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ELECTRIC FIELD CALCULATIONS BY NUMERICAL
TECHNIQUES
A THESIS SUBMITTED IN FUFILLMENT OF THE REQUIREMENTS FOR
THE DEGREE OF
BACHELOR OF TECHNOLOGY
IN
ELECTRICAL ENGINEERING
BY
BISWANATH MALIK
ROLL NO-
DEPARTMENT OF ELECTRICAL ENGINEERING
NATIONAL INSTITUTE OF TECHNOLOGY
ROURKELA-
ELECTRIC FIELD CALCULATIONS BY NUMERICAL
TECHNIQUES
A THESIS SUBMITTED IN FUFILLMENT OF THE REQUIREMENTS FOR
THE DEGREE OF
BACHELOR OF TECHNOLOGY
IN
ELECTRICAL ENGINEERING
BY
BISWANATH MALIK
ROLL NO-
UNDER THE GUIDANCE OF
PROF. SARADINDU GHOSH
& PROF. SANDIP GHOSH
DEPARTMENT OF ELECTRICAL ENGINEERING
NATIONAL INSTITUTE OF TECHNOLOGY
ROURKELA-
ACKNOWLEDGMENT
I would like to articulate our deep gratitude to our project guide Prof.Saradindu
Ghosh & Prof. Sandip Ghosh who has always been our motivation for carrying out
the project. I am thanking to C programmer Susanta Kumar Rout for his sincere
help to write programs. It is our pleasure to refer Microsoft Word exclusive of
which the compilation of this report would have been impossible. An assemblage
of this nature could never have been attempted with out reference to and
inspiration from the works of others whose details are mentioned in reference
section. We acknowledge out indebtedness to all of them. Last but not the least,
our sincere thanks to all of our friends who have patiently extended all sorts of help
for accomplishing this undertaking.
Biswanath Malik
Roll no.-
Contents
Chapter Topic Page
Abstract 6
Chapter 1 Introduction 7
Chapter 2 Finite difference method
2.1 fundamental of FDM 8
2.2 Two dimensional electric field calculations by FDM 12
Chapter 3 Finite elements method
3.1 Fundamentals of FEM 15
3.2 Two dimensional electric field calculations by FEM 20
Chapter 4 Three dimensional electric field calculations
4.1 Three-dimensional Laplace’s equation 34
4.2 Non-uniformly distributed dielectric of a capacitor 36
4.3 Electric fields near a dc busbar 39
Chapter 5 Finite Element Analysis using ANSYS
5.1Fundamentals of ANSYS 41
5.2 Motor analysis using finite elements methods in ANSYS workbench 53
Conclusion and future works 66
References 67
Appendix A: MATLAB Programming 68
Appendix B: C programming 73
Chapter 1
Introduction
Calculation of electric fields with the aid of an computer is now a inevitable tool in various
electricity-concerned technology, in particular, for analyzing discharge phenomenon and
designing high voltage equipments .Electric and magnetic fields comprises two components dealt
with in one of the classical physics, electromagnetism. Calculation of electric fields is usually
considered easier than that of magnetic ones from two reasons. First, the electric field is
expressed with a scalar potential at least in simple low frequency problems. Secondly, non linear
characteristics are more often involved in magnetic fields. Compared with magnetic field,
however, the calculation electric fields generally require higher accuracy, because the highest
electric field stress on insulator is usually the most important and decisive value in insulation
design or discharge study. This is one of reason why the boundary-dividing methods are
preferred to the region-dividing ones, such as finite difference method (FDM) or finite element
method (FEM). Usually former method does not need numerical differentiation to obtain field
values.
A fundamental equation for the electric field is Laplace’s equation or Poisson’s equation;
perhaps the simplest among many partial differential equations that express physical phenomena
among various numerical calculation methods, FDM and FEM is very unique as it is applied
exclusively to electric field calculations. Fundamental difference between FDM and FEM is that,
FDM can be used for calculation of potential at nodes only but FEM can be used for calculation
of potential at nodes as well as within the elements. Calculation of electric field in 3D
arrangement poses no essential problem by of the numerical methods if the field is given by
Laplace’s equation .The difficulty is that it usually required the tedious work preparing the input
of a large amount of errorless data associate with 3D conditions.
Numerical solution of EM problems started in the mid-1960s with the availability of modern
high-speed digital computers. Since then, considerable effort has been expended on solving
practical, complex EM-related problems for which closed form analytic solutions are either
intractable or do not exist. The numerical approach has the advantage of allowing the actual
work to be carried out by operates without a knowledge of higher mathematics, with a resulting
economy of labor on the part of the highly trained personnel.
