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Chapter 9
CAPACITOR
Objectives
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Describe the basic structure and
characteristics of a capacitor
Discuss various types of capacitors
Analyze series capacitors
Analyze parallel capacitors
Analyze capacitive dc switching circuits
Analyze pulse response of RC circuit
Basics of a Capacitor
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In its simplest form, a capacitor is an electrical
device constructed of two parallel plates
separated by an insulating material called the
dielectric.
In the neutral state, both plates have an equal
number of free electrons.
When a voltage source is connected to the
capacitor, electrons are removed from one plated
and deposited on the other.
No electrons flow through the dielectric.
Basics of a Capacitor
How a Capacitor Stores Energy
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A capacitor stores energy in the form of an electric field
that is established by the opposite charges on the two
plates.
A capacitor obeys Coulomb’s Law:
A force exists between two point-source charges that is
directly proportional to the product of the two charges and
inversely proportional to the square of the distance
between the charges.
F = k Q1Q2 / d2
C in DC circuit
Capacitors in DC Circuits
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A capacitor will charge up when it is connected
to a dc voltage source.
When a capacitor is fully charged, there is no
current.
There is no current through the dielectric of the
capacitor because the dielectric is an insulating
material.
A capacitor blocks constant dc.
Series Capacitors
When capacitors are connected in series, the total
capacitance is less than the smallest capacitance
value since the effective plate separation increases.
1/CT = 1/C1 + 1/C2 + 1/C3 + … + 1/Cn
Parallel Capacitors
The total parallel capacitance is the sum of all
capacitors in parallel.
CT = C1 + C2 + C3 + … + Cn
RC Time Constant
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The buildup of charge across the plates occurs
in a predictable manner that is dependent on
the capacitance and the resistance in a circuit.
The time constant of a series RC circuit is a time
interval that equals the product of the
resistance and the capacitance.
 = RC
VC  VF  (Vi  VF )e  t /
Charging and Discharging
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The charging curve is an increasing exponential.
The discharging curve is an decreasing
exponential.
VC  VF  (Vi  VF )e  t /
Transient time
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It takes 5 time constants to change the
voltage by 99% (charging or discharging),
this is called the transient time.
Pulse response of C
The Capacitor Voltage
THE RC INTEGRATOR
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Charging and Discharging of a capacitor
NOTE : 1. The capacitor appears as a short to
an instantaneous change in current
and as an open to dc.
2. The voltage across the capacitor
cannot change instantaneously; it can change
only exponentially.
SINGLE-PULSE RESPONSE OF RC
INTEGRATORS
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Two conditions of pulse
response must be
considered:
 When the input pulse
width (tw ) is greater
than five time
constants
 (tw  5
)
Insert pic. 15-5
SINGLE-PULSE RESPONSE OF RC
INTEGRATORS
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When the input pulse width is less than
five time constants ( tw  5 )
Insert pic. 15-6
Insert pic. 15-7
REPETITIVE-PULSE RESPONSE
OF RC INTEGRATORS
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If a periodic pulse waveform is applied to an
RC integrator, the output waveshape depends
on the relationship of the circuit time
constant and the frequency of the input
pulse.
Insert pic. 15-12
When a capacitor does not fully
charge and discharge
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When the pulse width and the time
between pulses are shorter than 5 time
constant the capacitor will not
completely charge or discharge. (exp(-1)=0.3679)
Figure 15-15
Steady-state response
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When a steady-state condition is
reached (t > 5 ), the output voltage
build up to a constant average value
Figure 15-16
The effect of an increase in time
constant
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As the time constant is
increased, the capacitor
charges less during a
pulse and discharges
less between pulses. If
it becomes extremely
long compared to the
pulse width, the output
voltage approaches a
constant dc voltage.
Figure 15-18
SINGLE-PULSE REPONSE OF RC
DIFFERENTIATORS
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The output voltage is taken across the
resistor.
The capacitor charges exponentially at
a rate depending on the RC time
constant.
