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RC Circuit: Charging Capacitor

Instantaneous charge q on a charging capacitor:. a. R. b. V. i. C. +. +. -. -. RC Circuit: Charging Capacitor. At time t = 0: q = CV(1 - 1); q = 0. At time t =  : q = CV(1 - 0); q max = CV. The charge q rises from zero initially to its maximum value q ma x = CV.

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RC Circuit: Charging Capacitor

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  1. Instantaneous charge q on a charging capacitor: a R b V i C + + - - RC Circuit: Charging Capacitor At time t = 0: q = CV(1 - 1); q = 0 At time t = : q = CV(1 - 0); qmax = CV The charge q rises from zero initially to its maximum value qmax = CV

  2. q Qmax Capacitor a R = 1400 W Rise in Charge b V i 4 mF Time, t 0.63 Q + + t - - Example 1. What is the charge on a 4-mF capacitor charged by 12-V for a time t = RC? The time t = RC is known as the time constant. e = 2.718; e-1 = 0.63

  3. q Qmax Capacitor a R = 1400 W Rise in Charge b V i 4 mF Time, t 0.63 Q + + t - - Example 1 (Cont.) What is the time constant t? The time t = RC is known as the time constant. In one time constant (5.60 ms in this example), the charge rises to 63% of its maximum value (CV). t = (1400 W)(4 mF) t = 5.60 ms

  4. As charge qrises, the current i will decay. a R b V i C + + - - RC Circuit: Decay of Current Current decay as a capacitor is charged:

  5. Capacitor i I 0.37 I Current Decay a R t b Time, t V i C + + Consider i when t = 0 and t =  . - - Current Decay The current is a maximum of I = V/R when t = 0. The current is zero when t =  (because the back emf from C is equal to V).

  6. Capacitor i a R = 1400 W I 0.37 I Current Decay b V i 4 mF t Time, t + + - - Example 2. What is the current i after one time constant (t = RC)? Given R and C as before. The time t = RC is known as the time constant. e = 2.718; e-1 = 0.37

  7. q Qmax Capacitor Capacitor i Rise in Charge I 0.37 I Current Decay Time, t t 0.63 I Time, t t Charge and Current During the Charging of a Capacitor. In a time t of one time constant, the charge q rises to 63% of its maximum, while the current i decays to 37% of its maximum value.

  8. a R a R b b V C V i C + + + + - - - - RC Circuit: Discharge After C is fully charged, we turn switch to b, allowing it to discharge. Discharging capacitor. . . loop rule gives: Negative because of decreasing I.

  9. a R b V i C + + - - Discharging Capacitor Note qo = CV and the instantaneous current is: dq/dt. Current i for a discharging capacitor.

  10. a R b V i C + + - - Prob. 45.How many time constants are needed for a capacitor to reach 99% of final charge? Let x = t/RC, then: e-x = 1-0.99 or e-x = 0.01 From definition of logarithm: 4.61 time constants x = 4.61

  11. a 1.4 MW i R b 1.8 mF C V 12 V + + - - Prob. 46. Find time constant, qmax, and time to reach a charge of 16 mC if V = 12 V and C = 4 mF. t = RC = (1.4 MW)(1.8 mF) t = 2.52 s qmax = 21.6 mC qmax = CV = (1.8 mF)(12 V); Continued . . .

  12. a 1.4 MW i R b 1.8 mF C V 12 V + + - - Prob. 46. Find time constant, qmax, and time to reach a charge of 16 mC if V = 12 V and C = 4 mF. Let x = t/RC, then: From definition of logarithm: x = 1.35 Time to reach 16 mC: t = 3.40 s

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