homework1_solution.pdf - Solutions to Homework 1 ECE 460...

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Solutions to Homework 1 ECE 460: Automatic Control Due: Wednesday, January 13, at 2:10pm Note 1. Your solutions should be neat and legible; always show your work. Problem 1. (40 Points) Using Euler’s formula, (a) Show that sin 2 θ = 1 2 (1 - cos 2 θ ) Solution: sin 2 θ = e - e - j 2 2 = e j 2 θ + e - j 2 θ - 2 e e - - 4 = e j 2 θ + e - j 2 θ - 2 - 4 = cos 2 θ + j sin 2 θ + cos 2 θ - j sin 2 θ - 2 - 4 = 2 cos 2 θ - 2 - 4 = 1 2 (1 - cos 2 θ ) (b) Show that 2 sin θ cos φ = sin( θ + φ ) + sin( θ - φ ). 1
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Solution: 2 sin θ cos φ = 2( e - e - j 2 )( e + e - 2 ) = e e + e e - - e - e - e - e - j 2 = e j ( θ + φ ) + e j ( θ - φ ) - e - j ( θ - ) - e - j ( θ + ) j 2 = e j ( θ + φ ) - e - j ( θ + ) + e j ( θ - φ ) - e - j ( θ - ) j 2 = e j ( θ + φ ) - e - j ( θ + ) j 2 + e j ( θ - φ ) - e - j ( θ - ) j 2 = = sin( θ + φ ) + sin( θ - φ ) Problem 2. (30 Points) Change the representation of complex numbers, i.e., if a complex number is in Cartesian coordinates, change it in polar coordinates, and vice versa. Plot the complex numbers and clearly label the values in the real-imaginary axes, magnitude, and angle of the numbers. (a) - 2 + j 3 (a) Problem 2a (b) Problem 2b Figure 1: Representations of complex numbers Solution: - 2 + j 3 = 4 + 3 e j atan( 3 / - 2) = 7 e j 140 Figure 1a shows the representation of the number in Cartesian coordinates and polar
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