Antero Arkkio29/03/2010

Mat-1.3656 Seminar on numerical analysis and computational science

Modelling of electrical machines

Small cage induction motor

Synchronous machines

8 MW diesel-generator 270 MVA turbo-generator

Over 99% of the electrical energy used in Finland is produced in rotating electrical machines.

Induction machines

30 kW cage induction motor 1.7 MVA slip-ring generator

About 60% of the electrical energy is consumed by rotating electrical machines.

Basic equations of electromagnetic field

Five field variables (E, D, H, B, J) are needed to present a complete electromagnetic field.

Maxwell’s equations Material equations

In addition, the boundary conditions are needed.

ρ∇⋅ =∇⋅ =

∂∇× = −

∂∂

∇× = +∂

0

t

t

DB

BE

DH J

εσµ

===

D EJ EB H

Aφ -formulation

Eddy-current problems: J is unknown.

=>

as identically . φ is the reduced scalar potential of electric field.

Current density and the partial differential equation of vector potential

ν= ∇× ∇⋅ = ∇× = =0 B A B H J H B

( )ν∇× ∇× =A J

( )∂ ∇×∂ ∂∇× = − = − = −∇×

∂ ∂ ∂t t tAB AE φ∂

= − −∇∂tAE

σ σ σ φ∂= = − − ∇

∂tAJ E ν σ σ φ∂

∇× ∇× = − − ∇∂

( )tAA

( )φ∇× ∇ ≡ 0

Aφ -formulation II

Eddy-current regions:

Source-current regions:

Combined equation:

Additional equation: or

Continuity: is continuous

is continuous

ν σ σ φ∂∇× ∇× + + ∇ =

∂( ) st

AA J

σ σ φ∂= − − ∇

∂tAJ

= sJ J

σ σ φ∂⎛ ⎞∇ ⋅ + ∇ =⎜ ⎟∂⎝ ⎠0

tA

∇⋅ = 0J

σ φ∂⎛ ⎞+∇ ⋅⎜ ⎟∂⎝ ⎠tA n

φ∂⎛ ⎞+∇ ⋅⎜ ⎟∂⎝ ⎠tA t ( )⋅E t

( )⋅J n

Alternating field Field in an electrical machine

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

-1500 -1000 -500 0 500 1000 1500

H [A/m]

B [T

]Magnetic characteristics of iron

Modelling motion I

The rotor of a high-speed induction motor is a non-slotted, copper-coated steel cylinder. The motion does not affect the material properties observed from the stator. Both the stator and rotor fields can be solved in the stator frame of reference after adding the motion voltage term vxB

σ σ φµ

⎛ ⎞ ∂⎛ ⎞∇× ∇× + − ×∇× + ∇ =⎜ ⎟⎜ ⎟ ∂⎝ ⎠⎝ ⎠

1 0tAA v A

( )σ σ

σ σ φ

= + ×

∂⎛ ⎞= − + ∇ − ×∇×⎜ ⎟∂⎝ ⎠

' '=

t

J E E v BA v A

Modelling motion IITypically in electrical machines, there is a slotting in both the stator and rotor, and the material properties seen from any single frame of reference are time dependent. The stator and rotor fields must be solved separately and forced to be continuous over the air gap.

σ σ φµ

σ σ φµ

⎛ ⎞ ∂∇× ∇× + + ∇ =⎜ ⎟ ∂⎝ ⎠

⎛ ⎞ ∂∇× ∇× + + ∇ =⎜ ⎟ ∂⎝ ⎠=

1 0

1 ' ' ' 0'

t

t

AA

A'A'

A A'In air gap:

The elements in the air gap are modified to allow continuous motion of the rotor. A typical time step in such a process is 30– 50 µs, i.e. one period of line frequency is divided in 400 –600 time steps.

Rotation within FEA

Belongs to a periodic boundary

Periodicity conditions for magnetic field

The geometry of an electrical machine typically repeats itself after one or two pole pitches.

