XIX International Conference on Electrical Machines - ICEM 2010, Rome
Advanced Calculation of Eddy Current Losses in PMSM with tooth windings C. Bode, W.-R. Canders
Abstract -- The paper deals with a new analytical approach for calculating eddy current losses in permanent magnet synchronous machines with tooth windings. The analytical method presented here differs from known approaches in two points: The field is calculated using an all-in-one method, so the simultaneous calculation of stator and no-load field is possible. To take finite magnet dimensions into account, the calculation of an easy-to-use correction factor for endless dimensions is introduced. Losses caused by stator slotting as well as losses caused by stator harmonics are included. To allow for several load cases the stator current phase angle and the eddy current reaction field is considered. Index Terms—eddy currents, permanent magnet, loss
Fig. 1: Coordinate System and Dimensions
A. I.
T
INTRODUCTION
HERE are three known principles for calculating eddy current losses in permanent magnets. One possibility is the use of a 3D-FEM approach [1]. A second method is the combination of FEM field calculation and an analytical approach using the results of the FEM calculation [2]. The third method is the analytical treatment of fields and losses [3], [4], [5]. The advantage of the third method is that it is possible to embed the calculation of the eddy current losses into the design procedure. The analytical method presented in this paper differs from known approaches in two points: The field is calculated using an all-in-one method, so the simultaneous calculation of stator and no-load field is possible. To take finite magnet dimensions into account, the calculation of an easy-to-use correction factor for endless dimensions is introduced. II.
CALCULATION OF AIR GAP FIELD
Normally the no-load flux density is calculated separately from the armature flux density. Using the described method, the air-gap field is the result of only one calculation. For calculating the air-gap filed the magnetic vector potential is used. To reduce the complexity, some simplifications are necessary: the relative permeability of the iron is assumed to be infinite and the error due to the use of a Cartesian coordinate system is neglected. The coordinate system and all relevant dimensions are shown in Fig. 1. The stator is represented by an equivalent current sheet. The magnetisation of the permanent magnets is replaced by an equivalent current density. The stator slotting is taken into account using a special boundary condition. 1
C. Bode is with Technical University Braunschweig, Institut for Electrical Machines, Traction and Drives, Braunschweig, Germany (e-mail:
[email protected]) W. R. Canders is with Technical University Braunschweig, Institut for Electrical Machines, Traction and Drives, Braunschweig, Germany (e-mail:
[email protected]) 978-1-4244-4175-4/10/$25.00 ©2010 IEEE
Representation of stator winding The stator currents are given by a symmetrical sinusoidal three-phase-system: i = Iˆ cos ( ω t − φ − 120° ) , u
s
s
s
iv = Iˆs cos ( ωs t − φs ) ,
(1)
iw = Iˆs cos ( ωs t − φs + 120° ) For the following steps it is appropriate to represent the current sheet by Fourier series. Using the real representation the current sheet is given by ∞
∑ Jˆ
J s ( x, t ) =
l =0 ∞
+
∑
s,ν
(
cos ωs t − ϕ s − ν ap x
(
)
Jˆs,ν cos ωs t − ϕ s + ν ap x
l =0
with the Fourier coefficients 12 Iˆs jνπ ⎛ ⎛τ b e sin ⎜ν an ⎜ n − ss Jˆs,ν = νπ bss ⎝ ⎝2 4 and with
ν = 3 l +1
)
(2)
ν = 3l + 2
⎞ ⎞ sin ⎛ν a bss ⎞ ⎟⎟ ⎜ n ⎟ 4 ⎠ ⎠⎠ ⎝
ap = π τ p .
(3)
(4)
For abbreviation the phase angle is written as
ϕ s = φs + 90° .
(5)
Current sheet waves of order ν=3l+1 run in direction of positive x-coordinate. Waves of the order of ν=3l+2 run in opposite direction. The following calculation gets easier by transforming (2) in a complex representation, thus the Fourier series of the current sheet is written as ∞
∞
∑ ∑ Jˆ
* s ,(ν 1 ,ν 2 )
e
( (
j ν 1 ωs t − ϕ s
) −ν a x ) 2
p
.
