Modeling of a permanent magnet synchronous machine with internal magnets using magnetic equivalent circuits

Transcription

1 This ocument contains a post-print version of the paper Moeling of a permanent magnet synchronous machine with internal magnets using magnetic equivalent circuits authore by W. Kemmetmüller, D. Faustner, an A. Kugi an publishe in IEEE Transactions on Magnetics. The content of this post-print version is ientical to the publishe paper but without the publisher s final layout or copy eiting. Please, scroll own for the article. Cite this article as: W. Kemmetmüller, D. Faustner, an A. Kugi, Moeling of a permanent magnet synchronous machine with internal magnets using magnetic equivalent circuits, IEEE Transactions on Magnetics, vol. 5, no. 6, 14. oi: 1.119/ TMAG BibTex entry: % This file was create with JabRef.9.. % Encoing: author = {Kemmetmüller, W. an Faustner, D. an Kugi, A.}, title = {Moeling of a permanent magnet synchronous machine with internal magnets using magnetic equivalent circuits}, journal = {IEEE Transactions on Magnetics}, year = {14}, volume = {5}, number = {6}, part = {}, oi = {1.119/TMAG } } Link to original paper: Rea more ACIN papers or get this ocument: Contact: Automation an Control Institute (ACIN) Internet: Vienna University of Technology office@acin.tuwien.ac.at Gusshausstrasse 7-9/E76 Phone: Vienna, Austria Fax: Copyright notice: c 14 IEEE. Personal use of this material is permitte. Permission from IEEE must be obtaine for all other uses, in any current or future meia, incluing reprinting/republishing this material for avertising or promotional purposes, creating new collective works, for resale or reistribution to servers or lists, or reuse of any copyrighte component of this work in other works.

2 The final version of recor is available at IEEE TRANSACTIONS ON MAGNETICS, VOL.??, NO.??, DECEMBER 1 1 Moeling of a permanent magnet synchronous machine with internal magnets using magnetic equivalent circuits Wolfgang Kemmetmüller 1 Member, IEEE, Davi Faustner 1, an Anreas Kugi 1 Member, IEEE 1 Automation an Control Institute, Vienna University of Technology, Vienna, Austria The esign of control strategies for permanent magnet synchronous machines (PSM) is almost exclusively base on classical q- moels. These moels are, however, not able to systematically escribe saturation or non-homogenous air gap geometries typically occurring in PSM. This paper eals with a framework for the mathematical moeling of PSM base on magnetic equivalent circuits. Different to existing works, the moel equations are erive by means of graph theory allowing for a systematic choice of a minimal set of state variables of the moel an a systematic consieration of the electrical connection of the coils of the motor. The resulting moel is calibrate an verifie by means of measurement results. Finally, a magnetically linear an a q-moel are erive an their performance is compare with the nonlinear moel an measurement results. Inex Terms magnetic equivalent circuit, electric motor, permanent magnet motors I. INTRODUCTION PERMANENT magnet synchronous motors (PSM) are wiely use in many technical applications. Numerous papers an books ealing with the esign of PSM have been publishe in recent years, see, e.g., [1], [], [], [4], [5], [6], [7]. The mathematical moels propose in these papers range from finite element analysis over reluctance moels to classical q-moels. Finite element (FE) moels exhibit a high accuracy for the calculate magnetic fiels an allow for an exact consieration even of complex geometries of the motor, see, e.g., [5], [6], [7], [8], [9], [1]. Due to their high (numeric) complexity, these moels are, however, harly suitable for ynamical simulations an a controller esign. The esign of control strategies for PSM is typically base on classical q-moels, which, in their original form, assume a homogenous air gap an unsaturate iron cores, see, e.g., [11], [1], [1], [14], [15], [16], [17], [18]. Many moern esigns of PSM (incluing e.g. PSM with internal magnets) exhibit consierable saturation an non-sinusoial fluxes in the coils. To cope with these effects, extensions of q-moels have been reporte in literature, which are all base on a heuristic approach an are limite to a very specific motor esign, see, e.g., [16], [17], [18]. In most cases, these moels are not able to accurately escribe the motor behavior in all operating conitions. Magnetic equivalent circuits have become very popular for the esign an the (ynamical) simulation of PSM in the recent years, see, e.g., [1], [], [19], [], [1], [], [], [4], [5], [6], [7], [8], [9], [], [1], [], [], [4], [5]. This is ue to the significantly reuce complexity in comparison to FE moels an their capability to systematically escribe saturation an non-homogenous air gap geometries. The accuracy an complexity of reluctance moels can be easily controlle by means of the choice of the reluctance network. While reluctance networks with a rather high com- Manuscript receive September 6, 1; revise??, 1. Corresponing author: W. Kemmetmüller ( kemmetmueller@acin.tuwien.ac.at). plexity are necessary to accurately escribe fiel profiles in the motor, moels of significantly reuce complexity alreay represent the behavior with respect to the torque, currents an voltages of the motor in sufficient etail. Thus, moels base on aequately chosen reluctance moels promise to be a goo basis for ynamical simulations an the (nonlinear) controller esign. In this paper, a framework for the systematic erivation of a state-space moel with a minimum number of nonlinear equations an state variables is presente. The main purpose of the erive moel is to provie a state-space representation for avance moel-base control strategies an thus to reprouce the ynamic input-to-output behavior of the motor in an accurate manner. The framework evelope here is universal, it is applie here to a specific internal magnet PSM that exhibits both large cogging torque an saturation. Section II presents the consiere moel an a complete reluctance moel of the motor. To obtain a minimal set of (nonlinear) equations that escribe the reluctance network, a metho base on graph theory is propose. This metho, well known from electrical networks, see, e.g. [6], [7], [8] is ajuste to the analysis of magnetic networks. Subsequently, the escription of the electrical connection of the coils an the choice of a suitable set of state variables for the ynamical system is outline. It shoul be note that the framework presente in this section can be applie to any PSM. Section III is concerne with a reuce moel base on finings of simulation results of the complete moel. Section IV shows the systematic calibration of the reuce moel an a comparison with measurement results. Starting from the nonlinear moel, a magnetically linear moel an a classical q-moel are systematically erive in Section V. Finally, the results of the nonlinear moel, the magnetically linear moel an the q-moel are compare with measurement results. II. CONSIDERED MOTOR AND COMPLETE MODEL The motor consiere in this paper is a permanent magnet synchronous motor with internal magnets. It comprises 1 Copyright (c) 14 IEEE. Personal use is permitte. For any other purposes, permission must be obtaine from the IEEE by ing pubs-permissions@ieee.org. with internal magnets using magnetic equivalent circuits, IEEE Transactions on Magnetics, vol. 5, no. 6, 14. oi: 1.119/TMAG

