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Impedance Young Won Lim 05/26/2015 Copyright (c) 2015 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 [email protected] This document was produced by using OpenOffice and Octave. Young Won Lim 05/26/2015 Sinusoidal voltage and current d vC iC = C⋅ dt d iL v L = L⋅ dt v C = A cos(ω t ) i L = A sin (ω t ) iC = −ω C A sin(ω t ) = A cos (ω t−π /2) = ω C A cos(ω t + π/ 2) v L = ω L A cos(ω t) vC cos(ω t) 1 = iC ω C cos(ω t + π/ 2) vL cos(ω t) =ωL iL cos(ω t −π / 2) [iC , v C ] [v L , i L ] leads by 90 Impedance leads by 90 3 Young Won Lim 05/26/2015 Leading and Lagging Current d vC iC = C⋅ dt d iL v L = L⋅ dt v C = cos(2 π t ) i L = sin (2 π t ) iC = −π sin (2 π t ) = cos(2 π t −π/ 2) = π cos (2 π t + π / 2) v L = π cos(2 π t) C=0.5 L=0.5 vL iC vC Impedance iL 4 Young Won Lim 05/26/2015 Leading and Lagging Current d vC iC = C⋅ dt d iL v L = L⋅ dt IC −j ωC VL VC jωL constant i vL iC vC Impedance IL iL 5 Young Won Lim 05/26/2015 Phasor and Ohm's Law sinusoidal i, v source sinusoidal i, v source sinusoidal i, v sinusoidal i, v IC VL VC IL Impedance 6 Young Won Lim 05/26/2015 Impedance : a complex scaling factor VC = A 0 IC = ωC A ZC = IL = A π/2 VC IC VL= ωLA phasor phasor ZL = a complex scaling factor V C = I C⋅Z C = I C ZC = Impedance −π /2 ( 1 ωC VL IL 0 phasor phasor a complex scaling factor −π / 2 ) V L = I L⋅Z L = I L ( ω L −j 1 = ωC j ωC +π/ 2 ) ZL = j ω L 7 Young Won Lim 05/26/2015 Ohm's Law VC = A IC = ωC A ZC = 0 IL = A π/2 VL= ωLA VC 1 = −π / 2 IC ωC −j 1 = = ωC j ωC ZL = VL = ωL IL 0 +π/2 = jω L vL iC vC Impedance −π /2 iL 8 Young Won Lim 05/26/2015 Phasor Example (1) A cos(ω t + 0) A 0 A +π/2 A cos(ω t + π /2) A π/2 A −π/2 A cos(ω t − π /2) A −π/2 A −π A A cos(ω t − π) −π Impedance 9 Young Won Lim 05/26/2015 Phasor Example (2) A cos(ω t + 0) A 0 2 A cos(ω t − π / 2) 2 A −π/2 Impedance 10 Young Won Lim 05/26/2015 Phasor A θ A cos(ω t + θ) = A cos(ω t) cos(θ) − A sin(ω t )sin(θ) = A cos(θ) cos(ω t) − A sin(θ)sin(ω t) = X cos(ω t) − Y sin(ω t ) A cos(θ) = X A sin(θ) = Y A = √ X 2+Y 2 tan θ = θ>0 Impedance ( X , Y ) = ( A cos θ , A sin θ) Y X leading X + j Y = A cos θ+ j A sin θ θ<0 lagging 11 Young Won Lim 05/26/2015 Linear Combination of cos(ωt) & sin(ωt) √ X 2 +Y 2 cos(ω t −θ) X cos(ω t )+Y sin(ω t ) X cos(ω t )+Y sin (ω t) [ X cos(ω t )+Y sin (ω t) = √ X 2 +Y 2 √ X X+Y cos(ω t )+ √ XY+ Y sin (ω t ) 2 2 2 2 ] = √ X 2 +Y 2 cos(ω t −θ) = √ A 2 + B 2 [ cos(θ)cos(ω t)+sin (θ)sin (ω t ) ] cos(θ) = = √ X 2 +Y 2 cos(θ−ω t ) sin (θ) = = √ X +Y cos(ω t −θ) 2 2 X √ X 2 +Y 2 Y √ X 2+Y 2 X cos(ω t )−Y sin(ω t ) √ X 2 +Y 2 cos(ω t +θ) X Y cos(ω t )− sin (ω t ) A A A cos(ω t +θ) cos (θ) cos(ω t )−sin (θ)sin (ω t ) Impedance 12 Young Won Lim 05/26/2015 RC Circuit e S (t) v C (t) v R (t ) v R (t ) I s (t) I s (t ) R = 6Ω v C (t ) XC = 8 Ω Impedance 13 Young Won Lim 05/26/2015 Current Source I s (t) 5 0.927 rad Impedance 14 Young Won Lim 05/26/2015 Resistor v R (t ) 30 v R = i R⋅R 0.927 rad [i R , v R ] same phase R = 6Ω Is (t) → 5 Is (t)⋅6 Ω → 30 Impedance 15 Young Won Lim 05/26/2015 Capacitor π /2 = 1.571 v s (t) 40 d vC iC = C⋅ dt −0.644 rad [iC , v C ] 1.571−0.927 = 0.644 leads by 90 XC = 8 Ω Is (t) → 5 Is (t)⋅8 Ω → 40 Impedance 16 Young Won Lim 05/26/2015 vR(t) + vC(t) v s (t) v R (t ) ` 30 0.927 rad Impedance 40 −0.644 rad 17 Young Won Lim 05/26/2015 Phasor Addition A 1 cos(ω t +θ1) = A 1 cos(ω t )cos(θ 1 ) − A 1 sin (ω t)sin (θ1 ) A 2 cos(ω t +θ2 ) = A 2 cos(ω t )cos(θ 2 ) − A 2 sin (ω t )sin (θ2 ) A 1 cos(ω t +θ1 ) + A 2 cos(ω t +θ2 ) = X cos(ω t ) − Y sin (ω t ) = A cos(ω + θ) X = A 1 cos(θ 1) + A 2 cos (θ2 ) A 1 cos(θ1 ) = 30 cos(0.927 ) = 18.0 Y = A 1 sin (θ1 ) + A 2 sin (θ2 ) A 1 sin (θ1 ) = 30 sin (0.927 ) = 24.0 A 2 cos(θ1 ) = 40 cos(−0.644 ) = 32.0 X = 50.0 A 2 sin (θ1 ) = 40 sin (−0.644) = −24.0 Y = 0.0 A = √ 50.02 +0.02 θ = tan−1 = 50.0 0.0 = 0 50.81 Impedance 18 Young Won Lim 05/26/2015 Is(t) and Es(t) R = 6Ω Z T = 10 Ω XC = 8 Ω ZT = 10 I s ZT = (5 Impedance 0.927)⋅(10 −0.927) = 50 19 0 −0.927 E s = I s ZT Young Won Lim 05/26/2015 Phasor v s (t) v s (t ) v R (t ) v R (t ) 30 30 40 0.927 rad 0.927 rad −0.644 rad 40 −0.644 rad 1.571−0.927 = 0.644 VR ES VC Impedance 20 Young Won Lim 05/26/2015 Phasor v R (t ) 30 0.927 rad Is ES ZT Impedance 21 Young Won Lim 05/26/2015 Sinusoidal Functions ● everlasting sinusoid function causal sinusoid function ● state at t = 0– = state at t = 0+ zero state at t = 0– continuous state between t = 0– & 0+ non-zero state at t = 0+ sin (ω t) sin (ω t)u (t) cos(ω t ) cos (ω t )u(t) zero conditions − at time t = 0 Impedance 22 creates non-zero conditions + at time t = 0 Young Won Lim 05/26/2015 Steady State Response Can apply impedance method Cannot apply impedance method IC VL VC IL Impedance 23 Young Won Lim 05/26/2015 References [1] http://en.wikipedia.org/ [2] J.H. McClellan, et al., Signal Processing First, Pearson Prentice Hall, 2003 Young Won Lim 05/26/2015