RC Circuits.pdf

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.

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

V
RC Circuit: Charging Capacitor
a
b
R
i
+
+
-
-
C
q
C
Rearrange terms to place in differential form:
Multiply by C dt :
R
V
dq
dt
q
C
V
iR
q
C
RCdq ( CV q)
dt
dq dt q dq
t dt
( CV q)
RC 0 ( CV q)
o RC

V
RC Circuit: Charging Capacitor
a
b
R
i
+
+
-
-
C
ln( CV q) ln( CV )
q
q dq
t dt
C 0 ( CV q)
o RC
ln( CV q) q
t
0
RC
t
RC
( )
ln CV q
t
CV RC
CV q CVe (1/ RC)
t
qCV 1e
t/
RC

RC Circuit: Charging Capacitor
V
a
b
R
i
+
+
-
-
C
q
C
Instantaneous charge q
on a charging capacitor:
q CV 1e t/
RC
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 to
its maximum value q max max
= CV CV

Example 1. What is the charge on a 4-F
capacitor charged by 12-V for a time t = RC
Q max
0.63 Q
q
Capacitor
Rise in
Charge
V
a
R = 1400
b
i
+
+
- -
4 F
Time, t
The time = RC is known
as the time constant
time constant.
t/
RC
qCV 1 e
e = 2.718; ; e -1 = 0.63
q
CV
10.37
qCV 1e
1
q
0.63CV

Example 1 (Cont.) What is the time constant
Q max
0.63 Q
q
Capacitor
Rise in
Charge
V
a
R = 1400
b
i
+
+
- -
4 F
Time, t
The time = RC is known
as the time constant.
= (1400 )(4
F)
= 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
V
a
b
R
i
+
+
-
-
C
q
C
As charge q rises, the
current i will decay.
q CV 1e t/
RC
dq d
CV
i CV CVe e
dt dt RC
t/ RC
t/
RC
Current decay as a
capacitor is charged:
i
V
R
e
t/
RC

Current Decay
V
a
b
R
i
+
+
-
-
C
q
C
0.37 I
I i
Capacitor
Current
Decay
Time, t
Consider i when
t = 0 and t = .
V
i
R
e
t/
RC
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).

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

Charge and Current During the
Charging of a Capacitor.
Q max
q
Capacitor
I i
Capacitor
0.63 I
Rise in
Charge
0.37 I
Current
Decay
Time, t
Time, t
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
V
After C is fully charged, we turn switch to
b, allowing it to discharge.
a R
a R
b
+
+
-
-
C
V
b
i
+
+
-
-
C
q
C
Discharging capacitor. . . loop rule gives:
E
iR;
q
C
iR
Negative because
of decreasing I.

Discharging From q 0 to q:
V
a
b
R
i
+
+
-
-
q
C
C
Instantaneous charge q
on discharging capacitor:
dq
q RCi;
q RC dt
dq
q
dt
RC
;
q
q
dq
q
0 0
t
dt
;
RC
ln
q
0
q
q
t
RC
t
0
ln
q
ln
q
0
t
RC
ln q
q
0
t
RC

Discharging Capacitor
V
a
b
R
i
+
+
-
-
q
C
C
ln q
q
0
q q e
0
t
RC
t/
RC
Note q o = CV and the instantaneous current is: dq/dt.
dq d
CV
i CVe e
dt dt RC
Current i for a
discharging capacitor.
t/ RC
t/
RC
i
V
C
e
t/
RC

V
Prob. 45. How many time constants are needed
for a capacitor to reach 99% of final charge
a R q
b
i
+
+
-
-
C
/
C
q qmax 1 e t RC
q
q
max
0.99 1e
t/
RC
Let x = t/RC, then: e -x = 1-0.991
or e -x = 0.01
1
x
0.01; e 100
x
e From definition
ln
e(100)
of logarithm:
t
x = 4.61
x
RC
4.61 time
constants
x

Prob. 46. Find time constant, q max , and time to
reach a charge of 16 C if V = 12 V and C = 4 F.
V
a
12 V
b
1.4 M
R
1.8 F
i
+
+
-
-
C
q q /
max 1 e t RC
= RC = (1.4 MW)(1.8 mF)
= 2.52 s
q max = CV = (1.8 F)(12 V);
q max max
= 21.6 C
q
q
max
16C
21.6C
1
e
t/
RC
/
Continued . . .
t RC
1e 0.741

Prob. 46. Find time constant, q max , and time to
reach a charge of 16 C if V = 12 V and C = 4 F.
V
a
12 V
b
1.4 M
R
1.8 F
i
+
+
-
-
C
t/
RC
1e 0.741
Let x = t/RC, then:
x
e 10.741 0.259
1
x
0.259; e 3.86
From definition
x
e of
ln
e(3.86)
logarithm:
t
x = 1.35 1.35; t (1.35)(2.52s)
RC
x
Time to reach 16 C:
t t = 3.40 s

CONCLUSION: Chapter 32B
RC Circuits