(6)
ν 1 =−∞ ν 2 = −∞
For the transformation the complex notation of cosine was used. The coefficients of the quasi-Fourier series are given by Jˆs , ν Jˆs*,(ν ,ν ) = (7) (η1 (ν 1 ,ν 2 ) + η 2 (ν 1 ,ν 2 ) ) . 2 The auxiliary functions are represented by 2
1
2
⎧1 if (ν 1 =1,ν 2 =3l +1) ∨ (ν 1 = − 1,ν 2 = − ( 3l +1) )
η1 (ν 1 ,ν 2 ) = ⎨
(8)
⎩0 other ⎧1 if (ν 1 =1,ν 2 = − ( 3l +2 ) ) ∨ (ν 1 = − 1,ν 2 =3l +2 ) (9) η 2 (ν 1 ,ν 2 ) = ⎨ ⎩0 other
Representation of magnets To take the no-load field into account, the magnets are replaced by an equivalent current density. Therefore, the edge current of the magnets is expanded to its complex Fourier series
B.
J M ( x, t ) =
∞
∑ Jˆ ν 1 =∞
M,ν 1
e
(
jν 1 ωs t − ap x
with the coefficients H ⎡ ⎛ απ JˆM,ν1 = c ⎢ 2sin ⎜ν 1 τp ⎣ 2 ⎝ To simplify the following rewritten to J M ( x, t ) =
∞
∞
∑ ∑ Jˆ
ν 1 =∞ ν 2 =∞
)
(10)
⎞ ⎛ π ⎞⎤ (11) ⎟ sin ⎜ν 1 ⎟ ⎥ . ⎠ ⎝ 2 ⎠⎦ steps the current density (10) is
* M,(ν1 ,ν 2 )
e
(
j ν1ωs t −ν 2 ap x
)
.
(12)
Because of the double sum a modification of (11) is necessary. The new coefficient is represented by * ˆ JˆM,( (13) ν1 ,ν 2 ) = η3 (ν 1 ,ν 2 ) J M,ν 1 with the auxiliary function ⎧1 if ν 1 = ν 2 η3 (ν 1 ,ν 2 ) = ⎨ . ⎩0 others
(14)
Region II reaches from y=0…hm and consist of magnetic material. The distribution of the magnetic vector potential Az is described by Poisson’s equation in Cartesian coordinates: ∂ 2 Az ∂ 2 Az + = jωe μ0 μ rσ Az − μ0 μr J z . (18) ∂x 2 ∂y 2 The first term on the right side of (18) describes the induced eddy current in a conductive medium. The second term of the right side refers to an external prescribed current density. Region I consists of air, thus σ=0, Jz=0. In this case (18) reduces to Laplace’s equation. In region II the conductivity is σ=σm and the current density is Jz=JM. A suitable approach for the separation of variables in (17) considers the spectrum of the slot harmonics and may be written as Az =
1 =∞
Orders of slot harmonics Before solving the Poisson-equation describing the airgap field it is helpful to know the expected orders of slot harmonics. The numbers of order are calculated by the method presented in [6]. Because only the orders of slot harmonics are of interest, a 1-dimensional field distribution is assumed. The effect of stator slotting is taken into account by using a relative magnetic conductance function, which is given by ∞ Λ λ ( x ) = 1 + ∑ n cos ( nNs ap x ) . (15) n =1 Λ 0
where Λ0 is the magnetic conductance in the case of a smooth air-gap and Ns is the number of slots per pole pair. The distribution of the harmonic of order ν1 of the flux density without presence of slotting is given by B = Bˆ cos ν (ω t − a x ) . (16) ν1
ν1
(
1
s
p
)
The resulting flux density distribution by taking the slotting into account is the result of multiplication of (15), (16) Bres,ν1 ( x ) = Bν1 λ ( x ) ∞ (17) . = ∑ Bˆν1 ,ν 2 cos (ν 1ωs t −ν 2 ap x ) n =−∞
ν 2 =ν 1 + nNs
Because of the stator slotting, harmonics of order ν2=ν1+nNs with n=0, 1, 2 are generated. Only the orders of harmonics are relevant, thus the amplitudes of (17) are not specified here. D.
Solving Poisson’s equation For calculating the vector potential the configuration is divided into two geometric regions as shown in Fig. 1. Region I reaches from y=hm…hm+δ and consists of air.
∞
j ν 1ωs t −ν 2 ap x
z,
2 =∞
1, 2
) ν 2 =ν1 + nNs
.