Chapter 2
THE Finite difference method
2.1 fundamentals of FDM
The finite difference method is a powerful numerical method for solving partial differential
equations. In applying the method of finite differences a problem is defined by:
- A partial differential equation such as Poisson's equation
- A solution region
- Boundary and/or initial conditions.
An FDM method divides the solution domain into finite discrete points and replaces the partial
differential equations with a set of difference equations. Thus the solutions obtained by FDM are
not exact but approximate. However, if the discretization is made very fine, the error in the
solution can be minimized to an acceptable level.
The Poisson's equation in 3-D is given by
For 2-D case, Poisson's equation simplifies to
In applying the methods of finite differences, we define the solution region into a finite number
of meshes as shown in Fig2.1.
.................................................................... (2.3e)
Let, P 1 , P 2 , P 3 andP 4 represent the midpoint of the arms as shown in Fig 2.2. In order to replace
the Poisson equation (2.2) by difference equations, we obtain the approximate first derivatives at
the points P 1 to P 2 and use these first derivatives to approximate the second derivative.
The first derivatives at P1and P2 are
.................................................................... (2.4a)
.................................................................... (2.4b)
In the same manner,
The first derivative at P1and P2 is
.................................................................... (2.7a)
.................................................................... (2.7b)
............................................ (2.8a)
In the same manner,
............................................ (2.8b)
Further, for Laplace equation, ρs and equation (2-8) simplifies to
Thus we see that voltage at the central node is the mean of the voltages at the other four nodes.
With reference to Fig2-1, equation (2-8) can be written as
Equation ( 2 .8) & equation (2.9) can be used to solve Poisson's and Laplace's equation
respectively when uniform grids are used. These equations, along with the specified boundary
conditions can be used to solve a problem.
MATLAB PROGRAM:
- V1 - 0 0 0 1.861 3.38 4.51 5.
- V2 0 0 0 0 3.125 6.645 9 10.6 11.
- V3 0 0 0 6.25 11.23 14.66 16.86 18.3 19.
- V4 0 0 12.5 19.14 23.044 25.3487 26.82 27.73 28.
- V5 0 25 32.81 36.22 38.08 39.1381 39.8 40.21 40.
- V6 50 56.25 58.2 59.055 59.52 59.79 59.95 60.05 60.
- V7 0 0 0 0 4.321 6.87 9.063 10.63 11.
- V8 0 0 0 6.25 13.49 17.95 21.1 23.15 24.
- V9 0 0 12.5 22.656 28.96 33.064 35.63 37.625 38.
- V10 0 25 37.5 44.726 48.655 51.25 52.85 53.92 54.
- V11 50 0 67.58 70.2365 71.684 72.6 73.1625 73.531 73.
- V12 0 0 0 11.035 12.1271 14.92 17 18.833 19.
- V13 0 0 12.5 23.8525 29.3303 33.23 35.72 37.625 38.
- V14 0 25 40.625 46.39 50.998 53.32 55.97 57.411 58.
- V15 50 62.5 69.53 72.78 74.9137 76.33 77.205 77.833 78.
- V16 0 0 12.5 20.3368 23.49 25.565 26.98 27.853 28.
- V17 0 25 37.5 45.3151 49.05 51.447 53.05 54.058 54.
- V18 50 62.5 69.535 72.926 75 76.2 77.255 77.867 78.
- V19 0 25 32.815 36.52 38.27 39.46 40.0325 40.5 40.
- V20 50 62.5 67.57 70.4587 72 73.3 73.5 73.73 73.
- V21 50 56.25 58.203 59.13 59.5675 59.85 60 60.125 60.
- V1 5.92 6.22 6.47 6.644 6.76 6.84 6.885 6.
- V2 12.46 12.94 13.27 13.52 13.7 13.77 13.81 13.
- V3 19.89 20.33 20.62 20.82 20.97 21.02 21.1 21.
- V4 28.76 29 29.21 29.357 29.4 29.46 29.5 29.
- V5 40.72 40.79 40.91 40.94 40.96 40.98
- V6 60.17 60.2 60.22 60.23 60.24 60.245 60.25 60.
- V7 12.44 12.95 13.3 13.51 13.65 13.77 13.82 13.
- V8 25.63 26.3 26.73 27 27.24 27.35 27.43 27.
- V9 39.62 40.16 40.55 40.8 40.89 41 41.11 41.
- V10 55.1 55.43 55.7 55.79 55.86 55.92 55.96
- V11 73.955 74 74.15 74.2 74.22 74.23 74.23 74.
- V12 19.94 20.36 20.66 20.82 20.95 21 21.1 21.
- V13 39.65 40.2 40.56 40.8 40.96 41 41.1 41.
