RC Circuit: Charging Capacitor R V C ++ -- a b i Instantaneous charge q on a charging capacitor: At...
12
RC Circuit: Charging Capacitor R V C + + - - a b i q C / 1 t RC q CV e Instantaneous charge q on a charging capacitor: At time At time t t = 0: = 0: q = CV(1 - 1); q = 0 q = CV(1 - 1); q = 0 At time At time t t = = : : q = CV(1 - 0); q q = CV(1 - 0); q max max = = CV CV The charge q rises from zero initially to its maximum value q max = CV
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Transcript of RC Circuit: Charging Capacitor R V C ++ -- a b i Instantaneous charge q on a charging capacitor: At...
RC Circuit: Charging Capacitor
R
V C
++
--
a
bi
q
C
/1 t RCq CV e
Instantaneous charge q on a charging
capacitor:
At time At time t t = 0: = 0: q = CV(1 - 1); q = q = CV(1 - 1); q = 00At time At time t t = = : : q = CV(1 - 0); qq = CV(1 - 0); qmaxmax = = CVCV
The charge q rises from zero initially to its maximum value qmax = CV
The charge q rises from zero initially to its maximum value qmax = CV
Example 1. What is the charge on a 4-F capacitor charged by 12-V for a time t =
RC?
Time, t
Qmaxq
Rise in Rise in ChargeCharge
Capacitor
0.63 Q
The time The time = RC = RC is is known as the time known as the time
constant.constant. /1 t RCq CV e
11q CV e
R = 1400
V 4 F
++
--
a
bi
e e = 2.718= 2.718; e; e-1-1 = 0.63 = 0.63
1 0.37q CV
0.63q CV0.63q CV
Example 1 (Cont.) What is the time constant ?
Time, t
Qmaxq
Rise in Rise in ChargeCharge
Capacitor
0.63 Q
The time The time = RC = RC is is known as the time known as the time
constant.constant.
R = 1400
V 4 F
++
--
a
bi
In one time In one time constant (5.60 ms constant (5.60 ms in this example), in this example), the charge rises to the charge rises to 63% of its 63% of its maximum value maximum value (CV).(CV).
= (1400 = (1400 )(4 )(4 F)F)
= 5.60 ms = 5.60 ms
RC Circuit: Decay of CurrentR
V C
++
--
a
bi
q
C
/1 t RCq CV e
As charge q rises, the current i will
decay.
/ /t RC t RCdq d CVi CV CVe edt dt RC
Current decay as Current decay as a capacitor is a capacitor is
charged:charged:
/t RCVi eR
/t RCVi eR
Current Decay
Time, t
I i
Current Current DecayDecay
Capacitor
0.37 I
R
V C
++
--
a
bi
q
C
The current is a The current is a maximum of I = V/R maximum of I = V/R
when t = 0.when t = 0.The current is zero The current is zero when t = when t = (because (because the back emf from C is the back emf from C is equal to V).equal to V).
/t RCVi eR
/t RCVi eR
Consider Consider ii when t = 0 when t = 0 and t = and t = . .
Example 2. What is the current i after one time constant (RC)? Given R and C as
before.
The time The time = RC = RC is is known as the time known as the time
constant.constant.
e e = 2.718= 2.718; e; e-1-1 = 0.37 = 0.37
max0.37 0.37V
i iR
max0.37 0.37V
i iR
R = 1400
V 4 F
++
--
a
bi
Time, t
I i
Current Current DecayDecay
Capacitor
0.37 I
/ 1t RCV Vi e eR C
Charge and Current During the Charging of a Capacitor.
Time, t
Qmaxq
Rise in Rise in ChargeCharge
Capacitor
0.63 I
Time, t
I i
Current Current DecayDecay
Capacitor
0.37 I
In a time In a time of one time constant, the of one time constant, the charge charge q q rises to 63% of its maximum, rises to 63% of its maximum, while the current while the current ii decays to 37% of its decays to 37% of its maximum value.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:Discharging capacitor. . . loop rule gives:
; q
iR iRC
E ; q
iR iRC
E
R
V C
++
--
a
bi
q
C
Negative Negative because of because of
decreasing decreasing II..
Discharging CapacitorR
V C
++
--
a
bi
q
C0
lnq t
q RC
/0
t RCq q e /0
t RCq q e
Note qNote qoo = CV and the instantaneous current is: = CV and the instantaneous current is: dq/dtdq/dt..
/ /t RC t RCdq d CVi CVe edt dt RC
/t RCVi e
C
/t RCVi e
CCurrent Current ii for a for a
discharging discharging capacitor.capacitor.
Prob. 45. How many time constants are needed for a capacitor to reach 99% of final charge?
R
V C
++
--
a
bi
q
C /max 1 t RCq q e /
max 1 t RCq q e
/
max
0.99 1 t RCqe
q
Let x = t/RC, Let x = t/RC, then:then:
ee-x-x = 1-0.99 or = 1-0.99 or ee-x-x = = 0.010.01
10.01; 100x
xe
e ln (100)e xFrom From
definition of definition of logarithm:logarithm:
xx = 4.61 = 4.61 t
xRC
4.61 time constants4.61 time constants
Prob. 46. Find time constant, qmax, and time to reach a charge of 16 C if V = 12 V and C = 4 F.
/max 1 t RCq q e /
max 1 t RCq q e R
V1.8 F
++
--
a
b i
1.4 M
C12 V = RC = (1.4 MW)(1.8 = RC = (1.4 MW)(1.8 mF)mF)
= 2.52 s = 2.52 s
qqmaxmax = CV = (1.8 = CV = (1.8 F)(12 F)(12 V);V);
qmax = 21.6 Cqmax = 21.6 C
/
max
16 C1
21.6 Ct RCq
eq
/1 0.741t RCe
Continued . . . Continued . . .
Prob. 46. Find time constant, qmax, and time to reach a charge of 16 C if V = 12 V and C = 4
F.
R
V1.8 F
++
--
a
b i
1.4 M
C12 V
/1 0.741t RCe
Let x = t/RC, Let x = t/RC, then:then:
1 0.741 0.259xe
10.259; 3.86x
xe
e ln (3.86)e xFrom From
definition of definition of logarithm:logarithm:
xx = 1.35 = 1.35 1.35; (1.35)(2.52s)t
tRC
t = 3.40 st = 3.40 sTime to reach 16 Time to reach 16 C:C:
