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Unformatted text preview: 1/16 EECE 260 Electrical Circuits Prof. Mark Fowler Complex Number Review 2/16 Complex Numbers Complex numbers arise as roots of polynomials. 1 ) )( ( 1 ) )( ( 1 1 2 = = = = j j j j j j Rectangular form of a complex number: } Im{ } Re{ z b z a jb a z = = + = real numbers Recall that the solution of differential equations involves finding roots of the characteristic polynomial Sodifferential equations often involve complex numbers Definition of imaginary # j and some resulting properties: ) ( ) ( ) )( ( : ) ( ) ( ) ( ) ( : bc ad j bd ac jd c jb a Multiply d b j c a jd c jb a Add + + = + + + + + = + + + The rules of addition and multiplication are straightforward: 3/16 Polar Form j re z = Polar form an alternate way to express a complex number Polar Form good for multiplication and division r &gt; 0 If r is negative then it is NOT in polar form!!! Note: you may have learned polar form as r we will NOT use that here!! The advantage of the re j is that when it is manipulated using rules of exponentials and it behaves properly according to the rules of complex #s: y x y x y x y x a a a a a a + = = / ) )( ( Dividing Using Polar Form ( ) ( ) ) ( 2 1 2 1 2 1 2 1 = j j j e r r e r e r 2 2 2 2 2 1 1 1 j j e r e r z = = Multiplying Using Polar Form ( ) ( ) ) ( 2 1 2 1 2 1 2 1 + = j j j e r r e r e r ( ) n j n n jn n n j n e r z e r re z / / 1 / 1 = = = { } { } { } 2 1 2 1 2 1 2 1 z z z z z z z z + = = 4/16 b a z = a + jb Im Re r Geometry of Complex Numbers We need to be able convert between Rectangular and Polar Forms this is made easy and obvious by looking at the geometry (and trigonometry) of complex #s: r a b cos sin r a r b = = = + = a b b a r 1 2 2 tan Conversion Formulas 5/16 omplex Exponentials vs. Sines and Cosines Eulers Equations: (A) (B) (C) (D) 6/16...
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 Spring '08
 FOWLER
 Exponential Function, Complex number, Euler's formula, polar form, Complex Number Review

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