Comparison of an Analytical Method and Matlab to Model Electromagnetic Distribution in a Trough

Transcription

1 Comparison of an Analytical Method and Matlab to Model Electromagnetic Distribution in a Trough JJ Bruyns University of Johannesburg, South Africa Jacob@uj.ac.za JC Greeff Tshwane University of Technology greeffjc@tut.ac.za SV Joubert Tshwane University of Technology joubertsv@tut.ac.za Abstract In designing devices that discharge electrostatic voltage it is important to know the radiation pattern. To interpret and visualize the result is just as important as being able to obtain a correct analytical result. This article discusses the traditional analytical method to determine the radiation pattern in a closed trough, as well as the use of a computer algebra system (CAS) to determine the radiation pattern. MATLAB is used as a computer algebra system to solve and visually display the electrostatic distribution in a trough. The results of the analytical and numerical solutions are compared to determine the accuracy of the CAS solution. The use of the CAS system and its educational advantages is discussed. Introduction Since all electric devices radiate a magnetic field, electromagnetic theory and principles are fundamental to the study of electrical systems. Electrostatic principles are used in various applications such as electrostatic filters, spraying, laser printers and many other applications. Although electrostatics has many useful applications, it also has many negative effects on humans and the environment. Because various electrostatic sources interact, the design of these devices requires thorough knowledge of the laws and principles of electrostatics in order to minimise their harmful effects. It is therefore essential to calculate, project and visualise electromagnetic and electrostatic fields in order to design applications and manage the effect on people and the environment. According to a study by Last (2006), strong electromagnetic fields (EMFs) of about 50 to 60 cycles per second (Hz) and the related electromagnetic radiation (EMR) are harmful to human beings. An article in Huisgenoot (2007) reports that a study conducted by the Californian Health Department found that the exposure to magnetic fields greatly increases the risk of serious illnesses. A number of studies discussing electromagnetic and electrostatic fields are available in the literature, of which the following are applicable to this study. O'Neil (1995) discusses the analytical solution of the Laplace equation to determine the electrostatic voltage due to a point charge, but does not cover any practical applications. Rothwell and Cloud (2001) discuss and explain the electrostatic potential but do not explain in detail how to calculate the potential or visualise the distribution. Although these studies provide essential principles, they do not address the difficulty of visualising the electrostatic voltage distribution in a closed environment.

2 To predict the magnetic field distribution in a closed environment, a closed metal trough will be used to simulate the existence of such fields that may be present in our immediate environments, incorporating the use of a computer algebra system (CAS) such as MATLAB. The application of MATLAB assists in the understanding and visualisation of electromagnetic voltage distribution in a trough and relates the mathematical theory to realworld applications. Objectives of this study The author proposes to use an analytical method (a mathematical quantitative method) and a mathematical technology program (a mathematical qualitative method) to solve Laplace's equation in order to determine the electromagnetic distribution in a trough and visually display the results. The quantitative and qualitative results will be compared and an error analyses conducted to evaluate the accuracy and usefulness of the qualitative method proposed. Analytical Solution The specific Laplace equation in rectangular coordinates in a two-dimensional system subject to certain boundary conditions is solved using the method of separation of variables. Problem statement Consider a rectangular trough of infinite length, of which the cross-section is illustrated in Figure 1. The trough is constructed of thin conducting sheets. Conducting material is used so that a voltage can be applied to the walls of the trough. A conducting lid is placed at y = b and is insulated from the other walls at the two gaps. The charge density, ρ(v), inside the trough is zero, V=V on the lid, and the other three sides are grounded, that is V=0. The challenge is to determine an expression for voltage V(x,y) within the trough, so that the electrostatic voltage inside the trough, due to the potential applied to the conducting cap, can be determined. y b Gap Conductor cap V=V 0 Gap V=0 Ρ v = 0 V=V(x,y) V=0 Conductor V=0 Figure 1: Cross-section of a rectangular metal trough grounded on three sides where the voltage applied to the cap is maintained at V. a x 2

