Chapter 9 Capacitors
MECH1100
Topics
Basic Capacitor
Types of Capacitors
Parallel Capacitors
Capacitors in DC Circuits
Capacitors in AC Circuits
Capacitor Applications
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Chapter 9 Capacitors
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The Basic Capacitor
Capacitor:
Electrical device composed of two parallel conductive
plates separated by an insulating material (dielectric).
The ability to store charge is the definition of capacitance.
Conductors
Dielectric
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Chapter 9 Capacitors
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The Basic Capacitor
VVSS
The charging
process…
Leads

Initially
Source
Fully
Charging
charged
removed
uncharged



++
+++
+++
+++
++
+++

 ++
+
++
+
++
+
+
+
AA +
A

 +
Dielectric


++
Plates


+
+

 

+

+  Electrons




+ B
BB




A capacitor with stored charge can act as a temporary battery.
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Chapter 9 Capacitors
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Capacitance
Capacitance is the ratio of charge to voltage
Q
C
V
C = Capacitance (ability to store charge)
Q = Charge (coulombs)
V = Volts
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Capacitance
Capacitance (farad)
Q
C
V
Farad
Amount of capacitance when one coulomb
of charge is stored with one volt across the
plates.
The amount of charge on a capacitor is determined
by the size of the capacitor (C) and the voltage (V).
Q  CV
If a 22 mF capacitor is connected to
a 10 V source, the charge is 220 mC
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Energy Storage
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
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Chapter 9 Capacitors
Electrical Characteristics
MECH1100
Voltage rating – amount of voltage that can be
withstood across the plates
Breakdown or working voltage - voltage at which
physical damage is done to the capacitor.
Dielectric strength – A measure of a compounds ability to
serve as an insulator. V/mil (.001” or 2.54 * 10-5 meter)
See Table 9-2 – Page 392
Temperature coefficient – indicates the amount and
direction of change in capacitance value with temperature
Leakage – amount of charge that leaks through
the dielectric
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Physical Characteristics
Plate area – Capacitance is
directly proportional to the
plate area.
Plate separation – Capacitance
is inversely proportional to the
distance between the plates.
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Dielectric constant (or relative
permittivity) – measure of a
material’s ability to establish an
electric field.
Capacitance is directly
proportional to the dielectric
constant.
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Capacitor types
Mica - very hard, heat resistant mineral
Mica capacitors are small with high working
voltage. er (5)
Foil
Mic a
Foil
Mic a
Foil
Mic a
Foil
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Chapter 9 Capacitors
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Capacitor types
Ceramic disk
Ceramic disks are small non-polarized capacitors
They have relatively high capacitance due to high er (1200)
Lead wire soldered
to silver elec trode
Solder
Ceram ic
dielec tric
Dipped phenolic c oating
Silv er elec trodes deposited on
top and bottom of c eram ic disk
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Chapter 9 Capacitors
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Capacitor types
Plastic Film
Plastic film capacitors are small and non-polarized.
They have relatively high capacitance due to larger
plate area. er (depends on the film used)
High-purity
foil elec trodes
Plastic film
dielec tric
Outer wrap of
polyester film
Capac itor sec tion
(alternate strips of
film dielec tric and
Lead wire
foil elec trodes)
Solder c oated end
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Chapter 9 Capacitors
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Capacitor types
Electrolytic (two types)
• Electrolytic capacitors have very high capacitance
1. not as precise as other types and
2. tend to have more leakage current.
• Electrolytic capacitors are the only ones polarized.
+
_
Aluminum electrolytic
Tantalum electrolytic
Symbol for any electrolytic capacitor
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Chapter 9 Capacitors
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Chapter 9 Capacitors
Power Factor Correction
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Defibrillator Capacitor can
deliver over 500 joules of
energy
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Capacitor types
Variable
• Variable capacitors:
1. have small capacitance values and
2. are adjusted manually
3. called Trimmers, padders or tuning capacitors
because they are used to make very fine
adjustments in a circuit.
• Varactor diode - A semiconductor that exhibits
capacitive characteristic; it is adjusted with an
electrical signal.
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Capacitance meter
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Chapter 9 Capacitors
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Capacitor labeling
Small capacitors values are frequently stamped on them
such as .001 or .01, which have implied units of
microfarads.
Electrolytic capacitors have larger values, so are
read as mF.
+ ++ +
47VTTMFVTT
Usually stamped as mF, but some older ones
may be shown as MF or MMF).
. 022
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Chapter 9 Capacitors
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Capacitors in series
1. Divide Voltage across them in proportion
to their capacitance
2. Current is same at all points while
charging; then goes to zero
Q
I
t
Q
V
C
Q  Q  Q  Qn
T
1
2
V  V  V  ...  V
S
1
2
N
Q is charge and C is capacitance
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I
Q
V
C
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Series capacitors
When connected in series, the total capacitance is
smaller than the smallest one.
Capacitance for any number of capacitors in series
CT 
1
1 1
1
1
   ... 
C1 C 2 C 3
CN
Capacitance for two capacitors in series
CC
C 
C C
1
2
T
1
2
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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
1000pF
C2
800 pF
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Chapter 9 Capacitors
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Voltage in Series Capacitors
• Voltage depends on capacitance value
• Smallest value capacitor will have the
largest voltage across it
• Largest value capacitor will have the
smallest voltage across it.
Q
V
C
C