3 The final version of recor is available at IEEE TRANSACTIONS ON MAGNETICS, VOL.??, NO.??, DECEMBER 1 stator coils, each woun aroun a single stator tooth, an 8 NFeB-magnets in the rotor, which are alternately magnetize. The setup of the motor is perioically repeate every 9 (number of pole pairs p = 4), such that only a quarter of the motor has to be consiere. Fig. 1 shows a sectional view of the PSM an the permeance network use to moel the stator an the rotor (air gap permeances are not inclue in this figure). G b1 G r magnet G m rotor coil G s u cs G l G r11 G s G u b ms G b G b1 G r1 G G m1 r1 u ms1 G b11 G b1 magnet 1 u cs G s G l1 G u s1 cs1 G l1 coil Fig. 1. Sectional view of the PSM with permeance network. G s1 stator coil 1 G s1 The motor is esigne to exhibit a large cogging torque by means of an inhomogeneous construction of the air gap, see Fig. 1. This is ue to the fact that the motor is use in an application where external torques beyon a certain limit shoul not yiel large changes in the rotor angle ϕ. The large cogging torque, however, makes the esign of highperformance control strategies more involve. Thus, tailore mathematical moels which accurately escribe the cogging torque an the saturation are require for the controller esign. A. Permeance network As alreay outline in the introuction, a network of nonlinear permeances is utilize for the erivation of a moel of the motor. Fig. epicts the propose permeance network of the motor. The permeances escribing the core of the stator an the rotor are approximate by cubois of length l an area A. To account for saturation effects in the core, the relative permeability µ r is efine as a function of the absolute value of the magnetic fiel strength H = u/l, i.e. µ r ( u /l), where u enotes the magnetomotive force. Fig. shows the relative permeability µ r for the applie core material M8-5A. The nonlinear permeances of the stator teeth then rea as ( ) usj A st µ µ r l st G sj (u sj ) =, j = 1,,, (1) l st φ m1 G s1 G s G s G a1 G a1 G a11 G m1 G s1 φ l1 G l1 G b11 G φ b1 b1 G r11 G r1 G l1 G s1 G b1 G s u cs1 u cs u cs u ms1 G l φ l φ b1 G b1 G b φ s1 φ s1 φ s1 φ s φ s φ s φ cs1 φ cs φ cs φ a11 φ a1 φa1 φ a φa1 φ a φ ms1 φ b11 φ r11 φ r1 φ b1 Fig.. Permeance network of the PSM. φ b φ b φ G r1 r1 φ r φ l1 G a1 G a G b G r G a φ m G m φ ms u ms with the area A st, the length l st an the magnetomotive force u sj of a stator tooth, an the permeability µ of free space. The permeances of the stator yoke can be foun analogously in the form G sjk (u sjk ) = ( ) usjk A sy µ µ r l sy l sy, jk {1,, 1}, () where A sy is the area, l sy escribes the length an u sjk is the corresponing magnetomotive force. The center of the rotor is ivie into 4 elements, which are escribe by the permeances G rjk (u rjk ) = ( ) urjk A r µ µ r l r l r, jk {11, 1, 1, }. Here, A r is the effective area, l r the effective length an u rjk the magnetomotive force of the rotor element. The permanent magnets are place insie the rotor of the motor. The resulting construction of the rotor exhibits parts, which have the form of very slener bars. The circumferential bars are escribe () Copyright (c) 14 IEEE. Personal use is permitte. For any other purposes, permission must be obtaine from the IEEE by ing pubs-permissions@ieee.org. with internal magnets using magnetic equivalent circuits, IEEE Transactions on Magnetics, vol. 5, no. 6, 14. oi: 1.119/TMAG

4 The final version of recor is available at IEEE TRANSACTIONS ON MAGNETICS, VOL.??, NO.??, DECEMBER 1 relative permeability µr magnetic fiel strength H (A/m) Fig.. Relative permeability µ r of the core material M8-5A. by G bjk (u bjk ) = ( ) ubjk A bc µ µ r l bc l bc, jk {11, 1, 1, }, with the area A bc, the length l bc an the magnetomotive force u bjk. The permeances of the raial bars rea as ( ) ubj A br µ µ r l br G bj (u bj ) =, j = 1,. (5) l br Again, A br enotes the area, l br the length an u brj the magnetomotive force of the raial bar element. The air gap of the motor is moele by two types of permeances: the permeances G ljk, jk {1,, 1}, escribing the leakage between ajacent stator teeth, an G ajk, jk {11, 1, 1,, 1, }, escribing the coupling between stator an rotor. The leakage permeances are efine as (4) G ljk = A lµ l l, jk {1,, 1}, (6) with the effective area A l an length l l. The air gap permeances G ajk are, of course, functions of the relative rotation ϕ of the rotor with respect to the stator. A geometric moel of these permeances using an approximate air gap geometry is possible but yiels inaccurate results ue to stray fluxes not covere by the approximate air gap geometry. Therefore, a heuristic approach, as has been propose in [], [19], is use to approximate the coupling between the rotor an stator, i.e. the air gap permeance G a. π 4 ϕ δ G G a (ϕ) = a,max ( ( 1 + cos π δ ϕ)) δ < ϕ δ (7) δ < ϕ π 4 Therein, ϕ is the relative rotation ϕ mappe to the interval ( π/4, π/4) by means of a moulo operation. Moreover, δ is a parameter which can be approximately etermine by the geometrical overlap between a permanent magnet an a stator tooth, an G a,max is the maximum value at ϕ =. Given G a of (7), the air gap permeances between the iniviual stator teeth an permanent magnets are efine as ( (j 1)π G ajk = G a ϕ 6 (k 1)π 4 ), (8) with j = 1,, an k = 1,. The NFeB-magnets exhibit an almost linear behavior in the operating range, which can be moele in the form of a constant magnetomotive force u msj, j = 1, an a linear permeance G mj (u mj ) = A mµ µ rm l m, j = 1,, (9) with the constant relative permeability µ rm, the effective area A m an the length l m. Given the coercive fiel strength H c of the magnets, their magnetomotive forces are escribe by u ms1 = u ms = H c l m. (1) The stator coils with N c turns are moele by u csj = N c i cj, j = 1,,, (11) with i cj being the electric current through the coil j. B. Balance equations Two approaches for the erivation of the balance equations (Kirchhoff s noe an branch equations) are typically use for magnetic reluctance networks: (i) mesh analysis [], [4], [5] an (ii) noe potential analysis [], [19], [], [1], [], [5], [6], [7], [8]. While a proper choice of meshes, yieling a set of inepenent equations might be tricky, the noe potential analysis automatically guarantees the inepenence of the resulting equations. Therefore, noe potential analysis is typically favore. In this paper an alternative approach for the systematic erivation of a minimal set of inepenent equations base on graph theory is propose. It uses a tree, which connects all noes of the network without forming any meshes. This approach is well known from electric network analysis, see, e.g., [6], [7], [8], an can be, as will be shown in this paper, irectly applie to magnetic permeance networks, see also [9]. The chosen tree has to connect all noes of the network without forming any meshes. Moreover, all magnetomotive force sources have to be inclue in the tree, which is always possible for non-egenerate networks. It further turns out to be avantageous to exclue as many air gap permeance from the tree as possible. One possible choice of a tree is given in Fig. by the components epicte in black. The co-tree is then compose of all components which are not part of the tree (epicte gray in Fig. ). Aing one co-tree element to the tree yiels a single mesh. For the subsequent erivation, it is useful to subivie the elements of the tree into magnetomotive force sources of the coils (inex tc), magnetomotive force sources of the permanent magnets (inex tm) an permeances (linear, nonlinear, angle Copyright (c) 14 IEEE. Personal use is permitte. For any other purposes, permission must be obtaine from the IEEE by ing pubs-permissions@ieee.org. with internal magnets using magnetic equivalent circuits, IEEE Transactions on Magnetics, vol. 5, no. 6, 14. oi: 1.119/TMAG

