• No results found

[NORMAL] New Scientific Contribution on the 2-D Subdomain Technique in Polar Coordinates: Taking into Account of Iron Parts

N/A
N/A
Protected

Academic year: 2020

Share "New Scientific Contribution on the 2-D Subdomain Technique in Polar Coordinates: Taking into Account of Iron Parts"

Copied!
29
0
0

Loading.... (view fulltext now)

Full text

(1)

Article

New Scientific Contribution on the 2-D Subdomain

Technique in Polar Coordinates: Taking into account

of Iron Parts

Frédéric Dubas1,* and Kamel Boughrara2

1 Département ENERGIE, FEMTO-ST, CNRS, Univ. Bourgogne Franche-Comté, F90000 Belfort, France 2 Laboratoire de Rcherche en Electrotechnique (LRE-ENP), Algiers, 10 av. Pasteur, El Harrach, BP 182, 16200,

Algeria; kamel.boughrara@g.enp.edu.dz

* Correspondence: frederic.dubas@univ-fcomte.fr; Tel.: +33-38-457-8203

Abstract:This paper presents a new scientific contribution on the two-dimensional (2-D) subdomain technique in polar coordinates taking into account the finite relative permeability of the ferromagnetic material. The constant relative permeability corresponds to linear part of the nonlinearB(H)curve. As in conventional technique, the method of separation of variables and the Fourier’s series are used for the resolution of magnetostatic Maxwell’s equations in each region. Although, the general solutions of magnetic field in the subdomains and boundary conditions (BCs) between regions are different in the conventional and proposed method. In this later, the magnetic field solution in each subdomain is a superposition of two magnetic quantities in the two directions (i.e.,r- andΘ-axis) and the BCs between two regions are also in both directions. For example, the scientific contribution has been applied to an air- or iron-cored coil supplied by a constant current. The distribution of local quantities (i.e., the magnetic vector potential and flux density) has been validated by a corresponding 2-D finite-element analysis (FEA). The obtained semi-analytical results are in very good agreement with those of numerical method.

Keywords: air-oriron-coredcoil;polarcoordinates; fourieranalysis;two-dimensional;subdomain technique

1. Introduction

The full calculation of magnetic field in electrical engineering applications is the first step for their design and optimization, the methods of magnetic field prediction can be classified in various categories [1]:

• Lehmann’s graphical [2];

• Numerical (i.e., finite-element, finite-difference, boundary-element, etc.) [3–5];

• Equivalent circuit (i.e., electrical, thermal, magnetic, etc.) [6–8];

• Schwarz-Christoffel mapping (i.e., conformal transformation, complex permeance model, etc.) [9]; • Maxwell-Fourier [10–15].

Currently, the works of design are based on (semi-)analytical models1(i.e., equivalent circuit, conformal transformation and Maxwell-Fourier methods). In comparison with the other methods,

(2)

under certain geometrical and physical assumptions, these models permit to obtain accurate analytical expressions of magnetic field and known as fast for the prediction of local and global electromagnetic performances. At present, Maxwell-Fourier methods are one of the most used semi-analytic approaches with very accurate results (i.e., error less than 5 %) on the electromagnetic performances calculation. These models are based on the formal resolution of Maxwell’s equations in Cartesian, cylindrical or spherical coordinates by using the method of seperation of variables and the Fourier’s series. Taking into account of iron parts and/or the effect of local/global saturation is still a scientific challenge in Maxwell-Fourier methods which is rarely explored in the literature [17,18]. Recently, Dubaset al.(2017) [1] realized an overview on the existing (semi-)analytical models in Maxwell-Fourier methods with the effect of local/global saturation, which can thus be classified as follows

• Multi-layers models (i.e., Carter’s coefficient [19,20], saturation coefficient [21,22], concept wave impedance [23–26], and convolution theorem [27–30]);

• Eigenvalues model, viz., the method of Truncation Region Eigenfunction Expansions (TREE) [31,32];

• Subdomain technique [1,33,34];

• Hybrid models, viz., the analytical solution combined with numerical methods [35,36] or (non)linear magnetic equivalent circuit [37–39].

The consideration of the effect of local/global saturation is appearing in hybrid models, where the solution is established analytically in concentric regions of very low permeability (e.g., air-gap and magnets) and other methods (e.g., numerical or magnetic equivalent circuit) are sought in regions where the saturation effect cannot be neglected. On other hand, the other models (i.e., multi-layers models, TREE method and sudomain technique) are more focused the global saturation. Some details and (dis)advantages of these techniques can be found in [1]. In most semi-analytical models based on the subdomain technique, the iron parts are considered to be infinitely permeability due the variation of material proprieties in the various directions, so that the saturation effect is neglected [16–18]. The first paper introducing the iron parts in the magnetic field calculation by using subdomain technique is [1], where the authors solve partial differential equations (EDPs) of magnetic potential vector in Cartesian coordinates in which the subdomains connection is performed directly in both directions (i.e., x- andy-edges). The 2-D magnetostatic model has been applied to an air- or iron-cored coil supplied by a constant current. In [33], the authors propose a 2-D semi-analytical model in spoke-type magnets synchronous machines based on the subdomain technique in polar coordinates with Taylor polynomial of degree 3 by focusing on the consideration of iron. The iron magnetic permeability is supposed constant corresponding to linear zone of the nonlinearB(H)curve. The subdomains connection is carried out in both directions (i.e.,r- andΘ-edges). The general solution of magnetic field is obtained by using the traditional BCs, in addition to new radial BCs (e.g., between the magnets and the rotor teeth, between the teeth and the slots of the stator) which are traduced into a system of linear equations according to Taylor series expansion. In [34], this semi-analytical model has been extended taking into account the initial magnetization curve in each soft-magnetic subdomain by an iterative procedure.

(3)

6

1

r

3

r

2

yr

0 x

1

3

4

5

4

r

2

r

Vacuumv0

CoilNtI ; S ;cc0

Air or Ironv0 iron 0ri

Figure 1.Physical and geometrical parameters (see Table1) of air- or iron-cored coil where⊗and

are respectively the forward and return conductor.

current. The iron magnetic permeability is constant corresponding to linear zone of the nonlinearB(H)

curve [1,33]. Nevertheless, as in [29,30,34], the saturation effect could be taken into account by an iterative calculation considering, at each iteration, a constant relative magnetic permeability according to the nonlinearB(H)curve. However, this is beyond the scope of the paper. In Section 3, in order to confirm the effectiveness of the proposed technique, all semi-analytical results are then compared to those found by 2-D FEA [40]. The comparisons are very satisfying in amplitudes and waveforms.

2. A 2-D Subdomain Technique of Magnetic Field in Polar Coordinates

2.1. Model Description and Assumptions

Figure1represents the physical and geometrical parameters of an air- or iron-cored coil withNt

turns of copper wire supplied by a constant currentI. The electromagnetic device is surrounded by an infinite box with null value of magnetic vector potential at it boundaries.

The analytical prediction of magnetic field based on the 2-D subdomain technique is done by solving magnetostatic Maxwell’s equations in polar coordinates(r,Θ)with the following assumptions: • The magnetic vector potential has only one component along thez-axis (i.e.,A={0; 0;Az}) and

then the end-effects are not considered;

• All materials are isotropic and the permeabilities are supposed constants in both directions (i.e.,r -andΘ-axis);

• All electrical conductivities of materials are supposed nulls (i.e., the eddy-currents induced in the copper/iron are neglected).

