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# Transformation Groups of the Doubly-Fed Induction Machine

Written By

Giovanni F. Crosta and Goong Chen

Reviewed: January 26th, 2022 Published: April 5th, 2022

DOI: 10.5772/intechopen.102869

From the Edited Volume

## Matrix Theory - Classics and Advances [Working Title]

Dr. Mykhaylo I. Andriychuk

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## Abstract

Three-phase, doubly-fed induction (DFI) machines are key constituents in energy conversion processes. An ideal DFI machine is modeled by inductance matrices that relate electric and magnetic quantities. This work focuses on the algebraic properties of the mutual (rotor-to-stator) inductance matrix Lsr: its kernel, range, and left zero divisors are determined. A formula for the differentiation of Lsr with respect to the rotor angle θr is obtained. Under suitable hypotheses Lsr and its derivative are shown to admit an exponential representation. A recurrent formula for the powers of the corresponding infinitesimal generator A0 is provided. Historically, magnetic decoupling and other requirements led to the Blondel-Park transformation which, by mapping electric quantities to a suitable reference frame, simplifies the DGI machine equations. Herewith the transformation in exponential form is axiomatically derived and the infinitesimal generator is related to A0. Accordingly, a formula for the product of matrices is derived which simplifies the proof of the Electric Torque Theorem. The latter is framed in a Legendre transform context. Finally, a simple, “realistic” machine model is outlined, where the three-fold rotor symmetry is broken: a few properties of the resulting mutual inductance matrix are derived.

### Keywords

• mutual inductance matrix
• Blondel-Park transformation
• exponential representation
• infinitesimal generator
• zero divisors
• circulants
• broken symmetry

## 1. Introduction

Three-phase, doubly-fed induction (DFI) machines have a long history [1, 2, 3, 4] and continue to be key constituents in energy conversion processes [5, 6]. Motivation for modeling a DFIgenerator comes from the need to deal with intermittency in the primary energy supply (e.g., the wind field) and with uncertainty in the load (i.e., the grid). Similarly, the modeling and control of a DFImotor can improve the efficiency and reliability of electric-to-mechanical work conversion. The equations modeling the ideal DFImachine have been studied for more than a century. Results in modeling and control [7, 8, 9, 10, 11, 12, 13], including those which derive from numerical simulation, demonstrate how attention to the DFImachine is being continuously paid. In essence, the ideal three-phase machine model centers on two matrices, on which this work focuses: the rotor-to-stator mutual inductance matrix, Lsr., which depends on the “rotor angle” θrand characterizes the machine itself, and the Blondel-Park transformation matrix, K., which depends on another angle and describes a change of variables, from the abcreference frame (Section 2) to the dq0reference frame (Section 4). Both matrices, Lsr.and K., appear in the Electric Torque Theorem (ETT) which relates mechanical to electrical variables and as such represents the raison d’êtreof the DFImachine. Stated in the abcframe, the ETTis a straightforward application of energy balance, once a Legendre transformation (Section 5.2) has been introduced and co-energy accordingly defined. Instead, the proof, in fact the translation of the ETTin the dq0frame (Section 5.3), requires all relevant properties of Lsr.and K.to be known. For this reason, in Section 3 the kernel, the range (Proposition 1), the classical adjoint and the left zero divisors (Proposition 3) of Lsr.are determined. Derivation benefits from Lsr.being a circulant matrix [14] (Lemma 1) and from its eigenvalues representing the discrete Fourier transform of a 3-sequence [15] (Proposition 2). Special attention need two constant matrices, A0and its square (Lemma 2), because they relate differentiation of Lsrwith respect to θrto multiplication (Theorem 1). In a suitable subspace of R3, Lsradmits an exponential representation (Theorem 2) with A0as infinitesimal generator. Section 4 is devoted to K.: its structure, as well as its exponential representation with generator A0, is inferred by satisfying, in sequence, a list of requirements (Proposition 4 and Theorem 4). The key formula for the product of matrices (Theorem 5) is then applied to prove the ETTin the dq0frame in one step (Theorem 6). An attempt is finally made in Section 6 to deal with a “realistic” machine model, where the three-fold rotor symmetry is broken: the band crotor axes are misaligned by angles ϵband ϵc. To second-order in ϵband ϵcthere exists a constant Bwhich relates differentiation to multiplication of the approximate inductance matrix (Proposition 7).