5 The final version of recor is available at IEEE TRANSACTIONS ON MAGNETICS, VOL.??, NO.??, DECEMBER 1 4 epenent, inex tg). Then, the overall vector of the tree fluxes φ t = [ φ T tc, φt tm, φt tg] T is efine by φ tc = [φ cs1, φ cs, φ cs ] T = [φ ms1, φ ms ] T φ tg = [φ s1, φ s, φ s, φ s1, φ s, φ b1, φ b, φ r11, φ r1, φ r1, φ m1, φ m, φ a11 ] T. (1a) (1b) (1c) The vector of the corresponing tree magnetomotive forces u t is efine in an analogous manner. The co-tree only comprises permeances such that the vector of the co-tree fluxes is given by φ c = [φ l1, φ l, φ l1, φ s1, φ b11, φ b1, φ b1, φ b, φ r, φ a1, φ a1, φ a, φ a1, φ a ] T (1) an the vector of co-tree magnetomotive forces u c is efine in the same way. Now, the following relations between the tree an co-tree fluxes an magnetomotive forces, respectively, can be formulate φ t = Dφ c u c = D T u t. (14a) (14b) The incience matrix D escribes the interconnection of the iniviual elements of the permeance network an its entries are either 1, or 1. It can be ecompose into a part D c linking the co-tree fluxes with the tree coil fluxes, a part D m linking the co-tree fluxes with the tree permanent magnet fluxes, an a part D g, which connects the co-tree fluxes with the fluxes of the tree permeances, i.e. D T = [ ] D T c, D T m, D T g. The constitutive equations of the permeances can be summarize in the form φ tg = G t φ c = G c u c, (15a) (15b) with the permeance matrices G t an G c of the tree an co-tree, respectively. Note that in general these matrices are functions of the corresponing magnetomotive forces (ue to saturation) an the isplacement of the rotor, i.e. G t (, ϕ) an G c (u c, ϕ). For the permeance network of Fig. these matrices rea as G t = iag [G s1, G s, G s, G s1, G s, G b1, G b, G r11, G r1, G r1, G m1, G m, G a11 ] G c = iag [G l1, G l, G l1, G s1, G b11, G b1, G b1, (16a) G b, G r, G a1, G a1, G a, G a1, G a ]. (16b) Inserting (15) into (14), we fin the following set of equations φ tc G t = DG c [ D T c, D T m, DT g ] u tc u tm. (17) If it is assume that the coil currents i c = [i c1, i c, i c ] T an thus the magnetomotive forces u tc are given, the unknown variables of (17) are φ tc, an. A simple reformulation of (17) yiels I D c G c D T g I D m G c D T φ tc g = G t + D g G c D T g u (18) tg ( DG c D T c u tc + D T ) mu tm with the ientity matrix I. It can be easily seen that a set of im ( ) = n = 1 nonlinear algebraic equations has to be solve for. All other quantities of the network can be calculate from simple linear equations. A proof of the existence an uniqueness of a solution of the nonlinear algebraic equations (18) is given in the Appenix A. C. Torque equation Starting from the magnetic co-energy of the permeance network, the electromagnetic torque of the motor is efine as τ = 1 ) (u p T G t tg ϕ + u T G c c ϕ u c, (19) with the number p of pole pairs, see, e.g., []. With the help of (14b) this equation can be reformulate in the form τ = 1 (u p T G t tg ϕ + u T t D G ) c ϕ DT u t, () with u T t = [ u T tc, ut tm, ut tg]. D. Voltage equation The mathematical moel (18) an () allows for a calculation of the magnetomotive forces, fluxes an the torque for given currents i c. This moel is useful for a static analysis of the motor. In a ynamical analysis, however, the coil voltages v c must be use as inputs. This relation is provie by Faraay s law ψ c = R c i c v c, (1) t with the flux linkage ψ c = N c φ tc, the wining matrix N c = iag [N c, N c, N c ], the electric resistance matrix R c = iag [R c, R c, R c ] an the electric voltages v c = [v c1, v c, v c ] T. Here, N c is the number of turns, R c the electric resistance an v cj the voltage of the respective coil j = 1,,. Eq. (1) links the fluxes φ tc of the coils with their currents i c. Thus, either φ tc has to be efine as a function of i c or vice versa. For nonlinear permeance networks, it proves to be avantageous to express the coil currents i c as functions of the fluxes by reformulating (18) in the form D c G c D T c D c G c D T g D m G c D T c I D m G c D T u tc g = D g G c D T c G t + D g G c D T g K 1 x φ () tc DG c D T m u tm. M M 1 Copyright (c) 14 IEEE. Personal use is permitte. For any other purposes, permission must be obtaine from the IEEE by ing pubs-permissions@ieee.org. with internal magnets using magnetic equivalent circuits, IEEE Transactions on Magnetics, vol. 5, no. 6, 14. oi: 1.119/TMAG

6 The final version of recor is available at IEEE TRANSACTIONS ON MAGNETICS, VOL.??, NO.??, DECEMBER 1 5 This means that the ynamical moel of the motor is given by a set of nonlinear ifferential-algebraic equations (DAE), i.e. (1) an (). Now, the following questions arise: 1) Do the state variables of (1) represent the minimum number of states or is it possible to reuce the number of states? ) Does the nonlinear set of equations () have a unique solution? ) How can the electric interconnection of the coils (e.g. elta or wye connection) be systematically taken into account? To answer the first two questions consier the matrix K 1 of (). Using the results of Appenix A, it turns out that K 1 is singular if the rows of D c are linearly epenent. Let us assume that D c R m n, m < n has m linear epenent rows. Then, the column space Dc I = span (D c) has imension m m an the orthogonal complement Dc = span ( a R m a T b =, b Dc) I has imension m. Let D I c be a matrix compose of m m inepenent vectors of Dc I (i.e. the image of D c ) an D c be compose of m inepenent vectors of Dc (i.e. the kernel of DT c ). Then, ( ) D T c Dc = hols an the nonsingular matrix with T 1c T 1 = I () I T 1c = [ (D ) T ] ( c ) D I T c (4) can be efine. Applying the transformation matrix T 1 in the form T 1 K 1 T 1 1 T 1 u tc = T 1 M 1 T 1 M (5) u K tg results in a matrix K with the structure [ ] K =, (6) K r where the number of zero rows an columns is m. To prove this statement, K is formulate as T 1c D c G c D T c T 1 1c T 1c D c G c D T g K = D m G c D T c T 1 1c I D m G c D T g. (7) D g G c D T c T 1 1c G t + D g G c D T g It can be seen that the prouct [ ] T 1c D c = ( ) D I T c Dc (8) gives m zero rows. Of course, the right-han sie multiplication with an arbitrary matrix oes not change the zero rows. To prove the zero columns in K, the prouct D T c T 1 1c is analyze. The matrix T 1c can be written in the form a T 1.. T 1c = a T m b T, (9) 1.. b T m m where a j Dc an b j Dc I. The inverse T 1 1c = [v 1,..., v m, w 1,..., w m m ] has to meet T 1c T 1 1c = I an therefore a T j v k = δ jk a T j w k = (a) b T j w k = δ jk b T j v k =, (b) with the Kronecker symbol δ jk, hols. Obviously, this means that v j Dc an w j Dc. I Base on this iscussion D T c T 1 1c = [, ] (1) hols, where the number of zero columns is equal to m an is a matrix with m m non-zero columns. Thus, K has m zero columns an rows. The application of T 1 to the vector of unknowns gives u tc = T 1cu tc, () T 1 the multiplication of M 1 with the transformation matrix results in T 1 φ tc = T 1cφ tc () an T 1 use in combination with M yiels T 1 DG c D T g u tm = T 1cD c D m G c D T g u tm, (4) D g which again has m zero rows. This iscussion shows two important results: (i) From (5) with () an (4) it can be seen that the part ( ) D T c utc of the coil currents cannot be calculate from the permeance network but has to be efine by the electrical connection of the coils. Only the part ( Dc) I T utc is etermine by the (reuce) set of nonlinear equations ( ) D I T ( ) c utc D I T c φtc K r = ( ) D I T (5) c Dc D m G c D T g u tm. D g (ii) Not the entire part of φ tc is inepenent but the components are restricte to fulfill ( D c ) T φtc =. (6) Copyright (c) 14 IEEE. Personal use is permitte. For any other purposes, permission must be obtaine from the IEEE by ing pubs-permissions@ieee.org. with internal magnets using magnetic equivalent circuits, IEEE Transactions on Magnetics, vol. 5, no. 6, 14. oi: 1.119/TMAG