3 The "trough problem" can be stated as follows: Find the potential V(x,y) inside the trough for 2 V V V = +, for 0 < x < a and 0 < y < b, with the boundary conditions as specified x y in Table 1: Table 1. Boundary conditions for the trough problem. V( 0,y) = 0 for 0< y<b V(a,y) = 0 for 0< y< b V(x, 0 ) = 0 for 0 < x < a V(x,b) = Vo for 0 < x < a. 2 V V The analytical solution of Laplace s two-dimensional equation V = + = 0, x y through the separation of variables method and after application of the boundary conditions, nπx nπy sin( )sinh( ) 4V yields o V ( x, y) = a a (Eq. 1) π 2 k 1 nπb nsinh( ) a The potential function V(x,y) satisfies the four boundary conditions and is the solution to the trough problem. Graphical illustration As an example, a truncated series is plotted for the values a = 20, b = 15 and V o = 100V. Figure 2 below represents the resultant potential distribution in the cross-section of the trough. The figure were produced with Scientific Workplace after obtaining the summation of the first 8 terms of the series solution of Equation (1). Figure 2 illustrates that the electrostatic voltage inside the trough is at its maximum close to the side to which the voltage is applied and distributed through the trough. The voltage decreases gradually from a maximum to zero volts against the sides of the trough that are grounded. Figure 2: The 3-D plot of the potential distribution with 100V applied to one side. 3

4 It is therefore possible to determine the electrostatic voltage distribution inside the trough analytically, and to plot the results using a computer algebra system. The solution, numerically produced simplified and plotted by Scientific Workplace, illustrates conveniently what is predicted by the analytical results. MATLAB Solution In the field of Engineering, MATLAB is one of the preferred and most commonly used software programs. The objectives of the MATLAB Partial Differential (PDE) Toolbox are to provide tools that will do the following: Numerically solve the PDE problem and produce an approximation to the solution. Represent the solution graphically The toolbox uses the Finite Element Method (FEM) to solve the defined problem, then provides a visual representation of the solution, which contributes to the important goal of understanding the result. The definition of the PDE problem Referring to the example above, the electrostatic potential in a trough of infinite length is determined as defined in the problem statement section. A square trough is used with Dirichlet-type boundary conditions, 100V on one side, while and the other three sides are kept at 0V. There is no volume charge in the domain, that is ρ =0. This leads to the problem of solving the Laplace equation 2 V = 0. In this case, ε is taken as ε = -1 since the relative coefficient of dielectricity in air is unity. V V The PDE to be solved then takes the form of + = 0. x y First select the application mode - Electrostatic. In the draw mode, draw the square geometry as depicted in Figure 3. Figure 3: Square trough with dimensions x = 20 and y = 15. 4

5 The PDE Toolbox solves the defined PDE numerically and the solution is then plotted using interpolated colouring where a colour bar is used to display the different colours and their numerical values. The solution in Figure 4 shows a high concentration of electrostatic voltage close to the boundary that had an initial charge of 100V. The voltage gradually decreases until it reaches 0V at the other three boundaries. Figure 4: Solution with colour bar. To view equipotential lines, a contour plot can be displayed as shown in Figure 5. Figure 5: Equipotential plot. To visualise the solution, a 3-D interactive plot is available. The mouse can be used to rotate this plot about the x-axis or the y-axis, and the angle from which the solution is viewed. Values at specific coordinates can be included in the plot as shown in Figure 6. The values in 5

