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OKORO, O. I.. Author 3. NWANGWU, E. O. ... *Nwangwu E.O., Okoro 0.1, **Oti S. E.* ..... (1980), Finite Elements in Electrical and Magnetic Field Problems. John.
University of Nigeria Virtual Library Serial No Author 1

OTI, Stephen Ejiofor

Author 2

OKORO, O. I.

Author 3

NWANGWU, E. O.

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A review of Application of the Lumped Parameter and FiniteElement Methods in the Thermal Analysis of Electrical Machines Lumped Parameter, Finite-Element, Temperature, Nodes, Fields, Potential, Discretization

A review of Application of the Lumped Parameter and FiniteElement Methods in the Thermal Analysis of Electrical Machines Electrical Engineering

ESITAEE 2008 NATIONAL CONFERENCE, UNIVERSlTY OF NIGERIA, N S U K K2008 A ~ ~ E ~ ~ ~ ~

A REVIEW OF THE APPLICATION OF THE LUMPED PARAMETER AND FINITE- ELEMENT METHODS IN THE THERMAL ANALYSIS OF ELECTRICAL MACHINES ?

*Nwangwu E.O.,

a.

*Department of Electrical/Electronics Engineering Technology Akmu Ibiam Federal Polytechnic, Unwana. Afikpo, Ebonyi State, Nigeria e-mail:

Okoro 0.1, **Oti S. E.*

**Department of Electrical Engineering, University of Nigeriq Nsukka, Enugu state, Nigeria. e-mail: [email protected] e-mail: [email protected]

ABSTRACT Determination of thermal stabilily of a.c machines is becoming increasingly important due to its eflect on the performance and ratings of machines. This paper reviews the application of lumped parameter and jnite-Element methods in the thermal analysis of machines. The origin, development, application and the merits and demerits of each method are reviewed. Special emphasis is laid on the jnite element analysis as a current technique in mathematid analysis in electrical engineering. The use of variational principle in Jnite element method is especially highlighted.

KEYWORDS:Lumped parameter, finite-element, temperature, nodes, fields, potential, discretization. and

simulation seem to be the most useful means for the proper design of a device cooling system. Engineering design is aided by engineering analysis and there are two main approaches to thermal analysis of electrical machines viz. the lumped parameter and the finite element approaches. The task in thermal analysis is to determine the temperalure

distribution

a

present.

INTRODUCTION Thermal modeling

in

the

machine. A non-uniform distribution means that temperature gradients and undesirable mechanical stresses are

In lumped parameter approach, average values of the temperature at different sections of the machine are determined by making an analogy between electrical and thermal systems and hence co,mputing approximate values

of

resistances

equivalent and

thermal

capacitances

of

different sections of the machines. In the finite element approach, however, calculations

of

the

temperature

distribution in a device are based on the

assumed

geometric

model,

materials, parameters and boundary

conditions. A correction related to cooling of the machine can be introduced and second temperature

Where a is the thermal diffusivity

distribution is calculated.

defined as a = k l p c

The governing equation for the

Where th&e is no dependence of

temperature distribution in a system is

temperature on time eqn (2) simplifies

such that for a complex system, an

to

analytical solution may not exist or may be very complex. Thus, for information

on

the

temperature

In the absence of internal heat

distribution in such a system, the finite

generation, equ (3) simplifies further to

element method is preferred over and

the laplace equation

above the lumped parameter approach Emphasis in this paper is therefore on the development and application of finite element method. The general expression for the heat conduction in a system [I] is given by

or

V2T = O

The determination of the. temperature fields in equations (I), (2), (3) and (4) will be the task of any thermal analysis. An electrical hachine has material

non-homogeneity

and

incorporates internal he& generation. Therefore, thermal analysis involves Wherep and C are the density and

seeking solution to equation (1).

specific heat of the material PC is the

The lumped parameter Approach:

thermal capacity of the system and K is

Lumped parameter models are

the material thermal conductivity, q, is

described by

the specific infernal heat generation

equations with only one independent

rate, T is the temperature; x, y, z are

variable[2]. In this approach, it was

the Cartesiaq coordinate system. For

previously generally assumed that

constant k throughout the material, equ

there is no interaction among various

(1) becomes

fields

for

ordinary differential

instance

between

the

electromagnetic and the thermal one. Single - field problems are solved by

means of approximate procedures,

Where p c = thermal heat capacity

generally

considerable

h

-

contact coefficient

simplifications of geometries and

A

=

cross - sectional area

requiring

materials. This

procedure largely

The approximate values of all

makes use of solutions previously

the pardmeters can be computed except

determined [3].

for the airgap which is difficult

For homogeneous structures, a

because there is no clear knowledge of

global volume method can be used but

the kind of airflow in it, laminar or

in the case of inhomogeneity, the

turbulent, since the airgap is not really

technique

of

smooth and there are very few

resistances

and

discrete

thermal

capacitances

is

scientific works on the heat exchange

normally preferred, with introduction

in such a geometry. Other phenomena

of some nodes as measuring points.

that

In discrete thermal resistances

cannot

be

easily

described

includes air sheets between magnetic

and capacitances techniques, current

circuits and the ferromagnetic end

supplies represent the heat injections

plates, thermal conductivity of the

(i.e. losses); the potential at the nodes

insulations around the copper wires

gives the value of the temperature.

and between the laminations. A more

Capacitances depend on the specific

complicated formula ,is required to

heat

describe radiation and free convection

and

elementary

the part.

proportional

of

Resistance

to

conductivity dimensions

density the

and for heat

each is

thermal geometrical

transfer by

conductivity. Hence the ratio

-

effects. The analysis will d s o require the separation of

iron,

copper,

and

mechanical losses. Iron and copper losses are computed using the directquadrature (d-q) models [4] with two real windings and two fictitious ones.

where R,= electrical resistance;

The two losses are assumed equal for

Rth = thermal resistance,

maximum

C,

losses are computed from the damping

.,

= electrical capacitance,

CU, = thermal capacitance. T = - CpV

hA

=time constant.....6

efficiency.

Mechanical

losses in the air in the airgap and other mechanical losses in equal parts from the two bearing coils. A thermal

simulation of a stepper motor using the

governing equations can be established

method described above was carried

with the necessary initial and boundary

out be Karmous et al [5] . The model

conditions often lend themselves easily

they got for a two-dimensional heat

to the finite element method. The name

conduction is shown in fig. 1.

"finite element" first appeared in 1960 in a paper by R.W Clough concerning was

plane elasticity problems. However,

established in a general form by using

the basis for the finite element method

loop incidence matrix. Thus in matrix

(FEM) goes back to the early 1940's in

form the equations are

the applied mathematics literature. The

The

equation

they

solved

... ... .(7)

P(t) + C a + G T = O dt

FEM found wide application and acceptability

With G =A'HA, C=AtC0,and P =Atp0

in

linear

structural

analysis and methods in the late 1950's and early 1960's. In 1965, Zienkiewicz

H and C, are diagonal matrix with Hk

and cheung illustrated the applicability

the

the

of the FEM to any field problem that

capacitance of the branch K. - Pok is

could be formulated by the variational

the current generator of branch K.

means.

Determining the conductances and

applications of FEM was in the area of

capacitances in the model is the

structural analysis. Thelate 1970's and

formidable

early 1980's saw more applications of

conductance

task

and

in

the

Cok

lumped

parameter method.

Up

to

mid-1970's

most

FEM in fluids problems, while the first finite element application in Electrical

The Finite - Element Method.

Engineering was in 1965 by A. M.

Origin:

Wislow in the analysis of saturation

The finite element method is il

effects in accelerator magnets.

numerical analysis procedure to obtain approximate solutions to problems posed in every field of engineering 161

Development:

The development of the f ~ t e

. The inability'to solve many complex

element concept requires the following

structural problems laid the foundation

information [6].

of the finite element method. Thus

1. Nodal point spatial locations

problems that cannot be easily solved

(geometry) 2. Elements connecting the nodal

using analytical approach but whose

defined in terms of the degrees of

points,

freedom allowed at the nodes and at

3. Mass properties,

4.

