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ECE201_Lecture30

ECE201_Lecture30 - ECE201 Linear Circuit Analysis I Lecture...

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1 ECE201 Linear Circuit Analysis I Lecture 30 Topic: Complex forcing function 1. Review of Complex Numbers Let j = 1 , any complex number can be represented in 3 ways: (1) In rectangular coordinates: a + jb a: real part Re{a+jb} b: imaginary part Im{a+jb} magnitude: a 2 + b 2 complex conjugate: a – jb (a + jb) (a – jb) = a 2 + b 2 = (magnitude) 2 Remember: j j = − 1 . (2) As a vector (in rectangular coordinates) (3) In polar coordinates: ρ e j θ ρ : magnitude θ : angle between the vector and the Re{ } axis, c.c.w. I m { } magnitude Re{ } a complex conjugate (mirror image w.r.t. Re{ } axis) -jb o jb I m { } Re{ } ρ θ
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2 Real part: ρ cos θ Imaginary part: ρ sin θ complex conjugate: ρ e j θ ρ e j θ ρ e j θ = ρ 2 = magnitude ( ) 2 short hand notation: ρ θ 2. Conversion between rectangular and polar representation of a complex number, (Euler Identity) Key: graphic representation of a vector (a, b) ( ρ , θ ) Euler Identity: ρ e j θ = a + jb = ρ cos θ + j ρ sin θ Cancel ρ : e j θ = cos θ + jsin θ We also get: cos θ = e j θ + e j θ 2 and sin θ = e j θ e j θ 2 j . (same #, 2 representations) I m { } jb b = ρ sin θ Re{} 0 a a = ρ cos θ ρ = a 2 + b 2 θ = tan 1 b a
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3 3. The Approach (1) Replace input excitation V s cos( ω t + θ ) or I s cos(
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