = 5 A were used for the identification. It can be seenthat the basic shape is equal to the nominal characteristics,only the maximum value is reduced and the transition phase isslightly changed. The identified shape seems to be reasonablesince the changes might account for unmodeled leakages.

In Fig. 11, the torque of the calibrated model is comparedwith measurement results. These results show a significantimprovement to the uncalibrated model in Fig. 9 and a verygood agreement in the complete operating range of the motor.Thus, it can be deduced that both the inhomogeneous air gapas well as saturation in the motor are adequately representedby the proposed model. It is worth noting that an even betteragreement between measurement and model could be achievedfor the cogging torque if the calibration would have beenperformed at a lower current, e.g., ics = 2.5 A. Then, however,the results for high currents would be worse such that thepresented results are a good compromise between the accuracyfor low and high currents.

Fig. 10. Comparison of the identified and the nominal air gap permeanceGa ().

The comparison of the induced voltages vcs j , j = 1, 2, 3 fora fixed angular velocity of 120 rad/s given in Fig. 12 furtherconfirm the high accuracy of the proposed model.

In conclusion, it was shown that a calibrated permeancemodel in form of a state-space representation with minimumnumber of states is suitable for the accurate description ofthe behavior of the motor in the complete operating range. Inthe subsequent section, a classical dq0-model of the motor,as it is typically employed in the controller design of PSM,will be derived. To do so, first a magnetically linear model isextracted from the nonlinear reduced model. It will be shownthat the simplifications associated with the magnetically linearand especially with the dq0-model result in rather large devi-ations from the measurement results. This also implies that acontroller design based on dq0-models is not able to exploit thefull performance of model based nonlinear control strategies.

V. SIMPLIFIED MODELSA. Magnetically Linear Model

If it is assumed that the relative permeability r of allpermeances is constant, then a magnetically linear permeancemodel is obtained. Starting from (18) (of course using theincidence matrix D and the tree and cotree magnetomotiveforces and fluxes of the reduced model of Section III), themagnetomotive forces utg of the tree permeances can becalculated in the form

utg = (

Gt + DgGcDTg)1

DgGc(

DTc utc + DTmutm)

(67)and the coil fluxes tc read as, see (17)

tc = Dc[

Gc GcDTg(

Gt + DgGcDTg)1

DgGc]

(

DTc utc + DTmutm)

. (68)Thus, the coil fluxes are given in the form of a superposition ofthe flux due to the coil currents utc = Nc ic and the permanentmagnets utm . Inserting (68) into the voltage equation (21), weget for the left-hand side

Ncddt

tc = Nc tc

J

+ Nc tc ic

L

ddt

ic. (69)

KEMMETMLLER et al.: MODELING OF A PSM WITH INTERNAL MAGNETS 8101314

Fig. 11. Comparison of the torque of the calibrated model with measurements(a) for zero currents, (b) for ic2 = ic3 = 2.5 A, ic1 = 0 A, and (c) foric2 = ic3 = 7.5 A, ic1 = 0 A.

The (symmetric) inductance matrix L can be formulated as

L = N2c Dc[Gc GcDTg

(Gt + DgGcDTg

)1DgGc

]DTc (70)

and the vector J reads asJ = NcDcTJ

(DTc utc + DTmutm

)(71)

with

TJ =[Gc

Gc

DTg H5DgGc GcDTg H5DgGc

+ GcDTg H5(

Gt

+ Dg Gc

DTg)

H5DgGc]

(72)

and

H5 =(

Gt + DgGcDTg)1

. (73)

Fig. 12. Comparison of the induced voltages of the calibrated model withmeasurements for = 120 rad/s.

Given these results, the voltage equation (21) can be formu-lated in the well-known form

L ()ddt

ic = J () + Rcic vc. (74)Note that the inductance matrix L and the vector J are bothnonlinear functions of the rotor angle . According to (20), thetorque of the motor in the magnetic linear case is given by

= 12

puTtcDcTJ DTc utc

r

+ 12

puTtmDmTJ DTmutm

c

+ puTtmDmTJ DTc utc p

. (75)

Here, three different parts can be distinguished: 1) for zero coilcurrents, i.e., utc = 0, the remaining part c represents the cog-ging torque of the motor. 2) Excluding the permanent magnetsof the motor, i.e., setting utm = 0, only the reluctance torquer due to the inhomogeneous air gap is present. 3) The part prepresents the main part of the torque. It is the only part whichcan be found in an ideal PSM with a homogenous air gap.

