mat-1.3656 seminar on numerical analysis and … 2010... · mat-1.3656 seminar on numerical...
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Antero Arkkio29/03/2010
Mat-1.3656 Seminar on numerical analysis and computational science
Modelling of electrical machines
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Small cage induction motor
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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.
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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.
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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
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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
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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
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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
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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
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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:
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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
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Periodicity conditions for magnetic field
The geometry of an electrical machine typically repeats itself after one or two pole pitches.
−A=A 0
A
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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
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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
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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
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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
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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
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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
]
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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.
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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
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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
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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
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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