EECE 200 Lab MATLAB Assignment, Fall 2011

American University of Beirut Department of Electrical and Computer Engineering EECE 200 – Introduction to Electrical and Computer Engineering Computer Assignment #1 Introduction to MATLAB The objective of this assignment is to familiarize students with the MATLAB software package. MATLAB is a powerful mathematical tool used in many engineering applications. Log on to any computer in the FEA Computer Labs and run MATLAB by opening the start menu and selecting All Programs – MATLAB – MATLAB R2008b. Once MATLAB starts, you will see the MATLAB desktop. The primary area of interest to us is the Command Window. In this window, the user enters the commands at the >> prompt. 1. Simple Mathematical Operations In the command window, type each of the following statements at the >> prompt, then press Enter or New Line: 1+1 3/7 8.2*2.5 3^2 sqrt(4) sin(pi/3) round(4.3) abs(-7.82) log(10) log10(10) exp(1) As you can see, simple computations are very easy. You can learn more about what the functions do by typing help followed by function name. Try: help sqrt We can assign variable names in order to save results of computations. If no variables are used, the answer is automatically assigned to the variable ans. Try the following: a = 2 b = 7 c = a + b You can view the list of variables using who. Now create the following variables: a = last five digits of your AUB ID number s = square root of a mob = your month of birth (Jan is 1, Feb is 2, Mar is 3, etc.) d = s×mob par = natural logarithm of d Q1: Report the value of par in the Moodle submission window. Use the following format: Q1: par = 2. Polynomial Operations Polynomials are defined in MATLAB using lists of numbers (also referred to as vectors) that correspond to the coefficients of the polynomial. As an example, the polynomial poly1 = x3 + 3x2 + 2x + par, where par is as calculated above, is represented as follows: poly1 = [1 3 2 par] To evaluate the polynomial at a certain value of x, use polyval(poly , x). Evaluate poly1 at x = 1.5: pv = polyval(poly1 , 1.5) Q2: Report the value of pv Polynomials can be added and subtracted using + and –, respectively. Try: newpoly = poly1 - [0 2 2 1] Note that the number of coefficients should be the same in both polynomials. Q3: Report the value of newpoly; write it as a polynomial Polyomials are multiplied using conv, and divided using deconv. As an example, try multiplying poly1 and x + 1 conv(poly1 , [1 1]) What is the resulting polynomial? Q4: Report the result; write it as a polynomial Note that deconv will return two polynomials, the first contains the coefficients of the quotient polynomial, and the second contains the coefficients of the remainder polynomial. Try dividing x3 + 3x2 + 3x + 3 by x + 2: [q , r] = deconv( [1 3 3 3] , [1 2]) What are the quotient and remainder polynomials? Q5: Report the result; write q and r as polynomials 3. Roots of Polynomials The MATLAB function roots finds the roots of polynomials. Find the roots of poly1 using: roots(poly1) How many roots are real? How many are complex? Q6: List the roots of poly1 4. Solving Equations in MATLAB MATLAB can find roots of functions using fzero. The arguments of fzero are the function itself, which can be defined using inline, and an initial guess or an interval where the root is located. As an example, we will find the root of the function 2 xe sin( ) x x near x = 6 : func1 = inline(' exp(x) * sin(x) / (x^2) ') fzero (func1, 6) Q7: Report the value of the root Verify that the function is in fact zero at the value found by MATLAB using: feval(func1, ans) You may not get a value of exactly zero, due to numerical approximations. Now we will solve a system of linear equations, such as those we encounter in circuit analysis: 7 v1 – 4 v2 – 2 v3 = 3 – 4 v1 + 9 v2 – 2 v3 = 0 – 2 v1 – 2 v2 + 5 v3 = – 12 There are several ways to solve this system of equations in MATLAB. The first method is to use solve: eq1 = '7*v1 - 4*v2 - 2*v3 = 3' eq2 ='-4*v1 + 9*v2 - 2*v3 = 0' eq3 = '-2*v1 - 2*v2 + 5*v3 = -12' [v1 , v2 , v3] = solve( eq1 , eq2 , eq3 ) Q8: Report the values of v1, v2, and v3 The second method uses matrix notation. We write the coefficients of v1, v2, and v3 in the equations in a table, referred to as matrix: G = [ 7 -4 -2 -4 9 -2 -2 -2 5] And the right hand side as a column vector: H = [ 3 0 -12] And use the \ operator to solve the system of equations: G\H Q9: Report the values of v1, v2, and v3 and compare with the previous result It is also possible to compute to inverse of the matrix G, and then multiply it by H to find the solution: inv(G) ans * H Q10: Report the values of v1, v2, and v3 and compare with the previous result Note also that the determinant of a matrix is calculated using det: det(G) Q11: Report the values of determinant of the matrix G 5. Derivatives and Integrals MATLAB provides an easy way to compute derivatives and integrals. To perform differentiation and integration, we have to declare a “symbolic” variable using syms. For example, the symbolic variable x is the variable that appears in dx inside the integral or in the denominator of the derivative dy/dx. Use the following to find the derivative of x3 + 2x2 + 5: syms x diff( x^3 + 2*x^2 + 5 ) Q12: Report the result To calculate the integral ( 2 sin( )) x x dx + ∫ , use: int ( x^2 + sin(x) ) Q13: Report the result We can also find the definite integral 2 0 ( sin( )) par x x dx + ∫ , by providing the limits of integration: int ( x^2 + sin(x) , 0 , par ) The result may be a fraction. To get a floating point approximation, use: double(ans) Q14: Report the result MATLAB can also solve differential equations using dsolve. As an example, we solve the differential equation y’ + b y = 0, where y is a function of x and b is a constant: dsolve('Dy + b*y = 0', 'x ') Note that capital D means “derivative of”. Solve the equation y’ + b y2 = 0. Q15: Report the result 6. Complex Numbers The quantities i and j are already defined in MATLAB as the square root of –1. Calculate the following: i*i j^3 Try the following operations on complex variables: z1 = 2 + 3j z2 = par – 4j z1 + z2 z1 – z2 z1 * z2 z2 / z1 The functions abs and angle allow us to convert the complex number from rectangular to polar form: abs(z1) angle(z2) The angle is given in radians. To convert to degrees, multiply by 180/π: 180/pi * angle(z2) Q16: Report the result in degrees The functions real and imag allow us to convert the complex number from polar to rectangular form. We will calculate the real part of 3eiπ: real( 3 * exp( i*pi ) ) Q17: Report the result 7. Plotting Functions and Data Functions can be easily plotted in MATLAB using ezplot! Let’s plot, as an example, the function 2 xe sin( ) x x from x = 0 to x = 9.5: ezplot( ' exp(x)*sin(x)/(x^2) ', [0 , 9.5] ) grid What did the grid command do? We can plot data points using plot: vd = [1 2 3 4 5 6 7 8 9] id = [2 4 par 8 10 12 15 19 22] plot(vd,id,'-o') grid The ‘-o’ in the plot command requests that the data points are connected by a straight line, and shown as small circles. Many other options are available; try help plot. On the Figure window, and using the Insert menu, add an x-axis label of (voltage in V) and a y-axis label of (current in mA). Also, add a title of (Device Characteristics). • Show the image to the Graduate Assistant