Capacitors in an AC circuit

Transcription

1 Capacitors in an AC circuit

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 Invertible Functions + ic v c (t ) d dt v c (t ) dt ic (t ) vc ic (t ), v c (0) 3

4 Ever-charging signal pairs decreasing d dt negative increasing dt positive decreasing charging negative increasing decreasing increasing + charging positive negative positive vc il ic vl charge discharge 4

5 Positive and Negative Charging + charging from 0 + charged charging from 0 discharged charged vc 5

6 Negative Charging from Vdd and 0 charging from 0 charging from Vdd + charged discharged charged vc 6

7 Positive and Negative Charging + charging from 0 charging from 0 discharged 0V +Vdd charged charging from +Vdd Vdd charged charging from 0 7

8 Time Reversal Vc cannot change abruptly +V dd 0 time reversal 0 +V dd +V dd 0 0 V dd V dd 0 time reversal V dd 0 8

9 (+) and ( ) Charging charging from +Vdd + charging from 0 0 +V dd +V dd 0 0 V dd V dd 0 charging from 0 + charging from Vdd 9

10 (+) charging / ( ) charging + charging charging from 0 from 0 vc charging charging from Vdd from 0 vc + charging + charging from Vdd from 0 10

11 Ideal Voltage Vc and Ic charging charging from Vdd from 0 vc + charging + charging from Vdd from 0 d dt ic 11

12 Sinusoidal VC and IC charging charging from Vdd from 0 vc + charging + charging from Vdd from 0 d dt ic + charging + charging from Vdd from 0 12

13 Sinusoidal VC and IC + charging from 0 + charged charging from 0 discharged charged vc d dt ic 13

14 Sinusoidal VC and IC charging from 0 charging from Vdd + charged discharged charged vc d dt ic 14

15 Three States 15

16 Capacitor Current positive charge (positive ions) insulator negative charge (free electrons) electrons leave electrons accumulate No actual electrons movement across insulator materials But, think as Displacement Current flows through the capacitor 16

17 Positive ions and free electrons positive charge (positive ions) insulator negative charge (free electrons) [[commons:user crap ]] (original work by commons:user:greg Robson) /thumb/f/f7/electron_shell_029_copper no_label.svg/200pxelectron_shell_029_copper_-_no_label.svg.png

18 Three States positive charge (positive ions) negative charge Negatively Charged State (free electrons) fully charged no current Positively Charged State fully charged no current Fully Discharged State possible large current 18

19 Currents in the Fully Discharged State large current Fully Discharged State Initially no current Fully Discharged State large current Fully Discharged State This state can flow large current in either direction depending on the voltage change 19

20 Inter-State Current Flowing Under Positively Charging Under Negatively Charging (+) current flow direction ( ) current flow direction electron flow direction electron flow direction 20

21 Inter-State Current Flowing Fully Discharged State Under Positively Charging (+) current flow direction (+) current flow direction electron flow direction electron flow direction large current Positively Charged State Crowded No more space no current 21

22 Inter-State Current Flowing Fully Discharged State Under Negatively Charging ( ) current flow direction ( ) current flow direction electron flow direction electron flow direction Initial large current Negatively Charged State Crowded No more space no current 22

23 An AC Voltage Source 23

24 Continuous (Ever-) Charing Operations Incremental Voltage Increment + Charging incrementally Incremental Voltage Decrement Charging incrementally + charging incrementally - charging incrementally - charging incrementally + charging incrementally + charging incrementally - discharging incrementally - charging incrementally + discharging incrementally 24

25 Superposition - charging incrementally + charging incrementally 25 + charging incrementally - charging incrementally

26 Superposition - Small Time Constant 26

27 Difference, Differentiation 27

28 Continuous Charing and Discharging Operations + charging incrementally - charging incrementally - charging incrementally 28 + charging incrementally

29 Incrementally Charging - charging incrementally + charging incrementally 29 - charging incrementally

30 An AC Voltage Source Fully Discharged State Under Positively Charging Positively Charged State Under Negatively Charging Fully Discharged State Under Negatively Charging Negatively Charged State Under Positively Charging Fully Discharged State 30

31 Fully Charged and Fully Discharged Fully + Charged Fully Discharged Fully Discharged Fully Discharged Fully Charged (+) Charging ( ) Charging ( ) Charging (+) Charging (+) Current ( ) Current ( ) Current (+) Current (+) Charging (+) Discharging ( ) Charging ( ) Discharging 31