2.2. Problem Discretization in Regions

In Figure2, we present the studied electromagnetic device which is divided into 7 regions with

µ=Cst, viz.,

• Region 1{∀Θ∧r∈[r1, r2]}withµ1=µv;

(4)

yr

0 x

Region 3 Region 4

Region 2

Region 1

Boundary Condition (i.e., Dirichlet)

Region 5

Figure 2.Definition of regions in the air- or iron-cored coil.

• Region 3{Θ∈[Θ1, Θ2]∧r∈[r2, r3]}withµ3=µv;

• Region 4{Θ∈[Θ5, Θ6]∧r∈[r2, r3]}withµ4=µv;

• Region 5 (i.e., the air or iron in the middle of the coil){Θ∈[Θ2, Θ3]∧r∈[r2, r3]}withµ5=µv

for the air orµ5=µironfor the iron;

• Region 6 (i.e., the forward conductor){Θ∈[Θ2, Θ3]∧r∈[r2, r3]}withµ6=µc;

• Region 7 (i.e., the return conductor){Θ∈[Θ4, Θ5]∧r∈[r2, r3]}withµ7=µc.

2.3. Governing EDPs in Polar Coordinates: Laplace’s and Poisson’s Equations

According to (A.1) (see Appendix A), the distribution of magnetic vector potential in polar coordinates(r,Θ)is governed by

∆Azj=

2Azj

r2 +

1 r·

Azj

r +

1 r2·

2Azj

Θ2 =0 for j={1, . . . , 5} (Laplace’s equation), (1a)

∆Azk=

2Azk

r2 +

1 r ·

Azk

r +

1 r2 ·

2Azk

Θ2 =−µk·Jzk for k={6, 7} (Poisson’s equation), (1b)

whereJzkis the current density of the coil defined by

Jzk=Ck·

Nt·I

Sc , (2)

in whichScis the conductor surface, andCk(withC6=1 andC7=−1) is the coefficient that represents

the current direction in the conductor.

According to Appendix A, the resolution of Laplace’s and Poisson’s equations by using the method of separation of variables and the Fourier’s series permit to obtain two potentials in both directions, viz.,AΘz•for theΘ-edges (A.2b) andArz•for ther-edges (A.2c). The spatial frequency (or

periodicity) of AΘz• andArz• are respectively defined byβ•h• andλ•n• withh•andn•the spatial

(5)

2.4. Definition of BCs

In electromagnetic, the general solutions of various regions depend on the BCs at the interface of two surfaces, which are defined by the continuity of the normal flux densityB⊥and parallel field

intensityHk[1]. On the outer BCs for(Θ1∧Θ6, ∀r)and(∀Θ, r1∧r4),Azsatisfies the Dirichlet BC

(see Figure2), viz.,Az=0.

Figure3represents the respective BCs at the interface between the various regions in both directions (i.e.,r- andΘ-edges).

2.5. General Solutions of Various Regions

2.5.1. Region 1

The solution ofAz1,Br1andBΘ1are determined by thecase-study no 1(i.e.,Az imposed on all

edges of a region) in Appendix B. The BCs on ther-edges of the region (see Figure3a) are met by posingcΘh =0 in (B.6). Therefore,Az1satisfying the BCs of Figure3aand solution of (1a) is given by

Az1=−

h1=1

d1Θh1· r2

β1h1

· Eσ

(β1h1,r,r1)

P

σ

(β1h1,r2,r1)

·sin[β1h1·(Θ−Θ1)], (3)

the components ofB1={Br1;BΘ1; 0}by

Br1=−

h1=1

d1Θh1·r2

r · E

σ(β1h1,r,r1)

P

σ(β1h1,r2,r1)

·cos[β1h1·(Θ−Θ1)], (4)

BΘ1=

h1=1

d1Θh1·r2

r · P

σ(β1h1,r,r1)

P

σ(β1h1,r2,r1)

·sin[β1h1·(Θ−Θ1)], (5)

whereE

σ

(w,x,y)andP

σ(w,x,y)are defined in (B.4),h1 the spatial harmonic orders in Region 1,d1 Θ

h1

the integration constant, andβ1h1=h1·πτΘ1withτΘ1=Θ6−Θ1.

Using a Fourier series expansion ofF1(Θ)(see Figure3a) over the intervalΘ = [Θ1, Θ6] = [Θ1, Θ1+τΘ1], the integration constantd1Θh1is determined in Appendix C with

d1Θh1= 2

τΘ1

·

Θ1+τΘ1 Z

Θ1

F1(Θ)·sin[β1h1·(Θ−Θ1)]·dΘ. (6)

2.5.2. Region 2

The same method than Region 1 is used to define the general solution in Region 2. By posing dΘh =0 in (B.6) (see Appendix B),Az2satisfying the BCs of Figure3band solution of (1a) is given by

Az2=

h2=1

c2Θh2· r3

β2h2

· Eσ

(β2h2,r4,r)

P

σ

(β2h2,r4,r3)

·sin[β2h2·(Θ−Θ1)], (7)

the components ofB2={Br2;BΘ2; 0}by

Br2=

h2=1

c2Θh2·r3

r · E

σ

(β2h2,r4,r)

P

σ(β2h2,r4,r3)

·cos[β2h2·(Θ−Θ1)], (8)

BΘ2=

h2=1

c2Θh2·r3

r · P

σ(β2h2,r4,r)

P

σ(β2h2,r4,r3)

(6)

whereh2 is the spatial harmonic orders in Region 2,c2Θh2the integration constant, andβ2h2=h2·πτΘ2

withτΘ2=Θ6−Θ1.

                                             

1 2 2 2 3 2

2 3 4 2

4 5 2 5 6 2

3 3 x , r r

6 6 , r r

1 r r 1 1 5 5 , r r

7 7 , r r

4 4 , r r

1 B

1 B

B F 1 B

1 B 1 B                             Region 3 Region 4 Region 1 Region 5 1 r 2 r 621345

 1

z1 r

A   0

  1

z1 r r

A 0

6

z1 r

A   0

(a) 6  3 r 21345  4 r Region 3 Region 4 Region 2 Region 5

6

z1 r

A   0

 1

z1 r

A   0

  4

z1 r r

A 0

                                              

1 2 3 2 3 3

3 3 4 3

4 5 3 5 6 3

3 3 x , r r

6 6 , r r

2 r r 2 2 5 5 , r r

7 7 , r r

4 4 , r r

1 B

1 B

B G 1 B

1 B 1 B                            (b) 21  3 r 2 r Region 3

 1

z3 r

A   0

     22

z3 r r z1 r r

AA       33

z3 r r z 2 r r

A A

2 2

z3 r z6 r

A   A  

(c) 65  3 r 2 r Region 46

z4 r

A   0

     22

z4 r r z1 r r

A A

     33

z4 r r z 2 r r

A A

5 5

z4 r z7 r

A   A  

(d) 43  3 r 2 r Region 5      22

z5 r r z1 r r

A A

     33

z5 r r z 2 r r

A A

 3   3

z5 r z6 r

A   A   4 4

z5 r z7 r

A   A  

(e) 32  3 r 2 r Region 6      22

z6 r r z1 r r

A A

     33

z6 r r z 2 r r

A A

3  3

r6 r 6 5 r 5 r

B     B  

2  2

r6 r 6 3 r 3 r

B     B  

(f) 54  3 r 2 r Region 7      22

z6 r r z1 r r

A A

     33

z6 r r z 2 r r

A A

5  5

r6 r 7 4 r 4 r

B     B  

4  4

r7 r 7 5 r 5 r

B     B  

(g)

