Download Capacitor Circuits

Chapter 26B - Capacitor
Circuits
AA PowerPoint
PowerPoint Presentation
Presentation by
by
Paul
Paul E.
E. Tippens,
Tippens, Professor
Professor of
of Physics
Physics
Southern
Southern Polytechnic
Polytechnic State
State University
University
©
2007
Objectives: After completing this
module, you should be able to:
• Calculate the equivalent capacitance of a
number of capacitors connected in series
or in parallel.
• Determine the charge and voltage across
any chosen capacitor in a network when
given capacitances and the externally
applied potential difference.
Electrical Circuit Symbols
Electrical circuits often contain two or more
capacitors grouped together and attached
to an energy source, such as a battery.
The following symbols are often used:
Ground
+ - + - + - + -
Battery
+
-
Capacitor
+
+
-
Series Circuits
Capacitors or other devices connected
along a single path are said to be
connected in series. See circuit below:
+
+
C1
- +
- +
-+
-+
C2
Battery
-
C3
Series connection
of capacitors.
“+ to – to + …”
Charge inside
dots is induced.
Charge on Capacitors in Series
Since inside charge is only induced, the
charge on each capacitor is the same.
Q1
+
+
C1
Q2
- +
- +
Q3
-+
-+
C2
Battery
-
C3
Charge is same:
series connection
of capacitors.
Q = Q1 = Q2 =Q3
Voltage on Capacitors in Series
Since the potential difference between
points A and B is independent of path, the
battery voltage V must equal the sum of
the voltages across each capacitor.
V1
+
+
C1
•A
V2
- +
- +
V3
-+
-+
C2
Battery
-
C3
•
B
Total voltage V
Series connection
Sum of voltages
V = V1 + V2 + V3
Equivalent Capacitance: Series
V1
+
+
V2
- +
- +
C1
C2
V3
-+
-+
-
C3
Q1= Q2 = Q3
1
1
1
1
 

Ce C1 C2 C3
Q
Q
C ; V 
V
C
V = V1 + V2 + V3
Q Q1 Q2 Q3



C C1 C2 C3
Equivalent Ce
for capacitors
in series:
n
1
1

Ce i 1 Ci
Example 1. Find the equivalent capacitance
of the three capacitors connected in series
with a 24-V battery.
Ce for
series:
n
1
1

Ce i 1 Ci
1
1
1
1



Ce 2  F 4  F 6  F
1
 0.500  0.250  0.167
Ce
1
1
 0.917 or Ce 
0.917
Ce
C1
C2
C3
+ - + -+ + - + -+ 2 F 4 F 6 F
24 V
CCee == 1.09
1.09 F
F
Example 1 (Cont.): The equivalent circuit can
be shown as follows with single Ce.
C1
C2
C3
+ - + -+ + - + -+ 2 F 4 F 6 F
n
1
1

Ce i 1 Ci
Ce
1.09 F
24 V
CCee == 1.09
1.09 F
F
24 V
Note that the equivalent capacitance Ce
for capacitors in series is always less than
the least in the circuit. (1.09 F < 2 F)
Example 1 (Cont.): What is the total charge
and the charge on each capacitor?
C1
C2
C3
+ - + -+ + - + -+ 2 F 4 F 6 F
Ce
1.09 F
24 V
24 V
QT = CeV = (1.09 F)(24 V);
For series circuits:
QT = Q1 = Q2 = Q3
CCee == 1.09
1.09 F
F
Q
C
V
Q  CV
QQTT == 26.2
26.2 C
C
QQ11 == QQ22 == QQ33 == 26.2
26.2 C
C
Example 1 (Cont.): What is the voltage across
each capacitor?
Q
Q
C ; V 
C1 C2 C3
V
C
+ - + -+ + - + -+ Q1 26.2  C
2 F 4 F 6 F
V1 