−A=A 0

A

Two-dimensional magnetic field

Current density and vector potential point in the same direction

According to the definition, the flux density is

( )( )

⎧ =⎪⎨

=⎪⎩

z

z

,

,

J x y

A x y

J

A

e

e

( )⎡ ⎤= + = ∇× = ∇× ⎣ ⎦

∂ ∂=> = = −

∂ ∂

z,

x x y y

x y

B B A x y

A AB By x

B Ae e e

Two-dimensional magnetic field II

Partial differential equation for a 2D vector potential

or

Similar equation as for the scalar potential of electric field, but the boundary conditions have a different meaning.

Dirichlet’s condition (A is constant) means that the field is parallel to the boundary.

Homogenous Neumann’s condition means that the field is perpendicular to the boundary.

ν ν⎡ ⎤⎛ ⎞∂ ∂ ∂ ∂⎛ ⎞− + =⎢ ⎥⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎣ ⎦

z zA Ax

Jx y y

e e ν∇ ⋅ ∇ = −( )A J

ν ∂=

∂0A

n

Ψ=

= =∑1

, 1,...,n

i ij jjG a i m

Ω

β Ω= = =∫T

d , 1,..., , 1,...,iij i j

i

K lG N i m j nC

Coupling circuit equations to field solution

Circuit equations for windings are

Ri is resistance of a windingLi is series inductanceΨi is flux linkage computed using FEM:

Ψ= + + =

d d , 1,...,d di i

i i i iiu R i L i mt t

Coupling circuit equations to field solution II

Voltage equations in matrix form

The field and circuit equations are solved together

This can be written as one matrix equation

= + +d dd dt tu Ri L i G a

( )

⎧ + − =⎪⎨

+ + =⎪⎩

dd

d dd d

0t

t t

S a a T a Fi

G a Ri L i u

( ) −⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤+ =⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦

0 0d0 dt

a T aS a Fi G L i uR

Coupling circuit equations to field solution IIITime discretisation tk+1 = tk + ∆t by Crank-Nicolson method leads to a system of equations

All the terms on the right hand side are known. If a sinusoidal time variation is assumed, the system of equations is

( )

( )

+ +∆

+∆ ∆

∆+

∆ ∆

⎡ ⎤ ⎡ ⎤+ −⎢ ⎥ =⎢ ⎥

+⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦⎡ ⎤ ⎡ ⎤− − ⎡ ⎤ ⎡ ⎤⎢ ⎥− + +⎢ ⎥ ⎢ ⎥ ⎢ ⎥

− −⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦⎣ ⎦⎣ ⎦

1 12

12 2

2

12 2

0 0

k kt

kt t

k kt

k kkt t

S a T F a

G R L i

S a T F au uG R L i

( )ω

−⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤+ =⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦

0 0j

0a T aS a Fi G L i uR

Electromagnetic analysis

-150

-100

-50

0

50

100

150

0 10 20 30 40

time [ms]

Cur

rent

[A]

050

100150200250300350

0 10 20 30 40

time [ms]

Torq

ue [N

m]

-600

-400

-200

0

200

400

600

0 10 20 30 40

time [ms]

Volta

ge [V

]

Loss distribution in a core

Time-discretised finite element analysis combined with a dynamic hysteresis model has been used to study an inverter-fed cage induction motor.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-1000

-500

0

500

1000C u rren t o f P h ase A

t [s ]

i as [A

]

M eas uredFE M

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-100

0

100

200

300

400An g u la r ve lo c ity o f th e ro to r

t [s ]

wrm

[rad

/s]

M eas uredFE M

Validation – Starting of the two-pole 30 kW motor

Magneto-mechanical characteristics of electrical steels

Aim – to develop a magneto-mechanical coupled model for electrical machines and study magnetostriction in electrical steel sheets

PhD thesis in pre-examinationKatarzyna FONTEYN

FundingTEKES, TKK

CollaborationSeveral units fromTKK and TTY

Aim – tools for thermo-mechanical analysis of a high-speed permanent-magnet machines

Tools for analysing high-speed PM motors

PhD thesis in pre-examinationZlatko KOLONDZOVSKI

FundingGraduate School ofEEE, TKK

Collaboration: LTY / Laboratory of Fluid Dynamics

End-winding fields, forces and vibrations

Aim – tools for analysing the magnetic field, forces and vibrations at the end windings of electrical machines.

PhD thesis in pre-examinationRanran LIN

FundingThe Finnish Technology Award Foundation,TKK