Intro ECE 2025
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This 0 page Class Notes was uploaded by Cassidy Effertz on Monday November 2, 2015. The Class Notes belongs to ECE 2025 at Georgia Institute of Technology - Main Campus taught by Biing Hwang Juang in Fall. Since its upload, it has received 10 views. For similar materials see /class/233877/ece-2025-georgia-institute-of-technology-main-campus in ELECTRICAL AND COMPUTER ENGINEERING at Georgia Institute of Technology - Main Campus.
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Date Created: 11/02/15
GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL and COMPUTER ENGINEERING ECE 2025 Spring 2012 Problem Set 1 Assigned 9Jan12 Due Date Week of 16Jan12 Reading In SP First App A on Complex Numbers pp 430451 and Ch 2 on Sinusoids pp 8413 The web site for the course uses t square httpstsquaregatechedu The login for t square is your GT login I1gt Please check t square daily All of cial course announcements will be posted there Turn in all STARRED problems Some of the problems have solutions that are similar to those found on the SPFirst CDROM After this assignment is handed in by everyone solutions will be posted to the web Your homework is due in recitation at the beginning of class After the beginning of your assigned recitation time the homework is considered late and will be given a zero Two Part Format for HW Solutions For each homework problem two distinct pieces of information are required for a complete solution a Approach Write a clear explanation of how you are going to solve the problem Write in complete sentences This explanation should be written with little or no mathematical formulas and it should also be written so that it is independent of the speci c numerical values in the problem b Details Carry out the solution of the particular problem Details mean getting the algebra correct making precise plots and doing the numerical calculations are the key Complex Numbers A complex number is just an ordered pair of real numbers Several different mathematical notations can be used to represent complex numbers In rectangular form we will use all of the following notations Z xay xjy where j39JTl RezjImZ Note that 139le 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 twodimensional space Addition of complex numbers is the same as vector addition ie add the real parts and add the imaginary parts In polar form we will use the complex exponential notation Z IzIeargz where IzI r Ixz y2 and 0 arctanyxargz In a vector drawing r is the length and 9 the direction of the vector measured from the positive xaxis The angle is often called the argument of the H 7 re complex number Here is another notation that is much less common 2 r4 0 Euler s Formula J re 6 rcos6jrsin6 can be used to convert between Cartesian and polar forms Some of these problems 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 11 Convert the following to polar form a z 7 c 2 71 J39JE e z 6745 b 2 5 1395 2 d 2 1703 f 2 2 13932 Give numerical values for the magnitude and the angle or phase in radians PROBLEM 12 Convert the following to rectangular form by using Euler s formula a z new c z 4429n3 b 2 glam d z 32 4713 Give numerical values for the real and imaginary parts PROBLEM 13 Evaluate the following and give the answer in both rectangular and polar form In all cases assume that 713 the complex numbers are 21 ilij and Z2 3811 a 22 b 222 c 2amp2 d 212 e 212zlzf Note 2 is the conjugate of Z Part e is the magnitudesquared which can also be written as the product of the complex number and its complex conjugate PROBLEM 14 Plot two periods of the following sinusoids over the timeinterval T S t S T where T is the period a A cosine wave with a period of 6 secs an amplitude of 2 and a phase of 77radians b xt cos077rt7037r c xt 3 cos207r t 7 002