B. Fundamental Wave ModelThe magnetically linear model of the previous section still

covers the complete nonlinearity due to the air gap perme-ances. In this subsection, a further simplification is made,where only the average values and the fundamental wavecomponents of the corresponding parts are considered.

Applying this approach to (70), the inductance matrix isgiven by

L =

Lm 12 Lm 12 Lm 12 Lm Lm 12 Lm 12 Lm 12 Lm Lm

(76)

with the constant main inductance Lm . The term J reduces to

J () = J

sin (p)

sin(

p 23)

sin(

p 43)

(77)

8101314 IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 6, JUNE 2014

Fig. 13. (a) Entry L11 of the inductance matrix L, (b) J1 of the vector J, and(c) Mcm,11 of the matrix Mcm for the magnetically linear and the fundamentalwave model.

and the torque can be formulated as = puTtmMcm () utc,where Mcm () reads as

M[

sin (p) sin(

p 23)

sin(

p 43)

sin (p) sin (p 23) sin (p 43

)]

. (78)

The coefficients Lm , J , and M can be obtained, e.g., by aFourier analysis of the corresponding entries of the magneti-cally linear model. Fig. 13 shows a comparison of the entriesof the inductance matrix L, the vector J, and the matrix Mcmbetween the magnetically linear model and the fundamentalwave model. It can be seen that a rather good approximationof the magnetically linear model can be obtained by means ofthe fundamental wave model.

The well-known dq0-representation of the fundamentalwave model can be found using the transformations

idiqi0

= K ()

ic1ic2ic3

,

vdvqv0

= K ()

vc1vc2vc3

(79)

with the transformation matrix K ()

K () =

cos (p) cos

(p 23

)cos

(p 43

)

sin (p) sin(

p 23)

sin(

p 43)

12

12

12

. (80)

Then, the dq0-model takes the form

ddt

id = 231

Lm

(3

2Lm piq + Rcid vd

)(81a)

ddt

iq = 231

Lm

(32

Lm pid 32 J + Rciq vq)

(81b)

and the torque is given by

= 2 pMums1 Nciq . (82)

C. Comparison of the ModelsUp to now three models of different complexity, i.e.,

a magnetically nonlinear model, a magnetically linear and afundamental wave model, were presented in this paper. In thissection, the torque calculated by these models is comparedwith measurement results, see Fig. 14.

The results for zero current [Fig. 14(a)] show that thecogging torque can be reproduced rather well by the nonlinearmodel. Even the magnetically linear model shows the basicbehavior of the cogging torque, however, with larger errorscompared to the nonlinear model. As a matter of fact, it is notpossible to reproduce the cogging torque with the fundamentalwave model. Thus, this model gives the worst results as it was,of course, expected.

For nominal and high currents shown in Fig. 14(b) and (c),respectively, this result is confirmed. Again the nonlinearmodel gives excellent agreement with the measurements whilethe performance of the magnetically linear model degradeswith increasing currents. This results from the fact that satu-ration is not included in the magnetically linear model. Thebasic shape is, however, much better reproduced than in thefundamental wave model.

This brief comparison shows that a controller designedusing a fundamental wave model cannot systematicallyaccount for the cogging torque and saturation. Using insteadthe nonlinear model for a controller design it can be expectedthat the control performance is superior to controllers basedon fundamental wave models. The obvious drawback of thenonlinear model is the increased complexity of the resultingcontrol strategy. Here, the magnetically linear model mightbe a good compromise between model complexity and modelaccuracy for the controller design and will yield significantimprovements in comparison to fundamental wave models.

KEMMETMLLER et al.: MODELING OF A PSM WITH INTERNAL MAGNETS 8101314

Fig. 14. Comparison of the measurement results with the nonlinear, themagnetically linear and the fundamental wave model for (a) for zero currents,(b) ic2 = ic3 = 2.5 A, ic1 = 0 A, and (c) for ic2 = ic3 = 7.5 A,ic1 = 0 A.

VI. CONCLUSIONA systematical modeling framework for PSM with internal

magnets was outlined in this paper. Different from existingworks, the balance equations were derived based on graph-theory, which allows for a systematic calculation of the mini-mum number of nonlinear equations. Furthermore, the choiceof a suitable state and the systematic consideration of the elec-trical connection of the coils were discussed. The quality of thecalibrated model was shown by a comparison with measure-ment results. Finally, a magnetically linear and a dq0-modelhave been derived and compared with the nonlinear model.

