GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL OF ELECTRICAL AND COMPUTER ENGINEERING

Transcription

1 GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL OF ELECTRICAL AND COMPUTER ENGINEERING ECE 2026 Summer 2018 Problem Set #1 Assigned: May 14, 2018 Due: May 22, 2018 Reading: Chapter 1; App. A on Complex Numbers, App. B on MATLAB; and Chapter 2 on Sinusoids. Notes: Only turn in starred problems. Complex Numbers: In Cartesian (rectangular) form, a complex number z can be written as: z = x + jy where j = 1, where x and y are each real numbers, referred to as the real and imaginary parts of z, respectively, and denoted x = Re{ z } and y = Im{ z }. Note that i = 1 is typical notation in most math courses. The pair (x, y) can be drawn as a vector, such that x is the horizontal coordinate and y the vertical coordinate in a two-dimensional space. Addition of complex numbers is the same as vector addition; i.e., add the real parts and add the imaginary parts. In polar form we can write the complex number z as z = re j where r is a real nonnegative number, referred to as the magnitude of z, and denoted r = z, and where is a real number, referred to as the angle (or argument) of the complex number z. In vector drawing, r is the length and is the direction of the vector measured from the positive x-axis. A less common way to write z = re j is z = r. Euler s Formula: re j = rcos( ) + jrsin( ) can be used to convert between Cartesian and polar forms, since it implies that r = x 2 + y 2 and tan( ) = y/x. Some of the problems below should be a review of complex numbers learned in high school. In these problems, a calculator will be useful for doing the complex arithmetic, especially if it is one that accepts both polar and cartesian formats. It is essential to learn how to use the polar format feature. However, it is also worthwhile to be able to do the calculations by hand, and visualize the calculation to understand what your calculator is doing. PROBLEM 1.1.* Convert the following to polar form: (a) z = 3 + 4j (b) z = j 3 + j (c) z = e j /3 + e j2 /3 (d) z = (1 + j) 8 (e) z = jcos( /6) sin( /6) (f) z = (1 + j) 2 (1 j) 2

2 PROBLEM 1.2.* The figure below shows the locations in the complex plane of z = re j when raised to the powers of k {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. In other words, it show the locations in the complex plane of z 0, z 1, z 2, z 3, z 4, z 5, z 6, z 7, z 8, z 9 and z 10 : Although the axes are not labeled, and you are not told which points correspond to which powers of k, you will still be able to accurately determine the value of z = re j : (a) Find r (an estimate is OK). (b) Find. PROBLEM 1.3.* Find all solutions to the equation: z 8 1 = 0. In other words, find the complex numbers z for which the above equation is true. (Hint: Since this is a 8-th order polynomial, there are 8 solutions.) These numbers are the so-called 8-th roots of unity. Plot them in the complex plane.

3 PROBLEM 1.4.* Plot two periods of the following sinusoids over the time-interval 0 t 2T, where T is the period: (a) A sinusoid having a period of 3 secs, an amplitude of 14, and a phase of radians. (b) x( t ) = 1.2cos( -- t + -- ). 5 2 (c) x( t ) = 2 cos(4 (t 0.05)). PROBLEM 1.5.* The waveform shown below can be represented by x( t ) = Re{Xe j 0 t }: x( t ) Time t (seconds) (a) Find 0. (b) Find the complex phasor X, expressed in polar form. (c) The waveform can also be written as x( t ) = Acos( 0 t + ). Find the amplitude A and phase.

4 (The problems below are not starred. Not to be turned in. They are provided for extra practice.) PROBLEM 1.6. Convert the following to rectangular form (by using Euler s formula): (a) z = e 1 + j (b) z = 6e j( 5 /6) (c) z = e j199.5 (d) z = e 1+ (e) z = ( /10) PROBLEM 1.7. Consider the following sum of n complex numbers: P n k =1 ejk /10. For what value or values of the integer n will the sum be zero? If there is more than one possible answer, be sure to specify them all. PROBLEM 1.8. Consider the sinusoidal waveform shown in the following figure: Time (Seconds) This waveform can be expressed in any of the following three equivalent forms: x( t ) = Acos( 0 (t t d )) = Acos( 0 t + ) = Acos(2 f 0 t + ). From the above waveform, determine the values of the parameters A, 0, f 0, t d, and. Choose the value of satisfying <.

5 PROBLEM 1.9. The following MATLAB code generates a plot of a sinusoidal signal. Derive a formula for the signal in the standard form Acos( t + ), where A is real and nonnegative, and <. Draw a sketch of the plot that will be done by MATLAB. dt = 1/1000; tt = -0.2 : dt : 0.2; Fo = 10; zz = 3*exp(j*(2*pi*Fo*(tt ))); xx = real( zz ); plot( tt, xx ); xlabel( TIME (sec) )

6 (Use this format for the cover page when you turn in your homework.) James A. Smith ECE 2026 HW #1 Due May 22, 2018 Section L02: Prof. Beck HW Grader: G. Burdell