Download Chapter 5 - Capacitors & Inductors (PowerPoint Format)

Chapter 5 – Capacitance & Inductance
• Capacitor, also called electrical condenser, device for storing
an electrical charge.
• In its simplest form a capacitor is two metal plates separated by
a non-conducting layer called the dielectric. Air, mica,
ceramics, paper, oil, and vacuums are used as dielectrics.
• When one plate is charged with electricity, the other plate
acquires an opposite charge. The amount of electric charge a
capacitor can hold is its capacitance.
• Because capacitors conduct direct current for only an instant,
they are used to prevent direct current from entering parts of
electric circuits. Large capacitors make it possible for power
lines to transmit more power.
(http://encarta.msn.com)
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ENGR201 Capacitance & Inductance
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Capacitance – Diagram and Symbol
Conductive
plates
Dielectric
C
Schematic Symbol
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ENGR201 Capacitance & Inductance
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Capacitor Types
•
•
•
•
There are many different styles of capacitors, generally
characterized by the type of dielectric material used and the
application for which the capacitor is intended:
General-purpose capacitors have dielectrics made of paper
impregnated with oil or wax, mylar, polyestyrene, glass, or
ceramic materials.
Mica, glass, ceramic dielectrics are often used in high-frequency
applications.
Aluminum electrolytic capacitors use aluminum plates separated
by a moistened borax paste and can provide relatively high
values of capacitance in small volumes. These capacitors are
often used in power supplies and motor-starting applications.
Tantalum electrolytic capacitors have lower losses and more
stable values (i.e. with respect to temperature) than aluminum
electrolytics.
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ENGR201 Capacitance & Inductance
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Capacitor Pictures
http://www.seacorinc.com
(Also see Figure 5.2)
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Capacitor Characteristics
εA
C
d
•
•
•
•
•
C is measured in Farads (F)
A is the area (in m2)of the parallel plates
d is the distance (in m) between the plates
 is permitivity of the dielectric
(0 is the permitivity of free space, and 0 =8.8510-12F/m
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Capacitor Characteristics
Two charged bodies separated by distance r will exert a force on
each other, attractive or repulsive depending on whether the
bodies have the opposite or same charge. This force is can be
written:
Q1Q 2
F
2
4πεr
Since the force is inversely proportional to  the permitivity of
the medium, the smaller the permitivity, the greater the force.
Therefore, materials with a high permitivity do not support a
strong (electrostatic) force field.
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Capacitor Characteristics
One farad is a large value. To illustrate, calculate the plate area
needed to construct a 1F capacitor if:
• the plates are separated by 0.1mm (1.010-4m) and
• using air as a dielectric ( = 0 = 8.8510-12F/m)
ε0A
C
d
dC
1104 m 1F
7
2
A


1.73

10
m
ε 0 8.85 1012 F / m
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ENGR201 Capacitance & Inductance
(2792 acres!)
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Capacitor Characteristics
• Typically, capacitors are measured in microfarads (F or uF),
nanofarads (nF) or picofarads (pf).
• Capacitance in a circuit may be deliberate (we insert a
capacitor) or unintentional and simply due to the way in which
a circuit is constructed (two long parallel conductors separated
by air or plastic).
• This unintentional capacitance is called stray capacitance, and
it may affect the operation of a circuit significantly.
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Capacitor Operation
The charge differential between the
plates of a capacitor stores energy.
+
-
-
-
• An electrostatic force field is created between the plates.
• The current which flows in the conductors to the capacitor does
not flow through the dielectric. Rather, the electrostatic field
(force field) set up by the differential charge causes charge
(electrons) to be removed from one plate as charge (electrons)
is deposited on the other.
• There appears to be a continuous flow of current through the
capacitor.
• Our primary interest is the terminal characteristics of a
capacitor, which can be described with the following equations.
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Capacitor Equations
The charge stored in a capacitor is proportional to the
voltage across the capacitor.
q(t )  Cv (t )
c
C
ic(t)
+
vc(t)
-
The current through a capacitor is
proportional to the rate of change of
the voltage
dv (t )
ic (t )  C
c
dt
Manipulating and integrating the previous equation allows us
to express the voltage in terms of the current:
t
t
1
1
vc (t ) 
ic ( x)dx   ic ( x)dx vc (t0 )

C t 
C t t0
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Capacitor Equations
C
q(t )  Cvc (t )
ic(t)
+
vc(t)
-
dvc (t )
ic (t )  C
dt
t
t
1
1
vc (t ) 
ic ( x)dx   ic ( x)dx vc (t0 )

