Chapter 32B - RC Circuits A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University © 2007

RC Circuits: The rise and decay of currents in capacitive circuits The calculus is used only for derivation of equations for predicting the rise and decay of charge on a capacitor in series with a single resistance. Applications are not calculus based. Check with your instructor to see if this module is required for your course. Optional: Check with Instructor

RC Circuit R V C a b RC-Circuit: Resistance R and capacitance C in series with a source of emf V. Start charging capacitor... loop rule gives: R V C a b i

RC Circuit: Charging Capacitor Rearrange terms to place in differential form: R V C a b i Multiply by C dt :

RC Circuit: Charging Capacitor R V C a b i

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

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

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

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:

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 =.

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

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.

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.

Discharging From q 0 to q: Instantaneous charge q on discharging capacitor: R V C a b i

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.

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

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...

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:

CONCLUSION: Chapter 32B RC Circuits