Boundary

Conditions

or

element. These degrees,of freedom are

restraints, 5. Forcing Functions (or loading) 6.

some other special points within the the generalized co-ordinates which

details,

represents some physical quantities

Analysis options.

such

as

displacements,

stresses,

The physical problem to be modeled is

temperatures, voltages, fluid potentials,

first carefully reviewed with all the

magnetic fields, Heat flux e.t.c. By

boundary conditions so as to select an

suitable, well-known techniques, the

appropriate and precise element. The

behaviour of the assembled structure

underlying theory of the types of

can then be described [7].

elements and their applications is most

Mathematically, the FEM is

relevant here. Also, the justification for

used to solve field problems that are

the use of FEM instead of other

usually

conventional methods should be that it

differential equations with proper

is the quickest and most cost -

boundary

effective solution to product design

extremum (maximum or minimum) of

and development i.e. when prototyping

a variational principle,, if it exists, or by

and testing is very costly or difficult.

some form of variational statements

The physical concept of FEM is based

on

an

approximate

and

described

by

conditions

a or

set by

of the

(incomplete variational principle). For example,

a

one. -

dimensional

systematic description of a continuous

transients heat conduction problem for

system (continuum) in terms of a

temperature distribution T, can be

finite, though large, number of co-

described by the diKerentia.1equation.

ordinates or degrees of freedom. This concept originated from the theory of structures where a physical structure is

Where a is the thermal d a s i v i t y and

usually built up from many structural

the boundary conditions are

elements. The elements are however,

T(O,t>O)

connected to each other at only a finite number of points. For each element, the

structural

properties

of

the

governing equations are uniquely

and the initial condition is T (x,O) = 0 or it can be formulated in terms of a

variational statement,

consists o f 1.

Discretization of the physical

SYstem,

2. Characterization of the element or

or in the form

elements (ie determination of the element coefficient matrices from the Where 6T = 0 at x = 0 for t >O; T(x,O)=OandT(O, t > O ) = l The two approaches of equ. (9) and equ (10) are mathematically equivalent The finite element method will use a local approximate solutions of the above within an element to build up a solution for an entire domain. It is worthy of note here that while a differential equation of a functional may be found, the reverse is not true.

A

differential

equation may be approximated over a set of discrete points using finite differences,

but

the

and

proper

interpolation procedures). 3.

Assembly of the local element

stiffness matrix and element force vector to obtain the global or master

under proper restrictions.

necessarily

variational statements

associated

functional am be minimized over a set of discrete domains by FEM. A summary of the frnite - element procedure according to tong et al [7]

matrix, 4. Application of appropriate constraints to

yield a solvable system of algebraic equations. After all these, well known numerical techniques are applied to yield the required solution.

Discretization/FiniteElementModel: The analysis begins by making a frnite element model of the device. The model is an asseinblage of finite elements, which are pieces of various sizes and shapes. Fig. 2 shows different types of elements used for finite element modeling in thermal analysis [8].

I

Fig.2. Examples of Engineering elements

I

A finite element thermal analysis of a typical static thermal or heat transfer problem

and the finite element model is shown in fig. 3.1 a and b.

RTTRRER S W A T H

COPPER WIRE

Fig.3.1a

Cross-section of current carrying copper wire with rubber

sheath.The wire and its current extend into and out of page. The finite element solution is shown in fig. 3.2 a and b. The finite elements are h7o

dimensional, consisting of triangles and quadrilaterals[9].

Fig.3.2 Sheathed wire, quadrilateral symmetry. (a) Finite element model (b) Isotherms, Free convection cooling. Discretization of the physical

means that the system is represented

problem into subdivisions or regions

by a discrete grid or node points

(elements) is the first step in any FEM

connected by elements. A theoretical

analysis. If a structure is discretized, it

basis for subdividing of a region into a

set pattern of elements is difficult to

coefficients of the function is gotten

realize. Engineering judgement is the

using the node co-ordinates. The

key at this stage of the analysis.

function is required to be interpolatory

Decisions must be made regarding the

on the vertices of the elements. The

number, size and shape of the

number

subdivisions. It is generally accepted

determine the order of the polynomial

that the greater the number of nodes

-

and elements, the more accurate the

order), cubic (fourth order) etc. The

solution.

terms chosen for the polynomial in a

However,

there

must

be

a

consideration for the computational time

and

efficiency.

