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DC/AC Fundamentals: A Systems
Approach
Thomas L. Floyd
David M. Buchla
Capacitors
Chapter 9
Ch.9 Summary
The Basic Capacitor
Capacitors are one of the fundamental passive
components. In its most basic form, it is composed of
two conductive plates separated by an insulating
dielectric.
Capacitance is the ability to store electrical charge.
Conductors
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
Dielectric
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Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.9 Summary
The Basic Capacitor
The charging
process…
Leads
Initially
uncharged
uncharged 
Charging
Charging
Source
removed
Source
removed
Fully
charged
Fully charged 


A
AA


+
++
++++
++
+
++
++++
+
+
+ +
+
+++
+
+
+ +
+++
++
VS VS
A +
 ++

Dielectric
 
++ 
 +
Plates

++ 
 +

  

+
+

 Electrons
+


++

BBB

 
B


A capacitor with stored charge can act as a temporary
battery.
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.9 Summary
Capacitance
Capacitance equals the ratio of charge to voltage
Q
C
V
Rearranging, the charge on a capacitor is found as
Q  C V
If a 22 mF capacitor is connected to a 10 V source, the
charge is 220 mC
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.9 Summary
Capacitance
Imagine you store rubber bands in a bottle that is
nearly full.
You could store more rubber bands
(like charge or Q) in a bigger bottle
(capacitance or C) or if you push them
in more (voltage or V). Thus,
Q  C V
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.9 Summary
Capacitance
A capacitor stores energy in the form of an electric field
that is established by the opposite charges on the two
plates. The energy of a charged capacitor is given by
the equation
W 
1
CV 2
2
where
W = the energy in joules
C = the capacitance in farads
V = the voltage in volts
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.9 Summary
Capacitance
The value of a capacitor depends on three physical
characteristics.
C is directly proportional to:
•the relative dielectric constant
•the plate area
C is inversely proportional to:
• the distance between the plates
The relationships listed
are given in the equation:
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
C  8.85  10
12
 r A 
F/m

 d 
© 2013 by Pearson Higher Education, Inc
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Ch.9 Summary
Capacitance
Find the capacitance (in farads) of a 4.0 cm diameter sensor
immersed in oil with plates that are separated by 0.25 mm.
(r = 4.0 for oil.)
The plate area is:
A  r 2    (0.02 m2 )  1.26  103 m2
The distance between the plates is 0.25 10-3 m, and:
3
2

(4.0)(1.26

10
m
)
12
  178 pF
C  8.85  10 F/m
3
0.25  10 m


DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.9 Summary
Capacitor Types: Mica
Mica capacitors are small with high working voltage
ratings. The working voltage is a component rating
that cannot be exceeded.
Foil
Mica
Foil
Mica
Foil
Mica
Foil
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.9 Summary
Capacitor Types: Ceramic disks
Ceramic disks are small non-polarized capacitors. They
have relatively high capacitance due to high dielectric
constants (r).
Lead wire soldered
to silver electrode
Solder
Ceramic
dielectric
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
Dipped phenolic coating
Silver electrodes deposited on
top and bottom of ceramic disk
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Ch.9 Summary
Capacitor Types: Plastic Film
Plastic film capacitors are small and non-polarized. They
have relatively high capacitance due to large plate area.
High-purity
foil electrodes
Plastic film
dielectric
Outer wrap of
polyester film
Capacitor section
(alternate strips of
film dielectric and
Lead wire
foil electrodes)
Solder-coated end
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.9 Summary
Capacitor Types: Electrolytic
Electrolytic capacitors have very high capacitance but they are not
as precise as other types and tend to have more leakage current.
Electrolytic capacitors are polarized; meaning polarity must be
observed when replacing an electrolytic capacitor.
+
Al electrolytic
_
Ta electrolytic
Component symbol
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.9 Summary
Capacitor Types: Variable
Variable capacitors typically have small capacitance
values and are usually adjusted manually.
The varactor diode is a solid-state device that is used as
a variable capacitor; it is adjusted with an electrical signal.
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.9 Summary
Capacitor Labeling
Capacitors use several labeling methods. Small
capacitors values are frequently stamped on them such
as .001 or .01, with implied units of microfarads.
Electrolytic capacitors have larger values, so are read
as mF. The unit is usually stamped as mF, but some
older ones may be shown as MF or MMF).
47
MF
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.9 Summary
Capacitor Labeling
Labels such as 103 or 104 are read as 10x103 (10,000 pF)
or 10x104 (100,000 pF) respectively. The third digit is the
multiplier.
When values are labeled as shown below, the units
are picofarads.
222
2200
What is the value of each capacitor shown above?
Each is 2200 pF
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.9 Summary
Series Capacitors
When capacitors are connected in series, the total
capacitance is lower than any component value. The
general equation for total series capacitance is:
CT 
1
1
1
1

 ... 
C1 C2
CT
The total capacitance of two series-connected capacitors is:
CT 
1
1
1

C1 C2
…or you can use the product-over-sum rule.
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.9 Summary
Series Capacitors
If a 0.001 mF capacitor is connected in series with an
800 pF capacitor, the total capacitance is:
444 pF
C1
0.001 µF
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
C2
800 pF
© 2013 by Pearson Higher Education, Inc
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Ch.9 Summary
Parallel Capacitors
When capacitors are connected in parallel, the total
capacitance equals the sum of the individual capacitor
values. The general equation for capacitors in parallel is
CT  C1  C2  ...  Cn
If a 0.001 mF capacitor is
connected in parallel with
an 800 pF capacitor, the
total capacitance is:
1800 pF
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
C1
C2
0.001 µF
800 pF
© 2013 by Pearson Higher Education, Inc
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Ch.9 Summary
Vfinal
RC Time Constant
When a capacitor is charged
through a series resistor by a dc
source, the charge curves are
exponential curves.
0
t
(a) Capacitor charging voltage
I initial
R
VS
C
0
(b) Charging current
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
t
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Ch.9 Summary
RC Time Constant
Vinitial
When a capacitor is discharged
through a resistor, the discharge
curves are also exponential curves.
t
0
(a) Capacitor discharging voltage
I
_
initial
R
C
0
t
(b) Discharging current
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.9 Summary
Universal Exponential Curves
100%
86%
80%
Rising exponential
63%
60%
40%
37%
Falling exponential
20%
t  RC
99%
98%
95%
Percent of final value
The exponential curves
show how the capacitor
in an RC circuit charges
(or discharges) over five
equal periods, called
time constants. For an
RC circuit, the length of
a time constant is
14%
5%
0
0
1t
2t
3t
2%
4t
1%
5t
Number of time constants
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.9 Summary
Universal Exponential Curves
The universal curves can be applied to general formulas for the voltage
(or current) curves for RC circuits. The general voltage formula is
v  VF  (Vi  VF )e t / RC
where
VF = final value of voltage
Vi = initial value of voltage
v = instantaneous value of voltage
e = Napier’s constant (approximately 2.71828)
The final capacitor voltage is greater than the initial voltage when the
capacitor is charging, and less that the initial voltage when it is
discharging.
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.9 Summary
Capacitive Reactance (XC)
Capacitive reactance is the opposition to ac provided
by a capacitor. The equation for capacitive reactance
is:
1
XC 
2fC
The reactance of a 0.047 mF capacitor when a
frequency of 15 kHz is applied is 226 W
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.9 Summary
Capacitive Reactances in Series
When capacitors are connected in series, the total reactance is
the sum of the individual reactances.
X C ( tot )  X C1  X C 2  ...  X Cn
Assume three 0.033 mF capacitors are in series with a 2.5
kHz ac source. What is the total reactance?
XC 
1
1

 1.93 kΩ
2fC 2(2.5 kHz)(0.033 μF)
X C(tot)  X C1  X C 2  X C 3
 1.93 kΩ  1.93 kΩ  1.93 kΩ  5.79 kW
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.9 Summary
Capacitive Reactances in Parallel
When capacitors are in parallel, the total reactance is the
reciprocal of the sum of the reciprocals of the individual
reactances.
1
X C ( tot ) 
1
1
1

 ... 
X C1 X C 2
X Cn
If three 0.033 mF capacitors are placed in parallel with a 2.5
kHz ac source, what is the total reactance?
The reactance of each capacitor is 1.93 kW, and
1
1
X C ( tot ) 

 643 W
1
1
1
1
1
1




X C1 X C 2 X C 3 1.93 kΩ 1.93 kΩ 1.93 kΩ
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.9 Summary
Capacitive Voltage Divider
Series capacitors are commonly used as a voltage divider. The capacitors
split the output voltage in proportion to their reactance (and inversely
proportional to their capacitance).
What is the output voltage from the capacitive voltage divider below?
1
 4.82 kΩ
2f(3 kHz)(1000 pF)
1

 482 Ω
2f(3 kHz)(0.01 mF)
 X C1  X C 2  5.3 kΩ
X C1 
X C2
X C(tot)
Vout
C1
1000 pF
1.0 V
f = 33 kHz
C2
Vout
0.01 µF
 XC 2 
 VS   482 W (1 V)  91 mV

X

 5.3 kW 
 C ( tot ) 
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.9 Summary
Capacitive Voltage Divider
Instead of using a ratio of reactances in the capacitor voltage divider
equation, you can use a ratio of the total series capacitance to the
output capacitance (multiplied by the input voltage). The result is the
same. For the problem presented in the last slide,
X C ( tot ) 
1
1
1