(19)
Insertion of (19) in (18) gets an ODE for the vector potential: ∂ 2 Aˆz,(ν1 ,ν 2 ) − γ 2 Aˆz,(ν1 ,ν 2 ) = − μ0 μ r J z,(ν1 ,ν 2 ) . (20) ∂y 2 The complex constant γ is given by γ 2 = ν 22 ap2 + jωe μ0 μ rσ
(21)
with the angular frequency of the induced eddy current ωe. A possible solution is given by a linear combination of hyperbolic function and a constant:
Aˆ z(ν ,ν ) ( y ) = A (ν 1 ,ν 2 ) sinh ( γ y )
. (22) + B (ν 1 ,ν 2 ) cosh ( γ y ) + C (ν 1 ,ν 2 ) The hyperbolic functions represent the homogenous part of the solution while the constant C describes the inhomogeneous part of solution. The constants A, B will be defined by boundary conditions. The constant C is defined by inserting C in (20), thus C is given by ⎧ μ 0 μ r Jˆz ,(ν ,ν ) if ν 1 = ν 2 ⎪ 2 C (ν 1 ,ν 2 ) = ⎨ (23) γ 1
C.
∞
( Aˆ (ν ν ) ( y ) e ∑ ∑ ν ν
2
1
2
⎪0 ⎩
others. In region I where no current density exists C disappears. In the absence of conductive material, γ becomes γ I 2 = ν 22 ap2 (24) and for the magnet material γ II 2 = ν 22 ap2 + jωe μ0 μrmσ m .
(25)
The unknown angular frequency ωe is defined by transformation from the stator coordinate system into the rotor coordinate system. The coordinate transformation is given by the substitution ap x → ωs t + ap x ' . (26) Thus, the angular frequency is defined by ωe = (ν 1 −ν 2 ) ωs .
(27)
To get the distribution of the flux density G G B = curl A (28) is used. The x-component of the magnetic flux density is given by Bx ( x, y , t ) =
∞
∞
∑ ∑ Bˆ
x ,(ν 1 ,ν 2 )
ν 1 = −∞ ν 2 = −∞
with the amplitude
( y)e (
j ν 1ωs t −ν 2 a p x
)
(29)
Bˆ x ,(ν ,ν 1
2
)
( y ) = γ [ A (ν
1
,ν 2 ) cosh ( γ y ) + B (ν 1 ,ν 2 ) sinh ( γ y )] .
(30)
The y-component is given by By ( x, y, t ) =
∞
∞
∑ ∑
ν1 =−∞ ν 2 =−∞
j (ν ω t −ν a x ) Bˆ y,(ν1 ,ν 2 ) ( y ) e 1 s 2 p
(31)
with the amplitude Bˆ y, (ν ,ν ) ( y ) = jν 2 ap [ A (ν 1 ,ν 2 ) sinh ( γ y )
(32) + B (ν 1 ,ν 2 ) cosh ( γ y ) + C (ν 1 ,ν 2 )] . The next steps are necessary to adapt the constants A, B to the boundary conditions respective transition conditions of the examined geometry. The first step refers to the boundary at y=0 and at the intersection at y=hm. The boundary at y=hm+δ will be considered in a separate step. The conditions are given by Bx,II ( x, y = 0, t ) = 0, (33) 1
2
μ rm Bx,I ( x, y = hm , t ) = Bx,II ( x, y = hm , t ) ,
(34)
By,I ( x, y = hm , t ) = By,II ( x, y = hm , t ) .
(35)
Inserting (29), (30), (31), (32) into (33), (34), (35) the constants are calculated to γ C − AI [γ II sinh ( γ I hm ) BI (ν 1 ,ν 2 ) = II II γ II cosh ( γ I hm ) (36) − μ rm γ I coth ( γ II hm ) cosh ( γ I hm )] − μ rm γ I coth ( γ II hm ) sinh ( γ I hm )
AII (ν 1 ,ν 2 ) = 0,
III. CALCULATING OF EDDY CURRENT LOSSES The induced electrical field in the magnetic material is given by ∂A Ez ( x , y , t ) = − z . (41) ∂t Transformed in the rotor coordinate system and by taking into account the harmonic time dependence, the electrical field is written as
Ez ( x , y , t ) = −
+CII sinh ( γ I hm ) ⎤⎦
e
{
PM =
Fig. 2: Boundary condition at y=hm+δ
1 2
=−
− ν 2 ) ωs Aˆ z ,(ν ,ν 1
2
)
( y)
(
j (ν 1 −ν 2 )ωs t −ν 2 ap x '
)
(42)
.