Figure 3.Boundary conditions (BCs) in both directions (i.e.,r- andΘ-edges):(a)Region 1,(b)Region 2,

(7)

Using a Fourier series expansion ofG2(Θ)(see Figure3b) over the intervalΘ = [Θ1, Θ6] = [Θ1, Θ1+τΘ2], the integration constantc2Θh2is determined in Appendix C with

c2Θh2= 2

τΘ2

·

Θ1+τΘ2 Z

Θ1

G2(Θ)·sin[β2h2·(Θ−Θ1)]·dΘ. (10)

2.5.3. Region 3

The solution ofAz3,Br3andBΘ3are determined by thecase-study no 1(i.e.,Az imposed on all

edges of a region) in Appendix B. The BCs on theΘ-edges of the region (see Figure3c) are met by posingern = 0 in (B.1) – (B.3). Therefore, Az3satisfying the BCs of Figure3cand solution of (1a) is

given by

Az3=AΘz3+Arz3, (11a)

z3= ∞

h3=1

"

c3Θh3·r2·

E

σ(β3h3,r3,r)

E

σ(β3h3,r3,r2)

+d3Θh3·r3·

E

σ

(β3h3,r,r2)

E

σ(β3h3,r3,r2)

#

·sin[β3h3·(Θ−Θ1)], (11b)

Arz3= ∞

n3=1

f3rn3·r2·sh

[λ3n3·(Θ−Θ1)]

sh(λ3n3·τΘ3)

·sin

λ3n3·ln

r r2

, (11c)

ther-component ofB3by

Br3=BΘr3+Brr3, (12a)

BrΘ3= ∞ h3=1

β3h3·

c3Θh3· r2

r · E

σ(β3h3,r3,r)

E

σ(β3h3,r3,r2)

+d3Θh3·r3 r ·

E

σ(β3h3,r,r2)

E

σ(β3h3,r3,r2)

·cos[β3h3·(Θ−Θ1)], (12b)

Brr3= ∞

n3=1

λ3n3· f3rn3·

r2

r ·

ch[λ3n3·(Θ−Θ1)]

sh(λ3n3·τΘ3)

·sin

λ3n3·ln

r r2

, (12c)

theΘ-component ofB3by

BΘ3=BΘΘ3+BrΘ3, (13a)

Θ3= ∞ h3=1

β3h3·

c3Θh3· r2

r · P

σ(β3h3,r3,r)

E

σ(β3h3,r3,r2)

−d3Θh3·r3 r ·

P

σ(β3h3,r,r2)

E

σ(β3h3,r3,r2)

·sin[β3h3·(Θ−Θ1)], (13b)

BrΘ3=−

n3=1

λ3n3· f3rn3·

r2

r ·

sh[λ3n3·(Θ−Θ1)]

sh(λ3n3·τΘ3)

·cos

λ3n3·ln

r r2

, (13c)

where h3 andn3 are the spatial harmonic orders in Region 3;c3Θh3,d3Θh3and f3rn3the integration constants;β3h3=h3·πτΘ3withτΘ3=Θ2−Θ1; andλ3n3=n3·π/τr3withτr3=ln(r3/r2).

Using Fourier series expansion of Az1|∀Θ∧r=r2and Az2|∀Θ∧r=r3(see Figure3c) over the interval

Θ= [Θ1, Θ2] = [Θ1, Θ1+τΘ3], the integration constantsc3Θh3andd3Θh3are determined in Appendix C

with

c3Θh3= 2

τΘ3

·

Θ1+τΘ3 Z

Θ1

Az1|r=r2

r2

·sin[β3h3·(Θ−Θ1)]·dΘ, (14a)

d3Θh3= 2

τΘ3

·

Θ1+τΘ3 Z

Θ1

Az2|r=r3

r3

·sin[β3h3·(Θ−Θ1)]·dΘ. (14b)

With a weighting functiong(r) =r−1and using a Fourier series expansion of Az6|Θ=Θ2∧∀r(see

Figure3c) over the intervalr= [r2, r3], the integration constant f3rn3is determined in Appendix C

with

f3rn3= 2

τr3

·

r3 Z

r2

1 r·

Az6|Θ=Θ2

r2

·sin

λ3n3·ln

r r2

(8)

2.5.4. Region 4

The solution in Region 4 is obtained using the same development than Region 3. By posingfnr =0

in (B.1) – (B.3) (see Appendix B),Az4satisfying the BCs of Figure3dand solution of (1a) is given by

Az4=AΘz4+Arz4, (16a)

z4= ∞

h4=1

"

c4Θh4·r2·

E

σ(β4h4,r3,r)

E

σ(β4h4,r3,r2)

+d4Θh4·r3·

E

σ

(β4h4,r,r2)

E

σ(β4h4,r3,r2)

#

·sin[β4h4·(Θ−Θ5)], (16b)

Arz4= ∞

n4=1

e4rn4·r2·sh

[λ4n4·(Θ6−Θ)]

sh(λ4n4·τΘ4)

·sin

λ4n4·ln

r r2

, (16c)

ther-component ofB4by

Br4=BrΘ4+Brr4, (17a)

BrΘ4= ∞ h4=1

β4h4·

c4Θh4· r2

r · E

σ(β4h4,r3,r)

E

σ(β4h4,r3,r2)

+d4Θh4·r3 r ·

E

σ(β4h4,r,r2)

E

σ(β4h4,r3,r2)

·cos[β4h4·(Θ−Θ5)], (17b)

Brr4=−

n4=1

λ4n4·e4rn4·

r2

r ·

ch[λ4n4·(Θ6−Θ)]

sh(λ4n4·τΘ4)

·sin

λ4n4·ln

r r2

, (17c)

theΘ-component ofB4by

BΘ4=BΘΘ4+BrΘ4, (18a)

Θ4= ∑∞ h4=1

β4h4·

c4Θh4· r2

r · P

σ(β4h4,r3,r)

E

σ(β4h4,r3,r2)

−d4Θh4·r3 r ·

P

σ(β4h4,r,r2)

E

σ(β4h4,r3,r2)

·sin[β4h4·(Θ−Θ5)], (18b)

BrΘ=−

n4=1

λ4n4·e4rn4·

r2

r ·

sh[λ4n4·(Θ6−Θ)]

sh(λ4n4·τΘ4)

·cos

λ4n4·ln

r

r2

, (18c)

where h4 and n4 are the spatial harmonic orders in Region 4; c4Θh4, d4Θh4 ande4rn4the integration constants;β4h4=h4·πτΘ4withτΘ4=Θ6−Θ5; andλ4n4=n4·π/τr4withτr4=ln(r3/r2).

