Transcript RC-Circuits
Chapter 32B - RC Circuits
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
Optional: Check with Instructor
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.
RC Circuit
RC-Circuit: Resistance R and capacitance C
in series with a source of emf V.
a
R
a
R
- -
C
V
b
i
+
+
+
+
V
b
- -
q
C
C
Start charging capacitor. . . loop rule gives:
q
E iR; V C iR
RC Circuit: Charging Capacitor
R
a
i
+
+
V
b
- -
q
C
C
q
V iR
C
dq
q
R
V
dt
C
Rearrange terms to place in differential form:
Multiply by C dt :
dq
dt
(CV q) RC
RCdq (CV q)dt
q
t dt
dq
0 (CV q) o RC
RC Circuit: Charging Capacitor
R
a
i
+
+
V
b
- -
q
C
C
t dt
dq
0 (CV q) o RC
t
q
ln(CV q) 0
RC
q
t
ln(CV q) ln(CV )
RC
CV q CVe
(1/ RC ) t
(CV q) t
ln
CV
RC
q CV 1 e
t / RC
RC Circuit: Charging Capacitor
R
a
i
+
+
V
b
- -
q
C
C
Instantaneous charge q
on a charging capacitor:
q CV 1 e
t / RC
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
Example 1. What is the charge on a 4-mF
capacitor charged by 12-V for a time t = RC?
Qmax
q
a R = 1400 W
Capacitor
Rise in
Charge
t
V
b
i
+
+
0.63 Q
- -
4 mF
Time, t
The time t = RC is known
as the time constant.
q CV 1 e
t / RC
q CV 1 e
1
e = 2.718; e-1 = 0.63
q CV 1 0.37
q 0.63CV
Example 1 (Cont.) What is the time constant t?
Qmax
q
Rise in
Charge
t
V
b
i
+
+
0.63 Q
a R = 1400 W
Capacitor
- -
4 mF
Time, t
The time t = RC is known
as the time constant.
t = (1400 W)(4 mF)
t = 5.60 ms
In one time constant
(5.60 ms in this
example), the charge
rises to 63% of its
maximum value (CV).
RC Circuit: Decay of Current
R
a
i
+
+
V
b
- -
q
C
C
As charge q rises, the
current i will decay.
q CV 1 e
t / RC
dq d
CV t / RC
t / RC
i
CV CVe
e
dt dt
RC
Current decay as a
capacitor is charged:
V t / RC
i e
R
Current Decay
R
a
i
+
+
V
b
- -
q
C
C
I
i
Capacitor
Current
Decay
0.37 I
t
Time, t
Consider i when
t = 0 and t = .
The current is a maximum
of I = V/R when t = 0.
V t / RC
i e
R
The current is zero when
t = (because the back
emf from C is equal to V).
Example 2. What is the current i after one time
constant (t RC)? Given R and C as before.
I
i
Capacitor
0.37 I
t
V
b
i
+
+
Current
Decay
a R = 1400 W
- -
4 mF
Time, t
The time t = RC is known
as the time constant.
V t / RC V 1
i e
e
R
C
e = 2.718; e-1 = 0.37
V
i 0.37 0.37imax
R
Charge and Current During the
Charging of a Capacitor.
Qmax
q
0.63 I
Capacitor
Rise in
Charge
t
Time, t
I
i
Capacitor
Current
Decay
0.37 I
t
Time, t
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.
RC Circuit: Discharge
After C is fully charged, we turn switch to
b, allowing it to discharge.
a
R
a
R
- -
C
V
b
i
+
+
+
+
V
b
- -
q
C
C
Discharging capacitor. . . loop rule gives:
E iR;
Negative because
q
iR of decreasing I.
C
Discharging From q0 to q:
R
a
i
dq
dt
;
q
RC
+
+
V
b
q
C
- -
C
Instantaneous charge q
on discharging capacitor:
dq
q RCi; q RC
dt
t dt
dq
q0 q 0 RC ;
q
t
ln q ln q0
RC
ln qq
q
0
t
t
RC
0
q
t
ln
q0 RC
Discharging Capacitor
R
a
i
+
+
V
b
q
C
- -
C
q
t
ln
q0 RC
q q0e
t / RC
Note qo = CV and the instantaneous current is: dq/dt.
dq d
CV t / RC
t / RC
i
CVe
e
dt dt
RC
V t / RC
Current i for a
i e
discharging capacitor.
C
Prob. 45. How many time constants are needed
for a capacitor to reach 99% of final charge?
a
R
q
i
+
+
V
b
C
q qmax 1 e
C
- -
q
qmax
Let x = t/RC, then:
1
x
0.01;
e
100
x
e
x = 4.61
t / RC
0.99 1 e
t / RC
e-x = 1-0.99 or e-x = 0.01
From definition
of logarithm:
t
x
RC
ln e (100) x
4.61 time
constants
Prob. 46. Find time constant, qmax, and time to
reach a charge of 16 mC if V = 12 V and C = 4 mF.
a 1.4 MW
V 12 V
1.8 mF
i
- -C
t = RC = (1.4 MW)(1.8 mF)
qmax = CV = (1.8 mF)(12 V);
q
qmax
+
+
bR
q qmax 1 e
t / RC
16m C
t / RC
1 e
21.6m C
t = 2.52 s
qmax = 21.6 mC
1 e
Continued . . .
t / RC
0.741
Prob. 46. Find time constant, qmax, and time to
reach a charge of 16 mC if V = 12 V and C = 4 mF.
a 1.4 MW
V 12 V
1.8 mF
i
- -C
1
x
0.259;
e
3.86
x
e
x = 1.35
0.741
Let x = t/RC, then:
+
+
bR
1 e
t / RC
e
x
1 0.741 0.259
From definition
of logarithm:
t
1.35;
RC
Time to reach 16 mC:
lne (3.86) x
t (1.35)(2.52s)
t = 3.40 s
CONCLUSION: Chapter 32B
RC Circuits