VectGUI: A MATLAB-Based Simulation Tool Ozlem Ozgun and Levent Sevgi
T
he vector analysis graphical user interface (VectGUI), a MATLABbased vector analysis visualization tool, can be used as a supplementary tool in the first course of electromagnetic education. The main purpose of this tool is to provide various visual aids that assist students in developing a mental image of some fundamental concepts in vector calculus and curvilinear coordinate systems. This tool is composed of seven subtools: two designed for the dynamic visualization of geometries in cylindrical and spherical coordinates; two used to display the scalar and vector fields; and three designed to demonstrate the gradient, curl, and divergence fields.
Vector Analysis Tools Teaching electromagnetics using novel modeling and simulation approaches has long been discussed, mostly in this magazine [1]–[6], including fundamental concepts and terms [7], [8]. Sevgi [9], [11] and Felsen et al. [10] discuss guided wave propagation via the well-known Sturm–Liouville problem and explain the connection between Green’s function and eigenvalue problems. The finite-difference timedomain method has been discussed, and MATLAB-based virtual tools and codes have been shared with readers [13]–[19]. Several antennas and propagation packages, from the very simple to the most comprehensive tools, have also been introduced [20]–[26]. Electromagnetic diffraction has been reviewed, and novel virtual tools have been introduced [27]–[29]. Virtual tools for teaching advanced-level mathematical functions and tools have also been reviewed with useful packages [30], [31]. Some Javabased applets for multivariable calculus, Digital Object Identifier 10.1109/MAP.2015.2463152 Date of publication: 21 August 2015
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editor’s Note This tutorial belongs to a simple MATLAB package that visualizes functions and operators (e.g., gradient of scalar fields and divergence of vector fields) in the cylindrical and spherical coordinate systems. These packages can be used in lectures such as vector algebra, mathematical foundations of electromagnetics, and mathematical methods in electromagnetics, together with the previous packages we introduced in this column [1], [2].
References
[1] L. Sevgi and Ç. Uluişik, “A tutorial on the Bessel functions and numerical evaluations of Bessel integrals,” IEEE Antennas Propagat. Mag., vol. 51, no. 6, pp. 222–233, Dec. 2009. [2] Ç. Uluişik and L. Sevgi, “A MATLAB-based visualization package for complex functions, and their mappings and integrals,” IEEE Antennas Propagat. Mag., vol. 54, no. 1, pp. 243–253, Feb. 2012.
Table 1. Differential Operations. Operation
Notation
Transformation
Gradient
df = grad ^ f h
Divergence
d $ F = div ^F h
Curl
d # F = curl ^F h
Input: scalar field Output: vector field Input: vector field Output: scalar field Input: vector field Output: vector field
including vector analysis, are available through the Internet [32]–[34]. Vision plays a vital role in electromagnetics because it removes the obstacle in understanding and perception of physical phenomena. New students often think that electromagnetics is a difficult course because some mathematical and geometrical concepts, as well as electromagnetic fields that are vector functions of space and time, are abstract and hard to visualize. Therefore, a wellestablished background on vector calculus and coordinate systems is essential in developing a better conceptual understanding of electromagnetic phenomena. Being motivated by this fact, this article presents a basic tutorial on the visualization of some fundamental concepts in the study of vector analysis. A MATLAB-based package (called VectGUI) is introduced with graphical user interface june 2015
(GUI) capabilities. As previously stated, this package includes seven subtools: cylGUI and sphGUI plot geometries in a dynamic manner by varying one of the coordinate variables in cylindrical and spherical coordinates, respectively; sfieldGUI and vfieldGUI plot user-defined scalar and vector fields, respectively, in a two-dimensional (2-D) Cartesian system; grad2dGUI and grad3dGUI plot and display the gradient of a user-defined scalar field, together with the field itself, in 2-D and three-dimensional (3-D) Cartesian systems, respectively; and, finally, curldivGUI plots and displays the curl and divergence of a user-defined vector field together with the field itself in both 2-D and 3-D Cartesian systems. The MATLAB-based VectGUI package can be downloaded from http://www.ee.hacettepe.edu. tr/~ozlem/. 113
Figure 1. The main window of the VectGUI package.
