EEE161 Applied Electromagnetics Laboratory 1

Transcription

1 EEE161 Applied Electromagnetics Laboratory 1 Instructor: Dr. Milica Marković Office: Riverside Hall milica@csus.edu Web: milica This laboratory exercise will introduce you to concepts in vectors and coordinate systems that will be used in electrostatics and magnetostatics. Read Chapter 3 from Ulaby book and Matlab help. Introduction 1. Find the Matlab program by going to Start, All Programs, Matlab 2. When the Matlab window opens, you will see three window: prompt, variables and edit. Later, when we make graphs, you will also see the graphics window. Do not type the program commands directly into the prompt window. Even though this is possible you want to have the entire program in front of you, so that you can debug the program if necessary. Matlab has its own program editor. 3. To write your code, from Matlab pulldown menu select File, New. 4. Save the new file as yourname.m. Be careful when you name a matlab file: (a) Avoid all special characters. (b) Do not use space (c) Do not start the file name with a number. (d) When you start writing your code, do not name any variables the same as the name of your file. 5. Start typing your code 6. It is a good practice to write comments for each line of your code. 7. The first line of code should have the explanation of what is the purpose of the program. 8. The second line of the code should be clear all command. This command clears workspace, command screen and all variables. 9. Matlab has excellent documentation. Look for Matlab help. 10. Laboratory report. Use Matlab to help you assemble the report! When you are done with one section of the lab, save the matlab file, then select File, Publish to, Word Document. Your matlab file, all the information from command window and all plots will be placed in the word document that has the same name as your m file. Be patient, as it takes about 30 seconds for matlab to open the word file. To submit your final report, you can edit this file and add comments, then put all the files that together that you worked on during the laboratory session. Using Matlab as a calculator Copy and paste the sample Matlab code below in the new file yourname.m you created in step above. The code below defines two constants a1 and a2 with values 8 and 4. Constants are then added, subtracted, multiplied, divided. Subsequently, exponent, logarithm, sine, cosine, tangent of numbers were found. % Create two scalars and add/subtract/multiply/divide them a1=8 a2=4 a3=a1+a2 a4=a1-a2 a5=a1*a2 a6=a1/a2 1

2 % Exponent, Logarthm, Sine, Cosine of a number a7=a1^a2 %Natural Logarithm a8=log(a1) %Base 10 logarithm a9=log10(a1) %Sine of a number a10=sin(a1) %Cosine of a number a11=cos(a1) %Tangent of a number a12=tan(a1) Problem 1a. Write a code in Matlab to find y = cos e x if x=5. Create a new matlab file, write comments for each line of code. Problem 1b. Explore Matlab help, and see how Matlab differentiates between angles given in degrees and radians. Specifically, write a code to find cosine of an angle of 45 o. Define a constant zdeg to be 45 o,then define another constant zrad to be π 4. The cosine value should be the same, whether you define an angle in degrees or radians. What did you have to do to make Matlab understand you defined an angle in degrees? Vectors Matlab code below creates vectors b, c and d. Vector b represents the vector in 3D b =3 x+4 y +5 z. Notethat we dont write anywhere explicitly x, y or z. We just keep track that we always write x component of b first in the Matlab vector followed by y and z components. Subsequently, vector b is multiplied with a constant a1. Finally we can check which number is at a certain location in vector b. %Create vector b b=[3, 4, 5] %Create two more vectors, these vectors could represent a 5D space. c=[6,2,1,9,7] d=[1,2,3,4,5] %Create a constant (scalar) a1 a1=5 %Multiply vector b with a scalar a1 b1=a1*b %We can check which number is in the second location in vector d b(2) Problem 2. Write a code in Matlab to add vectors b =3 x +4 y and a = x +2 y. Solve this problem first on paper. Create a new matlab file, write comments for each line of code. 2

3 Plotting vectors in x-y plane A function in Matlab that plots vectors is called quiver. Syntax of quiver function is as follows: quiver(x,y,δx,δy,const) (1) In the above expression, (X, Y )isthepointofvectorsorigin. δx, δy are the magnitudes of the vectors x and y component. Const at the end of expression will scale the vector. If you want to see the vector in its actual length, use const=1. For example, to plot a vector b =3 x +4 y in its actual length, write: quiver(0,0,3,4,1) Problem 3. Write a code in Matlab to plot vectors b =3 x +4 y and a = x +2 y. Solve this problem first on paper. Create a new matlab file, write comments for each line of code. Plotting vectors in 3D An equivalent function to plot vectors in 3D is called quiver3. Syntax of quiver3 function is as follows: Quiver3(X,Y,Z,δX,δY,δZ,const) (2) Problem 4. Write a code in Matlab to plot vectors b =3 x +4 y +3 z and a = x +2 y +3 z. Solve this problem first on paper. Create a new matlab file, write comments for each line of code. Creating Matrices Matrices are used extensively in electrical engineering. There are several ways to generate matrices in Matlab. In the example below several matrices are generated. The first matrix in the example Matlab file below is matrix g, followed by the explanation on how to generate matrices that contain all zeros. g = [ %Here we generate a matrix $g$. Note that the new row is labeled by a semicolon sign g=[1 2 3; 4 5 6] ] %Here is two simple ways to generate a matrix 5x5 with all elements 0 aa=zeros(5) %We can also generate a 2x5 matrix of zeros aa1=zero(2,5) %OR generate a random values in a matrix aa2=rand(5) %If a matrix row (or any other command in matlab that is too long) has too many %elements, then inste h1=[ ; ] %we can type 3

