Phasor Young Won Lim 05/19/2015

Transcription

1 Phasor

2 Copyright (c) Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License". Please send corrections (or suggestions) to youngwlim@hotmail.com. This document was produced by using OpenOffice and Octave.

3 Phase Lags and Leads d d x f x = cos x leads f x = sin x d d x f x = sin x leads f x = cos x f x dx = cos x C f x dx = sin x C lags lags f x = sin x f x = cos x d d x f x leads f x by 2 f x dx lags f x by 2 Phasors 3

4 Derivative of sin(x) f x = sin x slope leads d d x f x = cos x Phasors 4

5 Derivative of cos(x) f x = cos x slope leads d d x f x = sin x Phasors 5

6 Integral of sin(x) f x = sin x 0 / 2 sin t d t = 1 C = 1 C = area area - 1 lags f x dx = cos x C 0 x sin t d t 0 x sin t d t 1 = cos x Phasors 6

7 Integral of cos(x) f x = cos x 0 / 2 cos x d x = area lags 0 x cos t d t = sin x f x dx = sin x C Phasors 7

8 Sinusoid cos x sin x Same Amplitude Same Angular Frequency cos x sin x 1 cos 1 t 1 sin 1 t Acos t Phasors 8

9 Phasor Sinusoid (Sine Waves) Acos t Amplitude Angular Frequency Angle at t = 0 A 1. Representation using Euler s Formula Acos t = A t e i 2 A t e i 2 2. Representation using Real Part Acos t = Re {A e i t } = Re {A e i e i t } Ae i e i t Ae i A θ Phasors 9

10 Phasor A cos(ω t + θ) A cos(ω t + θ) = R{A e i(ωt + θ) } = R{e iω t Ae iθ } Ae iθ A = A cos θ + j A sin θ θ A π /4 ω rad /sec A π /4 A Phasors 10

11 Phasor Example (1) A 0 A cos(ω t + 0) A +π/2 A A cos(ω t + π/2) π/2 A π/2 A A cos(ω t π/2) π/2 A π π A A cos(ω t π) Phasors 11

12 Phasor Example (2) A 0 A cos(ω t + 0) 2 A π/2 2 A cos(ωt π/2) Phasors 12

13 Phasor A cos(ω t + θ) = A cos(ω t)cos(θ) A sin(ωt )sin(θ) = A cos(θ)cos(ω t) A sin(θ)sin(ω t) = X cos(ω t) Y sin(ωt ) A θ A cos(θ) = X A sin(θ) = Y A = X 2 +Y 2 tan θ = Y X ( X, Y ) = ( A cos θ, A sin θ) X + j Y = A cosθ+ j A sin θ θ > 0 leading θ < 0 lagging Phasors 13

14 Linear Combination of cos(ωt) & sin(ωt) X cos(ω t)+y sin(ω t) X cos(ωt )+Y sin(ω t) = X 2 +Y 2 [ X cos(ω t)+ Y sin(ωt X 2 +Y 2 X 2 +Y )] 2 = X 2 +Y 2 [cos (θ)cos(ωt )+sin(θ)sin (ω t) ] = X 2 +Y 2 cos(θ ω t) = X 2 +Y 2 cos(ωt θ) X cos(ω t) Y sin(ω t) X 2 +Y 2 cos(ω t θ) X cos(ωt )+Y sin(ω t) = X 2 +Y 2 cos(ωt θ) cos(θ) = sin(θ) = X X 2 +Y 2 Y X 2 +Y 2 X 2 +Y 2 cos(ω t +θ) Phasors 14

15 Phasor as a starting point (+1)cos(ωt) (0)sin(ωt ) A cos(ω t + 0) cos(0) sin(0) (0)cos(ω t) (+1)sin(ωt ) A cos(ω t + π/2) cos(π/2) sin (π/2) (0)cos(ω t) ( 1)sin (ω t) A cos(ω t π/2) cos( π/ 2) sin ( π/2) ( 1)cos(ω t) (0)sin(ω t) A cos(ω t π) cos( π) sin( π) Phasors 15

16 Phase Angles Phasors 16

17 Phasor Arithmetic A +π/4 2 A +0 A π/ 4 Phasors 17

18 Phasor Addition 1 +π/4 cos(ω t + π/ 4) 2 π/ 4 2cos(ω t π/ 4) cos(ωt + 0) Phasors 18

19 Phasor Addition Rule N x(t ) = k =1 N k=1 A k cos(ωt + θ k ) = A cos(ω t + θ) adding complex numbers A k e j(θ k ) = A e j θ a complex number N = k =1 N = R{ k=1 = R{ R{A k e j (ωt + θ k ) } N k=1 N = R{ k=1 N = R{ k=1 N = R{ k=1 A k e j(ωt) e j(θ k ) } A k e jθ k e j(ω t ) } A k e j θ k e j(ω t) } A e jθ e j(ωt) } A e j(ω t +θ) } = A cos(ω t + θ) Phasors 19

20 Phasor Multiplication & Division x(t ) = A 1 cos(ωt + θ 1 ) = R{A 1 e j(ωt + θ 1) } y(t) = A 2 cos(ω t + θ 2 ) = R{A 2 e j(ωt + θ 2) } x(t ) y (t ) = A 1 A 2 cos(ω t + θ 1 )cos(ω t + θ 2 ) = R{A 1 A 2 e j(2 ω t + θ 1 + θ 2 ) } different angular frequency! x(t ) y(t) = A 1 A 2 cos(ω t + θ 1 ) cos(ω t + θ 2 ) = R{ A 1 A 2 e j(θ 1 θ 2) } no more rotating! Phasors 20

21 Phasor Multiplication 1 +π/4 cos(ω t + π/ 4) 2 π/ 4 2cos(ω t π/ 4) cos(ω t + 0) Phasors 21

22 Phasor Scaling 1 +π/4 cos(ω t + π/ 4) 2 π/ 4 R{A 1 e j(ωt + θ 1) } R{A 2 e j(θ 2) } R{A 1 A 2 e j (ωt + θ 1 + θ 2 ) } 2 π/ 4 not a phasor just a scaling complex number cos(ω t + 0) Phasors 22

23 Vector Space V: non-empty set of objects defined operations: addition scalar multiplication u + v k u if the following axioms are satisfied for all object u, v, w and all scalar k, m V: vector space objects in V: vectors 1. if u and v are objects in V, then u + v is in V 2. u + v = v + u 3. u + (v + w) = (u + v) + w u = u + 0 = u (zero vector) 5. u + ( u) = ( u) + (u) = 0 6. if k is any scalar and u is objects in V, then ku is in V 7. k(u + v) = ku + kv 8. (k + m)u = ku + mu 9. k(mu) = (km)u 10. 1(u) = u Phasors 23

24 Basis Basis : a set of linear independent spanning vectors every complex number can be represented by e + j θ e j θ k 1 e + j θ + k 2 e + j θ linear combination of e + j θ and e + j θ which are one set of linear independent two vectors every complex number can also be represented by l 1 cosθ + l 2 j sin θ j sin θ e + j θ j sinθ cosθ cosθ e j θ Phasors 24

25 Basis (2) Basis : a set of linear independent spanning vectors e + j θ k 1 e + j θ + k 2 e + j θ k 1 e + j θ e + j θ k 2 e j θ e j θ e j θ l 1 cosθ + l 2 j sin θ l 1 cosθ j sinθ cosθ j sin θ l 2 j sin θ cosθ Phasors 25

26 C 1 and R 2 Spaces c 1 e +iω t i ωt + c 2 e c 3 cos(ω t) + c 4 i sin(ωt ) c 1 = (c 3 +c 4 )/2 c 2 = (c 3 c 4 )/2 real number real number c 3 = (c 1 + c 2 ) c 4 = (c 1 c 2 ) real number real number C 1 isin (ω t ) cos(ω t) c 1 e +iω t i ωt + c 2 e c 3 cos(ω t) + c 4 sin(ω t) c 1 = (c 3 c 4 i)/2 conjugate c 2 = (c 3 + c 4 i)/2 complex number +2 real part 2 imag part c 3 = (c 1 + c 2 ) c 4 = i(c 1 c 2 ) real number real number R 2 sin(ω t) cos(ω t) Phasors 26

27 Linear Combination of cos(ωt) & sin(ωt) X cos(ω t)+y sin(ω t) X 2 +Y 2 cos(ω t θ) X cos(ω t) Y sin(ω t) X 2 +Y 2 cos(ω t +θ) 20cos(ωt )+30 sin(ω t) cos(ω t 0.588) 20 cos(ωt ) 30sin(ω t) cos(ω t ) sin(ωt ) (20, 30) sin(ωt ) (20, 30) sin(ωt ) (20+ j 30) cos(ωt ) cos(ωt ) cos(ωt ) Phasors 27

28 References [1] [2] J.H. McClellan, et al., Signal Processing First, Pearson Prentice Hall, 2003