Using Fourier series expansion of Az1|∀Θ∧r=r2 and Az2|∀Θ∧r=r3 (see Figure3d) over the interval

Θ= [Θ5, Θ6] = [Θ5, Θ5+τΘ4], the integration constantsc4Θh4andd4Θh4are determined in Appendix C

with

c4Θh4= 2

τΘ4

·

Θ5+τΘ4 Z

Θ5

Az1|r=r2

r2

·sin[β4h4·(Θ−Θ5)]·dΘ, (19a)

d4Θh4= 2

τΘ4

·

Θ5+τΘ4 Z

Θ5

Az2|r=r3

r3

·sin[β4h4·(Θ−Θ5)]·dΘ. (19b)

With a weighting functiong(r) =r−1and using a Fourier series expansion of Az7|Θ=Θ5∧∀r(see

Figure3d) over the intervalr = [r2, r3], the integration constante4rn4is determined in Appendix C

with

e4rn4= 2

τr4

·

r3 Z

r2

1 r ·

Az7|Θ=Θ5

r2

·sin

λ4n4·ln

r r2

·dr. (20)

2.5.5. Region 5

For Region 5, the general solution is given according to the BCs ofcase-study no 1(i.e.,Azimposed

on all edges of a region) in Appendix B. Therefore,Az5satisfying the BCs of Figure3eand solution of

(1a) is given by

(9)

z5= ∞

h5=1

"

c5Θh5·r2·

E

σ(β5h5,r3,r)

E

σ(β5h5,r3,r2)

+d5Θh5·r3·

E

σ

(β5h5,r,r2)

E

σ(β5h5,r3,r2)

#

·sin[β5h5·(Θ−Θ3)], (21b)

Arz5= ∞ n5=1

n

e5rn5·sh[λ5n5·(Θ4−Θ)] sh(λ5n5·τΘ5) +f5

r n5·

sh[λ5n5·(Θ−Θ3)] sh(λ5n5·τΘ5)

o

·r2·sin

h

λ5n5·ln

r r2

i

, (21c)

ther-component ofB5by

Br5=BΘr5+BrΘ5, (22a)

BrΘ5= ∞ h5=1

β5h5·

c5Θh5· r2

r · E

σ(β5h5,r3,r)

E

σ(β5h5,r3,r2)

+d5Θh5·r3 r ·

E

σ(β5h5,r,r2)

E

σ(β5h5,r3,r2)

·cos[β5h5·(Θ−Θ3)], (22b)

Brr5= ∞ n5=1

λ5n5·

n

−e5rn5· ch[λ5n5·(Θ4−Θ)] sh(λ5n5·τΘ5) +f5

r n5·

ch[λ5n5·(Θ−Θ3)] sh(λ5n5·τΘ5)

o

·r2 r ·sin

h

λ5n5·ln

r r2

i

, (22c)

theΘ-component ofB5by

BΘ5=BΘΘ5+BrΘ5, (23a)

Θ5= ∑∞ h5=1

β5h5·

c5Θh5· r2

r · P

σ(β5h5,r3,r)

E

σ(β5h5,r3,r2)

−d5Θh5·r3 r ·

P

σ(β5h5,r,r2)

E

σ(β5h5,r3,r2)

·sin[β5h5·(Θ−Θ3)], (23b)

BΘr5=−

n5=1

λ5n5·

n

e5rn5·sh[λ5n5·(Θ4−Θ)] sh(λ5n5·τΘ5) +f5

r n5·

sh[λ5n5·(Θ−Θ3)] sh(λ5n5·τΘ5)

o

·r2 r ·cos

h

λ5n5·ln

r r2

i

, (23c)

whereh5 andn5 are the spatial harmonic orders in Region 5;c5Θh5,d5Θh5,e5rn5and f5rn5the integration constants;β5h5=h5·πτΘ5withτΘ5=Θ4−Θ3; andλ5n5=n5·π/τr5withτr5=ln(r3/r2).

Using Fourier series expansion of Az1|∀Θ∧r=r2 and Az2|∀Θ∧r=r3 (see Figure3e) over the interval

Θ= [Θ3, Θ4] = [Θ3, Θ3+τΘ5], the integration constantsc5Θh5andd5Θh5are determined in Appendix C

with

c5Θh5= 2

τΘ5

·

Θ3+τΘ5 Z

Θ3

Az1|r=r2

r2

·sin[β5h5·(Θ−Θ3)]·dΘ, (24a)

d5Θh5= 2

τΘ5

·

Θ3+τΘ5 Z

Θ3

Az2|r=r3

r3

·sin[β5h5·(Θ−Θ3)]·dΘ. (24b)

With a weighting functiong(r) =r−1and using a Fourier series expansion of Az6|Θ=Θ3∧∀r and

Az7|Θ=Θ4∧∀r(see Figure3e) over the intervalr= [r2, r3], the integration constantse5 r

n5and f5rn5are

determined in Appendix C with

e5rn5=

2

τr5

·

r3 Z

r2

1 r ·

Az6|Θ=Θ3

r2

·sin

λ5n5·ln

r r2

·dr, (25a)

f5rn5=

2

τr5

·

r3 Z

r2

1 r·

Az7|Θ=Θ4

r2

·sin

λ5n5·ln

r r2

·dr. (25b)

2.5.6. Region 6

For Region 6, the general solution is given according to the BCs ofcase-study no 2(i.e.,BrandAz

are respectively imposed onr- andΘ-edges of a region) in Appendix B. Therefore,Az6satisfying the

BCs of Figure3fand solution of (1b) is given by

(10)

z6=

c6Θ0 ·r2·lnln((rr33//rr2))+d6Θ0 ·r3·lnln((rr3//rr22))

· · ·+ ∞ h6=1

c6Θh6·r2· E

σ(β6h6,r3,r)

E

σ(β6h6,r3,r2)

+d6Θh6·r3· E

σ(β6h6,r,r2)

E

σ(β6h6,r3,r2)

·cos[β6h6·(Θ−Θ2)]

, (26b)

Arz6= ∑∞ n6=1

n

e6rn6·ch[λ6n6·(Θ−Θ2)] sh(λ6n6·τΘ6) −f6

r n6·

ch[λ6n6·(Θ3−Θ)] sh(λ6n6·τΘ6)

o

· r2 λ6n6 ·sin

h

λ6n6·ln

r r2

i

. (26c)

Considering (26b) and (26c) as well as the form of the current density distribution, i.e., (2), a particular solutionAzP6can be found. The following particular solution is proposed

AzP6=−1

4 ·r

2·

µ6·Jz6. (26d)

Ther-component ofB6is defined by

Br6=BΘr6+Brr6+BrP6, (27a)

r6=−

h6=1

β6h6·

c6Θh6·r2 r ·

E

σ(β6h6,r3,r)

E

σ(β6h6,r3,r2)

+d6Θh6· r3 r ·

E

σ(β6h6,r,r2)

E

σ(β6h6,r3,r2)

·sin[β6h6·(Θ−Θ2)], (27b)

Brr6= ∞ n6=1

n

e6rn6·sh[λ6n6·(Θ−Θ2)] sh(λ6n6·τΘ6) + f6

r n6·

sh[λ6n6·(Θ3−Θ)] sh(λ6n6·τΘ6)

o

·r2 r ·sin

h

λ6n6·ln

r r2

i

, (27c)

BrP6= 1

r ·

AzP6

Θ =0, (27d)

and theΘ-component ofB6by

BΘ6=BΘΘ6+BrΘ6+BΘP6, (28a)

Θ6=

c6Θ0 ·r2

r ·ln(r31/r2)−d6

Θ

0 ·rr3·ln(r31/r2)

· · ·+ ∞ h6=1

β6h6·

c6Θh6·r2

r · P

σ(β6h6,r3,r)

E

σ(β6h6,r3,r2)

−d6Θh6·r3

r · P

σ(β6h6,r,r2)

E

σ(β6h6,r3,r2)

·cos[β6h6·(Θ−Θ2)]

, (28b)

BΘr6=−

n6=1

n

e6rn6·ch[λ6n6·(Θ−Θ2)] sh(λ6n6·τΘ6) −f6

r n6·

ch[λ6n6·(Θ3−Θ)] sh(λ6n6·τΘ6)

o

·r2 r ·cos

h

λ6n6·ln

r r2

i

, (28c)

BΘP6=−

AzP6

r =

1

2·r·µ6·Jz6, (28d) where h6 andn6 are the spatial harmonic orders in Region 6; c6Θ0, d6Θ0, c6Θh6, d6Θh6, e6rn6 and f6rn6 the integration constants; β6h6 = h6·πτΘ6 with τΘ6 = Θ3−Θ2; and λ6n6 = n6·π/τr6 with

τr6=ln(r3/r2).

