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Finite Element Analysis of Electric Machines
------The Solver and Its Application
Danhong Zhong
Department of Electrical Engineering
The Pennsylvania State University
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Motivation
Steady-state finite element solver
Low rotor loss permanent magnetmachine design
Summary and future work
Outline
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Electric Machines in New Technologies
Flywheel for frequency regulation for
renewable and distributed generation
(Credit: Beacon Power)
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How do we design a machine that
suits our need?
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Flywheel energy storage system
Motor/Generator test setup for flywheel system
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Design Objectives
Flywheel motor/generator forsmall satellite application
Power level of 100W
Ultrahigh speed operation(150-300krpm)High frequency
electromagnetics
Continuously charging anddischarging
Thermal Constraints important
Electrical losses
Rotorlosses
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Design Methods
Rules-of-thumb, empirical tables, design equations, a designers
intuition are all valuable
Classic Way----prototyping
Large-scale numerical simulation
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Magnetic excitation is supplied by high-energy permanent magnet
No power loss is associated
with machine field excitation
High power/weight ratio
----Popular choice for
high-speed motor/generator
applications
Synchronous Permanent Magnet (PM) Machines
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Rotor losses
Rotor exposed to high electrical frequency harmonics
Heat generated
Rotor spins in vacuum, supported by magnetic bearings
Only method of heat transfer is through blackbody
radiation, which is a relatively poor heat transfer mechanism
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Temperature dependence of Neodymium Iron Boron
The properties of manymaterials in the flywheel rotor will
degrade with increasing
temperature
E.g., the intrinsic coercivity of
Neodymium-Iron-Boron decreases
significantly with temperature,
creating a risk of demagnetization
It is therefore important to
minimize rotor losses in this
application
Change of intrinsic coercivity of
Neodymium Iron Boron
with respect to temperature
(courtesy Dexter Magnetics)
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Solver overview
Programmed in MATLAB environment 2-D finite element analysis
Software capable of:
model building
mesh generation
steady-state solution solving (time stepping)
rotor losses calculation
Uses GMRES method to speed up steady-state solution
Allows study of rotor losses when insulating barriers existin the 2-D plane of the rotor (e.g., segmenting permanentmagnets)
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Road Map
1) Maxwells equation --The governing PartialDifferential Equation(PDE) of the problem
2) PDE - Finite Element Equations
3) Main techniques used in achieving the steady state
solution.
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Mathematical model
Maxwells equations:
t
BE
JH
B
0
Constitutive Laws:
MHB
BvEJ
00
)(
Governing Partial Differential Equation (PDE)
(1)
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Finite Element Method
The object is divided into FiniteElement mesh
A simple relationship is used torepresent the variables anywhere in
the element by variables on thenodes of the element.
Within the element, approximatefunctions in terms of nodal valuesare then derived from the PDE
Mesh (16761 points 33232 elements)
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element
The method of weighted residuals
Define a residual function:
(1)
The solution of the PDE should satisfy that, for a given weight function w,
(2)
(3)
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Finite Element Formulation
The finite element formulation is given as:
Where
D, K :global matrix decided by material properties and element shape
M, M : represents magnetization in ferromagnetic materials orpermanent magnets, can be a nonlinear function of x
I s : forced current flowing in stator windings
(4)
sysxs
yrxr
s
r
ssrs
srrr
s
rrr
IMM
MM
a
a
KK
KK
s
a
aS 0
00
0
3
~00
000
00
0~
3
10
(5)
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Steady-State Analysis
Under steady-state assumptions, after one period T we wish to achievethe same primary variables, i.e.
If we define the nonlinear state transition function as , this becomes:)(
(6)
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Steady-State Solution Techniques
The state transition function in (6) is determined by (5)
To achieve the steady-state solutions, we used the following numericaltechniques:
o Backward Euler Integration
o
Shooting-Newton Algorithmo Matrix-free GMRES
(5)(6)
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Shooting Newton Method
ModifyNO
Time integration of the differential equation
YES
Initial Value Final Value
Error < Tolerance?
Target Value (=Initial Value)End Iteration
Calculate
Modification
The steady-state solution in the time domain is obtained byusing a shooting Newton method:
Applying the Newton-Raphson method, we get:
WhereJis the Jacobian of the nonlinear state transition function
(8)
In our solver we define (7)
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Matrix-free GMRES Method
Generalized minimum residual method (GMRES) :
A Krylov-subspace method
When solving Ax=b, no direct access of matrix A is used, A onlyneed to be accessible via a subroutine that returns y=Az
The Jacobin 1
1
1
)0(
)(),0,)0((
j
jj
x
TxTxJ can not be explicitly written
(8)
A x b
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Efficiency Improvement
By using GMRES, the computation time of the shooting-Newtonmethod is dramatically reduced.
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Flux Density Distribution
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Rotor Loss Distribution
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Harmonic analysis
0 20 40 60 80 1000
0.05
0.1
0.15
0.2
0.25
harmonics
rotorlosses(w)
After we achieve the steady-state solution, we perform a Discrete FourierTransform of x(t) and calculate the eddy current rotor losses for eachharmonic.