Basic Concepts Measurable quantities in the study of physical phenomena are named either scalar or vector quantities. A scalar (e.g., charge and current) is a quantity that has only magnitude and can be described by a single numeric value. A vector (e.g., velocity, electric, and magnetic fields) is a quantity that has both magnitude and direction. Both scalars and vectors can be functions of position and time. A spatial distribution of a quantity is named the field. If the quantity is a
vector (scalar), the field is called a vector (scalar) field. Hence, scalar and vector fields are the major players of electromagnetics. To describe the variations of these quantities in space, a suitable coordinate system is needed. Although the laws of electromagnetism are valid in any coordinate system, a coordinate system provides a frame of reference and is chosen based on the symmetry of the geometry to ease its analysis. Therefore, the vector analysis has three main ingredients: 1) an orthogonal coordinate system (Carte-
Table 2. The subtools of the VectGUI package. Tool Name
Description
cylGUI
Displays geometries in a cylindrical coordinate system by changing one of the coordinate variables in a dynamic manner. Displays geometries in a spherical coordinate system by changing one of the coordinate variables in a dynamic manner. Displays a user-defined scalar field in 2-D. Displays a user-defined vector field in 2-D. Displays a scalar field and its gradient field in 2-D. Displays a scalar field and its gradient field in 3-D. Displays a vector field and its curl and divergence fields in 2-D and 3-D.
sphGUI
sfieldGUI vfieldGUI grad2dGUI grad3dGUI curldivGUI
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sian, cylindrical, and spherical); 2) vector algebra (addition, subtraction, and multiplication of vectors); and 3) vector calculus (differentiation and integration of vectors). Among them, the curvilinear coordinate systems and the concepts of gradient, divergence, and curl, which are based on the del differential operator, are sometimes hard to visualize and comprehend. In a 3-D coordinate system, a point is defined at the intersection of three surfaces, which can be denoted by u 1 = constant, u 2 = constant, a n d u 3 = constant surfaces. When these three surfaces are mutually orthogonal, we have an orthogonal coordinate system. In the Cartesian coordinate system ^ x, y, z, h u 1 = x, u 2 = y, and u 3 = z; in the cylindrical coordinate system ^ r, {, z h, u 1 = r, u 2 = { and u 3 = z; and in the spherical coordinate system ^ R, i, { h, u 1 = R, u 2 = i, and u 3 = {. Here, r is the radial distance of the point from the z-axis, R is the radial distance from the origin, { is the rotation angle around the z-axis and is between 60, 2r@, and i is the angle formed with the zaxis and is between 60, r@ . In addition, r = constant surface defines the surface of a circular cylinder with radius r, R = constant surface defines the surface of a sphere with radius R, { = constant surface defines a semi-infinite plane, and i = constant surface defines a cone surface. The identification of such surfaces is important especially in defining the differential quantities (i.e., the differential length, surface, and volume elements) that are used in the evaluation of line, surface, and volume integrals. Vector calculus deals with various differential operators defined in scalar and vector fields. These operators are usually expressed in terms of the del operator ^d h that is also known as the nabla operator. The del operator is defined in a general coordinate system as follows:
2 (.) t 1 2 (.) d = at u1 1 + a u2 h 2 2u 2 h 1 2u 1 (.) 2 1 , + at u3 h 3 2u 3
(1)
where ^ u 1, u 2, u 3 h are the coordinate variables, at u i, ^ i = 1, 2, 3 h is the base vector pointing in the direction of increasing u i, and h i is the metric IEEE Antennas & Propagation Magazine
coefficient to convert the differential change in the coordinate variable ^du i h into a differential change in length ^dl i h . If the coordinate variable is in length, the metric coefficient becomes unity. The metric coefficients, in three coordinate variables, are defined as follows: h 1 = 1, h 2 = 1, h 3 = 1 (Cartesian); h 1 = 1, h 2 = r, h 3 = 1 (cylindrical); and h 1 = 1, h 2 = R, h 3 = R sin i (spherical). The three most important differential operations are listed in Table 1. The gradient operator applies to a scalar field and returns a vector field. Its direction shows the direction of maximum increase of function, and its magnitude gives the rate of increase (or slope) along this maximal direction. In other words, it measures the direction and rate of change in a scalar field. In a general coordinate system, the gradient of a scalar field f ^ u 1, u 2, u 3 h is defined as follows:
2f 2f df = at u1 1 + at u2 1 h 2 2u 2 h 1 2u 1 2f + at u3 1 , h 3 2u 3
(a)
(b)
(c)
(d)
Figure 2. The cylGUI window: (a) the r-constant surfaces, (b) the {-constant surfaces, (c) the z-constant surfaces, and (d) the formed geometry.