- V14 58.93 59.42 59.67 59.85 59.9 60 60.05 60.
- V15 78.6 78.7 78.84 78.91 78.95 78.98 79 79.
- V16 28.84 29 29.227 29.34 29.42 29.47 29.5 29.
- V17 55.13 55.44 55.67 55.78 55.87 55.91 55.97
- V18 78.51 78.83 78.841 78.91 78.95 79 79 79.
- V19 40.72 40.81 40.87 40.93 40.96 40.98
- V20 74 74.06 74.13 74.18 74.2 74.25 74.24 74.
- V21 60.19 60.2 60.22 60.23 60.24 60.25 60.25 60.
Result:
Voltages =
x
0 0 0 0 0 0 0 0 50.
0 6.9652 13.9304 21.1778 29.5534 41.0167 60.2542 100.0000 0
0 13.9304 27.5788 41.2271 56.0194 74.2590 100.0000 0 0
0 21.1778 41.2271 60.1326 79.0380 100.0000 0 0 0
0 29.5534 56.0194 79.0380 100.0000 0 0 0 0
0 41.0167 74.2590 100.0000 0 0 0 0 0
0 60.2542 100.0000 0 0 0 0 0 0
0 100.0000 0 0 0 0 0 0 0
50.0000 0 0 0 0 0 0 0 0
y
distributions in various elements such that the potential is continuous across inter elements
boundaries. The approximate solutions for the whole region is
V(x, y) =∑ ᡈ
〲
〕
〲⢀⡨
Where N is the number of triangular elements into which the solution region is divided.
The most common from of approximation for ᡈ
〲
within an element is polynomial approximation,
namely,
〲
(x,y)=a+bx+cy …………….. (3.2)
For a triangular element and
〲
(x,y)=a+bx+cy+dx y…………………. (3.3)
for a quadrilateral element .The potential ᡈ
〲
in general is nonzero within element ‘e” but zero
outside “e” .It is difficult to approximate the boundary of the solution region with quadrilateral
elements; such elements are useful for problems whose boundaries are sufficiently regular .As
assumption of linear variation of potential within the triangular elements is same as assuming
that the electric field is uniform within the element; that is,
〲
= - V ᡈ
〲
=-(bax+cay)…………………. (3.3)
B. GOVERNING EQUATIONS OF EACH FINITE ELEMENT
Consider a typical triangular element, as shown in figure 3.2.The potentialᡈ
〲⡩
〲⡰
and ᡈ
〲⡱
at
nodes 1, 2, 3, respectively, are obtained by using eq. (3.3); that is
Ve
Ve
Ve
1 x 1 y 1
1 x 2 y 2
1 x 3 y 3
a
b
c
The coefficients a, b, c is determined from above equation as
a
b
1 x 1 y 1
1 x 2 y 2
1 x 3 y 3
⡹⡩
Ve
Ve
Ve
Fig3.2, typical triangular element.
Substituting this into above equation gives
Ve= 䙰1 ᡶ ᡷ䙱
x2y3 – ᡶ3y3 ᡶ3y3 – ᡶ1y3 ᡶ1y3 – ᡶ
y3 – y3 ᡷ3 − ᡷ1 ᡷ1 − ᡷ
Ve =∑
〶
〲〶
⡱
〶⢀⡩
And A is the area of the element “e”; that is,
2A=㘩
1 x1 y
1 x2 y
1 x1 y
The value of A is positive if the nodes are numbered counterclockwise. Above equation gives the
potential at any point within the element, provided the potentials at the vertices are known. This
is unlike the in finite difference analysis, where the potential is known at the grid points only.
Also note that αi are linear interpolation functions, and they have the following properties.
〶
⡱
〶⢀⡩
With respect to ᡈ
〳
, yields
[ ᠩ
〳〳
][ᠩ
ぃ
]= -[ᠩ
〳ぃ
][ᡈ
ぃ
] …………………………………… (3.9)
This can be written as
[A][V] = [B]
[V] = 䙰A䙱
⡹⡩
[B]
Where [V] = [ᡈ
〳
䙱, [A] = [ ᠩ
〳〳
] and [B] = - [ᠩ
〳ぃ
][ᡈ
ぃ
]
Since [A] is, in general, nonsingular, the potential at the free nodes can be found by using eq.
3.2 Typical two dimensional electric field calculations by finite elements
method
Here we have find the potential at the free nodes in the potential system using the finite elements
method.
The solution region is divided into 25 three-node triangular elements with the total number of
nodes being 21, shown in figure 3.3.
Figure 3.3, solution region divided into 25 triangular elements.
SOLUTION:
Determination of different elements coefficient matrix:
2
1
3
1 2
3 2
1
7
7
8
2