X C1 X C 2

1
1
1

1000 pF 0.01 μF
C 
 909 pF 
(1 V) 
Vout   ( tot )  V2  
 0.01 μF 
 C2 
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
 909 pF
C1
1000 pF
1.0 V
f = 33 kHz
91 mV
C2
Vout
0.01 µF
© 2013 by Pearson Higher Education, Inc
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Ch.9 Summary
Capacitive Phase Shift
When a sine wave is
applied to a capacitor,
there is a 90º phase shift
between voltage and
current. This phase shift
is described in one of
two ways:
VC
0
90
I
o
0
•Current leads voltage by 90o.
•Voltage lags current by 90o.
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.9 Summary
Power in a Capacitor
Energy is stored by the capacitor during a portion of
the ac cycle and returned to the source during
another portion of the cycle.
Voltage and current are always 90o out of phase. For
this reason, no true power is dissipated by a capacitor,
because stored energy is returned to the circuit.
The rate at which a capacitor stores or returns
energy is called reactive power. The unit for reactive
power is the VAR (volt-ampere reactive).
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.9 Summary
Power Supply Filtering
There are many applications for capacitors. One is in
filters, such as the power supply filter shown here.
Rectifier
60 Hz ac
C
Load
resistance
The filter smoothes the
pulsating dc from the
rectifier.
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.9 Summary
Coupling Capacitors
Coupling capacitors are used to pass an ac signal
from one stage to another while blocking dc.
3V
0V
The capacitor isolates dc
between the amplifier
stages, preventing dc in
one stage from affecting
the other stage.
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
0V
+V
C
Input
R1
Amplifier
stage 1
Amplifier
stage 2
Output
R2
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Ch.9 Summary
Bypass Capacitors
Another application is to bypass an ac signal to ground but
retain a dc value. This is widely done to affect gain in
amplifiers.
dc plus ac
0V
dc only
0V
The bypass capacitor
places point A at ac
ground, keeping only
a dc value at point A.
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
R1
A
Point in the circuit where
only dc is required
R2
C
© 2013 by Pearson Higher Education, Inc
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Ch.9 Summary
Key Terms
Capacitor
An electrical device consisting of two conductive
plates separated by an insulating material and
possessing the property of capacitance.
Dielectric
The insulating material between the conductive
plates of a capacitor.
Farad
RC time
constant
The unit measure for capacitance.
A fixed time interval set by the values of R and C,
that determines the response of a series RC
circuit to a change in dc input voltage. It equals
the product of R and C.
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.9 Summary
Capacitive
reactance
The opposition of a capacitor to sinusoidal
current; measured in ohms.
Instantaneous The value of power in a circuit at a given
power (p) instant of time.
True power
(Ptrue)
The power that is dissipated in a circuit
usually in the form of heat.
Reactive
power (Pr )
The rate at which energy is alternately
stored and returned to the source by a
capacitor. The unit is the VAR.
VAR
(volt-amperes
reactive)
The unit of measure for reactive power.
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.9 Summary
Quiz
1. The capacitance of a capacitor will be larger if
a. the spacing between the plates is increased
b. air replaces oil as the dielectric
c. the area of the plates is increased
d. all of the above
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.9 Summary
Quiz
2. The major advantage of a mica capacitor over
other types is
a. they have the largest available
capacitances
b. their voltage rating is very high
c. they are polarized
d. all of the above
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.9 Summary
Quiz
3. Electrolytic capacitors are useful in applications
where
a. a precise value of capacitance is required
b. low leakage current is required
c. large capacitance is required
d. all of the above
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.9 Summary
Quiz
4. If a 0.015 mF capacitor is in series with a 6800 pF
capacitor, the total capacitance is
a. 1568 pF
b. 4678 pF
c. 6815 pF
d. 0.022 mF
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.9 Summary
Quiz
5. Two capacitors that are initially uncharged are
connected in series with a dc source. Compared to
the larger capacitor, the smaller capacitor will have
a. the same charge
b. more charge
c. less voltage
d. the same voltage
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.9 Summary
Quiz
6. When a capacitor is connected through a resistor
to a dc voltage source, the charge on the capacitor
will reach 50% of its final charge in
a. less than one time constant
b. exactly one time constant
c. greater than one time constant
d. answer depends on the amount of voltage
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.9 Summary
Quiz
7. When a capacitor is connected through a series
resistor and switch to a dc voltage source, the voltage
across the resistor after the switch is closed has the
shape of
a. a straight line
b. a rising exponential
c. a falling exponential
d. none of the above
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.9 Summary
Quiz
8. The capacitive reactance of a 100 mF capacitor to
60 Hz is
a. 6.14 kW
b. 265 W
c. 37.7 W
d. 26.5 W
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.9 Summary
Quiz
9. If an sine wave from a function generator is applied
to a capacitor, the current will
a. lag voltage by 90o
b. lag voltage by 45o
c. be in phase with the voltage
d. none of the above
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.9 Summary
Quiz
10. A switched capacitor emulates a
a. smaller capacitor
b. larger capacitor
c. battery
d. resistor
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.9 Summary
Answers
1. c
6. a
2. b
7. c
3. c
8. d
4. b
9. d
5. a
10. d
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved