Modelling of eddy currents in steel laminations

M.Sc. (Tech.) Jenni Pippuri

2

Content

• Introduction• Methods

– General remarks on the three-dimensional (3-D) modelling of rotating electrical machines

– Inclusion of eddy currents in steel sheets in the two-dimensional (2-D) analysis of rotating electrical machines

• One-dimensional (1-D) diffusion equation• Mesh-free eddy-current modelling• Advanced hybrid method

• Results– Application: 12500-kW synchronous motor

• Conclusions

3

Introduction

• Careful analysis of a rotating electrical machine requires solving its magnetic field

• From the field formulation, the hysteresis, eddy-current and excess effects in the laminated iron core and the losses associated to those are typically excluded– Magnetic properties defined by a single-valued magnetisation

curve– Iron losses computed as post-processing by some statistical or

empirical method

• In this presentation, methods to include losses in steel laminations induced by eddy currents in the 2-D finite-element analysis (FEA) are presented– Applied to compute a synchronous machine

4

2-D analysis of electrical machines: Statistical theory of iron losses• The 2-D FEA often applied to study (radial flux) rotating

electrical machinery– Field and winding equations solved together– Laminated core modelled as a non-conducting, non-linear

medium• Single-valued magnetisation curve• Hysteresis, eddy-currents, and excess effects excluded from the

field solution• Losses from those phenomena post-processed

( ) ( ) ( )( )m

t h c e

2 1.52h s c s e s

1

N

m m mm

P P P P

C mf B C mf B C mf B=

= + +

= + +∑

5

2-D analysis of electrical machines: 1-D eddy-current model of laminations• Assuming that the width

and length of the sheets are much greater than their thickness

• Since coupled to the 2-D model the boundary conditions imposed by the 2-D field

( )( ) ( )

( )( ) ( )

,, 0

,, 0

xx

yy

A z tA z t

tA z t

A z tt

ν σ

ν σ

∂∇⋅ ∇ − =

∂∂

∇ ⋅ ∇ − =∂

z

y

x

2d

z

y

x

2d

z

y

x

2d

z

y

x

2d

6

2-D analysis of electrical machines: 1-D eddy-current model of laminations

• 2-D formulation in the domain of laminations

• Backward-Euler, finite-element, Newton-Raphson

y

x

νB2-D

H = ν1-DB1-D

Hc = H – νB2-D

y

x

νB2-D

H = ν1-DB1-D

Hc = H – νB2-D

( )( ) ( )( )( ) ( ) ( )

c

c

, , , , 0

, , , , ,z z

A x y t x y t

x y t d t x y t

ν

ν

∇ ⋅ ∇ − ∇× =

= − −

H

H H B

( ) ( ) ( )( ) ( ) ( )

, , , 0, 0

, , , 0, 0x y x

y x y

A d t dB x y t A t

A d t dB x y t A t

− = =

− = − =

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2-D analysis of electrical machines: Mesh-free eddy-current model of laminations• M.Sc. (Tech.) Paavo Rasilo• The mesh-free lamination model

( )( )

( )( ) ( )

( )( )

s 0 02

1 10 , d 2d

d

h t z b th z t z z d b t

t−

⎡ ⎤ ⎡ ⎤ ⎡ ⎤∂⎢ ⎥ ⎢ ⎥ ⎢ ⎥= +⎢ ⎥ ⎢ ⎥ ⎢ ⎥∂

⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦∫

M M M

αα σ

( ) ( ) ( )

( ) ( ) ( )

F 1

0

,

2cos , 2

N

n nn

n

b z t b t z

z nkz kd

α

πα

−

=

=

= =

∑

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2-D analysis of electrical machines: Mesh-free eddy-current model of laminations• 2-D equation in the domain of electrical steel sheets

• Crank-Nicolson, finite-element, Newton-Raphson

( )( )( ) ( )

( )( )

( ) ( ) ( )

