tutorial%20complex%201%20solution

tutorial%20complex%201%20solution - Solutions to Tutorial...

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Solutions to Tutorial Complex#1 Q1. This is fundamental, begin with the deMoivre relationships: cos sin cos sin j j ej θ =+ =− then add and subtract these to get the result. () 11 cos sin 22 j jj ee j j θθ −− = Q2 15 8 17 151.9 j −− =∠ D So 15 8 17 151.9 2 17 75.9 1 4 j j −− = ∠ = ∠ DD But there are other roots as well. In general: cos sin cos 2 sin 2 0, 1, 2. . j xj yr e r j rk j k k θπ += = + + + = ± ± So that, cos sin kk y r j ⎡⎤ ++ ⎛⎞ + ⎜⎟ ⎢⎥ ⎝⎠ ⎣⎦ The argument here is ; with 75.9 k π D k = 0 arg = -75.9 ° 14 j ⇒− k = +1 arg =-75.9+180 ° =104.1 ° j ⇒− + k = -1 arg =-75.9 ° -180 ° =-255.9 ° j k = ± 2 produces the same roots So roots are -1+4j and 1-4j. Q3 Dot and cross products are defined just as with real quantities. Some equivalent expressions are: {} 12 21 2 cos 3( 4) ( 4)(3) 24 Re 1 2 z zz xx yy ∗∗ ⋅= =+⇒ + = = 1 2 sin 3(3) ( 4)( 4) 7 Im 1 2 zz zz xy yx j ×= =−⇒− = =
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Q4 Trivial problem, but illustrates the mapping concept. (1 3 ) 8 6 (2 ) 3 4 wf j j j j =−= =− += The line joining the two points in the z- plane would be mapped into curves in the w- plane. To find the curve in the w-plane, use the procedure shown in the demo session. Obtain the equation in the
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This note was uploaded on 07/07/2009 for the course ELEC 3002 at Queensland.

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tutorial%20complex%201%20solution - Solutions to Tutorial...

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