There

nodes

can

linear (first order), quadratic (second

two - dimensional problem can be gotten from the Pascal's triangle. al(constant - 1 term)

are

however, different techniques for the

OF element

discretization or modeling through the

a3y (linear - 2 terms) (quadratic - 3 ax2 a ~ x y aY2

use of adaptive processes or mesh

terms)

refinements

a7x3 agx2y wx#

and

automatic

mesh

generation. The automatic grid

&X

(cubic - 4

terms) Evaluation of Element Matrices.

After

generation routine of Andersen [lo] is

discretizatip

using

software

element shapes, the approximating

application in finite element modeling.

function describing the element and the

A computer plot for the mesh can be

potential is substituted in ihe governing

gotten through. the use of rotational

equation. The source term is also

angular increments for the nodes. The

modeled and added to the governing

number of rotation depends on the

equation The function is evaluated at

number of nodes [g].

each element node. This yields the

a

typical

example

of

The element can be described

element coefficient matrices which can

by its shape function using it's vertex

be written in matrix form; The number

position in the w-ordinate system. The

of simultaneous equations for each

nodes of the ' element represents the

element depends on the number of

state variable or field of interest which

nodes in each element.

can be approximated by a piece-wise

Assembling the Element Matrices

but continuous functions. Polynomial

irito the Master or Global Matrix.

functions are often used and the

The

equations gotten above are

assembled using the global numbers of

Problems usually solved by

the elements. It is important to ensure

FEM are boundary-value problems in

good correspondence between the local

which certain values of the potential or

node numbers and the global numbers.

its derivative are given at more than

The

one point. This means that nodal

assembling

process

is

an

arithmetic operation.

values on specified boundary are

Two coordinate systems are employed

known. Inserting these values into the

in the FEM, the special co-ordinate

master matrix constrains it. The matrix

axes for the elements and the global

can then be re-ordered for partitioning.

axes for the entire system, The reasons

Finally, application of the variational

for special co-ordinate axes for the

principle

elements are

6.n = 0

1. To ease the construction of the trial

with respect to the potentials yields the

function (the interpolation function)

required system of equations for

and

evaluating the unknown potential

2. To ease the integration within the

values.

element.

This

means

additional

computations in the fonn of coordinate

A General Illustration of FEM

transformations since the elements are

Procedure.

to be assembled in the global frame.

The following illustration [l 11 will

Other coordinate systems known as

show clearly how the procedure is

natural coordinates (e.g Area or

applied.

Volume,coordinates) are often used in

Let u be an unknown potential

finite element analysis for ease of

(Temperature, magnetic flux etc.)

numerical integration Depending on

Let U be an approximating potential.

the element shape, a cartesian or

Within a typical triangular element

curvilinear axes can be chosen for the

(fig. 4)

element. Transformation from element coordinate coordinates

system to

the global

involves

rotations.

Mapping is used when a master element is employed. Constraining the Master Matrix.

4

Combining x, y and the elements of the inverted coeficient matrix into new I

Mg 4 Typical Triangular fimte ele&ent in x - y plane.

functions of position equation (1.3) may be written

it will be assumed that the potential is adequately

approximated

by

the

+

cy

Where

expression,

U

+

a

=

bx

...(1.1)

i.e a linear polynomial function And A = the surface area of the

This can be re-written as

triangle.

[U] = [1 x y] [a b c]' where t denotes transpose of

NOW, @(xi,yi)=O f o r i # j ..... (1.6)

0

The coefficients a, b, c, in equation (1.1)

fori=j

=1

matrix. can

be

independent

found

equations

from

the

which

are

obtained by requiring the potential to assume'vertex values UI, U2 and the three vertices, ie

Ul = a + b x , +cyl U2 = a + b x , +cy, U , = a + b x , +cy,

U3

at

Equalion (1.6) satisfies the condition that the b c t i o n must be interpolatory on the vertices oi-' the triangle. The potential gradient within the element may be found-from equ 1.4 as

The element energy becomes

In matrix form

or from equation (1.7)

Therefore

Thus for brevity, we define element matrix

[15] applied finite element to the modeling of transformers; Skorek et al

In matrix form

[16] equally used

finite element

1 w'"'= -uTseu ............(1.11) 2

method

For a given triangle, the matrix S is

phendmena in a thyristor. However,

readily evaluated. On substituting eqn.

few works have been done on using

(1.5) in equation (1. lo), a little algebra

finite element in the analysis of

yields

thermal

to

simulate the

phenomena

in

thermal

electrical

machines. FEM in Thermal Analysis.