⎧
}
∫
⎫
ℜ ⎨ Ez ( x, hm , t ) H x ( x, hm , t ) da ⎬ *
⎩A
⎭
2
ατ p 2μ0
⎧⎪
ℜ⎨
∞
∞
∑∑
⎪⎩ν1 =∞ ν 2 =∞
j (ν 1 −ν 2 )ωs Aˆ zI,(ν1 ,ν 2 ) ( hm ) (44)
}
* Bˆ xI, (ν1 ,ν 2 ) ( hm ) .
(38)
Using the described method, a system of linear equations for AI is attained: L (ν 2 ) AI (ν 2 ) = R (ν 2 ) . (40)
1
The calculation of eddy current losses may be based directly on (42). But the procedure is simplified considerably by using Poynting’s vector. The time average about a period of stator frequency is given by G 1 G G* S = ℜ E×H . (43) 2 The losses can be calculated by integration (43) over the surface of the magnet as depicted in Fig. 3. It can be shown, that only the integral over face A2 has to be taken into account. Thus, the eddy current losses in one magnet are given by
,
. − μ rmγ I cosh ( γ II hm ) sinh ( γ I hm ) The only unknown is the constant AI. It is calculated using the boundary condition at y=hm+δ. A possibility is the use of a 2D-permeance function described in [7]. In this paper the method proposed by [3], [8] is used. In this case the stator slotting is replaced by equivalent contour ys(x) representing the effect of slotting. Applying Ampere’s Law on the infinitesimal loop shown in Fig. 2, the following boundary condition is found: μ0 J s ( x, t ) = Bx,I ( x, hm + δ ) dx + By,I ( x + dx, hm + δ ) (39) ys ( x + dx ) − By,I ( x, hm + δ ) ys ( x ) .
∞
ν 1 =−∞ ν 2 = −∞
(37)
μ rmγ I [ AI BII (ν 1 ,ν 2 ) = γ II sinh ( γ II hm ) cosh ( γ I hm )
∞
∑ ∑ j (ν
So the cumulative losses of all magnets are given by PV = 2 pPM .
(45)
Fig. 3: Magnet surfaces for loss calculation
A.
Effect of finite length and width of magnets Until now, the magnets are assumed to be extended infinitely along the z-direction. In this case only a zcomponent of the electrical field exists. In reality there are finite dimensions along x- and z-direction, representing a subdivision into several regions, which are electrically insulated from each other. The resulting effect is similar to an increasing of the material resistivity and a decreasing of the losses. So it’s a desire to the real dimensions of the magnets along x- and z-direction into account. In this chapter an approach is presented for calculating factors for correction of finite dimensions. The calculation of the correction factor comprises two steps. In the first step, only the finite dimension in z-direction is taken into account, whereas the finite dimension along the x-coordinate is considered in the second step. Supposing only the dominating y-component of the flux density induces eddy currents in the magnets the calculation is simplified.
Moreover, the change of the eddy current reaction field compared to the 2D loss calculation is neglected. In the case of a low eddy current reaction field (at small angular speeds) this supposition achieves good results. For the first step the coordinate system shown in Fig. 4 is used. Because of the simplifications only electric field components in the x’z-plane are expected.
Eˆ x ( z ) = E sinh (ν ap z ) .
(58)
In the following step the subdivision of the magnets in xdirection is taken into account. It is impossible to use a boundary condition in this case because there is no mathematical bound parallel to the z-axis. Thus, an applicable approach of the stimulating flux density is necessary. Using Schwarz’ principle [9] (Fig. 5), a convenient description of the flux density is given by ∞
* * By ( x ', t ) = ∑ Bˆ y , n sin ( nam x ' )
(59)
n=0
with a m = nx π
τm .