Using Fourier series expansion of Az1|∀Θ∧r=r2 and Az2|∀Θ∧r=r3(see Figure3f) over the interval

Θ= [Θ2, Θ3] = [Θ2, Θ2+τΘ6], the integration constantsc6Θ0 &c6Θh6andd6Θ0 &d6Θh6are determined

in Appendix C with

c6Θ0 = 1

τΘ6

·

Θ2+τΘ6 Z

Θ2

1 r2

·hAz1|r=r2− AzP6|r=r2

i

·dΘ, (29a)

c6xh6= 2

τΘ6

·

Θ2+τΘ6 Z

Θ2

1 r2

·hAz1|r=r2− AzP6|r=r2

i

·cos[β6h6·(Θ−Θ2)]·dΘ, (29b)

d6Θ0 = 1

τΘ6

·

Θ2+τΘ6 Z

Θ2

1 r3

·hAz2|r=r3− AzP6|r=r3

i

(11)

d6hx6= 2

τΘ6

·

Θ2+τΘ6 Z

Θ2

1 r3

·hAz2|r=r3− AzP6|r=r3

i

·cos[β6h6·(Θ−Θ2)]·dΘ. (29d)

Using a Fourier series expansion ofµ6

µ5· Br5|Θ=Θ3∧∀rand µ6

µ3·Br3

Θ=Θ

2∧∀r(see Figure3f)

over the intervalr= [r2, r3], the integration constantse6rn6andf6rn6are determined in Appendix C

with

e6rn6= 2

τr6

· r3 Z r2 1 r2 · µ6 µ5

· Br5|Θ=Θ3−BrP6|Θ=Θ3

·sin

λ6n6·ln

r r2

·dr, (30a)

f6rn6= 2

τr6

· r3 Z r2 1 r2 · µ6 µ3

· Br3|Θ=Θ2−BrP6|Θ=Θ2

·sin

λ6n6·ln

r r2

·dr. (30b)

2.5.7. Region 7

The solution in Region 7 is using the same development than Region 6. Thus,Az7satisfying the

BCs of Figure3gand solution of (2) is defined by

Az7=AΘz7+Arz7+AzP7, (31a)

z7=

c7Θ0 ·r2·lnln((rr33//rr2))+d7Θ0 ·r3·lnln((rr3//rr22))

· · ·+ ∞ h7=1

c7Θh7·r2· E

σ(β7h7,r3,r)

E

σ(β7h7,r3,r2)

+d7Θh7·r3· E

σ(β7h7,r,r2)

E

σ(β7h7,r3,r2)

·cos[β7h7·(Θ−Θ4)]

, (31b)

Arz7= ∑∞ n7=1

n

e7rn7·ch[λ7n7·(Θ−Θ4)] sh(λ7n7·τΘ7) −f7

r n7·

ch[λ7n7·(Θ5−Θ)] sh(λ7n7·τΘ7)

o

· r2 λ7n7 ·sin

h

λ7n7·ln

r r2

i

, (31c)

AzP7=−1

4 ·r

2·

µ7·Jz7. (31d)

Ther-component ofB7is defined by

Br7=BΘr7+Brr7+BrP7, (32a)

r7=−

h7=1

β7h7·

c7Θh7·r2

r · E

σ(β7h7,r3,r)

E

σ(β7h7,r3,r2)

+d7Θh7· r3 r ·

E

σ(β7h7,r,r2)

E

σ(β7h7,r3,r2)

·sin[β7h7·(Θ−Θ4)], (32b)

Brr7= ∞ n7=1

n

e7rn7·sh[λ7n7·(Θ−Θ4)] sh(λ7n7·τΘ7) + f7

r n7·

sh[λ7n7·(Θ5−Θ)] sh(λ7n7·τΘ7)

o

·r2 r ·sin

h

λ7n7·ln

r r2

i

, (32c)

BrP7= 1

r ·

AzP7

Θ =0, (32d)

and theΘ-component ofB7by

BΘ7=BΘΘ7+BrΘ7+BΘP7, (33a)

Θ7=

c7Θ0 ·r2

r ·ln(r31/r2)−d7

Θ

0 ·rr3·ln(r31/r2)

· · ·+ ∞ h7=1

β7h7·

c7Θh7·r2

r · P

σ(β7h7,r3,r)

E

σ(β7h7,r3,r2)

−d7Θh7·r3

r · P

σ(β7h7,r,r2)

E

σ(β7h7,r3,r2)

·cos[β7h7·(Θ−Θ4)]

, (33b)

BΘr7=−

n7=1

n

e7rn7·ch[λ7n7·(Θ−Θ4)] sh(λ7n7·τΘ7) −f7

r n7·

ch[λ7n7·(Θ5−Θ)] sh(λ7n7·τΘ7)

o

·r2 r ·cos

h

λ7n7·ln

r r2

i

, (33c)

BΘP7=−

AzP7

r =

1

2·r·µ7·Jz7, (33d) where h7 andn7 are the spatial harmonic orders in Region 7; c7Θ0, d7Θ0, c7Θh7, d7Θh7, e7rn7 and f7rn7 the integration constants; β7h7 = h7·πτΘ7 with τΘ7 = Θ5−Θ4; and λ7n7 = n7·π/τr7 with

(12)

Using Fourier series expansion of Az1|∀Θ∧r=r2and Az2|∀Θ∧r=r3 (see Figure3g) over the interval

Θ= [Θ4, Θ5] = [Θ4, Θ4+τΘ7], the integration constantsc7Θ0 &c7Θh7andd7Θ0 &d7Θh7are determined

in Appendix C with

c7Θ0 = 1

τΘ7

·

Θ4+τΘ7 Z

Θ4

1 r2

·hAz1|r=r2− AzP7|r=r2

i

·dΘ, (34a)

c7xh7= 2

τΘ7

·

Θ4+τΘ7 Z

Θ4

1 r2

·hAz1|r=r2− AzP7|r=r2

i

·cos[β7h7·(Θ−Θ4)]·dΘ, (34b)

d7Θ0 = 1

τΘ7

·

Θ4+τΘ7 Z

Θ4

1 r3

·hAz2|r=r3− AzP7|r=r3

i

·dΘ, (34c)

d7hx7= 2

τΘ7

·

Θ4+τΘ7 Z

Θ4

1 r3

·hAz2|r=r3− AzP7|r=r3

i

·cos[β7h7·(Θ−Θ4)]·dΘ. (34d)

Using a Fourier series expansion ofµ7µ4· Br4|Θ=Θ5∧∀rand µ7

µ5·Br5

Θ=Θ

4∧∀r(see Figure3g)

over the intervalr= [r2, r3], the integration constantse7rn7andf7rn7are determined in Appendix C

with

e7rn7=

2

τr7

·

r3 Z

r2

1 r2

·

µ7

µ4

· Br4|Θ=Θ5−BrP7|Θ=Θ5

·sin

λ7n7·ln

r

r2

·dr, (35a)

f7rn7= 2

τr7

·

r3 Z

r2

1 r2

·

µ7

µ5

· Br5|Θ=Θ4−BrP7|Θ=Θ4

·sin

λ7n7·ln

r r2

·dr. (35b)

3. Validation of the Semi-Analytic Method with FEA

3.1. Introduction

The objective of this section is to validate the proposed 2-D subdomain method in polar coordinates(r,Θ)on the magnetic field distribution in relation to the numerical method. The physical and geometrical parameters of studied electromagnetic device are given in Table1.