V 
C
T
X
X
V


S
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Parallel capacitors
1.When capacitors are connected in parallel, the total
capacitance is the sum of the individual capacitors.
2.The voltage across capacitors is the same in each parallel
branch.
3.The total charge stored by each capacitor is proportional to
its capacitance. QT=Q1+Q2+Q3+…+QN
Q
I
t
VC2 = VC1
VC2 = VC1
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Parallel capacitors
The general equation for capacitors in parallel is
CT  C1  C2  C3  ...Cn
If a 0.001 mF capacitor is
connected in parallel with
an 800 pF capacitor, the
total capacitance is
1800 pF
C1
C2
0.001 µF
800 pF
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Capacitors in DC Circuits
A capacitor charges when connected to a DC voltage.
When fully charged, there is no current flow.
Except for leakage, a capacitor will remain charged for long
periods.
A capacitor appears as an “open” to a constant voltage.
A capacitor appears as a “short” to an instantaneous change
in voltage.
When discharging the flow of current is in the opposite
direction from the charging cycle.
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Chapter 9 Capacitors
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The RC time constant
RC Time Constant:
A fixed time interval that equals the product of the
resistance and the capacitance in a series capacitor
circuit.
  RC
τ  time(seconds)(tau)
R  resistance(ohms)
C  capacitance(farads)
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Chapter 9 Capacitors
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The RC time constant
When a capacitor is charged
through a series resistor and dc
source, the charging curve is
exponential.
R
Vfinal
0
t
(a) Capacitor c harging voltage
C
v
i
v


 I  ( I  I )e 
 V  1  e (
( instantaneuous)
 V  (V  V )e
F
i

Iinitial
t

F

t

( instantaneous )
F
i

(instantaneous)
F
F
t
RC
charging

from

0)
0
(b) Charging current
t
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The RC time constant
When a capacitor is discharged
through a resistor, the
discharge curve is also an
exponential. (Note that the
current is negative.)
Vinitial
t
0
(a) Capacitor disc harging voltage
R
Iinitial
C
v  Ve
i

t
RC
0
( discharging

from

0)
t
(b) Disc harging current
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Universal exponential curves
The general voltage formula is
v =VF + (Vi  VF)et/RC
VF = final value of voltage
Vi = initial value of voltage
v = instantaneous value of voltage
The final capacitor voltage is:
1. greater than the initial voltage when the capacitor is
charging, or
2. less than the initial voltage when it is discharging.
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Universal exponential curves
τ  RC
100%
95%
40%
37%
14%
Falling
(discharging)
exponential
5%
0
0
99%
Rising (charging)
exponential
63%
60%
20%
Any capacitor will
charge in 5 time
constants!!
98%
86%
80%
Percent of final value
Specific values for
current and voltage
can be read from a
universal curve. For
an RC circuit, the
time constant is
1
2%
2
3
4
Number of time constants
1%
5
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Chapter 9 Capacitors
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Universal exponential curves
Example
100%
τ  RC
20v
63%
60%
40%
37%
20%
14%
5%
0
0
99%
86%
80%
Percent of final value
How long will it
take to charge the
capacitor shown in
the circuit below to
16 volts? Use the
charging curve
shown.
95%
98%
1
2%
2
3
4
Number of time constants
1%
5
16 volts = 80% of final value
80% (16 v) of charge occurs at ~ 1.6 time constants
τ = (10*103Ω)*(.001x10-6F) = 10 x10-6 sec = 10µs
∴ Time to reach 16v = 1.6 x 10µs ~ 16 µs
Time to reach 17.2 volts?
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Capacitors in AC Circuits
When the source voltage is held at a constant amplitude
and the frequency is:
1. increased the amplitude of the current increases
which indicates a decrease in resistance to charging.
2. decreased the amplitude of the current decreases
which indicates an increase in resistance to charging
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Chapter 9 Capacitors
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Capacitive Reactance
The opposition to ac by a capacitor is known as
Capacitive reactance (XC). Unit is Ohms
Capacitive reactance is:
1. inversely proportional to frequency
2. inversely proportional to capacitance
1
XC 
2πfC
f – hertz
C - farads
The capacitive reactance of a 0.047 mF capacitor when a
frequency of 15 kHz is applied is 226 W
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Reactance for Series Capacitors
When capacitors are in series, the total capacitive
reactance is the sum of the individual reactance's.
X C(tot )  X C1  X C2  X C3    X Cn
Assume three 0.033 mF capacitors are in series with
a 2.5 kHz ac source. What is the total reactance?
The reactance of each capacitor is
XC 
1
1

 1.93 kW
2πfC 2π  2.5 kHz  0.033 μF 
X C(tot )  X C1  X C2  X C3
 1.93 kW  1.93 kW  1.93 kW 
5.79 kW
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Reactance for Parallel Capacitors
When capacitors are in parallel, the total reactance is the
reciprocal of the sum of the reciprocals of the individual
1
reactance's. X 
C(tot )
XC1 XC 2
1
1
1
1