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Rotor Loss Design Study
Design parameters are changed to study the effecton rotor losses
The stator current peak is adjusted to maintainconstant steady-state mechanical power
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Segmenting the PM poles
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Other Rotor Design Aspect
Laminated backiron
Different permanent magnet materials
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Stator Design: 36 Stator Slots
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Stator Design: Open Slot vs. Closed Slot
open slot
closed slot
0 20 40 60 80 1000
0.05
0.1
0.15
0.2
0.25
harmonics
los
s(w)
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Techniques for Reducing Rotor Losses
Laminating rotor backiron
Segmenting the Permanent magnet poles
Increasing slot number
Closing the stator slots
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Summary
Purpose of Finite element analysis for electric machines
The flywheel energy storage system
The steady-state nonlinear finite element solver
Its application
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Further research into numerical modeling of electric
machines
Special machine design and analysis
Future Work
2011 Chevrolet Volt Propulsion System
(Credit: GM)
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Thank you!
This work has been supported by NASA Grant NAG3-2598
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Currents flowing in permanent magnet are eddy currents. In machinedesign, magnets can be electrically insulated from each other and the
rotor backiron.
Eddy currents meet large impedance at the
end of the machine, surface charge will
accumulate and electric potential is built
across the magnet
Permanent Magnets
)(
1 2
0AMAt
A
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Electric Scalar Potential Dynamics
t
AJeddy
(2)s
dsJs
eddy
c
c
dst
AS
s
)(
The charge relaxation time constant / is extremely small inpermanent magnets, and so our system consists of a set of fast
dynamics (electric scalar potential) and slow dynamics (magneticvector potential). Therefore, singular perturbation techniques can
be used to analyze the system.
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Singular Perturbation Analysis Techniques
Under singular perturbation theory, when analyzing the fastdynamics, the slow variables can be assumed to be essentially
constant
If the fast dynamics are stable under this condition, the fast
variables will converge to a quasi-steady-state value, which is a
function of the slow variables When analyzing the slow dynamics, it can be assumed that the
fast variables have converged to their quasi-steady-state value
discussed above.
0
Sdst
A
S
m
n
Rzztztzxgz
Rxxtxtzxfx
,)(),,,,(
,)(),,,,(
00
0
0
small
),0,
~
,
~
(0 tzxg
(3)dst
AS
s
)(
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Applying Finite Element Method
To solve the Partial Differential Equations
use 2D finite element methodPrimary variable : A and Element type : triangular
Shape functions : linearProcedure used : weighted residuals
Error distribution principle :
Galerkins method
Boundary conditions:
0zA
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Newton-Raphson Method
Applying the Newton-Raphson method, we get:
WhereJis the Jacobian of the nonlinear state transition function
)()(11
1
jjj
xxfxx
x
fj
Define : and its called the Jacobian ofjxf xfJ
)(xf
(5)
In our solver we define
For a nonlinear equation, the Newton-Raphson Method provides thefollowing iterative procedure to solver for x:
0)( xf
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Equation solving
Now we look closely at the steady state iterative equation :
1
1
1
)0(
)(),0,)0((
j
jj
x
TxTxJ
In (5), is not known explicitly but is determinedthrough Backward Euler Integration. Thus:
11)(),0,)0(( jj TxTx
(5)
can not be explicitly written either and would have to be calculated using aform of numerical integration
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Equation solving (2)
One way of solving for the Jacobian is to differentiate both sides of (4)with respect to
J
Where and are matrices derived from (4). can then be computed by
repeatedly solving (6) starting from the initial conditionfC pC
J
Computation load analysis of this method:Matrix-Matrix multiplication on the right side and LU-factorization on the left
side. If the vector has components, the computation work per time step is
of at least orderN
2N
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The Backward Euler Integration Method calculate y at step
n+1 implicitly by :
Backward Euler Integration
Consider the problem:
)()(2
11 hOyhfyy nnn
In the solver we separate the period T into a number of time
steps h. Provided a solution at time t1, x(t1), we then cancompute the solution x(t1+h) at time t1+h, by applying the
backward Euler integration to (5) over the time interval h:
(5)
(7)
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Application of the GMRES
Computation load analysis of this method:
Now we consider solving the problem using GMRES method. By usingGMRES method, we dont need to provide explicitly. Instead, we only needto provide . We apply the procedure (6) to a vector
J
Matrix-vector multiplication on the right side and LU-factorization on the left
side. Typically the number of iterations required by GMRES to achieve a
sufficiently low relative error is substantially smaller than problem size. So by
using GMRES, the numerical efficiency is improved.
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The eddy currents in rotor are :Post-processing:
rotor loss calculation
After we achieve the steady-state solution at time 0, the entire response
x(t) can be calculated by integrating (5) for one period. We perform aDiscrete Fourier Transform of x(t) and calculate the eddy current rotorlosses for each harmonic in each element.
Se
iiis
Se
ie
s
ei
dSAjl
dSJl
P
2
2
)()(
)(
t
AJ