For this validation, the air- or iron-cored coil has been modeled using Cedrat’s Flux2D (Version 10.2.1., Altair Engineering, Meylan Cedex, France) software package (i.e., an advanced finite-element method based numeric field analysis program) [40]. The finite-element model is done with the same assumptions as in the semi-analytical model (see § 2.1. Model Description and Assumptions). The linear system (i.e., Cramer’s system), given in Appendix C, has been implemented in MatlabR (R2015a, Mathworks, Natick, MA, USA) by using the sparse matrix/vectors. A discussion on the numerical problems (viz., harmonics and ill-conditioned systems) of such semi-analytical models has been clarified in [1]. The Maxwell-Fourier methods exhibit a similar problem to the numerical methods due to the periodicity of Fourier series, and consequently to the finite number of harmonics. Hence, AzandB={Br;BΘ; 0}in the various regions (see § 2.5. General Solutions of Various Regions) have

been computed with a finite number of spatial harmonics terms H1max –H7max (for the Θ-edges)

andN3max–N7max(for ther-edges). As indicated in [41,42], these spatial harmonics terms, given in

Table1, have been imposed according to an optinal ratio, i.e., forH1maxgiven,

H•max = H1max·τΘ•

τΘ1

and N•max = H•max·τΘ•

(13)

Table 1.Physical and Geometrical Parameters of the Air- or Iron-Cored Coil.

Parameters, Symbols [Units] Values

Number of turns of the coil,Nt[–] 60

Supply current,I[A] 20

Conductor Surface,Sc[mm2] 120

Current density of the coil,Jzk[A/mm2] ±10

Effective axial length,Lz[mm] 60

Geometrical parameters in theΘ-axis,{Θ1;Θ2;Θ3;Θ4;Θ5;Θ6}[deg.] {0; 17; 21; 29; 33; 50}

Geometrical parameters in ther-axis,{r1;r2;r3;r4}[mm] {21; 81; 100; 160}

Relative magnetic permeability of the iron,µiron[–] 1,500

Number of harmonics for Region 1,H1max[–] 260

Number of harmonics for Region 2,H2max[–] 260

Number of harmonics for Region 3,{H3max; N3max}[–] {88; 124}

Number of harmonics for Region 4,{H4max; N4max}[–] {88; 124}

Number of harmonics for Region 5,{H5max; N5max}[–] {42; 124}

Number of harmonics for Region 6,{H6max; N6max}[–] {21; 124}

Number of harmonics for Region 7,{H7max; N7max}[–] {21; 124}

Coil

Air or Iron

Figure 4.2-D finite-element analysis (FEA) mesh for the air- or iron-cored coil.

The linear system size depends on the number of: (i) regions; (ii) BCs; and (iii) harmonics of each subdomain. In our study, the linear system (C.3) consists of 2,036 elements which is much smaller than the 2-D FEA mesh having 3,081 surfaces elements of second order (viz., the triangles number of system). For information, the 2-D FEA mesh for an air- or iron-cored coil is illustrated in Figure4. The personal computer used for this comparison has the following characteristics: HP Z800 Intel(R) Xeon(R) CPU@2.4 GHz (with 2 processors) RAM 16 Go 64 bits. The computation time of 2-D subdomain model is divided by 2 (viz., 0.5 sec for 2-D subdomain model and 1 sec for the 2-D FEA).

3.2. Results Discussion

The validation paths ofAzandB={Br;BΘ; 0}for the semi-analytic and numeric comparison are

given in Figure5.

The waveforms of global quantities are shown on different paths in Figure 6for Az and in

(14)

6

1 r

3 r

2

yr

 0 x

1

3

4

5

4 r

2 r

Path 2

Path 1

 

rr1r2 2

 

rr2r3 2

3 42     

2 32

    

1 22

    

Figure 5.Validation paths for the semi-analytic and numeric comparison.

it can be shown that a very good evaluation is obtained forAzand for the components ofB, whatever

the paths, for both air- and iron-core. This confirms that the effect of global saturation can be taken into account accurately. It is interesting to note that numerical peaks appear in the FEA results (see Figure6e, Figure7, Figure8band Figure11b) which are mainly due to the mesh. The relative error is less than 1.5 % for the various global quantities (see Figure6aand6cfor the maximum error).

4. Conclusion

It has been demonstrated that there exists no exact semi-analytical model based on the subdomain technique in polar coordinates taking into account of iron parts with(out) the nonlinearB(H)curve. An improved 2-D subdomain method in polar coordinates(r,Θ)to study the magnetic field distribution in the iron parts with a finite relative permeability have been presented in this paper. Nevertheless, the research work is an extension of [1] in polar coordinates(r,Θ).

The proposed new subdomain model is applied to an air- or iron-cored coil supplied by a constant current. The magnetic field solutions in the subdomains and interfaces conditions between regions are carried out in the two directions (i.e.,r- andΘ-axis). The iron relative permeability used in this model is constant and corresponds to the linear part of the nonlinearB(H)curve. However, the wholeB(H)

curve of the magnetic material can be applied with an iterative algorithm as in [29,30,34]. The proposed subdomain method in polar coordinates(r,Θ)takes less computing time than the FEA (approximately 2 fold versus to FEA). It is very suitable for design and optimisation of the electromechanical systems in general and electrical machines in particuler. The semi-analytical results have been validated with FEA and good agreement has been obtained in both amplitudes and waveforms.

(15)

0 17 21 29 33 50

Mag

n

etic

v

ecto

r

p

o

ten

tial

fo

r

Path

1

Mechanical angular position of Path 1 [deg.] FEA Subdomain model

0 2e-6

-2e-6

-1.2e-5

1.2e-5

Iron-cored coil

Air-cored coil

(a)

0 17 21 29 33 50

Mechanical angular position of Path 2 [deg.] FEA Subdomain model

Iron-cored coil

Mag

n

etic

v

ector

p

o

ten

tial

fo

r Path

2

0 1.23e-4

-1.23e-4

-7.4e-4

7.4e-4

Air-cored coil

(b)

21 81 100 160

Length of Path 3 [mm]

M

agn

etic

v

ector

p

o

tential

fo

r Path

3

0

-1e-4

2e-4

-2.5e-5

2.5e-5

FEA Subdomain model

Iron-cored coil

Air-cored coil

(c)

21 81 100 160

Length of Path 4 [mm]

FEA Subdomain model

Iron-cored coil

Mag

n

etic

v

ector

p

o

tential

fo

r Path

4

-1e-4

7e-4

-3.33e-5

3.33e-5

Air-cored coil

(d)

21 81 100 160

Length of Path 5 [mm]

Mag

n

etic

v

ecto

r

p

o

ten

tial

fo

r

Path

5

-1e-7

1.5e-7

-1.67e-8

4.17e-9

FEA Subdomain model

Iron-cored coil Air-cored coil

(e)

(16)

0 17 21 29 33 50 Mechanical angular position of Path 1 [deg.]

Iron-cored coil FEA Subdomain model The r -co m p o n en t o f th e m ag n etic flu x d en sity fo r Path 1 0 2.67e-4 -2.67e-4 -1.6e-5 1.6e-3 Air-cored coil (a)

0 17 21 29 33 50

Mechanical angular position of Path 1 [deg.]