(2)
which, in a Cartesian coordinate system, is
df = at x
2f 2f 2f .(3) + at y + at z 2x 2y 2z
The divergence operator acts on a vector field and yields a scalar field or quantity. It measures how much the vector field spreads out (or diverges) from the point of interest, i.e., the magnitude of a source or sink of a vector field at a given point. It represents the net outward flux of the vector field from an infinitesimal volume around the given point. In a general coordinate system, the divergence of a vector field F ^ u 1, u 2, u 3 h = at u1 Fu1 ^ u 1, u 2, u 3 h + at u 2 Fu 2 ^ u 1, u 2, u 3 h + at u 3 Fu 3 Fu3 ^ u 1, u 2, u 3 h is defined as
1 ; 2 ^ h 2 h 3 Fu1 h h1 h2 h3 2u 1 2 ^ h 3 h 1 Fu2 h 2 ^ h 1 h 2 Fu3 h E, + + 2u 2 2u 3 (4) d $ F =
which, in a Cartesian coordinate system, becomes
d$F=
2Fx 2Fy 2Fz .(5) + + 2x 2y 2z
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(a)
(b)
(c)
(d)
Figure 3. The sphGUI window: (a) the R-constant surfaces, (b) the i-constant surfaces, (c) the {-constant surfaces, and (d) the formed geometry. The curl operator applies to a vector field and returns a vector field. It measures how much the vector field rotates (curls) around a point of interest. It is a measure of the strength of a vortex source. Its magnitude represents the maximum circulation of a vector field within an infinitesimal area around the given point, and its direction is normal to the area in the sense of the june 2015
right-hand rule. In a general coordinate system, the divergence of a vector field F is defined as h 1 at u1 h 2 at u2 h 3 at u3 2 2 2 , 1 d # F = h 1 h 2 h 3 2u 1 2u 2 2 u 3 h 1 Fu1 h 2 Fu2 h 3 Fu3 (6) which, in Cartesian coordinate system, becomes 115
at x at y at z d # F = 2 2 2 .(7) 2x 2y 2z Fx Fy Fz
VectGUI Package VectGUI is a MATLAB-based tool for visualizing some fundamental concepts in vector analysis. The main window of the package is shown in Figure 1. The package includes seven subtools, which are summarized in Table 2. When the user clicks one of the subtools in the main window, a new window is opened corresponding to the clicked subtool. Each subtool in Table 2 is described below. The cylGUI tool (Figure 2) is designed to display geometries in a cylindrical coordinate system. The user enters the ranges of each coordinates ^r, {, z h . Once one of the plot buttons is clicked, the corresponding variable is sampled by a number of samples between the lower and upper limits, and the geometry is dynamically formed by drawing constant surfaces at every 0.1 s. For example, as shown in Figure 2(a), if the button Plot by varying r is clicked, five r- constant surfaces are plotted corresponding to r = 0, r = 0.5, r = 1, r = 1.5, and r = 2 surfaces at every 0.1 s. Similarly, the geometry is formed by varying { and z variables in Figure 2(b) and (c), respectively. If the number of samples is increased, the geometry is well formed, as shown in Figure 2(d). The sphGUI tool (Figure 3) is designed to display geometries in a spherical coordinate system by varying one of the coordinate variables ^ R, i, { h . This is similar to the cylGUI tool. The sfieldGUI tool (Figure 4) is designed to display the contour and surface plots of a scalar field in a 2-D Cartesian system. The function is symbolically entered by the user. The ranges of the function and the number of points along the x and y axes are also provided by the user. The vfieldGUI tool (Figure 5) is designed to plot a vector field in a 2-D Cartesian system. The scalar components of the vector field (i.e., Fx and Fy ) are entered symbolically by the
Figure 4. The sfieldGUI window (scalar field plotter).
Figure 5. The vfieldGUI window (vector field plotter). 116
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Figure 6. The grad2dGUI window (2-D gradient field plotter).
Figure 7. The grad3dGUI window (3-D gradient field plotter).
Figure 8. The curldivGUI window (curl and divergence plotter).
user. The ranges of the function and the number of points along the axes are defined by the user. The grad2dGUI tool (Figure 6) is designed to plot the gradient field of a scalar field in a 2-D Cartesian system. The field itself might also be plotted. The scalar field is symbolically entered by the user, together with its ranges. The components of the gradient vector are also displayed symbolically. The grad3dGUI tool (Figure 7) is designed to plot the gradient field of a scalar field in a 3-D Cartesian system. This tool is similar to the grad2dGUI tool. IEEE Antennas & Propagation Magazine
The curldivGUI tool (Figure 8) is designed to plot the divergence and curl of a vector field in a 2-D or 3-D Cartesian system. The components of the vector field are symbolically entered by the user. The expressions for curl and divergence are also displayed symbolically.