 13.1 V
C1
2 F
24 V
Q2 26.2  C
V2 

 6.55 V
C2
4 F
Q3 26.2  C
V3 

 4.37 V
VVTT == 24
24 VV
C3
6 F
Note:
Note: VVTT =
= 13.1
13.1 VV +
+ 6.55
6.55 VV +
+ 4.37
4.37 VV =
= 24.0
24.0 VV
Short Cut: Two Series Capacitors
The equivalent capacitance Ce for two series
capacitors is the product divided by the sum.
1
1
1
  ;
Ce C1 C2
C1C2
Ce 
C1  C2
Example:
(3  F)(6  F)
Ce 
3  F  6 F
C1
C2
+ - + + - + 3 F 6 F
CCee == 22 F
F
Parallel Circuits
Capacitors which are all connected to the
same source of potential are said to be
connected in parallel. See below:
Parallel capacitors:
“+ to +; - to -”
- -
C3
+
+
- -
C2
+
+
+
+
C1
- -
Voltages:
VT = V1 = V2 = V3
Charges:
QT = Q1 + Q2 + Q3
Equivalent Capacitance: Parallel
Q
C  ; Q  CV
V
Parallel capacitors
in Parallel:
- -
Ce = C1 + C2 + C3
C3
+
+
- -
C2
+
+
+
+
C1
- -
Q = Q1 + Q2 + Q3
Equal Voltages:
CV = C1V1 + C2V2 + C3V3
Equivalent Ce
for capacitors
in parallel:
n
Ce   Ci
i 1
Example 2. Find the equivalent capacitance
of the three capacitors connected in parallel
with a 24-V battery.
Ce for
parallel:
n
Ce   Ci
VT = V1 = V2 = V3
Q = Q1 + Q2 + Q3
i 1
24 V
Ce = (2 + 4 + 6) F
2 F
C1 C2
C3
4 F
6 F
CCee == 12
12 F
F
Note that the equivalent capacitance Ce for
capacitors in parallel is always greater than
the largest in the circuit. (12 F > 6 F)
Example 2 (Cont.) Find the total charge QT
and charge across each capacitor.
Q = Q1 + Q2 + Q3
24 V
2 F
C1 C2
C3
4 F
6 F
CCee == 12
12 F
F
V1 = V2 = V3 = 24 V
Q
C  ; Q  CV
V
QT = CeV
Q1 = (2 F)(24 V) = 48 C
QT = (12 F)(24 V)
Q1 = (4 F)(24 V) = 96 C
QQTT == 288
288 C
C
Q1 = (6 F)(24 V) = 144 C
Example 3. Find the equivalent capacitance
of the circuit drawn below.
24 V
C1
4 F
24 V
4 F
C1
C3,6
C2
3 F
C3
6 F
C3,6
(3 F)(6 F)

 2 F
3 F  6 F
Ce = 4 F + 2 F
CCee== 66 F
F
24 V
2 F
Ce
6 F
Example 3 (Cont.) Find the total charge QT.
CCee== 66 F
F
24 V
C1
4 F
24 V
4 F
C1
C2
3 F
C3
6 F
C3,6
Q = CV = (6 F)(24 V)
QQTT== 144
144 C
C
24 V
2 F
Ce
6 F
Example 3 (Cont.) Find the charge Q4 and
voltage V4 across the the 4F capacitor
VV44 == VVTT == 24
24 VV
24 V
4 F
C1
C2
3 F
C3
6 F
Q4 = (4 F)(24 V)
QQ44== 96
96 C
C
The remainder of the charge: (144 C – 96 C)
is on EACH of the other capacitors. (Series)
QQ33 == QQ66 == 48
48 C
C
This
This can
can also
also be
be found
found from
from
QQ == CC3,6
V 3,6 == (2
(2 F)(24
F)(24 V)
V)
3,6V3,6
Example 3 (Cont.) Find the voltages across
the 3 and 6-F capacitors
QQ33 == QQ66== 48
48 C
C
24 V
4 F
C1
C2
3 F
C3
6 F
48 C
 16.0V
V3 
3 F
48 C
 8.00V
V6 
6 F
Note:
Note: VV33 ++ VV66 == 16.0
16.0 VV ++ 8.00
8.00 VV == 24
24 VV
Use
Use these
these techniques
techniques to
to find
find voltage
voltage and
and
capacitance
capacitance across
across each
each capacitor
capacitor in
in aa circuit.
circuit.
Summary: Series Circuits
n
1
1

Ce i 1 Ci
QQ == QQ11 == QQ22 == QQ33
VV == VV11 ++ VV22 ++ VV33
For two capacitors at a time:
C1C2
Ce 
C1  C2
Summary: Parallel Circuits
n
QQ == QQ11 ++ QQ22 ++ QQ33
i 1
VV == VV11 == VV22 =V
=V33
Ce   Ci
For
For complex
complex circuits,
circuits, reduce
reduce the
the circuit
circuit in
in steps
steps
using
using the
the rules
rules for
for both
both series
series and
and parallel
parallel
connections
connections until
until you
you are
are able
able to
to solve
solve problem.
problem.
CONCLUSION: Chapter 26B
Capacitor Circuits