6 the figure are given accurate to four significant figures which are sufficiently accurate for most scientific calculations. Figure 6: 3-D plot with specific coordinate values. Conclusion It is easy for new users to learn how to use the interactive windows of MATLAB's PDE Toolbox to specify conditions and perform basic plots. With practice, advanced tasks can also be performed. The toolbox is versatile and can be used for a number of applications. The Graphical User Interface (GUI) is powerful enough for experienced users to perform advanced tasks and obtain a visual result without having to write any MATLAB code. MATLAB's PDE Toolbox has important advantages over traditional analytical methods. MATLAB solutions are faster and the results compare favourable with other CAS as well as analytical methods. The GUI provides solutions to the common PDE problems encountered in engineering courses. The PDE Toolbox has many uses throughout the design cycle, from conceptualisation and proof of concept through simulation. It can also play a major role in the following areas: Design and development Education Creative possibilities Accuracy Users often ignore the danger of assuming the accuracy of a PDE solver. This could be fatal in the field of engineering, where applications are often more important than theory. Joubert and Greeff (2006) warn against the use of available technology without questioning the accuracy of the solutions generated. To investigate the accuracy of the PDE Toolbox plot results, the point values obtained in Figure 6 at coordinates (10, 9.533) are substituted in the analytical solution given in Equation 1. The result is then compared to the voltage, V PDE = 48.79V, as calculated by the PDE Toolbox. The number of iterations necessary for an accurate result is also investigated. 6

7 Table 2 illustrates that eight terms is sufficient for four significant figure accuracy when approximating V(x,y) analytically using Equation 1. In the field of engineering and in this application, four significant accuracy is sufficient. Table 2: Eight terms in the solution of Equation (1) yield V(x,y) = V. N V N (x,y) V N+1 -V N In Table 3, the approximate analytical voltage, (V N ) is compared to the voltage obtained through MATLAB (V M ), at point (10, 9.533) where V M = 48.79V. Define the relative VM VN percentage difference (RPD) between V N and V M as RPD = 100%. For V 5, RPD of VM 0% between the analytical and MATLAB values indicates that the comparison between the two solutions is satisfactory and the PDE Toolbox can be regarded as "sufficiently accurate" under the specified conditions. In fact the calculations agree to 4 significant digits. Table 3: Comparison between analytical and MATLAB results. N V N V M -V N RPD % % % % % Conclusion Two-dimensional studies of complex electrostatic problems can be conveniently performed using the PDE Toolbox from MATLAB. The PDE Toolbox can also form a valuable tool for teaching concepts and visualising electrostatic voltage distribution. A survey showed that students find it easy to use the PDE Toolbox because of its graphical user interface (GUI) and the interactive nature of the program. The main advantage is that it is quick to find a graphical solution and interpret the effect of changes to conditions. The Toolbox can be used for educational purposes to complement analytical methods, allowing the student to better understand the theory and visualise the result. The module offers a useful way for students to set up models for electrostatics, magnetostatics and wave propagation. Theory and equations become alive, and students may experiment with geometries, field strengths, wavelengths, material properties and initial conditions. The PDE Toolbox will contribute to the development of modelling and has practical benefits in terms of visualisation and ease of use. However, a sound theoretical background is essential, especially for post-graduate studies and research. 7

8 It is proposed that a computer algebra system (CAS) such as MATLAB be used in conjunction with traditional mathematical methods to substantiate findings obtained through analytical methods. However, Computer Algebra Systems should not replace traditional methods but rather should be used to clarify and enhance qualitative results, and to illustrate and teach abstract and difficult concepts of electrostatic potential. The MATLAB PDE Toolbox can be used interactively by students and lecturers in other fields such as structural mechanics electrostatics magnetostatics AC power electromagnetics conductive media DC heat transfer diffusion References HUISGENOOT. (15 Februarie 2007). Koel op eie risiko. JOUBERT, S.V. & GREEFF J.C. (2006). Accuracy estimates for computer algebra systems IVP solvers. SAJS, 102, LAST, W. Electrostatic pollution. Available at: <( com/electropollution.html)>. Accessed 24 August MATLAB. Partial differential equation toolbox. Available at: < mathworks.com/ access/helpdesk/help/toolbox/pde>. Accessed 19 April O'NEIL, P.V. (1987). Advanced Engineering Mathematics. California: Wadsworth publishing company, USA. ROTHWELL, J.E. & CLOUD, J.M Electromagnetics. New York: CRC, USA. 8