University of Nigeria Virtual Library Serial No Author 1
OTI, Stephen Ejiofor
Author 2
OKORO, O. I.
Author 3
NWANGWU, E. O.
Title
Keywords
Description
Category Publisher Publication Date
Signature
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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