Capacitor ( 電容 ) : C. unit: Farad (F)

Transcription

1 Capacitor ( 電容 ) : C unit: Farad (F) 1

2 A C = ε A d ε : dielectric constant t A : area d : distance 2

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5 4 104 = pf = 0.1 μf = pf = μf 202 = Ω = 2k Ω = Ω = 100k Ω 5

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11 Q (charge) = C (capacitance) V (voltage) I (current) = dq dt -6 μ = 10-9 n=10 p =

12 Example 1: Direct Current (DC) V C V = 120 volts C = 25 pf Q = CV -12 Q = 120 x ( 25 x 10 ) -9 = 3 x 10 coulomb 12

13 Capacitors in series V1 V2 V3 Vn C1 C2 C3 Q Q Q Q1 Q2 Q3 Cn Q Qn Ce E E 13

14 Q Q Q Q3 Q E = = V1+ V2 + V Vn = C C C C C e n Q = Q = Q = = Q = Q n Q Q Q Q Q E = = C C C C C e n = C C C C C e n 14

15 Capacitors in parallel V1 V2 V3 Vn E C1 C2 C3 Cn Q Q Q Q Q1 Q2 Q3 Qn E Ce 15

16 V1 = V2 = V3 =... = Vn = E EC = Q = Q + Q + Q Q = V C + V C + V C V C e n n n EC = EC + EC + EC + + EC e n C = C + C + C + + C e n 16

17 Example 2: Direct Current (DC) and charging + V R _ R V C + _ V C 17

18 Example 2: Direct Current (DC) and charging ir + V R _ 0 = V VR VC V R C + _ ic V C ir = ic = i(t) i 18

19 Example 2: Direct Current (DC) and charging ir + V R _ 0 = V VR VC V R C + _ ic V C ir = ic = i(t) V VR = ir R ir Q(t) = C VC(t) 19

20 VR + VC = V Q ir R + = V C Q RQ + = V C Q Q + = RC V R 20

21 VR + VC = V Q ir R + = V C y + ay = b -at y(t) = c1e + c2 Q RQ + = V C Q Q + = RC V R 21

22 VR + VC = V Q ir R + = V C y + ay = b -at y(t) = c1e + c2 Q If Q(t=0) = 0 RQ + = V C -t/rc Then, Q(t) = VC(1- e ) Q V Q + = V RC R i(t) = Q (t) = e -t/rc R 22

23 Question: If R = 1000, C = , V = 5, What are V(t) and i(t) at t = 0, τ, and 5τ? 23

24 clf; clear; close all; v = 5; r = 1000; c = ; tau = r * c; time = ones(1000,1); voltage = time; current = time; figure(1) subplot(2,1,1); t(21 1) plot(time/tau,voltage/v,'color','r','linewidth',2); ylim([ ]) title('voltage'); subplot(2,1,2);, plot(time/tau,current/(v/r),'color','g','linewidth',2); ylim([ ]) title('current'); end for i = 1:1:1000 time(i) = i*tau/100; voltage(i) = v*(1-exp(-time(i)/tau)); ( (i)/t )) current(i) = v/r*(exp(-time(i)/tau)); end 24

25 t t V solution : VC () t = V 1 e τ, i() t = e τ, where RC R τ = V V R τ τ 25

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27 Michael Faraday, FRS (September 22, 1791 August 25, 1867) was an English chemist and physicist who contributed to the fields of electromagnetism and electrochemistry. From: 27

28 Faraday studied the magnetic field around a conductor carrying a DC electric current, and established the basis for the magnetic field concept in physics. He discovered electromagnetic induction, diamagnetism and electrolysis. He established that magnetism could affect rays of light and that there was an underlying relationship between the two phenomena. His inventions of electromagnetic rotary devices formed the foundation of electric motor technology, and it was largely due to his efforts that electricity became viable for use in technology. From: 28

29 Although Faraday received little formal education and knew little of higher mathematics, such as calculus, he was one of the most influential scientists in history. Some historians of science refer to him as the best experimentalist in the history of science. The SI unit of capacitance, the farad, is named after him, as is the Faraday constant, the charge on a mole of electrons (about 96,485 coulombs). Faraday's law of induction states that a magnetic field changing in time creates a proportional electromotive force. From: 29

30 The Royal Institution Christmas Lectures have been held in London annually since They serve as a forum for presenting complex scientific issues to young people p in an informative and entertaining manner. In the mid 1820s Michael Faraday, a former Director of fthe Royal Institution, initiated the first Christmas Lecture series at a time when organised education for young people was scarce. He presented a total of 19 series, establishing an exciting new venture of teaching science to young people that was eventually copied by other institutions internationally. The Christmas Lectures have continued annually since this time, stopping only during World War II. From: 30

31 Michael Faraday, nineteenth century scientist and electrician, shown delivering the British Royal Institution's Christmas Lecture for Juveniles during the Institution's Christmas break in From: 31

32 Example 3: Alternating Current (AC) A ic E = Vm sin(ωt) ~ C V C i 32

33 Example 3: Alternating Current (AC) i A ic E = Vm sin(ωt) ~ C + _ V C i vc(t) = E = Vm sin(ωt) 33

34 VC(t) = E = Vm sin(ωt) q(t) = C VC(t) dq(t) dvc(t) d ic(t) = = C = C [ Vm sin(ω t) ] dt dt dt π = Vm ω C cos(ω t) = Vm ω C sin(ω t + ) 2 34

35 VC(t) = E = Vm sin(ωt) π π ic(t) = Vm ω C sin(ω t + ) = Im sin(ω t + ) 2 2 here Im = Vm ω C The capacitive reactance is defined as: XC Vm 1 1 XC = = = Im ω C (2πf) C 35

36 π VC(t) = E = Vm sin(ωt), ic(t) = Vm ω C sin(ω t + ) 2 Question: E = 10 sin( 200π t) C = 2000 μ F Please calculate: i(t) and Xc 36

37 π ic(t) = Vm ω C sin(ω t + ) 2 π = 10 * (200π) * (0.002) sin(ω t + ) 2 π = sin(ω t + ) Xc = = ω C (2πf) C = Ω 37

38 π VC(t) = E = Vm V sin(ωt), ic(t) = Vm ω Csin(ω t+ ) 2 ωt 38

39 vm = 10; w = 200*pi; c = 0.002; cycle = 2; dt = cycle*(2*pi)/w; time = ones(1000,1); voltage = time; current = time; end for i = 1:1:1000 time(i) = i*dt/1000; voltage(i) = (vm)*sin(w*time(i)); current(i) = (vm*w*c)*cos(w*time(i)); end figure(1) plot(time*w,voltage,'color','r','linewidth',2); hold on; plot(time*w,current,'color','g','linewidth',2); legend('v(t)', 'i(t)'); set(gca,'xtick',0:pi:4*pi), p set(gca,'xticklabel',{'0','pi','2pi','3pi','4pi'}) xlim([0 4*pi]) 39

40 VC(t) = Vm sin(ωt) π ic(t) = Vm ω C sin(ω t + ) 2 = Vm ω C sin(ω t + θ ) 在電容器上, 電流波形領先電壓波形 π/2 ( 相位 )=+θ θ = π/2 40