danhong presentation

8/2/2019 Danhong Presentation

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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

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