Conclusions A MATLAB-based simulation package, VectGUI, has been developed for vector analysis. The intent is to develop a better conceptual understanding of curvilinear coordinate systems, scalar and vector fields, gradient, curl, and june 2015
divergence. The tool may be used in undergraduate electromagnetics courses and, perhaps, in other disciplines that use vector analysis.
Author Information Ozlem Ozgun (ozgunozlem@gmail. com;
[email protected]) is with the Electrical and Electronics Engineering Department, Hacettepe University, Beytepe, Ankara, Turkey. Levent Sevgi (
[email protected];
[email protected]) is with the Electrical and Electronics Engineering Department, Okan University, Akfirat, Tuzla, Istanbul, Turkey. 117
References
[1] L. Sevgi and C. Goknar, “An intelligent balance between numerical and physical experimentation,” IEEE Potentials Mag., vol. 23, no. 4, pp. 40–44, Oct./Nov. 2004. [2] L. B. Felsen and L. Sevgi, “Electromagnetic engineering in the 21st century: Challenges and perspectives,” ELEKTRIK, Turk. J. Electr. Eng. Comput. Sci., vol. 10, no. 2, pp. 131–145, Feb. 2002. [3] L. Sevgi, “A new electromagnetic engineering program and teaching via virtual tools,” Progr. Electromagn. Res. B (EWS 2006 Special Issue), vol. 6, pp. 205–224, 2008. [4] L. Sevgi, “Teaching electromagnetic modeling and simulation as a graduate-level course,” IEEE Antennas Propagat. Mag., vol. 54, no. 5, pp. 261–269, Oct. 2012. [5] L. Sevgi, Complex Electromagnetic Problems and Numerical Simulation Approaches. Hoboken, NJ: IEEE Press—John Wiley, 2003. [6] L. Sevgi, Electromagnetic Modeling and Simulation. Hoboken, NJ: IEEE Press—John Wiley, 2014. [7] L. Sevgi, “Innumeracy: The meaning of the numbers we use,” IEEE Antennas Propagat. Mag., vol. 49, no. 2, pp. 195–190, Apr. 2007. [8] L. Sevgi, “Numerical fourier transforms DFT and FFT,” IEEE Antennas Propagat. Mag., vol. 49, no. 3, pp. 238–243, June 2007. [9] L. Sevgi, “Guided waves and transverse fields: Transverse to what?” IEEE Antennas Propagat. Mag., vol. 50, no. 6, pp. 221–225, Dec. 2008. [10] L. B. Felsen, F. Akleman, and L. Sevgi, “Wave propagation inside a two-dimensional perfectly conducting parallel plate waveguide: Hybrid ray-mode techniques and their visualizations,” IEEE Antennas Propagat. Mag., vol. 46, no. 6, pp. 69–89, Dec. 2004. [11] L. Sevgi, “Sturm-Liouville equation: The bridge between eigenvalue and Green’s function problems,” ELEKTRIK, Turk. J. Electr. Eng. Comput. Sci., vol. 14, no. 2, pp. 293–311, June 2006. [12] L. Sevgi, “Guided waves and eigenvalue extraction from propagation characteristics,” IEEE Antennas Propagat. Mag., vol. 50, no. 2, pp. 222–234, Apr. 2008.