7 The final version of recor is available at IEEE TRANSACTIONS ON MAGNETICS, VOL.??, NO.??, DECEMBER 1 6 This implies that the set of ifferential equations (1) for the flux linkage can be reuce to m m ifferential equations, where ( Dc) I T φtc is a possible choice of inepenent states. Remark 1: To systematically obtain the reuce set of nonlinear equations from the transforme set of equations (5), the reuction matrix H 1, H 1 = H 1r I, (7) I H 1 H T 1 = I with H 1r = [, I] R (m m ) m, is introuce. Multiplying (5) with H 1 from the left sie irectly yiels the reuce equations H 1 K H T H 1rT 1c u tc 1 = H 1rT 1c φ tc H 1 T 1 M u K tg r (8) with H 1r T 1c D c G c D T c T 1 1c HT 1r H 1r T 1c D c G c D T g K r = D m G c D T c T 1 1c HT 1r I D m G c D T g D g G c D T c T 1 1c HT 1r G t + D g G c D T g (9) an the new vector of unknowns H 1 T 1 u tc = H 1rT 1c u tc. (4) In a further step, the electrical connection of the coils will be consiere by means of the interconnection matrix V c, i.e. u tc = V c ū tc. (41) Here, ū tc correspons to the inepenent currents of the coils. Using e.g. a wye connection of the three coils, the constraint reas as i c1 + i c + i c =, which can be accounte for by the interconnection matrix i c1 i c = 1 [ ] 1 ic1. (4) i 1 1 c i c Thus, ū tc = N c [i c1, i c ] T has been chosen as the vector of inepenent currents. Replacing u tc by (41) in the reuce vector of unknowns (4) results in H 1rT 1c u tc = H 1rT 1c V c ū tc. (4) If the matrix H 1r T 1c V c is nonsingular, ū tc can be use as the new vector of inepenent unknown coil currents an no further action is necessary. In cases where the matrix is not square, the resulting nonlinear set of equations is overetermine, i.e. there are more equations than unknowns. This can irectly be seen by calculating the left-han sie of the reuce [ūt set of equations (8) in the form K tc, φ T tm, utg] T T, with K given by S 11 H 1r T 1c D c G c D T g K = S 1 I D m G c D T g, (44) S 1 G t + D g G c D T g where S 11 = H 1r T 1c D c G c D T c T 1 1c HT 1rH 1r T 1c V c S 1 = D m G c D T c T 1 1c HT 1r H 1rT 1c V c S 1 = D g G c D T c T 1 1c HT 1r H 1rT 1c V c. (45a) (45b) (45c) Uner the previous assumption that H 1r T 1c V c is not square, the matrix K has more rows than columns, which implies that not all components of the reuce flux vector H 1r T 1c φ tc in (8) can be arbitrarily assigne an therefore use as state variables in (1). Thus, a part of the reuce flux vector has to be ae to the vector of unknowns. Let us assume that the upper-left entry S 11 of K has n epenent rows. The transformation T = [ S 11, S11] I, where S I 11 is the column space of S 11 an S 11 is the orthogonal complement to S I 11, is use to introuce a transforme vector φ tc in the form T φtc = S 11 [I, ] +S H r I 11 [, I] φ tc = H 1r T 1c φ tc. (46) H 4r It can be seen that aing the first n elements of φ tc to the vector of unknowns results in a set of nonlinear equations with a unique solution. To o so, (46) is inserte into (8) with (41) an (44), (45) resulting in [ ] S K H r φtc ū tc = SI 11 H φ 4r tc H 1 T 1 M, (47) [ (S ) ] T T with S = 11,,. Obviously, Hr φtc is obtaine as a solution of (47) an H 4r φtc has to be use as inepenent state in the ynamical equation (see (1), (8) an (41)) t H 4r φ tc = H 4r T 1 H 1rT 1c N 1 ( c Rc N 1 ) c V c ū tc v c. (48) As a result of this moeling framework we get the DAE system (47), (48) which is of minimal imension an systematically accounts for the electric interconnection of the coils. E. Simulation results To evaluate the behavior of the PSM, simulations of the mathematical moel were performe. In a first step, the torque an the magnetomotive forces for fixe currents were investigate using (18) an (). Fig. 4(a) shows the cogging torque, i.e. the torque for zero currents i cj =, j = 1,,. It can be seen that a pronounce cogging torque with a perioicity of 15 is present in the motor. The results given in Fig. (b)- () were obtaine for i c = i c =.5 A, i c1 = A, which approximately correspons to the nominal value. A closer look at the torque in Fig. 4(b) shows that the characteristics of the torque is far from being sinusoial, which woul be expecte for an ieal PSM. The magnetomotive forces in the stator teeth an yoke epicte in Fig. 4(c)-() further reveal that the magnetomotive forces in the yoke are much smaller than for the teeth. Copyright (c) 14 IEEE. Personal use is permitte. For any other purposes, permission must be obtaine from the IEEE by ing pubs-permissions@ieee.org. with internal magnets using magnetic equivalent circuits, IEEE Transactions on Magnetics, vol. 5, no. 6, 14. oi: 1.119/TMAG

8 The final version of recor is available at IEEE TRANSACTIONS ON MAGNETICS, VOL.??, NO.??, DECEMBER 1 7 torque τ (mn m) torque τ (N m) (a) This fact gives rise to the evelopment of a simplifie permeance network, which covers the essential effects of the complete moel. A simplifie moel of reuce imension an complexity is especially esirable for a prospective controller esign. Thus, the following simplifications will be mae: (i) The permeances G s1, G s an G s1 of the stator yoke are neglecte, i.e. set to. (ii) Simulations show that the fluxes in the raial rotor bars are very small compare to the fluxes in the rest of the motor. Thus, the simplification G b1 = G b = is use. (iii) With the last simplification, the circumferential rotor bars an the centre of the rotor can be moele by a single equivalent permeance G b an G r, respectively. In the subsequent section, the simplifie moel will be presente in more etail. A comparison of simulation results of the complete with the reuce moel will justify the simplifying assumptions being mae. A. Permeance network III. REDUCED MODEL (b) Fig. 5 shows the reuce permeance network. Therein, the effective permeances of the circumferential bars an the center of the rotor are given by magnetomotive force (A m) u s1 u s u s (c) G b = G r = ( ) A bc µ µ ub r l bc l bc ( ) A r µ µ ur r l r (49a) l r, (49b) while all other components remain the same as for the complete moel. Given the tree in Fig. 5, the flux vector of the tree permeances φ tg reas as magnetomotive force (A m) u s1 u s u s angle ϕ ( ) () Fig. 4. Simulation results of the complete moel (a) for zero currents an (b)-() for i c = i c =.5 A, i c1 = A. φ tg = [φ s1, φ s, φ s, φ b, φ m1, φ m, φ a11 ] T (5) an the vector of the co-tree fluxes is given by φ c = [φ r, φ l1, φ l, φ l1, φ a1, φ a1, φ a, φ a1, φ a ] T. (51) The magnetomotive forces are efine accoringly an the remaining fluxes an magnetomotive forces are equal to the complete moel. The permeance matrices of the tree an cotree reuce to G t = iag [G s1, G s, G s, G b, G m1, G m, G a11 ] G c = iag [G r, G l1, G l, G l1, G a1, G a1, G a, G a1, G a ] (5a) (5b) Copyright (c) 14 IEEE. Personal use is permitte. For any other purposes, permission must be obtaine from the IEEE by ing pubs-permissions@ieee.org. with internal magnets using magnetic equivalent circuits, IEEE Transactions on Magnetics, vol. 5, no. 6, 14. oi: 1.119/TMAG