## 2. The ideal doubly-fed induction machine

Definition 1. (Three-phase, ideal DFI machine) [3, 4]. A three-phase DFImachine is said ideal whenever its stator and rotor windings exhibit three-fold symmetry. Moreover, magnetomotive forces and flux waves created by the windings are sinusoidally distributed and windings give rise to a linear electric network.

Remark 1. (Neglected phenomena). Higher harmonics, hysteresis, and eddy currents are excluded by the model. Deviations from three-fold symmetry will be addressed in Section 6.

Notation. (abcframes). The most natural frames where three-phase stator and rotor voltages and currents can be represented are the abcframes. For example, the stator currents are an ordered triple which one agrees to represent as a vector

jabcs=jasjbsjcsTrsR3,E1

whose components are functions of time tT. A similar notation will hold for other electric quantities. Unless otherwise specified, all vectors are understood in R3.

Hp. (Function class). Dependence of all quantities of interest on time is assumed as smooth as required.

Definition 2. (Balanced triple). An abccurrent triple is balanced or is a trivial zero sequence, whenever

ja+jb+jc=0,tT.E2

Such sequences define the subspace BR3, a plane through the origin; the corresponding notation is jBB.

Notation.

θrt02πis the electric rotor angle at a time t, formed by the rotor araxis with respect to the stator asaxis.

βrt02πis the electric rotor angle at a time t, formed by daxis with respect to the rotor araxis (Section 4).

βst02πis the electric rotor angle at a time t, formed by daxis with respect to the stator asaxis (Section 4).

jabcrNrNsjabcris the stator-referred (ständer-bezogen) vector of rotor currents, where Nrand Nsare the rotor and stator turns.

A boldface, roman capital denotes a matrix in MNrowNcolof Nrow2rows ×Ncol2columns.

am,nis the cofactor of entry mnin AMNN, where Nrow=Ncol=N2.

am,nis the corresponding matrix.

adjAis the matrix adjoint to AMNN, obtained by transposing am,n.

u)(vis the dyadic product of the column vector uby the row vector v, both R3.

13is the 3×3identity matrix.

The end of a Proofis marked by , that of a general statement or of a Remark by ◊.

The electrical network equations which describe the dynamics of the three-phase, ideal DFImachine are well-known [3, 4], as a consequence they are omitted.

## 3. Group properties of the mutual inductance matrix

By letting φ23π, the rotor-referred (läufer-bezogene) form of the mutual inductance matrix is [3, 4]

Lsrθr=cosθrcosθr+φcosθrφcosθrφcosθrcosθr+φcosθr+φcosθrφcosθrE3

Given Lsrθrone defines the stator-referred (ständer-bezogen) rotor-to-stator mutual inductance matrix

LsrθrNrNsLmsLsrθr,E4

where Lms>0is a constant parameter. Hereinafter, the dependence of the involved matrices and of related quantities on θrwill be shown only if mandatory. The following properties hold because rows two and three of Lsrare right shift-circular permutations of the first row and because all row-wise (and column-wise) sums of Lrsvanish.

Proposition 1. (Eigenvalues ofLsrand their implications).

• The eigenvalues of Lsrare μ0=0, μ±=32e±iθr.

• detLsr=0, θr,

• dimKerLsr=1and KerLsr=jR3j1=j2=j3Ksr, θr.

• dimrangeLsr=2and rangeLsr=B, θr,

Remark 2. (Orthogonal decomposition ofR3). The last two properties in the list translate the orthogonal decomposition

R3=KerLsrrangeLsr=KsrBj=jKsr+jB,E5

where the straight line Ksr:j1=j2=j3is the normal to the plane B:j1+j2+j3=0of Eq. (2).