0 02

1 1

z

, ,, , , d 2 , ,

, , , , , ,

d

d

n n n

z x y tx y z t z z d x y t

t

x y t x y t A x y t

−

⎛ ⎞⎡ ⎤ ⎡ ⎤⎜ ∂ ⎟⎢ ⎥ ⎢ ⎥∇× + =⎜ ⎟⎢ ⎥ ⎢ ⎥∂⎜ ⎟⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎝ ⎠

⎡ ⎤= ∇× = ∇× ⎣ ⎦

∫B

H B 0

B A u

M M

αα σ

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2-D analysis of electrical machines: Advanced hybrid model of laminations• Dr. (Tech.) Emad Dlala• Advanced hybrid model of electrical steel sheets• The hysteresis part defined by any suitable static

hysteresis model• Scaling function applied in the eddy-current part in

order to approximate the skin-effect

( ) ( ) ( ) ( )

( )( ) ( ) ( )

0.5

´h c e

2

0 1 2 0 0 1 revs s

d d, ,

d dB t B t

H t B H t B C Ct t

B t B tB t a a a a B

B B

= + +

⎛ ⎞ ⎛ ⎞= + + = +⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

γ

δ δ

γ δ α α

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2-D analysis of electrical machines: Advanced hybrid model of laminations• 2-D formulation in the domain of steel laminations

• Crank-Nicolson, finite-element, fixed-point• Locally convergent fixed-point method proposed by Dr.

Dlala with the fixed-point coefficient

• n time-step, k iteration• Converges fast, in 10 to 20 iterations

( ) ( ) ( )( )( ) ( )FP

FP

, , , , , ,

, , , ,

x y t x y t x y t

x y t x y t

= +

∇× ∇× = −∇×

H B M

A M

ν

ν

( )

( ) ( )11

FP 2

kkyx

n x y

HHB B

C

== ∂∂+

∂ ∂=ν

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Test case: 12500-kW Synchronous motor

• Slide by M.Sc. (Tech.) Paavo Rasilo• A synchronous motor

for an extruder

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Test case: 12500-kW Synchronous motor

• Test motor simulated at the rated load with the discussed models

• Both sinusoidal and an input voltage produced by pulse-width modulation applied

• Particularly, the eddy-current losses in the core of the machine of interest

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Test case: 12500-kW Synchronous motor

• Sinusoidal supply voltage• First order 2-D elements• 400 time-steps per

period

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Test case: 12500-kW Synchronous motor

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Test case: 12500-kW Synchronous motor

• PWM supply• First order 2-D elements• 1600 time-steps per

period

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Test case: 12500-kW Synchronous motor

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Conclusions

• Methods for modelling the steel laminations in the 2-D field analysis were presented and utilised to study a synchronous motor– Active research going on in the field– Simulation results should be verified by measurements

• 1-D model– Advantage: Field in the lamination solved– Disadvantage: Possible convergence problems, computational

time

• Mesh-free model– Advantage: Field in the lamination solved, stable– Disadvantage: Computational time

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Conclusions

• Hybrid model– Advantage: Stable, computationally efficient, hysteresis and

excess losses included in the 2-D field solution– Disadvantage: Field in the lamination thickness not solved

• Further reading:– Dlala E., “Comparison of Models for Estimating Magnetic Core Losses in

Electrical Machines Using the Finite-Element Method”, IEEE Trans. on Magn., Feb. 2009, Vol. 45, No. 2, pp. 716–725.

– Dlala E., Belahcen A., and Arkkio A., “Locally Convergent Fixed-Point Method for Solving Time-Stepping Nonlinear Field Problems”, IEEE Trans. on Magn., Nov. 2007, Vol. 43, No. 11, pp. 3969–3975.

– Rasilo P. and Arkkio A., “A Mesh-Free Model of Eddy-Current Losses for 2D Analysis of Ferromagnetic Laminations”, Compumag 2009, Brazil

– Pippuri J. and Arkkio A., “Inclusion of Eddy Currents in Laminations in Two-Dimensional Finite Element Analysis”, Compumag 2009, Brazil