Capacitor chapter 10

Copyright ©2011 by Pearson Education, Inc.
publishing as Pearson [imprint]
Introductory Circuit Analysis, 12/e
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Chapter 10
Capacitors

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OBJECTIVES
• Become familiar with the basic construction of a
capacitor and the factors that affect its ability to
store charge on its plates.
• Be able to determine the transient (time-varying)
response of a capacitive network and plot the
resulting voltages and currents.
• Understand the impact of combining capacitors in
series or parallel and how to read the nameplate
data.
• Develop some familiarity with the use of computer
methods to analyze networks with capacitive
elements.

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INTRODUCTION
• The capacitor has a significant impact on
the types of networks that you will be able
to design and analyze.
• Like the resistor, it is a two-terminal device,
but its characteristics are totally different
from those of a resistor.
• In fact, the capacitor displays its true
characteristics only when a change in the
voltage or current is made in the network.

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THE ELECTRIC FIELD
FIG. 10.1 Flux distribution from an isolated positive charge.
• Electric field (E) ⇨ electric flux lines ⇨ to
indicate the strength of E at any point
around the charged body.
• Denser
flux lines ⇨
stronger
E.

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THE ELECTRIC FIELD
FIG. 10.2 Determining the force on a unit
charge r meters from a charge Q of
similar polarity.

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THE ELECTRIC FIELD
FIG. 10.3 Electric flux distributions: (a) opposite charges; (b) like charges.
• Electric flux lines always extend from a
+ve charged body to a -ve charged body,
⊥ to the charged surfaces, and never
intersect.

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CAPACITANCE
FIG. 10.4 Fundamental charging circuit.
⇨V=IR

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CAPACITANCE
FIG. 10.7 Effect of a dielectric on the field distribution between the plates of a
capacitor: (a) alignment of dipoles in the dielectric; (b) electric field components
between the plates of a capacitor with a dielectric present.

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CAPACITANCE
TABLE 10.1 Relative permittivity (dielectric constant) Σr of various
dielectrics.

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CAPACITOR Construction
FIG. 10.9 Example 10.2.
⇨ R =ρL/A

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CAPACITORS
Types of Capacitors
• Capacitors, like resistors, can be listed
under two general headings: fixed and
variable.
FIG. 10.11 Symbols for the
capacitor: (a) fixed; (b) variable.

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CAPACITORS
Types of Capacitors
FIG. 10.12 Demonstrating that, in
general, for each type of
construction, the size of a capacitor
increases with the capacitance
value: (a) electrolytic; (b) polyester-
film; (c) tantalum.

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CAPACITORS
Types of Capacitors
FIG. 10.20 Variable capacitors: (a) air; (b) air trimmer; (c) ceramic dielectric
compression trimmer. [(a) courtesy of James Millen Manufacturing Co.]
• Variable Capacitors
– All the parameters can be changed to create a variable
capacitor.
– For example; the capacitance of the variable air capacitor
is changed by turning the shaft at the end of the unit.

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CAPACITORS
Leakage Current and ESR
FIG. 10.21 Leakage current: (a) including the leakage resistance in the equivalent
model for a capacitor; (b) internal discharge of a capacitor due to the leakage current.

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CAPACITORS
Capacitor Labeling
FIG. 10.23 Various marking schemes for small capacitors.

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CAPACITORS
Measurement and Testing of Capacitors
• The capacitance of
a capacitor can be
read directly using
a meter such as
the Universal LCR
Meter.
FIG. 10.24 Digital reading capacitance
meter. (Courtesy of B+K Precision.)

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TRANSIENTS IN CAPACITIVE
NETWORKS: THE CHARGING PHASE
• The placement of
charge on the
plates of a
capacitor does not
occur
instantaneously.
• Instead, it occurs
over a period of
time determined by
the components of
the network.
FIG. 10.26 Basic R-C charging network.

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FIG. 10.27 vC during the charging phase.
The current ( ic )
through a
capacitive
network is
essentially zero
after five time
constants of the
capacitor
charging phase.

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FIG. 10.28 Universal time constant chart.

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TABLE 10.3 Selected values of e-x.

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• The factor t, called the time constant of
the network, has the units of time, as
shown below using some of the basic
equations introduced earlier in this text:

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FIG. 10.29 Plotting the equation yC = E(1 – e-t/t) versus time (t).

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FIG. 10.31 Demonstrating that a capacitor has the
characteristics of an open circuit after the charging phase
has passed.

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FIG. 10.32 Revealing the short-circuit equivalent for the
capacitor that occurs when the switch is first closed.

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TRANSIENTS IN CAPACITIVE NETWORKS: THE
CHARGING PHASE
Using the Calculator to Solve Exponential Functions
FIG. 10.35 Transient network for Example 10.6.

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CHARGING PHASE
FIG. 10.36 vC versus time for the charging network in Fig. 10.35.

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CHARGING PHASE
FIG. 10.37 Plotting the waveform in Fig. 10.36 versus time (t).