Lemma 1 (Eigenvalues of a permutation matrix, pp. 65–66 of M. Marcus and H. Minc’s textbook [14]). For a general N2and for an N×Nmatrix P, a.k.a. “circulant”, which results from the right shift-circular permutation of the first row c0cN1cN2c1, one denotes ϵei2π/Nand introduces the polynomial ψ.of degree N1in the complex variable ζ

ψζn=0N1cnζn.E6

The possibly multiple eigenvalues μk0kN1of Pare obtained by letting ζ=ϵmand evaluating ψϵmfor m=1,2,,N. Since ϵmei2πm/N, there exists only one value of m, denoted by , at which all powers of ϵappearing in ψ.are equal: ϵ=ϵ2=ϵ3==ϵN1=1. Such value is =N. Therefore

ψϵN=n=1N1cn=ψϵ0when=N.E7

If the additional property

n=1N1cn=0E8

is exhibited by the rows of P, then

ψϵN=0=μ0.E9

Such eigenvalue is algebraically (and geometrically) simple.

Remark 3. (Features ofψ., kandn). The polynomial ψ.shall not be confused with any of the polynomials annihilated by P. There is no correspondence between the eigenvalue label kand the ordering of powers induced by m.

Proofof Proposition 1. Since Lsrθris a circulant matrix of the sequence c0c2c1

Lsrθ=c0c2c1c1c0c2c2c1c0E10

and

c0+c1+c2=0,θr02π,E11

then Lemma 1 applies with N=3. Hence =3. In the first place, μ0is independent of θr. Next, one verifies the other two eigenvalues, μ±=32e±iθr, are respectively returned by ψϵand by ψ[ϵ2] and do instead depend on θr. The property dimKerLsr=1derives from the algebraic simplicity of μ0=0. From Eq. (11) one deduces KerLsr=Ksr. The properties of rangeLsrare not independent of those of KerLsr: namely, they follow from orthogonality, as highlighted by Eq. (5).

Proposition 2. (Eigenvalues of a circulant as the discreteFourier transform of a 3-sequence [15]). Let ϵbe as in Lemma 1. The discrete Fourier transform b3b0b1b2of a 3-sequence c3c0c1c2is obtained, in terms of row vectors, by

b0b1b2=c0c1c21111ϵϵ21ϵ2ϵc0c1c2T.E12

The circulant Pc3assembled from c3is diagonalized by Taccording to

Pc3=T1b0000b1000b2T.E13

Application to Lsrof Eq. (10) requires a permutation of the first row:

μ0μ1μ2=c0c2c1100001010T.E14

Lemma 2 (The matrixA0and its properties). Let the matrix A0be defined by

A013011101110.E15

Its properties are the following.

• (Determinant, rank, eigenvalues, eigenspaces).

detA0=0;rankA0=2,E16
λ0=0,λ1=i,λ2=+i.E17

There is an eigenspace of dimension one, X0A0, which corresponds to λ0:

X0A0=KerA0=KerLsr=Ksr.E18

• (Left zero divisors). Theconstant, nontrivial left zero divisors of A0are given by dyads

Zc=c)(1E19

where ck, k=1,2,3, are real constants with ck0for at least one k.

• (Dyadic representation). The matrix A0admits no“algebraic” dyadic representation of the form

A0=a)(bfalseE20

with constant ak, bkR, k=1,2,3.

• (Sign reversal). There exists no left zero divisors of A0which, added to A0, reverses its sign.

• (Recurrent formula for powers ofA0). Given A0and

A02=13+131)(1,E21

the powers of A0are obtained from

A0n=11+n3%4/2A02n2%2,n3,E22

where n3%4stands for the remainder from integer division of n3by 4 and the “/” (slash) denotes division between integers; similarly, n2%2stands for the remainder from integer division of n2by 2.

Proofof Lemma 2. The properties described by Eqs. (16)-(19) are immediately verified, as well as the nonexistence of an algebraic dyadic representation. The statement about sign reversal is proved by contradiction. To obtain the recurrent formula for powers of A0one computes A03(=A0) and A04(=A02), then one examines the higher powers A0and the sequence formed by their signs. Since the sequence has period 4 and reads ++++, then the exponents of both A0and 1in Eq. (22) can be determined.

Remark 4, to Lemma 2.

• Let a)in Eq. (20) be replaced by ), then there exists a C1, divergence-free vector field fgiving rise to the “differential” dyadic representation of A0

A0=13)(f.E23

The system of first-order linear partial differential equations to which fis the solution is obtained by comparing like terms in the arrays.

• As already noticed in the proof, A0cannot be a left zero divisor of itself. (In fact, A02is given by Eq. (21)).

• Obviously, there is no way of including n=0in any recurrent formula for the powers of A0, of which Eq. (22) is an example because detA0=0.

• As one can easily verify, the eigenvalues of A02are λ0=0, λ1=1. The latter has algebraic multiplicity α1=2and geometric multiplicity γ1=1. As a consequence, its eigenspace, X1A02, not only has a dimension α1γ1+1=2but complies with

X1A02=BE24

as well. In other words, by recalling Eqs. (5), (18), and (21),

A02=13BorA02ψ=ψB,ψR3E25

i.e., A02coincides with 13restricted to the subspace B. ◊

Proposition 3. (The matrixLsr.: trigonometric decomposition and classical adjoint; the left zero divisors ofLsr., their kernel and range).

• (The matrices C and S). Lsrθris a linear combination of trigonometric functions according to

Lsrθr=Ccosθr+Ssinθr,E26

where Cand Sare the constant, 3×3matrices

C=3213121)(1;S=01110111032.E27

• (Relations betweenA0, CandS).

C=32A02,S=32A0.E28

• (The classical adjoint matrix). The classical adjoint to Lsrθr(transpose of the cofactor matrix) is

• (Left divisors as dyads). If fk., k=1,2,3, denote real-valued functions of class CM02π(for some M), one of which, at least, does not vanish identically, then a left zero divisor Z(linker Nullteiler) of Lsris a rank one dyad

Zfθ=fθ)(1E30

forming an algebra Z.

• (Kernel of theZ’s).

KerZ=B,Z.E31

Proofof Proposition 3. The identification of Cand Sfollows from expanding the cos.±23πentries in Lsr. In order to determine adjLsrone starts from a relation which is one of the many formulas due to Laplace [16]

and holds for a general AMN×N; then one recalls detLsr=0and the θr-invariance of μ0of Eq. (9): one can thus compute the classical adjoint to either constant matrix, Cor S, whichever is simpler to deal with; the result is the dyad on the right side of Eq. (29), a result which holds θr. The search for left zero divisors of Lsras dyads like that of Eq. (30) is suggested by Eqs. (29) and (32) because adjLsrmust be a zero divisor of Lsr. The most general form of a left zero divisor Zis inferred from Eq. (11): since all column-wise sums of Lsrvanish, the columns of Zmust be equal. Therefore such divisor, if non-trivial, has rank one and is obtained from the dyadic product of Eq. (30). Finally, Eq. (32) and the orthogonal decomposition Eq. (5) imply

Remark 5. (Duality; divisors; other properties ofLsr).

• From Eqs. (3234) one says Lsrand adjLsraredualto each other.

• Theconstant, nontrivial left zero divisors of Lsrare those of Eq. (19). By consistency, the matrices Cand Sof Eq. (26) not only have the same classical adjoint as Lsrθrhas but have all and the same zero divisors, because Eq. (26) holds θr.

• Right zero divisors are obtained by transposing the left ones.

• Nonexistence of the representation of Eq. (20) prevents A0it from being a left zero divisor of Lsr.. Nor can A0be, as Eqs. (28) and (22) show, a divisor of either Cor Staken separately.

• If Lsrstands for the subspace of functions f.C002π3complying with Lsrθrfθrdθr=0, and if amk, m=0,1,2,, bmk, m=1,2,, are the cosine and, respectively, the sine Fourier coefficients of the (real-valued) components fk., k=1,2,3of f., then

fLsra11=a12=a13andb11=b12=b13.E35

Remark 6. (The physical meaning ofLsr, CandS). As Eq. (5) suggests, given any instantaneous current vector jtR3, left multiplication by Lsrθrtreturns a balanced current triple jBtB. This follows from the three-fold symmetry of the ideal DFImachine, mirrored by the structure of Lsr.. Moreover, the Cterm of Eq. (26) represents the opposite of reactive torque, whereas the Sterm represents active torque.◊

Theorem 1. (Differentiation ofLsrwith respect toθr). Let A0and Zcbe respectively given by Eqs. (15) and (19). Then the derivative of Lsrwith respect to θris the set-valued map

A0+ZcLsrθrLsrθrθr.E36

Notation: a convenient notation is LsrθrθrMθr.

Proofof Theorem 1. Differentiation of Lsrθras represented by Eq. (26), and the use of Eq. (28) yield

Lsrθrθr=32A0cosθr+32A02sinθr.E37

One seeks for a constant matrix B1which complies with

32A0cosθr+32A02sinθr=B1Ccosθr+B1Ssinθr.E38

The application of Eq. (22) leads to

B1=A0.E39

Then, the whole set of constant matrices Bcomplying with LsrθrBLsris obtained by adding to B1a left zero divisor Zcas of Eq. (19)

B=A0+Zc.E40

The result justifies the notation for Lsrθrof Eq. (36) as a set-valued map.

The above Eq. (36) means Lsrθrθrj=A0+ZcLsrθrj, jR3. In spite of this last relation, the search for an exponential representation of Lsr.and for a one-parameter (θr) group acting on the whole of R3is ill-posed. Namely, Lsr0is not invertible, hence one cannot normalize Lsrby Lsr0and no unit element of the group can be defined. To a greater extent, the search for a generator for the group would make no sense. Nonetheless, an exponential representation is obtained in the subspace B.

Theorem 2. (Exponential representations onB). If jBthen the following hold.

Lsrθrj=32eθrA0j,jBE41

and

Mθrj=M0eθrA0j,jBE42

with

M0=32A0.E43

Proofof Theorem 2. A matrix Jθris sought for, which, like Lsrθr, splits into a cos.and a sin.term as in Eq. (26) and, unlike Lsr., satisfies J0=13. As Eqs. (28) suggest, one solution is

Jθr=13cosθr+23Ssinθr=23C+121)(1cosθr+23Ssinθr.E44

Next, one requires θr-differentiation to coincide with the multiplication of J.by a constant matrix H

Jθrθr=HJθr.E45

By identifying terms like trigonometric functions one obtains the pair

H=23S23HS=13i.e.H2j=13jjB.E46

One solution, moduloleft zero divisors, comes from the properties of A02in Eq. (25):

H=A0.E47

Hence Jθr=eθrA0. The proposed representation of Lsr.is

Lsrθr=32eθrA0121)(1cosθr.E48

Consistency with Lsrθr.implies

Lsrθrθr=32A0Jθr+121)(1sinθr.E49

Since jB, then the rightmost dyads in Eqs. (48) and (49) return 0when right multiplied by j. Replacing A0in Eq. (47) by the Bof Eq. (40) does not change the results (Eqs. (48) and (49)) because

eθrZcj=0,jB,Zc.E50

Theorem 3. (Symmetry properties).

LsrTrsθr=LsrθrE51
MTrsθr=MθrE52
LsrA0=0.E53

Proofof Theorem 3. The first two relations are immediate. The third one follows from the representation of Lsras linear combination of powers of A0according to Eqs. (26) and (28).

## 4. The Blondel-Park transformation and the rotation group

### 4.1 Axiomatics of the transformation

The Kirchhoff voltage and current laws bring redundancy into the abcframe representations. In order to remove said redundancy, another frame, called dq0, is introduced, where only two components of a vector shall matter, the direct one, d, and the quadrature component, q.

Definition 3. (Thedq0frame). Let dq0denote a reference frame for electric quantities of axes dand q, subject to five specifications.

d.1)The new frame shall be suitable to represent both stator-referenced and rotor-referenced quantities.

d.2)The component of a stator-referenced quantity with respect to both the director daxis and the stator asaxis shall be represented by the same function of angle, evaluated at arguments which differ by βs. Similarly for variables pertaining to the rotor ar: the phase difference shall be βr.

d.3)The above angles are related by

βs=βr+θr.E54

d.4)The quadratureor qaxis shall be orthogonal to din the L202πsense: if wd.and wq.are the d- and q-components of a (generally complex-valued) signal w.which depends on η, then: 02πwdηwqηdη=0.

d.5)The third entry w0.of a vector w.in the dq0frame shall be equal to the sum of its abccomponents. (For this reason, such a sum is called “zero sequence”, or Nullfolge, and may be trivial or not).

Problem 1. (Theabctodq0transformation problem[3, 4, 6]). Find a transformation K.that maps a vector wabc(a physical quantity) from the abcsand, respectively, the abcrframes to a vector wdq0in the dq0frame, as specified by Definition 3 and

K.1)is invertible and linear;

K.2)conserves instantaneous electric power;

K.3)has the same functional form for both stator and rotor quantities,

K.4)depends at most on one real parameter, an “electric angle”, which may be different for stator or rotor quantities;

K.5)is of class C1at least with respect to that parameter;

K.6)magnetically decouples flux linkages [6].

Proposition 4. (Matrix representation). A solution to Problem 1 which applies to a three-phase machine exists and is the Blondel-Park [1, 2, 7] transformation K.

Kη23cosηcosηφcosη+φsinηsinηφsinη+φ121212,E55

where ηstands for an electrical angle. One has

wdq0t=KηtwabctE56

and w0t=wa+wb+wct,tand, if wthas a period 2π, wdtwqtdt=0.

Proofof Proposition 4. The structure of K.can be inferred by satisfying, in sequence, requirements K.1, K.2, K.6, d.4, d.5. The Ansatz

Kη=K0eηF,E57

where K0K0and Fis a constant matrix, is shown to be consistent with all requirements, hence the entries of K0and Fcan be identified. No further details can be provided for reasons of space.

Remark 7 to Proposition 4. (On the exponential representation ofKη). As a result of work at proving Proposition 4, Kηdefines a one-parameter (η) group of unitary (power preserving) transformations, represented by Eq. (57). Since K0is invertible, then

RηK01KηE58

and R0is the unit element. The existence of the composition law is implied by the Ansatz. Obviously, detKη=1implies detRη=1,η. .

Theorem 4. (Infinitesimal generator). The infinitesimal generator Fof K.is K0-similar to the opposite of the infinitesimal generator A3of rotations about the x̂3axis of R3according to

F=K01A3K0E59

and is related to the A0of Eq. (41) by

F=A0.E60

Proofof Theorem 4. From Eq. (55)

dKηdη=23sinηsinη2π3sinη+2π3cosηcosη2π3cosη+2π3000E61

and the constant matrix Bsatisfying dKηdη=BKηreads

B=010100000=A3.E62

Next, the infinitesimal generator of R.Eq. (58) is identified according to

dRηdη=K01BKη=K01BK0RηFRη.E63

In other words, the matrix FK01BK0is the sought for infinitesimal generator of the group R.. This proves Eq. (59). The relation between Fand Bis to be expected (e.g., §2.5 of Altmann’s textbook [17]). Finally, Eq. (60) follows from direct verification.

### 4.2 The product of matrices formula

Theorem 5. (The formula). Equations (54), (57) and (42) imply

KβsLsrθrθrKβr1=K0M0K01=32A3.E64

Proofof Theorem 5. The proof branches out according to which current triple is being dealt with.

• (Balanced current tripletrivial zero sequence). Let jabcB, then, by Eqs. (42), (57), and (60) and applying transposition

KβsLsrθrθrKβr1=KβsMθrKβr1=K0eβsA0Mβr+θrK01==K0Mβs+βr+θrK01=K0M0K01=32A3.E65

• (General current triple). For general jabcR3no exponential representation is available. In analogy with Eq. (26) one identifies the constant matrices P,Qand Rgiving rise to Kη=23Pcosη+23Qsinη+13R. TheproductMθrK1βr, after simplification, turns out to be an affine function of cosθr+βrand sinθr+βrwhich in turn depend on anglesums: products of the involved matrices hide anaddition formulafor angles on which trigonometric functions depend. Taking Eq. (54) into account, left multiplication by Kβsleads to a polynomial in cosβsand sinβswith coefficients like PCQTrs, RCPTrsand so forth. All βs-dependent terms in the polynomial disappear. Eventually, the only non-zero term is 23PCQTrs=32A3, a constant.

Remark 8. (Prior results). To the best of the authors’ knowledge, the role of the Blondel-Park transformation in realization theory was pointed out by J.L. Willems [8], who derived the exponential representation of K.while obtaining a time-invariant system from time-varying electric machine equations. The group properties of K.have been known for some time (e.g., [9], p. 1060). Instead, the relation of Fto A0, at least in the form of Eq. (60), the relation between exponential representations of K.and Lsr., and the roles played by the left zero divisors of Lsr.and by the subspace B, seem to have been overlooked so far.◊

## 5. Electric torque

### 5.1 The electric torque law in the abcframe

From the principles of analytical mechanics, the following relation can be deduced [3, 4] for the ideal DFImachine in generator mode. The relation involves previously defined quantities, namely stator and rotor currents and a machine parameter, the Lsr.of Eq. (4), and a quantity, the electric torque Tel,g, which has not yet been mentioned herewith. As a consequence, the relation can be regarded as the physical “law” whichdefinesTel,g.

