3-Lesson_Notes_Lecture_19 - EE 204 Lecture 19 Review of...

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EE 204 Lecture 19 Review of Complex Numbers Complex Numbers: Any complex number A can be expressed as Aaj b = + Where: a = real part of A b = imaginary part of A 1 j ≡− Both a & b are real numbers The bar on the top is used to denote that A is a complex number Basic Complex Number Operations: A brief review of basic complex number operations will be presented next Powers of j : () 2 2 11 j =− = 32 3 1 j j =−×− = 42 2 (1 ) (1 ) 1 jj j =×= × = 54 (1) ( ) j j j = 64 2 (1) ( 1) 1 j =×=× = and so on
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The Complex Conjugate of a Complex Number: Given the complex number Aaj b =+ The complex conjugate of A is given by * Aa j b = Where the superscript * denotes the complex conjugate. to obtain the complex conjugate replace j by j Addition and Subtraction of Complex Numbers: Given the two complex numbers 11 1 Aa j b = + & 22 2 j b The sum and difference is obtained as follows: 12 1 2 () A a j b b ±=± + ± Example 1: Given 1 34 Aj & 2 25 =− + Calculate: a) AA + b) A A Solution: a) (3 2) (4 5) 1 9 j j + =−+ +=+ b) (3 ( 2)) (4 (5)) 5 ( 1) 5 j j j −= −+ = + = Multiplication of Complex Numbers:
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Given the two complex numbers 11 1 Aa j b = + & 22 2 Aa j b =+ Their product is given by: 2 1 2 1 1 2 2 12 21 () ( ) ( ) ( ) ( ) A A a jb a jb a a ja b ja b j b b a a b b j a b a b = + + =+++ =− + + If we multiply a complex number by its conjugate, we always obtain a real number *2 2 2 2 2 ( ) ( ) AA a jb a jb a jab jab j b a b = + = + Example 2: Given 1 34 Aj & 2 25 =− + Calculate: a) AA b) * A A Solution: a) (3 4)( 2 5) ( 6 20) ( 8 15) 28 13 j j j j =+ −+ = −− +−+ = −+ b) 2 (3 4)(3 4) (3 4 ) 25 j j − = + = Division of Complex Numbers: Given the two complex numbers 1 A aj b = + & 2 A b The division 11111 1 2 1 2 2 1 1 2 2 2 2 2 2 A a jb a jb a jb a a b b j a b a b A a jb a jb a jb a b ++ + + ÷= = = × = + Where we use the complex conjugate of the denominator to facilitate the division Example 3:
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Given 1 34 Aj =+ & 2 25 =− + Calculate 12 A A ÷ Solution: 3 4 3 4 2 5 ( 6 20) ( 8 15) 14 23 14 23 2525 4 2 5 2 9 2 9 2 9 jjj j j AA j jj j + + −− −+ + −− ÷= = × = = = + Magnitude of a Complex Number: The magnitude of a complex number A is defined as: * AA A A == (the bar is removed from A in order to denote the magnitude) Given Aaj b 22 () Aa j b a j ba b −=+ Euler’s Formula: Taylor series expansions of cos θ
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This note was uploaded on 01/16/2010 for the course EE ee204 taught by Professor Profosama during the Spring '09 term at King Fahd University of Petroleum & Minerals.

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3-Lesson_Notes_Lecture_19 - EE 204 Lecture 19 Review of...

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