9 The final version of recor is available at IEEE TRANSACTIONS ON MAGNETICS, VOL.??, NO.??, DECEMBER 1 8 G l1 φ l1 φ s1 φ s φ s by x = [u cs1, u cs, u cs, φ ms1, φ ms, u s1, u s, u s, u b, u m1, u m, u a11 ] T M 1 = [φ cs1, φ cs, φ cs,,,,,,,,, ] T. (54a) (54b) G s1 G s G s φ cs1 φ cs φcs u cs1 u cs ucs φ m1 G a1 G a11 G m1 G a1 u ms1 G l1 φ l1 G b G r G l φ l G a1 φ a11 φ a1 φa1 φ a φ a1 φa φ ms1 φ b φ r Fig. 5. Reuce permeance network of the PSM. G a φ ms G a φ m an the components of the incience matrix rea as D c = G m u ms (5a) [ ] 1 D m = (5b) D g = (5c) The balance an the torque equations are efine equally to the complete moel an are, therefore, not repeate here. In the subsequent section, however, the erivation of the voltage equations accoring to Section II-D is carrie out for the reuce moel. B. Voltage equation Following () the vector of unknowns x an the right-han sie M 1 for the reuce permeance network of 5 are given The column space D I c of D c from (5a) reas as D I c = 1 1 (55) 1 1 with the orthogonal complement D c = [1, 1, 1] T. Thus, the transformation matrix T 1c accoring to (4) is given by T 1c = , (56) 1 1 an the matrix H 1r, see (7) reas as [ ] 1 H 1r =. (57) 1 The linear combination of coil currents which can be calculate from the set of equations are efine by [ ] ucs1 u H 1r T 1c u tc = cs (58) u cs u cs an the sum of the currents ( ) D T c utc = u cs1 + u cs + u cs, see (6), cannot be euce from the permeance network. This is immeiately clear, since applying the same current to all three coils oes not change the fluxes in the machine. The vector of inepenent coil fluxes is then given by [ ] φcs1 φ H 1r T 1c φ tc = cs (59) φ cs φ cs an the constraint ( ) D T c φtc = φ cs1 + φ cs + φ cs = has to be met. v b i b v c1 i c i c1 v c v a i a v c i c i c Fig. 6. Electrical connection of the motor coils (elta). The coils of the motor are connecte in elta connection, see Fig. 6, which oes not irectly imply an aitional constraint v c Copyright (c) 14 IEEE. Personal use is permitte. For any other purposes, permission must be obtaine from the IEEE by ing pubs-permissions@ieee.org. with internal magnets using magnetic equivalent circuits, IEEE Transactions on Magnetics, vol. 5, no. 6, 14. oi: 1.119/TMAG

10 The final version of recor is available at IEEE TRANSACTIONS ON MAGNETICS, VOL.??, NO.??, DECEMBER 1 9 on the currents. The electric voltages, however, have to meet v c1 + v c + v c =. Using this constraint in the oe N c t (φ cs1 + φ cs + φ cs ) = R c (i c1 + i c + i c ) + v c1 + v c + v c, (6) i c1 + i c + i c = can be irectly euce. Finally, the set of inepenent ifferential equations is given by N c t (φ cs1 φ cs ) = R c (i c1 i c ) + v c1 v c N c t (φ cs φ cs ) = R c (i c i c ) + v c v c. (61a) (61b) Remark : Note that the electrical interconnection of the coils oes not have to be consiere since H 1r T 1c V c = I, with V c = (6) 1 1 an ū tc = N c [i c1 i c, i c i c ] T. C. Comparison with complete moel an measurements To prove that the reuce moel captures the essential behavior of the complete moel with sufficient accuracy, a comparison of the torques for zero current (see Fig. 7(a)) an for i c = i c =.5 A, i c1 = A (see Fig. 7(b)) is given. It can be seen that almost perfect agreement between the two moels can be achieve. The comparison of the magnetomotive force u s in Fig.7(c) shows some minor ifferences between the complete an reuce moel, which, however, o not significantly influence the torque. Therefore, the simplifications of the reuce moel can be consiere feasible. For the evaluation of the moel quality in comparison with the behavior of the real motor, measurements at a test bench were performe. The test bench given in Fig. 8 is compose of (i) the PSM, (ii) a torque measurement shaft, (iii) a highly accurate resolver, (iv) an inertia isk an (v) a harmonic rive. The PSM is connecte to a voltage source, where the terminal voltage v c is ajuste to obtain a esire current i c while the terminal voltages v a an v b are set to zero, see Fig. 6. To measure the torque τ as a function of the angle ϕ, the PSM is riven by a harmonic rive motor at a constant rotational spee of n = rpm. Fig. 9 epicts a comparison of the measure an simulate torque of the PSM. It can be seen that a rather goo agreement between measurement an reuce moel is given, which is remarkable, since the moel has only been parameterize by means of geometrical an nominal material parameters. The reuce moel, however, is not accurate enough for a high precision control strategy. Therefore, the next section is concerne with the calibration of certain moel parameters to further improve the moel accuracy. torque τ (mn m) torque τ (N m) magnetomotive force us (A m) (a) (b) complete reuce angle ϕ ( ) (c) Fig. 7. Comparison of the complete with the reuce moel (a) for zero currents an (b)-(c) for i c = i c =.5 A, i c1 = A. PSM torque sensor angle sensor inertia Fig. 8. Test bench for the PSM. harmonic rive Copyright (c) 14 IEEE. Personal use is permitte. For any other purposes, permission must be obtaine from the IEEE by ing pubs-permissions@ieee.org. with internal magnets using magnetic equivalent circuits, IEEE Transactions on Magnetics, vol. 5, no. 6, 14. oi: 1.119/TMAG

11 The final version of recor is available at IEEE TRANSACTIONS ON MAGNETICS, VOL.??, NO.??, DECEMBER 1 1 torque τ (mn m) torque τ (N m) torque τ (N m) (a) (b) measurement simulation angle ϕ ( ) (c) Fig. 9. Comparison of the reuce moel with measurements (a) for zero currents, (b) for i c = i c =.5 A, i c1 = A an (c) for i c = i c = 7.5 A, i c1 = A. IV. MODEL CALIBRATION The main reason for the moel errors are the inaccuracies in the air gap permeances G ajk. Thus, the following strategy is introuce for the ientification of the air gap permeances: (i) The torque is measure for fixe currents i cs1 =, i cs = i cs = i cs an a fixe step size in the angle ϕ, resulting in a measurement vector τm k, k = 1,..., N ϕ with the corresponing angles ϕ k = k ϕ, the number of measurements N ϕ an ϕn ϕ = π/. (ii) It is assume that G a = G a,nom + G a, with the nominal value G a,nom an the corrective term G a to be ientifie. Of course, the corrective term has to meet the symmetry conition (8), G alm = G a (ϕ (l 1)π/6 (m 1)π/4), l = 1,, an m = 1,. For fixe angles ϕ k the corresponing values are given by G γlm alm, where the inex γ lm is efine as γ lm = mo ( k (l 1) N ) ϕ (m 1)N ϕ 1, N ϕ + 1. (6) (iii) For each angle ϕ k, the relation ( Gt + D g G c D T ) ( g u k tg = D g G c D T c u tc + D T m u ) tm (64) with G t (ϕ k, u k tg) an G c (ϕ k, u k tg) has to be fulfille, see (18). (iv) The erivation G a / ϕ, neee for the calculation of the torque, see (), is approximate by G a ϕ ϕ k + ϕ = Gk+1 a G k a. (65) ϕ The corresponing magnetomotive forces of the air gap also have to be evaluate at ϕ k + ϕ/. Since they are calculate from (64) at the angles ϕ k, these values are obtaine by averaging the magnetomotive forces at the angles ϕ k an ϕ k+1, i.e. u alm ϕk + ϕ = uk+1 alm + uk alm, (66) with l = 1,, an m = 1,. With these prerequisites, N ϕ torque equations in the form τ k = τm k an 7N ϕ nonlinear equations efine by (64), are given. The N ϕ unknown values of G k a an the 7N ϕ unknown vectors u k tg are given as the solution of this set of equations. This solution is foun numerically using e.g. MATLAB an results in the esire corrective term G k a as a function of ϕ. permeance Ga (µv s/a m) nominal ientifie angle ϕ ( ) Fig. 1. Comparison of the ientifie an the nominal air gap permeance G a (ϕ). Fig. 1 shows a comparison of the nominal air gap permeance G a (ϕ) moel aopte from [] with the ientifie values, where measurements with fixe currents i cs1 =, i cs = i cs = i cs = 5 A were use for the ientification. It can be seen that the basic shape is equal to the nominal characteristics, only the maximum value is reuce an the transition phase is slightly change. The ientifie shape seems to be reasonable since the changes might account for unmoele leakages. Copyright (c) 14 IEEE. Personal use is permitte. For any other purposes, permission must be obtaine from the IEEE by ing pubs-permissions@ieee.org. with internal magnets using magnetic equivalent circuits, IEEE Transactions on Magnetics, vol. 5, no. 6, 14. oi: 1.119/TMAG

