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Complex Numbers Practice _1

# Complex Numbers Practice _1 - Assigned Due EE301 Fall 2001...

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Assigned 8/31/01 EE301 Due 9/12/01 Fall 2001 Prof. Fowler EE 301 HW#1 Complex Numbers & Complex Functions NOTE: Show Steps Used – don’t just use your calculator’s complex number functions!! 1. For each of the following complex numbers, show its graphical representation on the complex plane and convert it into polar form. a. 2– j 3 b. 6 c. 2 j d. 2+ j 4 e. –3– j 4 f. –4+ j 4 2. For each of the following complex numbers, show its graphical representation on the complex plane and convert it into rectangular form (show steps used – don’t just use your calculator’s polar-to- rectangular function). a. 3 e j π /8 b. 4 e j π c. 3 e j 2 π /3 d. 3 e j 0 e. 3 e j π /2 f. 3 e j 4 π 3. Find the following products a. (2– j 3) 3 e j π /8 b. (2+ j 4) 3 e j 2 π /3 4. Find the following quotients a. (2– j 3)/3 e j π /8 b. (2+ j 4)/3 e j 2 π /3 5. For each of the following complex numbers, simplify the result as appropriate (note: in each case the number is rewritten in a form that is helpful to find the answer) a. je -j π /2 = j × e -j π /2 b. -e j 3 π /4 = -1 × e j 3 π /4 c. 4 / 4 / π π j j e j je × = 6. For each of the following complex-valued sequences of numbers, plot the magnitude and angle versus the integer variable } , 3 , 2 , 1 , 0 , 1 , 2 , 3 , { n . a. z ( n ) = 4 + j 3 n b. z ( n ) = (4 + j 3 n )/(4 n + j 3) c. z ( n ) = e j 4 π n 7. For each of the following functions, plot the magnitude and angle versus the real variable . ) , ( −∞ x a. z ( x ) = 10/(4x + j 3) b. z ( x ) = e j 4 π x c. Plot the real and imaginary parts of the function in 7b

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EE 301 HW#1 Solutions Note: ALWAYS state units of quantities . Here the magnitudes are unitless because we don’t know what the complex numbers physically represent (i.e. voltage, etc.) BUT the angles are in radians – so state their units as “rad” 1. Rect to Polar: For each case you need to compute the magnitude and angle – sometimes it is best to use the formulae but other times a bit of thinking provides an easy way to the answer – always look for these things to save time on exams!!! a. 2– j 3 rad 983 . 0 ) 2 / 3 ( tan 13 ) 3 ( 2 | | 1 2 2 = = = + = z z 983 . 0 13 z j e = Sanity Check: -0.983> π /2 so the angle is in 4 th Quadrant as shown b. 6 This one is easiest done by inspection of the graph: 6 0 6 | | = = = z z z Note: for this one, polar = rect Any positive real number has an angle of zero Any negative real number has an angle of ±π c. 2 j This one is easiest done by inspection of the graph: 2 / 2 rad 2 / 2 | | π = π = = j e z z z Any positive imaginary number has an angle of π /2 Any negative imaginary number has an angle of - π /2 d. 2+ j 4 107 . 1 1 2 2 20 rad 107 . 1 ) 2 / 4 ( tan 20 4 2 | | j e z z z = = = = + = 107 . 1 20 z j e = Sanity: 1.107 < π /2, which matches plot!!! e. –3– j 4 rad 069 . 4 927 . 0 ) 3 / 4 ( tan 5 25 ) 4 ( ) 3 ( | | 1 2 2 = π + = = = = + = z z 069 . 4 5 j e z = Note: the calculator gave the wrong quadrant so I had to adjust by adding π . You could also subtract π to get an equivalent answer: 21 . 2 5 j e z = Re Im 2 -3 Re Im 6 Re Im 2 Re Im 2 4 Re Im -3 -4 Calc thinks its here!
f. –4+ j 4 2 4 16 2 ) 4 ( ) 4 ( | | 2 2 = × = + = z Angle from inspection of graph: 4 / 3 π = z

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