Definition 4. (Electric torque in theabcframe). Let the ideal DFImachine have Ppoles and be described by current vectors jabcsand jabcr. The electric torque ingeneratormode is defined by

Tel,g=P2jabcsTrsLsrθrθrjabcr.E66

The relevance of Eq. (66) sits in the link it establishes between electric quantities and a mechanical one: in generator mode, it is the torque produced by, usually a working fluid, on the DFImachine shaft which, through a suitably excited rotor, gives rise to electric currents in the stator coils; in motor mode power flow from machine coils to the shaft is reversed. All machine control laws rely on Eq. (66) in order to be implemented.

### 5.2 Co-energy and the Legendre transform

Ansatz. (Internal energy). If Sis entropy and Tis temperature, then the first differential of internal energy Uof an electric machine with one mechanical degree of freedom, θm, at constant volume Vand numbers of moles N, is

dU=TdS+jdλTel,mdθm,E67

where λis the vector of flux linkages, Tel,mis mechanical torque inmotormode (opposite to that in generator mode), and θmis the shaft angle.

A consequence of the Ansatz is the following.

Proposition 5. (Relation between motor torque and internal energy).

Tel,m=UθmS,V,N,λ.E68

Definition 5. (Legendre transform of energy with respect to flux linkage). Let ΛIR3be a subset where Uis at least twice differentiable and convex with respect to λand let pdenote the variable conjugate to λ. Then the Legendre transform Yof energy Uwith respect to λis defined by

YSVNpθmsupλΛpλUSVNλθm.E69

Remark 9. (Conjugate variables; motor torque; differential geometric setting).

• pcoincides with jand one has

Y+U=jλ.E70

• Motor torque can thus be rewritten as

Tel,m=YθmS,V,N,j.E71

• The extensive variables on which energydepends are S, N, λand θm, and as such are coordinates of the dynamical system’s manifold N. Instead, the intensive variables T, μ(vector of chemical potentials), jand Tel,mbelong to the system’s co-tangent bundle TN[11, 13]. Upon a multivariate Legendre transformation, as many extensive variables can be replaced by their conjugates, which are intensive variables.◊

Remark 10. (Yvs.Wfld'). By identifying Uwith “the energy Wfldstored in the coupling fields” of an electric machine having Ppoles, the rotor of which forms the mechanical angle θmin the stator frame, one has the following relations:

• the electrical angle θris related to the mechanical angle θmby θr=P2θm(multiplier effect of P);

• usually [3, 4] Tel,mis related to “co-energy” Wfld'jabcsθrby

Tel,m=W'fldjabcsθrθm=P2W'fldjabcsθrθr.E72

In other words,

Wfld'=YS,V,N,j.E73

Remark 11. (Models of real machines). The relation between Yand torque applies to any machine and can, in principle, deal with any functional dependence between λand j. Nonlinear λjrelations [9, 10, 12] become of interest when saturation of the magnetic circuit has to be modeled. Hysteresis and the related energy losses pose further difficulties. ◊

### 5.3 The electric torque theorem in the dq0frame

Translating Eq. (66) into the dq0relies on relations between K.-transformed current vectors which involve all three angles, βs,βr,θr. Translation is made remarkably simpler by Theorem 5.

Theorem 6. (Electric torque in thedq0frame). For an ideal DFImachine, the electric torque in generator mode and in the dq0-frame is the following bilinear form for the matrix A3:

Tel,g=P232NrNsLmsjdsjqsA3jdrjqrE74

which simplifies to

Tel,g=+P232NrNsLmsjdsjqrjqsjdr.E75

Proofof Theorem 6. It suffices to combine Eqs. (66), (56) and (64). The matrix A3makes the 3rdentries of current vectors not relevant ().