12 The final version of recor is available at IEEE TRANSACTIONS ON MAGNETICS, VOL.??, NO.??, DECEMBER 1 11 torque τ (mn m) torque τ (N m) torque τ (N m) (a) (b) measurement simulation angle ϕ ( ) (c) Fig. 11. Comparison of the torque of the calibrate moel with measurements (a) for zero currents, (b) for i c = i c =.5 A, i c1 = A an (c) for i c = i c = 7.5 A, i c1 = A. In Fig. 11, the torque of the calibrate moel is compare with measurement results. These results show a significant improvement to the uncalibrate moel in Fig. 9 an a very goo agreement in the complete operating range of the motor. Thus, it can be euce that both the inhomogeneous air gap as well as saturation in the motor are aequately represente by the propose moel. It is worth noting that an even better agreement between measurement an moel coul be achieve for the cogging torque if the calibration woul have been performe at a lower current, e.g. i cs =.5 A. Then, however, the results for high currents woul be worse such that the presente results are a goo compromise between the accuracy for low an high currents. The comparison of the inuce voltages v csj, j = 1,, for a fixe angular velocity of 1 ra/s given in Fig. 1 further confirm the high accuracy of the propose moel. voltage vcs (V) v cs measurement simulation v cs angle ϕ ( ) v cs Fig. 1. Comparison of the inuce voltages of the calibrate moel with measurements for ω = 1 ra/s. In conclusion, it was shown that a calibrate permeance moel in form of a state-space representation with minimum number of states is suitable for the accurate escription of the behavior of the motor in the complete operating range. In the subsequent section, a classical q-moel of the motor, as it is typically employe in the controller esign of PSM, will be erive. To o so, first a magnetically linear moel is extracte from the nonlinear reuce moel. It will be shown that the simplifications associate with the magnetically linear an especially with the q-moel result in rather large eviations from the measurement results. This also implies that a controller esign base on q-moels is not able to exploit the full performance of moel base nonlinear control strategies. V. SIMPLIFIED MODELS A. Magnetically linear moel If it is assume that the relativ permeability µ r of all permeances is constant, then a magnetically linear permeance moel is obtaine. Starting from (18) (of course using the incience matrix D an the tree an co-tree magnetomotive forces an fluxes of the reuce moel of Section III), the magnetomotive forces of the tree permeances can be calculate in the form = ( G t + D g G c D T ) 1 ( g Dg G c D T c u tc + D T m u tm) (67) an the coil fluxes φ tc rea as, see (17) φ tc = D c [G c G c D T ( g Gt + D g G c D T ) 1 ] g Dg G c ( D T c u tc + D T ) mu tm. (68) Thus, the coil fluxes are given in the form of a superposition of the flux ue to the coil currents u tc = N c i c an the permanent Copyright (c) 14 IEEE. Personal use is permitte. For any other purposes, permission must be obtaine from the IEEE by ing pubs-permissions@ieee.org. with internal magnets using magnetic equivalent circuits, IEEE Transactions on Magnetics, vol. 5, no. 6, 14. oi: 1.119/TMAG

13 The final version of recor is available at IEEE TRANSACTIONS ON MAGNETICS, VOL.??, NO.??, DECEMBER 1 1 magnets u tm. Inserting (68) into the voltage equation (1), we get for the left-han sie N c t φ φ tc φ tc tc = N c ω + N c ϕ i }{{ c t i c. (69) } J L The (symmetric) inuctance matrix L can be formulate as [ L = Nc D c G c G c D T ( g Gt + D g G c D T ) 1 ] g Dg G c D T c (7) an the vector J reas as with an T J = J = N c D c T J ( D T c u tc + D T mu tm ) [ Gc ϕ G c ϕ DT g H 5 D g G c G c D T G c g H 5 D g ϕ ) ] H 5 D g G c + G c D T g H 5 ( Gt ϕ + D g G c ϕ DT g (71) (7) H 5 = ( G t + D g G c D T ) 1 g. (7) Given these results, the voltage equation (1) can be formulate in the well-known form L (ϕ) t i c = J (ϕ) ω + R c i c v c. (74) Note that the inuctance matrix L an the vector J are both nonlinear functions of the rotor angle ϕ. Accoring to (), the torque τ of the motor in the magnetic linear case is given by τ = 1 put tc D ct J D T c u tc + 1 put tm D mt J D T m u tm τ r τ c + pu T tm D mt J D T c u tc. τ p (75) Here, three ifferent parts can be istinguishe: (i) For zero coil currents, i.e. u tc =, the remaining part τ c represents the cogging torque of the motor. (ii) Excluing the permanent magnets of the motor, i.e. setting u tm =, only the reluctance torque τ r ue to the inhomogeneous air gap is present. (iii) The part τ p represents the main part of the torque. It is the only part which can be foun in an ieal PSM with a homogenous air gap. B. Funamental wave moel The magnetically linear moel of the previous section still covers the complete nonlinearity ue to the air gap permeances. In this subsection, a further simplification is mae, where only the average values an the funamental wave components of the corresponing parts are consiere. Applying this approach to (7), the inuctance matrix is given by L = L m 1 L m 1 L m 1 L m L m 1 L m, (76) 1 L m 1 L m L m with the constant main inuctance L m. The term J reuces to sin (pϕ) J (ϕ) = Ĵ sin ( pϕ π) sin ( (77) pϕ 4 π) an the torque can be formulate as τ = pu T tmm cm (ϕ) u tc, where M cm (ϕ) reas as [ ( ) sin (pϕ) sin pϕ π ˆM sin ( ) pϕ 4π ] sin (pϕ) sin ( ) pϕ π sin ( ) pϕ 4π. (78) The coefficients L m, Ĵ an ˆM can be obtaine e.g. by a fourier analysis of the corresponing entries of the magnetically linear moel. Fig. 1 shows a comparison of the entries of the inuctance matrix L, the vector J an the matrix M cm between the magnetically linear moel an the funamental wave moel. It can be seen that a rather goo approximation of the magnetically linear moel can be obtaine by means of the funamental wave moel. The well-known q-representation of the funamental wave moel can be foun by using the transformations i = K (ϕ) i c1, v = K (ϕ) v c1 (79) i q i i c i c v q v with the transformation matrix K (ϕ), K (ϕ) = cos (pϕ) cos ( ) pϕ π sin (pϕ) sin ( ) pϕ π 1 1 v c v c cos ( ) pϕ 4π sin ( ) pϕ 4π. Then, the q-moel takes the form t i = 1 ( ) L L mpωi q + R c i v m t i q = ( 1 L mpωi ) Ĵω + R ci q v q L m an the torque is given by C. Comparison of the moels 1 (8) (81a) (81b) τ = p ˆMu ms1 N c i q. (8) Up to now three moels of ifferent complexity, i.e. a magnetically nonlinear moel, a magnetically linear an a funamental wave moel, were presente in this paper. In this section, the torque τ calculate by these moels is compare with measurement results, see Fig. 14. The results for zero current (Fig. 14(a)) show that the cogging torque can be reprouce rather well by the nonlinear moel. Even the magnetically linear moel shows the basic behavior of the cogging torque, however, with larger errors compare to the nonlinear moel. As a matter of fact, it is not possible to reprouce the cogging torque with the funamental wave moel. Thus, this moel gives the worst results as it was, of course, expecte. For nominal an high currents epicte in Fig. 14(b) an (c), respectively, this result is confirme. Again the nonlinear moel gives excellent agreement with the measurements while Copyright (c) 14 IEEE. Personal use is permitte. For any other purposes, permission must be obtaine from the IEEE by ing pubs-permissions@ieee.org. with internal magnets using magnetic equivalent circuits, IEEE Transactions on Magnetics, vol. 5, no. 6, 14. oi: 1.119/TMAG