Future work will deal with the application of the methodol-ogy to other motor designs as, e.g., PSM with surface magnets,reluctance machines or asynchronous machines. In addition,

the use of the models derived in this paper for nonlinear andoptimal controller design is an ongoing topic of research.Here, first results show a high potential of the modelingapproach and a significant improvement in comparison tocontrol strategies using classical dq0-models.

APPENDIXEXISTENCE AND UNIQUENESS OF SOLUTION

The set of nonlinear equations in (18) has to be solvednumerically. Thus, it is interesting to examine if a solutionexists and if it is unique. The matrices Gt and Gc are positivesemidefinite matrices for all utg and . This can be easilyseen since the entries of these diagonal matrices are positiveexcept for the air gap permeances, which can become zerofor certain angles . In addition, a suitable construction ofthe permeance network ensures that Dg has independent rowssuch that DgGcDTg is also positive semidefinite. To show thatthe sum of this term with Gt is even positive definite, considerthe vector x which fulfills

xT Gt x = 0. (83)The only possible solution x of (83) is equal to x =[0, . . . , ]T , R. It is then rather simple to show that

xT DgGcDTg x > 0 utg, (84)which implies that F = Gt + DgGcDTg is positive definite. Inthe magnetic linear case, the permeances are independent ofthe magnetomotive force and therefore, the positive definite-ness of F is sufficient for the existence and uniqueness of asolution of (18). In the nonlinear case, however, it has to beshown the Jacobian of F(utg, )utg is positive definite [39].The Jacobian can be written in the form

F +n

j=1

Futg, j

utg (85)

where utg, j describes the j th entry of utg . Using the fact that

r (H ) + H r (H ) H

> 0 (86)

holds, it can be shown that the Jacobian (85) indeed is positivedefinite for all utg and . It can be further shown that

limutgF (utg

)utg = (87)

holds, which implies that there exists a unique solution of theset of nonlinear (18), see [39].

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Wolfgang Kemmetmller (M04) received the Dipl.-Ing. degree in mecha-tronics from Johannes Kepler University Linz, Linz, Austria, in 2002, and thePh.D. degree in control engineering from Saarland University, Saarbrcken,Germany, in 2007.

He was a Research Assistant with the Chair of System Theory andAutomatic Control, Saarland University, from 2002 to 2007. From 2007to 2012, he was a Senior Researcher with the Automation and ControlInstitute, Vienna University of Technology, Vienna, Austria, and he has beenan Assistant Professor since 2013. His current research interests includephysics based modeling and the nonlinear control of mechatronic systemswith a special focus on electrohydraulic and electromechanical systems.

Dr. Kemmetmller is an Associate Editor of the IFAC JournalMechatronics.

David Faustner received the Dipl.-Ing. degree in electrical engineering fromthe Vienna University of Technology, Vienna, Austria, in 2011.

He has been a Research Assistant with the Automation and ControlInstitute, Vienna University of Technology, since 2011. His current researchinterests include physics based modeling and the nonlinear control of electricalmachines.

Andreas Kugi (M94) received the Dipl.-Ing. degree in electrical engineeringfrom the Graz University of Technology, Graz, Austria, in 1992, and the Ph.D.degree in control engineering and the Habilitation degree in automatic controland control theory from Johannes Kepler University (JKU), Linz, Austria, in1995 and 2000, respectively.

He was an Assistant Professor with JKU from 1995 to 2002. In 2002, hewas a Full Professor with Saarland University, Saarbrcken, Germany, wherehe was the Chair of system theory and automatic control in 2007. Since 2007,he has been a Full Professor of complex dynamical systems and the Headof the Automation and Control Institute, Vienna University of Technology,Vienna, Austria. He is involved in several industrial research projects in thefield of automotive applications, hydraulic and pneumatic servo-drives, smartstructures, and rolling mill applications. His current research interests includephysics-based modeling and control of (nonlinear) mechatronic systems,differential geometric and algebraic methods for nonlinear control, and thecontroller design for infinite-dimensional systems.

Prof. Kugi is an Editor-in Chief of the IFAC Journal Control EngineeringPractice and he has been a Corresponding Member of the Austrian Academyof Sciences since 2010.

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