(60)
The Fourier coefficients are calculated as τ ⎞ ⎤ ⎛ ⎡ 2n ⎢ν a + na cos ⎜ ω t − (ν a + na ) 2n ⎟ ⎥ ⎝ ⎠ ⎥ ⎢ ⎢ ⎥ τ ⎞ ⎛ sin ⎜ (ν a + na ) ⎢ ⎥ ⎟ 2n ⎠ Bˆ y ⎢ ⎝ * ⎥. Bˆ y , n = − ⎢ τm 2n τ ⎞⎥ ⎛ cos ⎜ ω t − (ν a − na ) ⎢+ ⎟⎥ 2n ⎠ ⎥ ⎝ ⎢ ν a − na ⎢ ⎥ τ ⎞ ⎛ ⎢ ⎥ sin ⎜ (ν a − na ) ⎟ 2n ⎠ ⎝ ⎣⎢ ⎦⎥ x
m
0
p
p
m
m
x
m
p
m
x
Fig. 4: Finite dimensions in z-direction are taken into account
x
The induced electric field is described by Faraday’s law G G ∂B curl E = − . (46) ∂t An additional condition is G div E = 0 . (47) Using (46), (47) the PDE of the induced electric field is written as 2
∂ Ez
m
0
p
p
m
m
x
m
p
m
x
(61)
2
∂ By
2
∂ Ez
+ = . (48) 2 2 ∂z ∂x ' ∂t ∂x ' To calculate the correction factor the flux density is assumed as B ( x ', t ) = Bˆ cos ( ω t − ν a x ' ) . (49) y
y
0
p
Thus, the electric field is of the form Ez ( x ', z , t ) = Eˆ z ( z ) cos ( ω0 t − ν ap x ' ) .
(50)
Inserting (49), (50) in (48) a PDE of second order for the electric field is given by 2 ∂ Eˆ z ( z ) 2 − (ν ap ) Eˆ z ( z ) = ω0ν ap Bˆ y (51) 2 ∂z with the general solution Eˆ z ( z ) = D sinh (ν ap z ) + E cosh (ν ap z ) + F . (52) The constants D, E are defined by the boundary condition while constant F is obtained by insertion of F in (51) F = − ω0 Bˆ y ν ap . (53) No electric field exists on the boundaries at z=-lm/2, z=lm/2 Eˆ z ( z = − lm 2 ) = 0, (54) Eˆ z ( z = lm 2 ) = 0, thus
D = 0,
(55)
E = − F cosh (ν ap lm 2 )
(56)
Using (47), the x-component of the electric field is written as E ( x ', z , t ) = Eˆ ( z ) sin ( ω t − ν a x ' ) (57) x
with
x
0
p
Fig. 5: Using Schwarz’ principle to account for finite x-dimensions
Insert (59), (61) in (50), (52), the z-component of the electric field is given by ∞
Ez ( x ', z , t ) = ∑ ( E1 cosh ( nam z ) + F1 ) n=0
(
sin ω0 t − (ν ap + nam ) 2 n − nam x ' τm
x
+ ( E2 cosh ( nam z ) + F2 )
(
sin ω0 t − (ν ap + nam ) 2 n + nam x ' τm
x
+ ( E3 cosh ( nam z ) + F3 )
(
sin ω0 t − (ν ap − nam ) 2 n − nam x ' τm
x
+ ( E4 cosh ( nam z ) + F4 )
(
sin ω0 t − (ν ap − nam ) 2 n + nam x ' τm
x
whereas the x-component can be calculated as
) ) (62) ) )
⎧ sin(x x ) si ( x ) = ⎨ ⎩1
∞
Ex ( x ', z , t ) = ∑ E1 sinh ( nam z ) n=0
(
cos ω0t − (ν ap + nam ) 2 n − nam x ' τm
x
+ E2 sinh ( nam z )
(
cos ω0t − (ν ap + nam ) 2 n + nam x ' τm
x
+ E3 sinh ( nam z )
(
cos ω0t − (ν ap − nam ) 2 n − nam x ' τm
x
+ E4 sinh ( nam z )
(
cos ω0t − (ν ap − nam ) 2 n + nam x ' τm
x
) ) (63)
if x ≠ 0
. (77) if x = 0 Using (76) a correction factor for different orders of the stimulating flux density waves and number of magnet subdivisions can be calculated. Fig. 6 shows the correction factor βp as a function of the order of harmonics for different subdivisions nx, nz.
) ).
The constants are written as
(
lm
E1 = F1 cosh nam
2 nx
),
(64)
ω nx 1 F1 = Bˆ y sin namτ m ν ap + am n
( ( na
+ ν ap ) 2 n
τm
m
x
),
(65)
E2 = E1 ,
(66)
F2 = F1 ,
(67)
(
),
lm
E3 = F3 cosh nam
2 nx
(68)
ω nx 1 F3 = Bˆ y sin namτ m ν ap − am n
( ( na
− ν ap ) 2 n
τm
m
x
Fig. 6: Correction factor for finite magnet dimensions depending on nx, nz
IV. RESULTS
),
(69)
E 4 = E3 ,
(70)
F4 = F3 .
