R02 - ENEE 241 02* READING ASSIGNMENT 2 Thu 02/05 Lecture...

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ENEE 241 02* READING ASSIGNMENT 2 Thu 02/05 Lecture Topic: n th root of a complex number; continuous-time sinusoids; phasors Textbook References: sections 1.3 and 1.4 Key Points: The equation z n = e , where φ is a given angle, has n roots of the form z = e ; these are obtained by setting θ = ( φ + 2 ) /n , where k = 0 ,...,n - 1. The functions cos θ and sin θ are both periodic with period 2 π (radians), and are shifted versions of each other. The generic time-dependent sinusoid A cos(Ω t + φ ) has three parameters: amplitude A , an- gular frequency Ω (rad/sec) and initial phase φ . The cyclic frequency f (Hz) and period T (sec) are related to Ω by f = 1 /T = Ω 2 π The stationary phasor of A cos(Ω t + φ ) is the complex number Ae . The sum of two (or more) sinusoids of arbitrary amplitudes and phases but of identical frequency Ω is a sinusoid of frequency Ω. Sinusoids of identical frequency can be added together by taking the complex sum of their stationary phasors. Theory and Examples: 1 . The equation z n = v , where z is a complex variable and v is a complex constant, has n complex roots (this is true for any polynomial of degree n ). These roots can be found using the following observations:
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R02 - ENEE 241 02* READING ASSIGNMENT 2 Thu 02/05 Lecture...

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