The Q -co m p o n en t o f the m ag n etic flu x d en sity fo r Path 1 0 2.67e-4 -2.67e-4 -1.6e-5 1.6e-3 FEA Subdomain model Iron-cored coil Air-cored coil (b)

Figure 7.Waveform ofBfor Path 1: (a)r- and (b)Θ-component.

0 17 21 29 33 50

Mechanical angular position of Path 2 [deg.]

The r -co m p o n en t o f th e m ag n etic flu x d en sity fo r Path 2 0 0.025 -0.025 -0.2 0.1 Iron-cored coil FEA Subdomain model Air-cored coil (a)

0 17 21 29 33 50

Mechanical angular position of Path 2 [deg.]

The Q -co m p o n en t o f the m ag n etic flu x d en sity fo r Path 2 0 6.67e-4 -6.67e-4 -4e-3 4e-3 FEA Subdomain model Iron-cored coil Air-cored coil (b)

Figure 8.Waveform ofBfor Path 2: (a)r- and (b)Θ-component.

21 81 100 160

Length of Path 3 [mm]

The r -co m p o n en t o f th e m ag n etic flu x d en sity fo r Pa th 3 FEA Subdomain model -2e-3 0.018 -7.5e-4 5e-4 Iron-cored coil Air-cored coil (a)

21 81 100 160

Length of Path 3 [mm] Iron-cored coil The Q -co m p o n en t o f th e m ag n etic flu x d en sity fo r Path 3 5e-4 -1e-3 -0.01 8e-3 FEA Subdomain model Air-cored coil (b)

(17)

21 81 100 160 Length of Path 4 [mm]

FEA Subdomain model

Iron-cored coil

The

r

-co

m

p

o

n

en

t

o

f

th

e

m

ag

n

etic

flu

x

d

en

sity

fo

r

Pa

th

4

-0.015 0.025

-1.67e-3

1.65e-3

Air-cored coil

(a)

The

Q

-co

m

p

o

n

en

t

o

f the

m

ag

n

etic

flu

x

d

en

sity

fo

r

Path

4

0

-0.08 0.08

-0.013 0.013

21 81 100 160

Length of Path 4 [mm]

FEA Subdomain model

Iron-cored coil Air-cored coil

(b)

Figure 10.Waveform ofBfor Path 4: (a)r- and (b)Θ-component.

21 81 100 160

Length of Path 5 [mm]

The

r

-co

m

p

o

n

en

t

o

f

th

e

m

ag

n

etic

flu

x

d

en

sity

fo

r

Pa

th

5

-0.12 0.02

-8e-3

6e-3

FEA Subdomain model

Iron-cored coil Air-cored coil

(a)

21 81 100 160

Length of Path 5 [mm]

The

Q

-co

m

p

o

n

en

t

o

f the

m

ag

n

etic

flu

x

d

en

sity

fo

r

Path

5

0

-6e-4

6e-4

-1.2e-4

1.2e-4

FEA Subdomain model

Iron-cored coil Air-cored coil

(b)

Figure 11.Waveform ofBfor Path 5: (a)r- and (b)Θ-component.

Author Contributions:The work presented here was carried out in cooperation among all authors, which have

written the paper and have gave advice for the manuscripts.

Conflicts of Interest:The authors declare no conflict of interest.

Appendix A The 2-D General Solution of EDPs (i.e., Laplace’s and Poisson’s equations) in Polar Coordinates

Using the magnetostatic Maxwell’s equations (viz., the Maxwell-Ampere law, the Maxwell-Thomson law, and the magnetic material equation) [1], the general EDPs in terms of magnetic vector potentialA={0; 0;Az}withµ=Cstcan be expressed in polar coordinates(r,Θ)by

∆Az=

2A

z

r2

+1

r ·

Az

r +

1 r2·

2Az

Θ2

=ES, (A.1a)

ES=−

µ·Jz+µ0

r ·

MΘ+r·

r − Mr

Θ

. (A.1b)

(18)

material in whichµ0andµrare respectively the vacuum permeability and the relative permeability of

the magnetic material (withµr =1 for the vacuum orµr 6=1 for the magnets/iron).

The magnetic vector potentialAzis governed by Poisson’s equation (i.e.,ES6=0) or Laplace’s

equation (i.e.,ES=0). Using the method of separation of variables, the 2-D general solution ofAzin

both directions (i.e.,r- andΘ-edges) can be written as Fourier’s series

Az =AΘz +Arz+AzP, (A.2a)

AΘz =

C0Θ+D0Θ·ln(r)

· E0Θ+F0Θ·Θ

· · ·+ ∞ h=1

h ·rβh

· · ·+DΘh ·r−βh !

·

"

h ·cos(βh·Θ)

· · ·+FhΘ·sin(βh·Θ)

#

, (A.2b)

Arz=

Cr0+D0r·ln(r)

· E0r+F0r·Θ

· · ·+ ∞ n=1

(

Crn·cos[λn·ln(r)]

· · ·+Drn·sin[λn·ln(r)]

)

·

"

Enr ·ch(λn·Θ)

· · ·+Fnr·sh(λn·Θ)

#

, (A.2c)

whereAzPis the particular solution ofAzrespecting the second memberESin (A.1),C0Θ–FhΘ&C0r–Fhr

the integration constants,βh&λnthe spatial frequency (or periodicity) ofAzΘ &Arz, andh&nthe

spatial harmonic orders.

UsingB=∇ ×A, the components of magnetic flux densityB={Br;BΘ; 0}can be deduced by

Br = 1

r ·

Az

Θ and BΘ=− Az

r (A.3)

which leads to

Br =BΘr +Brr+

1 r·

AzP

Θ , (A.4a)

BΘr =

F0Θ r ·

C0Θ+D0Θ·ln(r)

· · ·+ ∑∞ h=1

βh

r ·

h ·rβh

· · ·+DhΘ·r−βh !

·

"

−EΘh ·sin(βh·Θ)

· · ·+FhΘ·cos(βh·Θ)

#

, (A.4b)

Brr =

F0r r ·

C0r+Dr0·ln(r)

· · ·+ ∞ n=1

λn

r ·

(

Crn·cos[λn·ln(r)]

· · ·+Dnr ·sin[λn·ln(r)]

)

·

"

Ern·sh(λn·Θ)

· · ·+Fnr·ch(λn·Θ)

#

, (A.4c)

and

BΘ=BΘΘ+BrΘAzP

r , (A.5a)

BΘΘ=−

0

r · E0Θ+F0Θ·Θ

· · ·+ ∞ h=1

βh

r ·

ChΘ·rβh

· · · −DΘh ·r−βh !

·

"

h ·cos(βh·Θ)

· · ·+FhΘ·sin(βh·Θ)

#

, (A.5b)

BrΘ=−

Dr

0

r · E0r+F0r·Θ

· · ·+ ∞ n=1

λn

r ·

(

−Cnr ·sin[λn·ln(r)]

· · ·+Dnr ·cos[λn·ln(r)]

)

·

"

Enr ·ch(λn·Θ)

· · ·+Fnr·sh(λn·Θ)

#

(19)

  &

t

z r r

A F

ll r t r   & l

z r r

A G   &

l  

z r

A  L r

  &

r  

z r

A   R r

yr

 0 x rz A

 0

(a)   & & 0 t t r zr r

z r r

A A F           ll r t r   & & 0 l l r zr r

z r r

A A G             & & 0       l l r z r z r

A L r

A        & & 0       r r r z r z r

A R r

A   

 

yr

 0 x rr z z

AA

  0

(b)

Figure B.1.Azimposed on all edges of a region:(a)General and(b)Principle of superposition.