  
XC ( tot) 
X C1 X C2 X C3
X Cn
XC1  XC 2
If the three 0.033 mF capacitors from the last example are
placed in parallel with the 2.5 kHz ac source, what is the
total reactance?
The reactance of each capacitor is 1.93 kW
X C(tot ) 
1
1


1
1
1
1
1
1


+
+
X C1 X C2 X C3 1.93 kW 1.93 kW 1.93 kW
643 W
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Ohms Law for Capacitors
The capacitive reactance of a capacitor is the same as a
resistor
V
I
Xc same as resistance
XC
Current and volts must be in the same units but it does matter
which
Calculate I
(i.e. rms, peak, p to p)
1
XC 
2πfC 7957.7Ω
Irms= 2vrms/7957.7Ω = 0.25mA
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Capacitive Voltage Divider
Two capacitors in series are used as a capacitive voltage
divider.
The capacitors split the output voltage:
1. in proportion to their reactance and
2. inversely proportional to their capacitance.
C1
 XCX 
VX  
VS
1000 pF
 XC ( tot) 
1.0 V
f = 33 kHz
 Ctot 
VX  
VS
 CX 
XCX is the reactance of the capacitor CX
XC(tot) – total capacitive reactance
VX – voltage across capacitor CX
C2
0.01 µF
Vout
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Capacitive Voltage Divider
What is the output voltage for the capacitive voltage
divider?
X C1 
XC2 
1
1

 4.82 kW
2πfC1 2π  33 kHz 1000 pF 
1
1

 482 W
2πfC2 2π  33 kHz  0.01 μF 
1.0 V
f = 33 kHz
C1
1000 pF
C2
V
0.01 µF out
X C (tot )  X C1  X C 2
 4.82 kW  482 W  5.30 kW
Vout
 XC2 
 482 W 

Vs  

1.0 V =
X

5.30
k
W


 C (tot ) 
91 mV
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Capacitive Voltage Divider
Two capacitors in series are commonly used as a capacitive voltage
divider. The capacitors split the output voltage in proportion to their
reactance (and inversely proportional to their capacitance).
What is the output voltage for the capacitive voltage divider?
1
1

 4.82 kW
2πfC1 2π  33 kHz 1000 pF 
1
1


 482 W
2πfC2 2π  33 kHz  0.01 μF 
X C1 
XC2
X C (tot )  X C1  X C 2
 4.82 kW  482 W  5.30 kW
Vout
 XC2 
 482 W 

Vs  

1.0 V = 91 mV
X

 5.30 kW 
 C (tot ) 
1.0 V
f = 33 kHz
C1
1000 pF
C2
Vout
0.01 µF
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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,
C(tot ) 
Vout
1000 pF 0.01 μF  909 pF
C1C2

C1  C2
1000 pF  0.01 μF
C 
 909 pF 
  (tot ) Vs  
1.0 V = 91 mV
C
0.01
μF


 2 
C1
1000 pF
1.0 V
f = 33 kHz
C2
Vout
0.01 µF
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Capacitive phase shift
When a sine wave
is applied to a
capacitor, there is a
phase shift between
voltage and current
such that current
always leads the
voltage by 90o.
VC
0
90o
I
0
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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.
Power is maximum
where the V and I
curves cross
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Power in a capacitor
1. Voltage and current are always 90o out of phase.
2. No true power is dissipated by a capacitor, because
stored energy is returned to the circuit.
3. The rate at which a capacitor stores or returns
energy is called reactive power.
4. The unit for reactive power is the VAR (voltampere reactive).
Pr  VrmsIrms
2
V rms
Pr 
XC
2
Pr  I rmsXC
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Rectifiers
Convert a sinusoidal wave (ac) to a pulsing dc wave
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Power supply filtering
Pulsing voltage and current are not usually desirable
for operating equipment
A capacitor adds a smoothing effect to the wave
by discharging on the down slope of the curve.
Rectifier
60 Hz ac
C
Load
resistance
The filter smoothes
the pulsating dc from
the rectifier.
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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 The unit of capacitance.
RC time A fixed time interval set by the R and C values,
constant that determine the time response of a series RC
circuit. It equals the product of the resistance
and the capacitance.
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Key Terms
Capacitive The opposition of a capacitor to sinusoidal
reactance current. The unit is the ohm.
Instantaneous The value of power in a circuit at a given
power (p) instant of time.
True power The power that is dissipated in a circuit
(Ptrue) usually in the form of heat.
Reactive The rate at which energy is alternately stored
power (Pr ) and returned to the source by a capacitor. The
unit is the VAR.
VAR The unit of reactive power.
(volt-ampere
reactive)
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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.
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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.
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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.
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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.
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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.
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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
100%
95%
c. greater than one time constant.
Percent of final value
b. exactly one time constant.
63%
60%
40%
37%
20%
14%
5%
0
0
99%
86%
80%
a. less than one time constant.
98%
1
2%
2
3
4
Number of time constants
1%
5
d. answer depends on the amount of voltage.
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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
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