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CHARGING PHASE
FIG. 10.38 iC and yR for the charging network in Fig. 10.36.

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TRANSIENTS IN CAPACITIVE NETWORKS:
THE DISCHARGING PHASE
• How to discharge a capacitor and how long
the discharge time will be.
• You can, of course, place a lead directly
across a capacitor to discharge it very
quickly—and possibly cause a visible
spark.
• For larger capacitors such those in TV
sets, this procedure should not be
attempted because of the high voltages
involved.

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NETWORKS: THE DISCHARGING PHASE
FIG. 10.39 (a) Charging network; (b)
discharging configuration.
• For the voltage across
the capacitor that is
decreasing with time,
the mathematical
expression is:

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FIG. 10.40 yC, iC, and yR for 5t switching between contacts in Fig. 10.39(a).

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FIG. 10.41 vC and iC for the network in Fig. 10.39(a) with the
values in Example 10.6.

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The Effect of on the Response

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FIG. 10.43 Effect of increasing values of C (with
R constant) on the charging curve for vC.

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FIG. 10.44 Network to be analyzed in Example 10.8.

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FIG. 10.45 vC and iC for the network in Fig. 10.44.

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FIG. 10.46 Network to be analyzed in
Example 10.9.
FIG. 10.47 The charging phase for the
network in Fig. 10.46.

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FIG. 10.48 Network in Fig. 10.47 when the switch
is moved to position 2 at t = 1t1.

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FIG. 10.49 vC for the network in Fig. 10.47.

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FIG. 10.50 ic for the network in Fig. 10.47.

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INITIAL CONDITIONS
• The voltage across the capacitor at this
instant is called the initial value, as shown
for the general waveform in Fig. 10.51.
FIG. 10.51 Defining the regions associated with a
transient response.

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INITIAL CONDITIONS

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INITIAL CONDITIONS
FIG. 10.53 vC and iC for the network in Fig. 10.52.

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INITIAL CONDITIONS
FIG. 10.54 Defining the parameters in Eq. (10.21)
for the discharge phase.

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THÉVENIN EQUIVALENT: t =RThC
• You may encounter instances in
which the network does not have the
simple series form in Fig. 10.26.
• You then need to find the Thévenin
equivalent circuit for the network
external to the capacitive element.

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FIG. 10.57 Applying Thévenin’s theorem to
the network in Fig. 10.56.

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FIG. 10.58 Substituting the Thévenin equivalent for the

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FIG. 10.59 The resulting
waveforms for the network in Fig.
10.56.

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FIG. 10.61 Network in Fig. 10.60
redrawn.

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FIG. 10.62 yC for the network in Fig. 10.60.

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THE CURRENT iC
• There is a very special relationship between the
current of a capacitor and the voltage across it.
• For the resistor, it is defined by Ohm’s law: iR =
vR/R.
• The current through and the voltage across the
resistor are related by a constant R—a very simple
direct linear relationship.
• For the capacitor, it is the more complex
relationship defined by:

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THE CURRENT iC
FIG. 10.64 vC for Example 10.14.

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THE CURRENT iC
FIG. 10.65 The resulting current iC for the applied voltage in Fig. 10.64.

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CAPACITORS IN SERIES AND IN
PARALLEL
• Capacitors, like resistors, can be
placed in series and in parallel.
• Increasing levels of capacitance can
be obtained by placing capacitors in
parallel, while decreasing levels can
be obtained by placing capacitors in
series.

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PARALLEL
FIG. 10.66 Series capacitors.

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PARALLEL
FIG. 10.67 Parallel capacitors.

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PARALLEL
FIG. 10.68 Example 10.15. FIG. 10.69 Example 10.16.

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PARALLEL
FIG. 10.71 Reduced equivalent for the

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PARALLEL
FIG. 10.73 Determining the final
(steady-state) value for yC.

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PARALLEL

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ENERGY STORED BY A CAPACITOR
• An ideal capacitor does not dissipate any of the
energy supplied to it.
• It stores the energy in the form of an electric field
between the conducting surfaces.
• A plot of the voltage, current, and power to a
capacitor during the charging phase is shown in
Fig. 10.75.
• The power curve can be obtained by finding the
product of the voltage and current at selected
instants of time and connecting the points
obtained.
• The energy stored is represented by the shaded
area under the power curve.

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ENERGY STORED BY A CAPACITOR
FIG. 10.75 Plotting the power to a capacitive element
during the transient phase.

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APPLICATIONS
Touch Pad
FIG. 10.77 Laptop touch pad.

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APPLICATIONS
Touch Pad
FIG. 10.78 Matrix approach to capacitive sensing in a touch pad.

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APPLICATIONS
Flash Lamp
FIG. 10.81 Flash camera: general appearance.

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APPLICATIONS
Flash Lamp
FIG. 10.82 Flash camera: basic circuitry.

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APPLICATIONS
Flash Lamp
FIG. 10.83 Flash camera:
internal construction.

Capacitor chapter 10

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