Figure 15-21
Pulse Response of the differentiator
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The response to the rising pulse edge
The response between the rising and
the falling edges
The response to the falling pulse edge
Pulse Response of the differentiator
Summary of RC Differentiator
Response to a Single Pulse
Figure 15-23
REPETITIVE-PULSE RESPONSE OF
RC DIFFERENTIATORS
Figure 15-28
Analysis of a Repetitive waveform
Figure 15-30-31
Capacitor Applications
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Capacitors are used in timing circuits to
generate time delays, based on the RC time
constant.
Dynamic memories used in computers are
simply very tiny capacitors used as a storage
element.
Pulse signal to DC converter
RC Circuit Application
RC time delay
RC-DC converter
C in AC circuit
(Sine wave)
Capacitive Reactance, XC
The relationship between capacitive
reactance, capacitance and frequency
is:
XC = 1/(2 f C)
where: XC is in ohms ()
f is in hertz (Hz)
C is in farads (F)
Capacitive Reactance, XC
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Capacitive reactance (XC) is the opposition to
sinusoidal current, expressed in ohms.
The rate of change of voltage is directly
related to frequency.
As the frequency increases, the rate of
change of voltage increases, and thus current
( i ) increases.
An increase in i means that there is less
opposition to current (XC is less). XC is
inversely proportional to i and to frequency.
Capacitors in ac Circuits
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The instantaneous capacitor current is equal
to the capacitance times the instantaneous
rate of change of the voltage across the
capacitor.
This rate of change is a maximum positive
when the rising sine wave crosses zero.
This rate of change is a maximum negative
when the falling sine wave crosses zero.
The rate of change is zero at the maximum
and minimum of the sine wave.
Analysis of Capacitive ac
Circuit
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The current leads the voltage by 90 in a
purely capacitive ac circuit.
The resulting current can be expressed in
polar form as I90 or in rectangular form as
jI.
Power in a Capacitor
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Energy is stored by the capacitor during a portion of
the voltage cycle; then the stored energy is returned
to the source during another portion of the cycle.
Instantaneous power is the product of v and i .
True power (Ptrue) is zero, since no energy is
consumed by the capacitor.
The rate at which a capacitor stores or returns
energy is called reactive power (Pr); units: (VAR).
Pr=VrmsIrms ; Pr = V2rms / Xc ; Pr = I2rms Xc
Capacitor Applications
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Capacitors are used for filtering in power supplies.
Since capacitors do not pass dc, they are used for dc blocking
and ac coupling.
For power line decoupling, capacitors are connected between the
dc supply and ground, to suppress unwanted voltage spikes that
occur on the dc supply voltage due to fast switching.
Capacitors are used to bypass an ac voltage around a resistor
without affecting the dc resistance.
Capacitors are used in filters, to select one ac signal with a
certain specified frequency from a wide range of signals with
many different frequencies. For example, the selection of one
radio station and rejecting the others.
Filtering in Power supply
DC Blocking or AC coupling
Summary
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Capacitance is directly proportional to the
plate area and inversely proportional to the
plate separation.
Dielectric constant is an indication of the
ability of a material to establish an electric
field.
Dielectric strength is one factor that
determines the breakdown voltage of a
capacitor.
A capacitor blocks constant dc.
Summary
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A capacitor is composed of two parallel
conducting plates separated by a dielectric
insulator.
Energy is stored by a capacitor in the electric
field between the plates.
One farad is the amount of capacitance when
one coulomb of charge is stored with one volt
across the plates.
Summary
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The time constant for a series RC circuit is
the resistance times the capacitance.
In an RC circuit, the voltage and current in a
charging or discharging capacitor make a
63% change during each time-constant
interval.
5 time constants are required of a capacitor
to fully charge or to discharge fully. This is
called the transient time.
Charging and discharging are exponential
curves.
Summary
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Total series capacitance is less than that of
the smallest capacitor in series.
Capacitance adds in parallel.
Current leads voltage by 90 in a capacitor.
XC is inversely proportional to frequency and
capacitance.
The true power in a capacitor is zero; that is,
there is no energy loss in an ideal capacitor.