14 The final version of recor is available at IEEE TRANSACTIONS ON MAGNETICS, VOL.??, NO.??, DECEMBER 1 1 inuctance L11 (mh) J1 (V s) Mcm,11 (mn m/a ) (a) (b) linear funamental angle ϕ ( ) (c) torque τ (mn m) torque τ (N m) torque τ (N m) (a) (b) measure nonlinear linear fun. wave angle ϕ ( ) (c) Fig. 1. (a) Entry L 11 of the inuctance matrix L, (b) J 1 of the vector J an (c) M cm,11 of the matrix M cm for the magnetically linear an the funamental wave moel. Fig. 14. Comparison of the measurement results with the nonlinear, the magnetically linear an the funamental wave moel for (a) for zero currents, (b) i c = i c =.5 A, i c1 = A an (c) for i c = i c = 7.5 A, i c1 = A. the performance of the magnetically linear moel egraes with increasing currents. This results from the fact that saturation is not inclue in the magnetically linear moel. The basic shape is, however, much better reprouce than in the funamental wave moel. This brief comparison shows that a controller esigne using a funamental wave moel cannot systematically account for the cogging torque an saturation. Using instea the nonlinear moel for a controller esign it can be expecte that the control performance is superior to controllers base on funamental wave moels. The obvious rawback of the nonlinear moel is the increase complexity of the resulting control strategy. Here, the magnetically linear moel might be a goo compromise between moel complexity an moel accuracy for the controller esign an will yiel significant improvements in comparison to funamental wave moels. VI. CONCLUSION A systematical moeling framework for PSM with internal magnets was outline in this paper. Different to existing works, the balance equations were erive base on graph-theory, which allows for a systematic calculation of the minimum number of nonlinear equations. Further, the choice of a suitable state an the systematic consieration of the electrical connection of the coils were iscusse. The quality of the calibrate Copyright (c) 14 IEEE. Personal use is permitte. For any other purposes, permission must be obtaine from the IEEE by ing pubs-permissions@ieee.org. with internal magnets using magnetic equivalent circuits, IEEE Transactions on Magnetics, vol. 5, no. 6, 14. oi: 1.119/TMAG

15 The final version of recor is available at IEEE TRANSACTIONS ON MAGNETICS, VOL.??, NO.??, DECEMBER 1 14 moel was shown by a comparison with measurement results. Finally, a magnetically linear an a q-moel have been erive an compare with the nonlinear moel. Future work will eal with the application of the methoology to other motor esigns as, e.g. PSM with surface magnets, reluctance machines or asynchronous machines. Moreover, the use of the moels erive in this work for nonlinear an optimal controller esign is an ongoing topic of research. Here, first results show a high potential of the moeling approach an a significant improvement in comparison to control strategies using classical q-moels. APPENDIX A EXISTENCE AND UNIQUENESS OF SOLUTION The set of nonlinear equations in (18) has to be solve numerically. Thus, it is interesting to examine if a solution exists an if it is unique. The matrices G t an G c are positive semi-efinite matrices for all an ϕ. This can be easily seen since the entries of these iagonal matrices are positive except for the air gap permeances, which can become zero for certain angles ϕ. Moreover, a suitable construction of the permeance network ensures that D g has inepenent rows such that D g G c D T g is also positive semi-efinite. To show that the sum of this term with G t is even positive efinite, consier the vector x which fulfills x T G t x =. (8) The only possible solution x of (8) is equal to x = [,..., α] T, α R. It is then rather simple to show that x T D g G c D T g x >,, ϕ, (84) which implies that F = G t + D g G c D T g is positive efinite. In the magnetic linear case, the permeances are inepenent of the magnetomotive force an therefore, the positive efiniteness of F is sufficient for the existence an uniqueness of a solution of (18). In the nonlinear case, however, it has to be shown the Jacobian of F(, ϕ) is positive efinite, see, e.g., [9]. The Jacobian can be written in the form F + n j=1 F,j, (85) where,j escribes the j-th entry of. Using the fact that µ r (H) + H µ r(h) H > (86) hols, it can be shown that the Jacobian (85) inee is positive efinite for all an ϕ. It can be further shown that lim F () = (87) hols, which implies that there exists a unique solution of the set of nonlinear equations (18), see [9]. REFERENCES [1] S.-C. Yang, T. Suzuki, R. D. Lorenz, an T. M. Jahns, Surfacepermanent-magnet synchronous machine esign for saliency-tracking self-sensing position estimation at zero an low spees, IEEE Transactions on Inustry Applications, vol. 47, pp , 11. [] V. Ostovic, Dynamics of Saturate Electric Machines. Springer, [] J. R. Henershot an T. J. E. Miller, Design of Brushless Permanent- Magnet Machines. Motor Design Books LLC, 1. [4] D. Hanselman, Brushless Permanent Magnet Motor Design. Magna Physics Publishing, 6. [5] G. Y. Sizov, D. M. Ionel, an N. A. O. Demerash, Multi-objective optimization of pm ac machines using computationally efficient - fea an ifferential evolution, in Proceeings of the International Electric Machines an Drives Conference, 11. [6] Y. Perriar, Reluctance motor an actuator esign: Finite-element moel versus analytical moel, IEEE Transactions on Magnetics, vol. 4, pp , 4. [7] K. Reichert, A simplifie approach to permanent magnet an reluctance motor characteristics etermination by finite-element methos, International Journal for Computation an Mathematics in Electrical an Electronic Engineering, vol. 5, pp , 6. [8] B. Tomczuk, G. Schröer, an A. Wainok, Finite-element analysis of the magnetic fiel an electromechanical parameters calculation for a slotte permanent-magnet tubular linear motor, IEEE Transactions on Magnetics, vol. 4, pp. 9 6, 7. [9] D. M. Ionel an M. Popescu, Finite-element surrogate moel for electric machines with revolving fiel: Application to ipm motors, IEEE Transactions on Inustry Applications, vol. 46, pp. 44 4, 1. [1] M. A. Jabbar, H. N. Phyu, Z. Liu, an C. Bi, Moeling an numerical simulation of a brushless permanent-magnet c motor in ynamic conitions by time-stepping technique, IEEE Transactions on Inustry Applications, vol. 4, pp , 4. [11] W. Leonhar, Control of Electrical Drives, n e. Springer, [1] P. C. Krause, O. Wasynczuk, an S. D. Suhoff, Analysis of Electric Machinery an Drive Systems. IEEE Press,. [1] C. Geraa, K. J. Braley, M. Sumner, an G. Asher, Non-linear ynamic moelling of vector controlle pm synchronous machines, in Proceeings of the European Conference on Power Electronics an Applications, 5, pp. 1 P.1. [14] A. E. Fitzgeral, C. Kingsley, an S. D. Umans, Electric Machinery. McGraw-Hill,. [15] D. M. Dawson, J. Hu, an T. C. Burg, Nonlinear Control of Electric Machinery. Marcel Dekker, [16] F. Morel, J.-M. Retif, X. Lin-Shi, an C. Valentin, Permanent magnet synchronous machine hybri torque control, IEEE Transactions on Inustrial Electronics, vol. 55, pp , 8. [17] T. Geyer, G. A. Beccuti, G. Papafotiou, an M. Morari, Moel preictive irect torque control of permanent magnet synchronous motors, in Proceeings of the Energy Conversion Congress an Exposition, 1, pp [18] C.-K. Lin, T.-H. Liu, an L.-C. Fu, Aaptive backstepping pi sliingmoe control for interior permanent magnet synchronous motor rive systems, in Proceeings of the American Control Conference, 11, pp [19] V. Ostovic, A novel metho for evaluation of transient states in saturate electric machines, IEEE Transactions on Inustrial Applications, vol. 4, pp. 96 1, [] L. Zhu, S. Z. Jiang, Z. Q. Zhu, an C. C. Chan, Analytical moeling of open-circuit air-gap fiel istributions in multisegment an multilayer interior permanent-magnet machines, IEEE Transactions on Magnetics, vol. 45, pp. 11 1, 9. [1] A. R. Tariq, C. E. Nino-Baron, an E. G. Strangas, Iron an magnet losses an torque calculation of interior permanent magnet synchronous machines using magnetic equivalent circuit, IEEE Transactions on Magnetics, vol. 46, pp , 1. [] J. Tangu, T. Jahns, an A. El-Refaie, Core loss preiction using magnetic circuit moel for fractional-slot concentrate-wining interior permanent magnet machines, in Proceeings of the IEEE Energy Conversion Congress an Exposition, 1, pp [] B. Sheikh-Ghalavan, S. Vaez-Zaeh, an A. H. Isfahani, An improve magnetic equivalent circuit moel for iron-core linear permanent-magnet synchronous motors, IEEE Transactions on Magnetics, vol. 46, pp. 11 1, 1. [4] S. Serri, A. Tani, an G. Serra, A metho for non-linear analysis an calculation of torque an raial forces in permanent magnet multiphase bearingless motors, in Proceeings of the International Symposium on Copyright (c) 14 IEEE. Personal use is permitte. For any other purposes, permission must be obtaine from the IEEE by ing pubs-permissions@ieee.org. with internal magnets using magnetic equivalent circuits, IEEE Transactions on Magnetics, vol. 5, no. 6, 14. oi: 1.119/TMAG

