(Capacitor) Chapter32B 1

7/29/2019 (Capacitor) Chapter32B 1

1/19

Chapter 32B - RC Circuits

A PowerPoint Presentation by

Paul E. Tippens, Professor of Physics

Southern Polytechnic State University

2007


2/19

RC Circuits: The rise and decayof currents in capacitive circuits

The calculus is used only for derivation of

equations for predicting the rise and decayof charge on a capacitor in series with asingle resistance. Applications are notcalculus based.

Check with your instructor to see if thismodule is required for your course.

Optional: Check with Instructor


3/19

RC Circuit

R

V C

+

+

--

a

b

RC-Circuit: Resistance R and capacitance Cin series with a source of emf V.

Start charging capacitor. . . loop rule gives:

;q

iR V iR

C

E

R

V C

+

+

--

a

bi

q

C


4/19

RC Circuit: Charging Capacitor

Rearrange terms to place in differential form:

qV iRC

R

V C

+

+

--

a

bi

qC

dq q

R Vdt C

( )RCdq CV q dt

( )

dq dt

CV q RC

0 ( )

q t

o

dq dt

CV q RC

Multiply by C dt :


5/19


R

V C

+

+

--

a

bi

qC 0 ( )

q t

o

dq dt CV q RC

0ln( )

q t

CV q RC

(1/ )RC tCV q CVe

ln( ) ln( )t

CV q CV

RC

( )ln

CV q t

CV RC

/1 t RCq CV e


6/19


R

V C

+

+

--

a

bi

qC

/1

t RCq CV e

Instantaneous charge qon a charging capacitor:

At time t= 0: q = CV(1 - 1); q = 0

At time t= : q = CV(1 - 0); qmax= CV

The charge qrises from zero initially toits maximum value qmax = CV


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Example 1. What is the charge on a 4-mFcapacitor charged by 12-V for a time t = RC?

Time, t

Qmax q

Rise inCharge

Capacitor

t

0.63 Q

The time t = RCis known

as the time constant. /1 t RCq CV e

11q CV e

R =1400 W

V 4 mF

+

+

--

a

bi

e= 2.718; e-1= 0.63

1 0.37q CV

0.63q CV


8/19

Example 1 (Cont.) What is the time constant t?

Time, t

Qmax q

Rise inCharge

Capacitor

t

0.63 Q


as the time constant.

R =1400 W

V 4 mF

+

+

--

a

bi

In one time constant

(5.60 ms in thisexample), the chargerises to 63% of itsmaximum value (CV).

t = (1400 W)(4 mF)

t = 5.60 ms


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RC Circuit: Decay of Current

R

V C

+

+

--

a

bi

qC

/1 t RCq CV e

As charge qrises, thecurrent iwill decay.

/ /t RC t RC dq d CV

i CV CVe e

dt dt RC

Current decay as acapacitor is charged:

/t RCVi e

R


10/19

Current Decay

Time, t

I i

CurrentDecay

Capacitor

t

0.37 I

R

V C

+

+

--

a

bi

qC

The current is a maximum

of I = V/R when t = 0.The current is zero whent = (because the backemf from C is equal to V).

/t RCVi e

R

Consider i when

t = 0 and t = .


11/19

Example 2. What is the current iafter one timeconstant (t RC)? Given Rand Cas before.


as the time constant.

e= 2.718; e-1= 0.37

max0.37 0.37V

i iR

R =1400 W

V 4 mF

+

+

--

a

bi

Time, t

I i

CurrentDecay

Capacitor

t

0.37 I

/ 1t RCV Vi e e

R C


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Charge and Current During theCharging of a Capacitor.

Time, t

Qmaxq

Rise in

Charge

Capacitor

t

0.63 I

Time, t

Ii

Current

Decay

Capacitor

t

0.37 I

In a time t of one time constant, the charge qrises to 63% of its maximum, while the current idecays 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 tob, allowing it to discharge.

Discharging capacitor. . . loop rule gives:

;q

iR iR

C

E

R

V C

+

+

--

a

bi

q

C

Negative becauseof decreasing I.


14/19

Discharging From q0 to q:

;dq

q RCi q RC

dt

Instantaneous charge qon discharging capacitor:

R

V C

+

+

--

a

bi

qC

;dq dt

q RC

0 0

;q t

q

dq dt

q RC

00

ln

tq

q

tq

RC

0ln lnt

q qRC

0

lnq t

q RC


15/19

Discharging Capacitor

R

V C

+

+

--

a

bi

q

C0

ln

q t

q RC

/

0

t RCq q e

Note qo = CV and the instantaneous current is: dq/dt.

/ /t RC t RC dq d CV

i CVe edt dt RC

/t RCVi e

C

Current ifor a

discharging capacitor.


16/19

Prob. 45. How many time constants are neededfor a capacitor to reach 99% of final charge?

R

V C

+

+

--

a

bi

qC /max 1 t RCq q e

/

max

0.99 1t RCq

eq

Let x = t/RC, then: e-x = 1-0.99 or e-x = 0.01

1 0.01; 100xx

ee

ln (100)e xFrom definitionof logarithm:

x= 4.61t

x

RC

4.61 timeconstants


17/19

Prob. 46. Find time constant, qmax, and time toreach a charge of16 mC ifV= 12 V and C= 4 mF.

/max 1 t RCq q e R

V

1.8 mF

+

+

--

a

bi

1.4 MW

C12 V

t = RC = (1.4 MW)(1.8 mF)

t = 2.52 s

qmax = CV = (1.8 mF)(12 V); qmax = 21.6 mC

/

max

16 C1

21.6 C

t RCqe

q

m

m

/1 0.741t RC

e

Continued . . .


18/19

Prob. 46. Find time constant, qmax, and time toreach a charge of16 mC ifV= 12 V and C= 4 mF.

R

V

1.8 mF

+

+

--

a

bi

1.4 MW

C12 V

/1 0.741t RCe

Let x = t/RC, then:

1 0.741 0.259xe 1

0.259; 3.86x

xe

e ln (3.86)e x

From definitionof logarithm:

x= 1.35 1.35; (1.35)(2.52s)t tRC

t= 3.40 sTime to reach 16 mC:


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CONCLUSION: Chapter 32B

RC Circuits

(Capacitor) Chapter32B 1

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