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[13] L. Sevgi and Ç. Uluisik, “A MATLAB-based transmission line virtual tool: Finite-difference time-domain reflectometer,” IEEE Antennas Propagat. Mag., vol. 48, no. 1, pp. 141–145, Feb. 2006. [14] G. Çakır, M. Çakır, and L. Sevgi, “A multipurpose FDTD-based two dimensional electromagnetic virtual tool,” IEEE Antennas Propagat. Mag., vol. 48, no. 4, pp. 142–151, Aug. 2006. [15] M. Çakır, G. Çakır, and L. Sevgi, “A two-dimensional FDTD-based virtual metamaterial—Wave interaction visualization tool,” IEEE Antennas Propagat. Mag., vol. 50, no. 3, pp. 166–175, June 2008. [16] G. Toroglu and L. Sevgi, “FDTD MATLAB codes for the first and second order electromagnetic models,” IEEE Antennas Propagat. Mag., vol. 56, no. 2, pp. 221–239, Apr. 2014. [17] A. Pekmezci and L. Sevgi, “FDTD based metamaterial modeling,” IEEE Antennas Propagat. Mag., vol. 56, no. 5, pp. 289–303, Oct. 2014. [18] G. Çakır, M. Çakır, and L. Sevgi, “A novel virtual FDTD-based microstrip circuit design and analysis tool,” IEEE Antennas Propagat. Mag., vol. 48, no. 6, pp. 161–173, Dec. 2006. [19] G. Çakır, M. Çakır, and L. Sevgi, “Radar cross section modeling and simulation: Part II—A novel FDTD-based RCS prediction virtual tool,” IEEE Antennas Propagat. Mag., vol. 50, no. 2, pp. 81–94, Apr. 2008. [20] L. Sevgi, “A ray shooting visualization MATLAB package for 2D groundwave propagation simulations,” IEEE Antennas Propagat. Mag., vol. 46, no. 4, pp. 140–145, Aug. 2004. [21] L. Sevgi and Ç. Uluisik, “A MATLAB-based visualization package for planar arrays of isotropic radiators,” IEEE Antennas Propagat. Mag., vol. 47, no. 1, pp. 156–163, Feb. 2005. [22] L. Sevgi, Ç. Uluisik, and F. Akleman, “A MATLAB-based two-dimensional parabolic equation radiowave propagation package,” IEEE Antennas Propagat. Mag., vol. 47, no. 4, pp. 164– 175, Aug. 2005. [23] F. Akleman and L. Sevgi, “A novel MoM- and SSPE-based groundwave propagation field strength prediction simulator,” IEEE Antennas Propagat. Mag., vol. 49, no. 5, pp. 69–82, Oct. 2007. [24] O. Ozgun, G. Apaydin, M. Kuzuoglu, and L. Sevgi, “PETOOL: MATLAB-based one-way
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and two-way split step parabolic equation tool for radiowave propagation over variable terrain,” Comput. Phys. Commun., vol. 182, no. 12, pp. 2638–2654, 2011. [25] L. Sevgi, “A mixed-path groundwave field strength prediction virtual tool for digital radio broadcast systems in medium and short wave bands,” IEEE Antennas Propagat. Mag., vol. 48, no. 4, pp. 19–27, Aug. 2006. [26] G. Apaydin and L. Sevgi, “Numerical investigations of and path loss predictions for surface wave propagation over sea paths including hilly island transitions,” IEEE Trans. Antennas Propag., vol. 58, no. 4, pp. 1302–1314, Apr. 2010. [27] F. Hacıveliog˘ lu, L. Sevgi, and P. Y. Ufimtsev, “Electromagnetic wave scattering from a wedge with perfectly reflecting boundaries: Analysis of asymptotic techniques,” IEEE Antennas Propagat. Mag., vol. 53, no. 3, pp. 232–253, June 2011. [28] F. Hacıveliog˘ lu, M. A. Uslu, and L. Sevgi, “A MATLAB-based simulator for the electromagnetic wave scattering from a wedge with perfectly reflecting boundaries,” IEEE Antennas Propagat. Mag., vol. 53, no. 6, pp. 234–243, Dec. 2011. [29] M. A. Uslu and L. Sevgi, “MATLAB-based virtual wedge scattering tool for the comparison of high frequency asymptotics and FDTD method,” Int. J. Appl. Comput. Electromagn., vol. 27, no. 9, pp. 697–705, Sept. 2012. [30] L. Sevgi and Ç. Uluisik, “A tutorial on the bessel functions and numerical evaluations of bessel integrals,” IEEE Antennas Propagat. Mag., vol. 51, no. 6, pp. 222–233, Dec. 2009. [31] Ç. Uluisik and L. Sevgi, “A MATLAB-based visualization package for complex functions, and their mappings and integrals,” IEEE Antennas Propagat. Mag., vol. 54, no. 1, pp. 243–253, Feb. 2012. [32] (2015, May 8). Multivariable calculus. [Online]. Available: https://multivariablecalculus. wordpress.com/video-demonstrations/ [33] (2015, May 8). Math applets for calculus at SLU. [Online]. Available: http://www.slu.edu/ classes/maymk/MathApplets-SLU.html [34] (2015, May 8). Exploring multivariable calculus. [Online]. Available: http://web.monroecc. edu/calcNSF/
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