## 6. A “realistic” machine model

Real machines deviate from the hypotheses which have led to the relatively simple form of the equations discussed so far. A satisfactory model shall account for one or more of the following features:

1. (a)the effects of tooth saliency and slots on the linked fluxes,

2. (b)deviations from three-fold symmetry,

3. (c)the instantaneous dependence of self-and mutual inductances on current, when the magnetic material is not linear,

4. (d)memory effect in a non-linear, hysteretic magnetic material.

Models which, step-wise, account for features ato dare “realistic” in the sense of Fitzgerald and Kingsley [3]. Features listed under aare relatively simple to model if three-fold symmetry is assumed: very briefly, higher harmonics are introduced which, because of linearity, can be dealt with separately. Instead, broken symmetry may be of some interest: the model outlined herewith focuses on feature band consists of constructing a “realistic” mutual inductance matrix, then determining its algebraic (determinant, eigenvalues) and analytical (θr-derivative) properties.

Definition 6. (Broken symmetry in the rotor). At fixed θrthe rotor araxis forms angles θr, θrφand θr+φwith the as, bsand csaxes respectively. With ϵband ϵcsatisfying

03ϵb2π,3ϵc2π<<1E76

the rotor braxis forms angles θr+φ+ϵb, θr+ϵband θrφ+ϵbwith the as, bsand csaxes respectively. Similar relations hold for the rotor craxis in terms of ϵc.

As a consequence the mutual inductance matrix is

Lsrθrϵbϵc=cosθrcosθr+φ+ϵbcosθrφ+ϵccosθrφcosθr+ϵbcosθr+φ+ϵccosθr+φcosθrφ++ϵbcosθr+ϵc.E77

Because of broken symmetry, Lsrθrϵbϵcis no longer circulant. However, its column-wise entries add to zero and the following properties hold.

Proposition 6. (Kernel, adjoint, zero divisors for generalϵbandϵc).

To second order in ϵband ϵc, Lsrθrϵbϵcis approximated by

LsrθrϵbϵcGϵb,ϵc2==Cϵb,ϵc2cosθr+Sϵb,ϵc2sinθr+Cϵb,ϵc1cosθr+Sϵb,ϵc1sinθr,E79

where the four new matrices have to be defined. Cϵb,ϵc2and Sϵb,ϵc2are obtained from Cand Sof Eq. (27) when their second columns are multiplied by 112ϵb2and their third columns are multiplied by 112ϵc2. Similarly, Cϵb,ϵc1and Sϵb,ϵc1derive from splitting the cosθrand sinθrterms in the following matrix

0ϵbsinθr+φϵcsinθrφ0ϵbsinθrϵcsinθr+φ0ϵbsinθrφϵcsinθrCϵb,ϵc1cosθr+Sϵb,ϵc1sinθr.E80

As a consequence, a property can be stated about the derivative of Gϵb,ϵc2.

Proposition 7. (Differentiation as multiplication). At least to second order in ϵband ϵc, there exists a matrix B, independent of θr, by which the differentiation of Gϵb,ϵc2is represented as multiplication

Gϵb,ϵc2θrθrj=BGϵb,ϵc2θrj,jR3.E81

Such matrix complies with

B2Cϵb,ϵc2+Cϵb,ϵc1=Cϵb,ϵc2+Cϵb,ϵc1.E82

In particular, to 1st order in ϵband ϵc

B2C+Cϵb,ϵc1=C+Cϵb,ϵc1.E83

## 7. Conclusion

In view of the large amounts of power converted from electric to mechanical or vice-versa, mathematical methods for electric machinery have to undergo continuous investigation and, possibly, improvement. Model errors, although “small” in relative terms, may translate into large amounts of mishandled power. To date, control methods and the corresponding algorithms are satisfactory in the low frequency (tens of Hz) range: better performance is needed to deal with the higher (thousands of Hz) frequency components of a transient [18]. This work has focused on the basics of the ideal DFImachine model, where linearity and three-fold symmetry are the main features. As a result, the electric torque theorem has been stated in the dq0frame without any restriction on the j’s. The product of matrices formula has accordingly simplified the proof. Some of the properties derived in the ideal case have been shown to hold even if symmetry is broken.

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Written By

Giovanni F. Crosta and Goong Chen

Reviewed: January 26th, 2022 Published: April 5th, 2022