4 h2=[ , ; ] %We can check which number is in a certain location h2(2,2) %We can also change the value at a location h2(2,2)=10000 %We can also create a column vector from the matrix h2 using the colon operator x=h2(:,1) %or a row vector x=h2(1,:) Matrix multiplication To multiply matrices, matrix dimensions must agree. For example, if we have two matrices A (2 x 3)and B (3 x 2) as given below: A = [ ] a11 a 12 a 13 B = b 11 b 12 b a 21 a 22 a 21 b b 31 b 32 Then the product of the two matrices C = AB is equal to C = [ ] a11 b 11 + a 12 b 21 + a 13 b 31 a 21 b 12 + a 22 b 22 + a 23 b 32 In Matlab the multiplication sign is used to multiply scalars, vectors or matrices. Problem 5. Multiply two matrices A and B given below, with normal matrix multiplication rules on paper. Then run the bottom Matlab program and compare results. %Define two matrices A and B A=[1 2 3; 4 5 6; 7 8 9] B=[10 11; 12 13; 14 15] %Multiply matrices C=A*B Multiplication of matrices element by element Often we want to multiply only element a 11 in matrix A with the element at the same position of matrix B b 11. In that case we have to write a dot in front of the multiplication sign.. The matrices have to be of the same size, for example 3x5. 4

5 [ ] [ ] a11 a A = 12 a 13 b11 b B = 12 b 13 a 21 a 22 a 23 b 21 b 22 b 23 Then the product of two matrices element-by-element D = A. B will be D = [ ] a11 b 11 a 12 b 12 a 13 b 13 a 21 b 21 a 22 b 22 a 23 b 23 Problem 6. Multiply two matrices A and B given below, with element-by-element matrix multiplication on paper. Then run the bottom Matlab program and compare results. %Define two matrices A and B A=[1 2 3; 4 5 6; 7 8 9] B=[ ; ; ] %Multiply matrices D=A.*B Making Plots It is often useful to represent data in graphical form. Data can be either the result of a simulation or actual data from the laboratory exercise. For example, you can measure voltage of a node in an electrical circuit over the range of frequencies. For example, you may get data that looks like this: Frequency [MHz] voltage [V] It is very useful to see this data in a graphical form. If you specify the frequency as a vector b, and the voltage as a vector c, you can use the program below to plot voltage as a function of frequency. %%MAKING PLOTS %Plot vector c on y-axis as a function of b vector on x-axis b=[ ] c=[ ] plot(b,c) It is usually required that the plot is labeled: x-axis, y-axis, title etc. With the code below you can make the line width thicker, change the x-axis to log scale, add title etc. In addition, sometimes, it is easier to define a vector by defining start and end point and step. See how vector x is defined below. Start point of the vector is 0, end point is 2π and step is π/100. You can further explore Matlab by pressing F1 or selecting help from the bar, then select Demos tab, select Graphics and then 2-D plots. 5

6 %Let s make the line of the plot thicker plot(b,c, LineWidth,4) %Let s change the line color plot(b,c, Color, red ) %Let s make a log scale on the x-axis semilogx(b,c) %Let s add a title to the plot and the labels for x and y axis and a grid plot(b,c), title( Experiment 1 ), xlabel( frequency ), ylabel( resistance ),grid %Plotting functions %first generate the vector of x-axis points from 0 to 2*pi with step pi/100 x=0:pi/100:2*pi %define a sin function of x y=sin(x) %plot the function plot(x,y) %Let s see some other interesting functions Matlab can make such as square %wave y1=square(x) plot(x,y1) %tangent y2=atan(x) plot(x,y2) %tangent in a better range of x-values x1=0:pi/100:pi/2-0.01*pi y3=tan(x1) plot(x1,y3) %What if we want to plot two functions on the same graph? plot(x,y) hold on plot(x,y1) %or to do two plots on the same graph we can write the following plot(x,y,x,y1) %above we can see that the matlab takes care of the graph colors Part 1 Write Matlab code to find the distance vector R between the sphere q located at a point P (x c,y c,z c ), as shown in Figure 1. Define the points P and P e at the beginning of the code (pick two specific points, such as [1,2,3] and [4,5,6]), followed by the general expression for the distance vector between these two points. Use Matlab to plot these two points and the distance vector between them. See hints below. 6

7 Figure 1: Part 1 1. In order to solve this problem, you first have to know how to define a vector in Matlab. A vector a =2 x +1 y +5 z in Matlab is written as follows a =[2, 1, 5]. 2. Magnitude of a vector is calculated as ma = norm(a) a 3. Unit vector in a direction of the vector a is written as ua = norm(a) 4. Refer to the attachment to find addition, subtraction, multiplication and division of vectors in Matlab. Instead of my guide, you can also use Matlab help. 5. To plot the field use quiver function B=quiver(X,Y,U,V) This function will plot the vector defined by U and V at a point X and Y. 6. To label the points P c and P e, after quiver command use annotation( ellipse,[x,y,a,b]) Where x and y are the position of the point and a and b are the height and width of an ellipse. Conclusion Solve all of the above examples by hand and compare the results with the ones you found during the lab. In addition, for each of the problems, write what you have learned. Do not write an essay, just in a few sentences write what you have done in each problem, and specifically what you have learned from the work done. Due Date Submit the printout of the lab work and the conclusion by NEXT FRIDAY. LAB IS DUE BY noon (stamped) next Friday in the main office. No late labs. Do not bring previous lab writeup for the next Lab session. 7