  &

t

z r r

A F

ll r t r   & l

z r r

A G

& 0

l  

z r

A 

& 0

r  

z r

A 

yr

 0 x rz A

 0

Figure B.2.Particular case:Az=0 onΘ-edges andAzimposed onr-edges of a region.

Appendix B Simplification of Laplace’s Equations according to imposed BCs

Appendix B.1 Case-Study no 1: "Azimposed on all edges of a region"

FigureB.1ashows a region (forΘ∈[Θr,Θl]andr∈[rl,rt]) whoseAzimposed on all edges. By

respecting the BCs and applying the principle of superposition on the magnetic quantities, FigureB.1a is redefined by FigureB.1b.

In the case-study no 1,Az= AΘz +Arz, i.e., (A.2), is redefined by

AΘz =

h=1

" cΘh ·rl·

E

σ(βh,rt,r)

E

σ(βh,rt,rl)

+dΘh ·rt·

E

σ

(βh,r,rl)

E

σ

(βh,rt,rl)

#

·sin[βh·(Θ−Θr)], (B.1a)

Arz=

n=1

ern·

sh[λn·(Θl−Θ)]

sh(λτΘ) +f

r n·

sh[λn·(Θ−Θr)]

sh(λτΘ)

·rl·sin

λn·ln

r rl

, (B.1b)

the componentBr =BΘr +BrrofB, i.e., (A.4), by

BrΘ= ∞

h=1

β

" cΘh ·rl

r · E

σ

(βh,rt,r)

E

σ(βh,rt,rl)

+dΘh ·rt

r · E

σ

(βh,r,rl)

E

σ

(βh,rt,rl)

#

(20)

Brr =

n=1

λ

−ern·

ch[λn·(Θl−Θ)]

sh(λτΘ) +f

r n·

ch[λn·(Θ−Θr)]

sh(λτΘ)

· rl

r ·sin

λn·ln

r rl

, (B.2b)

and the componentBΘ=BΘΘ+BΘr ofB, i.e., (A.5), by

Θ=−

h=1

β

"

−cΘh ·rl

r · P

σ

(βh,rt,r)

E

σ(βh,rt,rl)

+dΘh · rt

r · P

σ

(βh,r,rl)

E

σ(βh,rt,rl)

#

·sin[βh·(Θ−Θr)], (B.3a)

BrΘ=−

n=1

λ

ern·

sh[λn·(Θl−Θ)]

sh(λτΘ) + f

r n·

sh[λn·(Θ−Θr)]

sh(λτΘ)

· rl

r ·cos

λn·ln

r rl

, (B.3b)

where cΘh, dΘh, ern and fnr are new integration constants; βh = h·π/τΘ with τΘ = Θl −Θr;

λn = n·π/τrwithτr=ln(rt/rl); andE

σ

(w,x,y)&P

σ

(w,x,y)are [44]

E

σ(w,x,y) =

x y

w

−y

x w

and P

σ(w,x,y) =

x y

w

+y

x w , (B.4) with E

σ(w,x,y)

x =

w x ·Pσ

(w,x,y) and Eσ

(w,x,y)

y =−

w y ·Pσ

(w,x,y), (B.5a)

P

σ(w,x,y)

x =

w x ·Eσ

(w,x,y) and Pσ

(w,x,y)

y =−

w y ·Eσ

(w,x,y). (B.5b) WhenAz=0 onΘ-edges andAzimposed onr-edges (see FigureB.2),AzwithArz=0 in (B.1) is

expressed by

Az =

h=1

" cΘh ·rl·

E

σ(βh,rt,r)

E

σ(βh,rt,rl)

+dΘh ·rt·

E

σ

(βh,r,rl)

E

σ

(βh,rt,rl)

#

·sin[βh·(Θ−Θr)], (B.6a)

ther-component ofBwithBrr=0 in (B.2) by

Br =

h=1

β

" cΘh ·rl

r · E

σ

(βh,rt,r)

E

σ

(βh,rt,rl)

+dΘh ·rt

r · E

σ

(βh,r,rl)

E

σ

(βh,rt,rl)

#

·cos[βh·(Θ−Θr)], (B.6b)

theΘ-component ofBwithBΘr =0 in (B.3) by

BΘ=− ∞

h=1

β

"

−cΘh ·rl

r · P

σ

(βh,rt,r)

E

σ(βh,rt,rl)

+dΘh ·rt

r · P

σ(βh,r,rl)

E

σ(βh,rt,rl)

#

·sin[βh·(Θ−Θr)]. (B.6c)

Appendix B.2 Case-Study no 2: "Brand Azare respectively imposed on r- andΘ-edges of a region"

FigureB.3ashows a region (forΘ∈ [Θr,Θl]andr∈ [rl,rt]) whoseBrandAzare respectively

imposed onr- andΘ-edges. By respecting the BCs and applying the principle of superposition on the magnetic quantities, FigureB.3ais redefined by FigureB.3b.

In the case-study no 2,Az= AΘz +Arz, i.e., (A.2), is redefined by

AΘz =

c0Θ·rl·lnln((rrtt//rr)

l)+d

Θ

0 ·rt·

ln(r/rl)

ln(rt/rl)

· · ·+ ∞ h=1

h ·rl·

E

σ(βh,rt,r)

E

σ(βh,rt,rl)

+dΘh ·rt· E

σ(βh,r,rl)

E

σ(βh,rt,rl)

·cos[βh·(Θ−Θr)]

, (B.7a)

Arz=

n=1

ern·

ch[λn·(Θ−Θr)]

sh(λτΘ) −f

r n·

ch[λn·(Θl−Θ)]

sh(λτΘ)

· rl

λn

·sin

λn·ln

r

rl

Figure

Figure 1. Physical and geometrical parameters (see Table 1) of air- or iron-cored coil where ⊗ and ⊙are respectively the forward and return conductor.
Figure 2. Definition of regions in the air- or iron-cored coil.
Figure 3. Boundary conditions (BCs) in both directions (i.e.,(c) r- and Θ-edges): (a) Region 1, (b) Region 2, Region 3, (d) Region 4, (e) Region 5, (f) Region 6, and (g) Region 7.
Figure 4. 2-D finite-element analysis (FEA) mesh for the air- or iron-cored coil.
+7

References

Related documents

Reciprocating pumps are NOT acceptable. • Cargo manifolds, bunker connections and lifting equipment MUST meet OCIMF ‘Recommendations for Oil Tanker Manifolds and

The most recent release of the OmniPCX Office RCE offering is appliance-based and provides support for the UC functionality: voice; mobility (mobile UC, secure access control

The other term can be found from the series solution either by an appropriate superposition of the two solutions we found, or by going back to the series coefficients and choosing

Using figure 10.1.3 as an example, the point shown has rectangular coordinates... This makes it very easy to convert equations from rectangular to

Easy to follow guitar diagrams will be used to show how to play exercises, scales, chords, and arpeggios.. The diagram on the next page represents a picture of the

As always, to evaluate the double integral, we need to rewrite it as an iterated integral (this time, in polar coordinates).. Let’s make slices where θ

[10%] Suppose that u(r, θ) is harmonic in the open disc of radius 2 and continuous on the closed disc of radius 2, where (r, θ) are polar coordinates... In that case, the

attempts, the Idea Ontology related research does not go into details of how producing structured metadata could aid Idea Management nor does it propose any data linking solutions