Other element of the matrix S is

As pointed out earlier, finite

similarly gotten. The total energy W is

element method is used mostly in the

given by

solution of boundary value problems. The use of FEM in thermal analysis

The variational principle can now be applied to equ. (1.13) with respect to each potential. i.e

recognizes the ana,logy that exists between

thermal

analysis.

The

and

general

structural conduction

equation is given [7] in matrix form as '

[B]$}+ [K]{T)= p)+ { N ) ......11 Where

@)= the vector of grid point Applicdion of FEM

temperatures

Several applications of FEM in Electrical Engineering abound. It has been used in the modeling of electrical machines for different purposes. For instance, Vassent

et al [12], and

Kanerva [13] used finite element method to simulate the operation of induction machine; Long et a1 [I41 also used finite element to model skewed slots in D.C machines; Dedulle et al

$)=the vector of time derivates of temperatures

{P) = the vector of known time fimctions of applied heat flow

{N)

=

the vector of temperature-

dependent nonlinear heat flow [K]

=

the constant heat conduction

coefficient matrix. [B]

=

the constant thermal capacity

coefficient matrix.

Convection heat flux on the surface is

In

linear transient

analysis, the

given by

equation to be solved is expressed by the Newrnan p method as:

Q, = ha(Tf - T ) ........12

Where h is the heat transfer coefficient A is the surface area exposed to fluid or air Tf is the temperature of fluid or air

Where

M a t i o n heat flux from a surface depends on surface emissivity, the

fourth

power

of

E,

and

absolute

temperature hence,

6t

=

a user - specified time

step,

p

=

a user - specified integration

stability factor

(0 I

p

5 1; 0.5

recommended)

q = OE(T+T,) ........13 Where o is the Stefan

n = nh time step -

Boltzman

Unlike in the finite differences, the

constant

time step size is not restricted in FEM

T, is a constant to express absolute

solution to eqn(l7).

temperature. Inclusion of radiation in

When thermal conductivity or heat

the general conduction equation results

transfer coefficients are temperature

in a nonlinear finite element model and

dependent a non-linear analysis is

equation (11) becomes

indicated. For nonlinear steady state

[B$}+ [K](T)+[R](T+ qJ = @) ....14

analysis, the equation to be solved is

*

[ K ] ( ~ > + [ R ] @ +=K@'>+ ~ (N)......18

FEM thermal analysis is grouped into linear eteady state and transient) and

Equation (18) requires an iterative

nonlinear (steady state and transient)

solution

analysis.

technique is emp!oyed.

and

Newton-Raphson

Linear steady state thermal analysis involves solution of the laplace or Poisson equations;

FEM Softwares.

There are softwares already developed

for

Electromagnetic

structural finite

and

element

Laplace

V 2=~0...........15

analysis. The structural analysis finite

Poisson's

q g ............ 16 v 2 T = -...

element programs are also suited for

k

thermal analysis and there are three

world-class programs for this namely

temperature distribution is described in

ABAQUS,

and

terms of the generalized coordinates

MSC/NASTRAN. Steady - state finite

only which in turn is in terms of

element thermal analysis can also be

position only. However, formulation of

carried out using two electromagnetic

the flnite element equations presents a

finite

namely

formidable task. Nevertheless, there

and

are algorithms that help in the difficult

ANSYS,

element

programs

AOSNAGNETIC AOSNAGNUM [9].

task of discretization of the structure

On the other hand, there is also the

into domains. In all, the solution by

possibility of developing one's own

finite

Program using language. This

information on the actual performance

the FORTRAN involves many

element

method

gives

of the device. Close approximate

The

solutions are obtained and their

SIMULINK toolbox of Mathlab can

accuracy depends on the ability of the

as

user. Hence, the new method of

subroutine also be

subprograrns[lO]. used

illustrated by

analysis is worth investing the time

Kanerva[ 1 31.

and energy to acquire the tool and the

CONCLUSION.

competency.

We have attempted to review the approaches

employed

in

thermal

6

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C.

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ISIMULINK

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Figure 1 .

on

Modeling

and