32 A Cycle Fully Discharged State Fully Discharged State 32

33 State Transition Diagram Fully Discharged State Fully Discharged State 33

34 Current Flow Positive Charged State Fully Discharged State Fully Discharged State Negative Charged State 34

35 Fully Discharged : Large Current Fully Discharged State large current Fully Discharged State large current Enough space for large movement of charges This state can flow large current in either direction Fully + Charged Fully Discharged Fully Discharged Fully Discharged dv c = 1 (max value) dt Fully Charged 35

36 Fully Charged : Zero Current Positively Charged State Negatively Charged State fully charged no current fully charged no current Crowded No more space no current Fully + Charged Fully Discharged Fully Discharged Fully Discharged dv c = 0 (min value) dt Fully Charged 36

37 Incrementally, Charging Positively v (t 1 ) < 0 v (t 2) < 0 v (t 3) < 0 v (t 4 ) > 0 v (t 5) > 0 v ' (t 1) > 0 v ' (t 2 ) > 0 v ' (t 3 ) > 0 v ' (t 4 ) > 0 v ' (t 5 ) > 0 Δ v (t 1 ) > 0 Δ v (t 2 ) > 0 Δ v (t 3 ) > 0 Δ v (t 4) > 0 Δ v (t 5 ) > 0 t1 t2 t3 37 t4 t5

38 Incrementally, Charging Positively v (t 1 ) < 0 v (t 2) < 0 v (t 3) < 0 v (t 4 ) > 0 v (t 5) > 0 v ' (t 1) > 0 v ' (t 2 ) > 0 v ' (t 3 ) > 0 v ' (t 4 ) > 0 v ' (t 5 ) > 0 Δ v (t 1 ) > 0 Δ v (t 2 ) > 0 Δ v (t 3 ) > 0 Δ v (t 4) > 0 Δ v (t 5 ) > 0 excess positive electrons ions Δ v (t 1 ) > 0 equilibrium Δ v (t 2 ) > 0 Δ v (t 3 ) > 0 38 positive ions Δ v (t 4 ) > 0 excess electrons Δ v (t 5 ) > 0

39 Incrementally, Charging Negatively v (t a ) > 0 v (t b ) > 0 v (t c ) = 0 v (t d ) < 0 v (t e ) < 0 v ' (t a ) < 0 v ' (t b ) < 0 v ' (t c ) < 0 v ' (t d ) < 0 v ' (t e ) < 0 Δ v (t a ) < 0 Δ v (t b ) < 0 Δ v (t c ) < 0 Δ v (t d ) < 0 Δ v (t e ) < 0 ta 39 tb tc td te

40 Incrementally, Charging Negatively v (t a ) > 0 v (t b ) > 0 v (t c ) = 0 v (t d ) < 0 v (t e ) < 0 v ' (t a ) < 0 v ' (t b ) < 0 v ' (t c ) < 0 v ' (t d ) < 0 v ' (t e ) < 0 Δ v (t a ) < 0 Δ v (t b ) < 0 Δ v (t c ) < 0 Δ v (t d ) < 0 Δ v (t e ) < 0 positive ions excess electrons Δ v (t a ) < 0 equilibrium Δ v (t b ) < 0 Δ v (t c ) < 0 40 excess positive electrons ions Δ v (t d ) < 0 Δ v (t e ) < 0

41 Difference of Samples y (t ) = sin(t ) y [ n] = sin(nt ) y [ n] y [n+1] = sin(n T ) sin ((n+1)t ) y [n] y [n+1] T dy dt 41

42 Fully Charged and Fully Discharged y [ n] y [n+1] y [ n] h = bar(t1, [y1' y2'], "stacked") set(h(1), "facecolor", "g"); set(h(2), "facecolor", "y"); hold on plot(t1, y1) axis([0 pi]); y [ n] y [n+1] = y (n T ) y ((n+1)t )=sin(n T ) sin ((n+1)t ) 42

43 Fully Charged and Fully Discharged Fully + Charged Fully Discharged y (t)=sin (t) h = bar(t1, y2/t(2), "hist") set(h(1), "facecolor", "y"); hold on plot(t1, y1) axis([ ]); Fully Discharged Fully Discharged Fully Charged 43 y [n] y [n+1] T dy dt