C t 
C t t0
The change in the voltage across a capacitor is proportional to
the area under the current (which is the amount of charge
added to or subtracted from the capacitor).
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Capacitor Power & Energy Equations
Since p(t) = v(t) i(t) 
C
ic(t)
+
vc(t)
-
Energy, w(t), is the integral of
p(t), which yields:
 dvc (t ) 
p (t )  vc (t ) C
dt 

1 2
q 2 (t )
wc (t )  Cvc (t ) 
2
2C
Unlike a resistor, which always absorbs power, a capacitor may
absorb or deliver power (store energy as we store charge and
increase the strength of the electrostatic field and release energy
as we remove charge and the electrostatic field collapses).
That is, p(t) may be positive or negative for a capacitor.
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Inductance
• Like the capacitor, an inductor is a circuit element capable of
storing energy.
• Generally, an inductor is made by forming coils of wire around a
core. The core may be air, nonmagnetic material, or magnetic
material such as iron.
Coils
Core
L
Magnetic
Flux Lines
Schematic Symbol
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Inductance
• Current flowing through the wire creates an electromagnetic
(force) field around the wire. The strength of the field is affected
by the number of coils, the amount of current, and the magnetic
properties of the core.
• Inductance may occur in a circuit unintentionally due to the way
the circuit is constructed. Such inductance is called stray
inductance.
Coils
Core
L
Magnetic
Flux Lines
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Inductance Equations
iL(t)
+
vL(t)
-
The voltage across an inductor is
proportional to the rate of change of
inductor current.
L
di L (t)
v L (t)  L
dt
Manipulating and integrating the previous equation allows us
to express the current in terms of the voltage:
t
t
1
1
i L (t)   v L (x)dx   v L (x)dx i L (t 0 )
L x 
L x t 0
The change in current through an inductor is proportional to the area
under the inductor voltage.
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Inductance Power Equations
Since p(t) = v(t) i(t) 
 diL (t ) 
p (t )   L
iL (t )

dt 

iL(t)
+
vL(t)
-
L
Energy, w(t), is the integral of p(t),
which yields:
1 2
wc (t )  LiL (t )
2
Unlike a resistor, which always absorbs power, an inductor may
absorb or deliver power (store energy as we increase the strength of
the electromagnetic field or release energy as the field collapses).
That is, p(t) may be positive or negative for an inductor.
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Capacitors in Parallel
ix(t)
+ i1(t)
vx(t) C1
-
ix(t)
i2(t)
C2
in(t)
Cn
+
= v (t)
Ceq
x
-
ix (t )  Ceq
dvx (t )
dt
Since all the capacitors are in parallel they have a common
voltage, vx(t).
i (t )  i (t )  i (t )   i (t )
x
1
2
n
dvx (t )
dvx (t )
dvx (t )
 C2
  Cn
Applying KCL yields: ix (t )  C1
dt
dt
dt
dvx (t )
ix (t )   C1  C2   Cn 
dt
In order for these two circuits to be equivalent we conclude:
Ceq  C1  C2 
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 Cn (Capacitors in parallel act like resistors in
series.)
ENGR201 Capacitance & Inductance
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ix(t)
C1
+
+
v1(t)
+-
Capacitors in Series
ix(t)
Since all the capacitors are in
series they have a common
current, ix(t).
+
C2 v (t) = vx(t) Ceq
2
vx(t)
t
+
1
Cn v (t)
vx (t ) 
ix (t )dt

n
Ceq 
-Applying KVL yields:
vx (t )  v1 (t )  v2 (t )   vn (t )
t
1
1
vx (t ) 
ix (t )dt 

C1 
C2
ix (t )   C11  C21 
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t
 i (t )dt 
x

1

Cn
t
 i (t )dt
x

t
 Cn1   ix (t )dt

ENGR201 Capacitance & Inductance
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Capacitors in Series
ix(t)
C1
+
C2
vx(t)
Cn
ix (t )   C11  C21 
+
ix(t)
v1(t)
+
+v2(t) = vx(t)Ceq
+
vn(t)
-
t
 Cn1   ix (t )dt

Ceq1  C11  C 21 
 C n1
-
Capacitors in series act like resistors in parallel – we add
the reciprocals of the individual values.
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Inductors in Series and Parallel
Using similar reasoning that was used for capacitors, we can
find equivalents for inductors in series or in parallel.
If n inductors are connected in series, they can be replaced
by an equivalent inductance Leq, where
Leq  L1  L2 
 Ln
(Inductors in series act like resistors
in series.)
If n inductors are connected in parallel , they can be replaced
by an equivalent inductance Leq, where
Leq1  L1 1  L21 
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 Ln1
(Inductors in parallel act
like resistors in parallel.)
ENGR201 Capacitance & Inductance
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