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Published byStephanie Harper Modified over 2 years ago

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RC Circuit: Charging Capacitor R V C a b i Instantaneous charge q on a 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 max = CV

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

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

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

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Current Decay Time, t I i Current Decay Capacitor 0.37 I R V C a b i 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). Consider i when t = 0 and t =.

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

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

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

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

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

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Prob. 46. Find time constant, q max, and time to reach a charge of 16 C if V = 12 V and C = 4 F. R V 1.8 F a b i 1.4 M C 12 V = RC = (1.4 MW)(1.8 mF) = RC = (1.4 MW)(1.8 mF) = 2.52 s q max = CV = (1.8 F)(12 V); q max = 21.6 C Continued...

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Prob. 46. Find time constant, q max, and time to reach a charge of 16 C if V = 12 V and C = 4 F. R V 1.8 F a b i 1.4 M C 12 V Let x = t/RC, then: From definition of logarithm: x = 1.35 t = 3.40 s Time to reach 16 C:

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