44 y[n+1] y[n] t = linspace(0, pi*2, 50); t1 = t; t2 = t + t(2); y1 = sin(t1); y2 = sin(t2) - sin(t1); stem(t1, y2) hold on plot(t1, y1) y (t)=sin (t) y [n ] y [n+1] = y (n T ) y ((n+1)t )=sin(n T ) sin ((n+1)t ) 44

45 Fully Charged and Fully Discharged clf t = linspace(0, pi*2, 50); t1 = t; t2 = t + t(2); y1 = sin(t1); y2 = sin(t2) - sin(t1); y3 = e.^(-20*t); y4 = conv(y2, y3); y5 = y4([1:length(t1)]); subplot(3, 1, 2); stem(t1, y2) subplot(3, 1, 1); hold on plot(t1, y1); plot(t1, y3); subplot(3, 1, 3); stem(t1, y5); 45

46 Pulse vc ic ic ω vc vc ic ic 46 d vc = C dt ic XC

47 Time Constants ic τ = RC small time constant τ = RC medium time constant τ = RC large time constant 47

48 Time Constants ic τ 1 < τ2 < τ 3 a1 > a2 > a3 t τ e = e τ = RC = t RC = e a t 1 a 48

49 Time Constants ic τ = RC e t τ τ = RC = e t RC e small τ small C large τ large C 1 large R ωc small Fully Capacitative Fully Resistive v C (t ) v C (t) ic (t) ic (t) 49 t τ = e t RC 1 R ωc

50 Time Constants ic τ = RC e t τ τ = RC = e t RC e small τ small C large τ large C 1 large R ωc small Fully Capacitative Fully Resistive 50 t τ = e t RC 1 R ωc

51 Superposition - Small Time Constant 51

52 Small Time Constants 52

53 Superposition Large Time Constant 53

54 Large Time Constants 54

55 Time Constants ic τ = RC e t τ τ = RC = e t RC e small τ small C large τ large C 1 large R ωc small Fully Capacitative Fully Resistive 55 t τ = e t RC 1 R ωc

56 Plotting superposition results clf t = linspace(0, pi*2, 50); tt= linspace(0, pi*2, 500); N = length(t); NN= length(tt); t1 = t; t2 = [t(2:n), t(n)]; y1 = sin(t1); y2 = sin(t2) - sin(t1); yy = [y1; zeros(nn/n-1, N)]; yy2= yy(:)'; a = 1/300; yy3= e.^(-a*tt); yy3 =yy3 - [zeros(1, NN/N), e.^(-a*tt)](1:nn); svec = zeros(1, NN); for i = 1:NN; tvec = zeros(1, NN); tvec = [zeros(1, i-1), yy3]; tvec = yy2(i) * tvec(1:nn); svec = svec + tvec; endfor yy4 = svec; % yy4= conv(yy2, yy3); y5 = yy4([1:nn/n:nn]); yy5= yy4([1:nn]); 56 subplot(4, 1, 2); stem(t1, y2) subplot(4, 1, 1); hold on plot(t1, y1); plot(tt, yy3); subplot(4, 1, 3); stem(t1, y5); hold on plot(tt, yy5) subplot(4, 1, 4); plot(yy4);

57 Small Time Constant yy = [y1; zeros(nn/n-1, N)]; yy2= yy(:)'; a = 300; yy3= e.^(-a*tt); yy3 =yy3 [zeros(1, NN/N), e.^(-a*tt)](1:nn); τ = RC e t τ = e t RC small τ small C large 57 1 ωc

58 Large Time Constant yy = [y1; zeros(nn/n-1, N)]; yy2= yy(:)'; a = 1/300; yy3= e.^(-a*tt); yy3 =yy3 [zeros(1, NN/N), e.^(-a*tt)](1:nn); τ = RC e t τ = e t RC large τ large C small 58 1 ωc

59 Envelope of the samples v C (t ) v C (t) ic (t ) ic (t) v C (t) v C (t) ic (t) ic (t ) 59

60 Evercharging signal pairs charge discharge charge discharge 60

61 I leads V by 90 Initial charge Full charge SHORT OPEN V=0 I=0 I : peak V : peak 61 I V

62 Evercharging signal pairs charge discharge charge discharge 62

63 Evercharging signal pairs charge discharge charge discharge 63

64 Evercharging signal pairs decreasing increasing decreasing increasing decreasing increasing d dt negative positive charge discharge negative positive vc il ic vl charge discharge 64

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