16 The final version of recor is available at IEEE TRANSACTIONS ON MAGNETICS, VOL.??, NO.??, DECEMBER 1 15 Power Electronics, Electrical Drives, Automation an Motion, 1, pp [5] H. Seok-Hee, T. Jahns, an W. Soong, A magnetic circuit moel for an ipm synchronous machine incorporating moving airgap an crosscouple saturation effects, in Proceeings of the IEEE International Electric Machines & Drives Conference, 7, pp [6] T. Raminosoa, J. Farooq, A. Djerir, an A. Miraoui, Reluctance network moelling of surface permanent magnet motor consiering iron nonlinearities, Energy Conversion an Management, vol. 5, pp , 9. [7] T. Raminosoa, I. Rasoanarivo, an F.-M. Sargos, Reluctance network analysis of high power synchronous reluctance motor with saturation an iron losses consierations, in Proceeings of the Power Electronics an Motion Control Conference, 6, pp [8] Y. Kano, T. Kosaka, an N. Matsui, Simple nonlinear magnetic analysis for permanent-magnet motors, IEEE Transactions on Inustry Applications, vol. 41, pp , 5. [9] A. Ibala, A. Masmoui, G. Atkinson, an A. G. Jack, On the moeling of a tfpm by reluctance network incluing the saturation effect with emphasis on the leakage fluxes, International Journal for Computation an Mathematics in Electrical an Electronic Engineering, vol., pp , 11. [] M.-F. Hsieh an Y.-C. Hsu, A generalize magnetic circuit moeling approach for esign of surface permanent-magnet machines, IEEE Transactions on Inustrial Electronics, vol. 59, pp , 1. [1] M. Amrhein an P. T. Krein, - magnetic equivalent circuit framework for moeling electromechanical evices, IEEE Transactions on Energy Conversion, vol. 4, pp , 9. [], Force calculation in - magnetic equivalent circuit networks with a maxwell stress tensor, IEEE Transactions on Magnetics, vol. 4, pp , 9. [] J. D. Law, T. J. Busch, an T. A. Lipo, Magnetic circuit moeling of the fiel regulate reluctance machine part i: Moel evelopment, IEEE Transactions on Energy Conversion, vol. 11, pp , [4] M. L. Bash, J. M. Williams, an S. D. Pekarek, Incorporating motion in mesh-base magnetic equivalent circuits, IEEE Transactions on Energy Conversion, vol. 5, pp. 9 8, 1. [5] M. L. Bash an S. D. Pekarek, Moeling of salient-pole woun-rotor synchronous machines for population-base esign, IEEE Transactions on Energy Conversion, vol. 6, pp. 81 9, 11. [6] N. Christofies, Graph Theory: An Algorithmic Approach. Acaemic Press, [7] L. O. Chui, C. A. Desoer, an E. S. Kuh, Linear an Nonlinear Circuits. McGraw-Hill, [8] A. Kugi, Non-linear Control Base on Physical Moels. Springer, 1. [9] F. Wu an C. Desoer, Global inverse function theorem, IEEE Transactions on Circuit Theory, vol. 19, pp , 197. Wolfgang Kemmetmüller (M 4) receive the Dipl.-Ing. egree in mechatronics from the Johannes Kepler University Linz, Austria, in an his Ph.D. (Dr.-Ing.) egree in control engineering from Saarlan University, Saarbruecken, Germany, in 7. From to 7 he worke as a research assistant at the Chair of System Theory an Automatic Control at Saarlan University. From 7 to 1 he has been a senior researcher at the Automation an Control Institute at Vienna University of Technology, Vienna, Austria an since 1 he is Assistant Professor. His research interests inclue the physics base moeling an the nonlinear control of mechatronic systems with a special focus on electrohyraulic an electromechanical systems where he is involve in several inustrial research projects. Dr. Kemmetmüller is associate eitor of the IFAC journal Mechatronics. Davi Faustner receive the Dipl.-Ing. egree in electrical engineering from the Vienna University of Technology, Austria, in October 11. Since November 11 he works as a research assistant at the Automation an Control Institute at Vienna University of Technology. His research interests inclue the physics base moeling an the nonlinear control of electrical machines. Anreas Kugi (M 94) receive the Dipl.-Ing. egree in electrical engineering from Graz University of Technology, Austria, an the Ph.D. (Dr. techn.) egree in control engineering from Johannes Kepler University (JKU), Linz, Austria, in 199 an 1995, respectively. From 1995 to he worke as an assistant professor an from to as an associate professor at JKU. He receive his Habilitation egree in the fiel of automatic control an control theory from JKU in. In, he was appointe full professor at Saarlan University, Saarbrücken, Germany, where he hel the Chair of System Theory an Automatic Control until May 7. Since June 7 he is a full professor for complex ynamical systems an hea of the Automation an Control Institute at Vienna University of Technology, Austria. His research interests inclue the physics-base moeling an control of (nonlinear) mechatronic systems, ifferential geometric an algebraic methos for nonlinear control, an the controller esign for infinite-imensional systems. He is involve in several inustrial research projects in the fiel of automotive applications, hyraulic an pneumatic servo-rives, smart structures an rolling mill applications. Prof. Kugi is Eitor-in Chief of the IFAC journal Control Engineering Practice an since 1 he is corresponing member of the Austrian Acaemy of Sciences. Copyright (c) 14 IEEE. Personal use is permitte. For any other purposes, permission must be obtaine from the IEEE by ing pubs-permissions@ieee.org. with internal magnets using magnetic equivalent circuits, IEEE Transactions on Magnetics, vol. 5, no. 6, 14. oi: 1.119/TMAG