(71)
In this paragraph the calculated magnetic field and the eddy current losses are discussed for the treated machine shown in Fig. 7.
The average over time and volume of the square of the electric field is given by
E
2
(n
x
T
1 n x nz
, nz ) =
τm
lm
nx
2
∫ ∫ ∫ (E
T τ m lm
2 x
2
)
+ Ez dz dx ' dt .
(72)
0 0 − lm 2
In case of nx=0, nz=0 the electric field is written as ω Bˆ y Ez ( x ', t ) = − cos ( ω0 t − ν ap x ' ) . ν ap
(73)
Thus the average over time and volume is E
2
T
1
( 0, 0 ) = ∫ E T
2 z
dt .
(74)
0
Because PM ~ E2 the correction factor is defined by
β p ,( n , n ) (ν ) = E x
z
2
(n
x
, nz ) E
2
( 0, 0 ) .
(75)
Using (73) and (74), (75) is written as
β p ,( n , n ) (ν ) = x
y
1
∞
∑ 2 n=0
ν ap na m
⎡⎣si 2 ( ( nam + ν ap ) 2τn
m
+si
2
( (na
m
− ν ap ) 2 n
2 nz
nam l m
)
2
x
( (na
τm
m
+ ν ap ) 2 n
x
)
τm
2
m
p
2 nx
lm
m 2 nz
(76) with
Relevant geometrical and electrical data are listed in Tab. 1. The analytical results are compared with the numerical results calculated with Flux©. To get an impression of the accuracy of the analytical calculations the saturation of the iron has been ignored. Tab. 1: Geometrical and electrical data of example machine
( (na − ν a ) )⎤⎦ tanh ( na )) ⋅si
(
)
τm
+2 cos ( nπ ) si
⋅ 1−
x
Fig. 7: Treated machine
Description height of magnet width of slot opening pole pitch iron length number of pairs of poles remanent flux density rel. permeability of magnets electric conductivity of magnets factor of pole coverage nominal slot current phase of nominal current
symbol hm bss τp le p Br
μrm σm α Is,n
ϕs,n
value 5.00 mm 7.00 mm 46.33 mm 100.00 mm 8 1.00 T 1.05 2⋅106 S/(Ohm m) 0.70 1500 A 0°
The armature field and the no-load field at y=hm for n=6000/ min are shown in Fig. 8. In both cases a good agreement is achieved. Fig. 9 shows the eddy current losses referred to the different origins. The subdivision of magnets is neglected, thus nx=0, nz=0. For the losses generated by the no-load field very good agreement between analytical and numerical calculation is achieved. The analytically calculated losses caused by the armature field differ from the numerically calculated. This difference increases with the speed. The resulting losses for no-load field, stator field and total air-gap field (n=6000/ min) are listed in Tab. 2. They are specified for different magnet division nx, nz using (76). Obviously, the losses decrease with increasing numbers of nz as may be expected. subdivisions nx,
Fig. 10: Eddy current losses vs. phase angle (n=6000/ min) Tab. 2: Eddy current losses for n=6000/ min
nx 1 4
Pv/W (no-load field) 5623.3 (nx=0, nz=0) nz=1 nz=2 4482.7 3708.0 2749.6 2465.8
Pv/W (stator field) 21257.3 (nx=0, nz=0) nz=1 nz=2 12814.3 8414.2 3532.8 3132.1
Pv/W (total field) 26208.0 (nx=0, nz=0) nz=1 nz=2 16720.1 11621.7 5817.4 5227.7
VI. REFERENCES
Fig. 8: Calculated flux densities for y=hm, n=6000/ min
Fig. 9: Eddy current losses vs. speed
As mentioned earlier, the presented model is also applicable to the treatment of different phase angles of the stator current. In Fig. 10 the total losses are plotted vs. the phase angle and compared to the sum of the separately calculated losses (n=6000/ min). For small phase angles good agreement between analytical and numerical calculation is achieved. V.
CONCLUSION
Using the presented model it is possible to embed the calculation of eddy current losses into the design process. This is important to prevent undue heating and demagnetization, especially for high-speed machines. The necessary subdivisions of magnets can be easily identified using the presented correction factor.
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