Electromagnetism Lecture#12-13 MUHAMMAD MATEEN YAQOOB THE
UNIVERSITY OF LAHORE SARGODHA CAMPUS
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THE SUPERPOSITION THEOREM Some circuits require more than one
voltage or current source. For example, most amplifiers operate
with two voltage sources: an ac and a dc source. Additionally, some
amplifiers require both a positive and a negative dc voltage source
for proper operation. When multiple sources are used in a circuit,
the superposition theorem provides a method for analysis. The
superposition method is a way to determine currents in a circuit
with multiple sources by leaving one source at a time and replacing
the other sources by their internal resistances. Recall that an
ideal voltage source has a zero internal resistance and an ideal
current source has infinite internal resistance. All sources will
be treated as ideal in order to simplify the coverage. MATEEN
YAQOOB DEPARTMENT OF COMPUTER SCIENCE
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THE SUPERPOSITION THEOREM A general statement of the
superposition theorem The current in any given branch of a
multiple-source circuit can be found by determining the currents in
that particular branch produced by each source acting alone, with
all other sources replaced by their internal resistances. The total
current in the branch is the algebraic sum of the individual
currents in that branch. MATEEN YAQOOB DEPARTMENT OF COMPUTER
SCIENCE
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THE SUPERPOSITION THEOREM The steps in applying the
superposition method are as follows: Step 1. Leave one voltage (or
current) source at a time in the circuit and replace each of the
other voltage (or current) sources with its internal resistance.
For ideal sources a short represents zero internal resistance and
an open represents infinite internal resistance. Step 2. Determine
the particular current (or voltage) that you want just as if there
were only one source in the circuit. Step 3. Take the next source
in the circuit and repeat Steps 1 and 2. Do this for each source.
MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE
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THE SUPERPOSITION THEOREM Step 4. To find the actual current in
a given branch, algebraically sum the currents due to each
individual source. (If the currents are in the same direction, they
are added. If the currents are in opposite directions, they are
subtracted with the direction of the resulting current the same as
the larger of the original quantities.) Once you find the current,
you can determine the voltage using Ohm's law MATEEN YAQOOB
DEPARTMENT OF COMPUTER SCIENCE
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The approach to superposition is demonstrated in the following
Figure for a series-parallel circuit with two ideal voltage
sources. MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE
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Circuit Excitation Source-free circuits (free of independent
sources) DC Source excitation (independent sources) MATEEN YAQOOB
DEPARTMENT OF COMPUTER SCIENCE
Slide 9
FIRST-ORDER CIRCUIT Three passive elements (resistors,
capacitors, and inductors) individually, Circuits having various
combinations of two or three of the passive elements. RC and RL
circuits. Analysis of RC and RL circuits by applying Kirchhoffs
laws. The differential equations resulting from analyzing RC and RL
circuits are of the first order. Hence, the circuits are
collectively known as first-order circuits. MATEEN YAQOOB
DEPARTMENT OF COMPUTER SCIENCE
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THE SOURCE-FREE RC CIRCUIT MATEEN YAQOOB DEPARTMENT OF COMPUTER
SCIENCE
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The natural response of a circuit refers to the behavior (in
terms of voltages and currents) of the circuit itself, with no
external sources of excitation. The voltage response of the RC
circuit. MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE
Slide 13
The time constant of a circuit is the time required for the
response to decay by a factor of 1/e or 36.8 percent of its initial
value. The voltage response of the RC circuit. MATEEN YAQOOB
DEPARTMENT OF COMPUTER SCIENCE
Slide 14
The Key to Working with a Source - free RC Circuit is Findin g
: 1. The initial voltage v(0) = V 0 across the capacitor. 2. The
time constant . v C (t) = v(t) = v(0)e t/ other variables Capacitor
current i C Resistor voltage v R Resistor current i R can be
determined. MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE
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THE SOURCE FREE RL CIRCUIT MATEEN YAQOOB DEPARTMENT OF COMPUTER
SCIENCE
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The smaller the time constant of a circuit, the faster the rate
of decay of the response. The larger the time constant, the slower
the rate of decay of the response. At any rate, the response decays
to less than 1 percent of its initial value (i.e., reaches steady
state) after 5. The current response of the RL circuit MATEEN
YAQOOB DEPARTMENT OF COMPUTER SCIENCE
Slide 18
The Key to Working with a Source - free RL Circuit is to Find :
1. The initial current i(0) = I 0 through the inductor. 2. The time
constant of the circuit. i L (t) =i(t) = i(0)e t/. other variables
Inductor voltage v L Resistor voltage v R Resistor current i R can
be obtained. MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE
Slide 19
SECOND-ORDER CIRCUITS A second-order circuit is characterized
by a second-order differential equation. It consists of resistors
and the equivalent of two energy storage elements.
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Circuit Excitation Two ways of excitation 1. By initial
conditions of the storage elements (These source free circuits will
give natural responses as expected) 2.By step inputs: Circuits are
excited by independentsources. These circuits will give both the
natural response and the forced response
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Finding Initial and Final Values We begin by learning how to
obtain the initial conditions for the circuit variables and their
derivatives, as this is crucial to analyze second order circuits.
Perhaps the major problem students face in handling second-order
circuits is finding the initial and final conditions on circuit
variables. Students are usually comfortable getting the initial and
final values of v and i but often have difficulty finding the
initial values of their derivatives: dv/dt and di/dt.
Slide 22
There are two key points to keep in mind in determining the
initial conditions. Firstas always in circuit analysiswe must
carefully handle the polarity of voltage v(t) across the capacitor
and the direction of the current i(t) through the inductor. Keep in
mind that v and i are defined strictly according to the passive
sign convention. One should carefully observe how these are defined
and apply them accordingly.
Slide 23
Second, keep in mind that the capacitor voltage is always
continuous so that v(0 + ) = v(0 ) (a) and the inductor current is
always continuous so that i(0 + ) = i(0 ) (b) where t = 0 denotes
the time just before a switching event and t = 0 + is the time just
after the switching event, assuming that the switching event takes
place at t = 0. Therefore, in finding initial conditions, we first
focus on those variables that cannot change abruptly, capacitor
voltage and inductor current, by applying Eq. (a & b).
Slide 24
Applications of RC and RL circuits RC and RL Circuits have many
applications in the field of Electrical, Electronics,
Communication, Computer Engineering, Signal Processing and so on
These circuits have many practical applications, some of their
major applications are listed below: 1.Amplifiers 2.Oscillators
3.Filters 4.Switching Regulator 5.Tuned Amplifiers 6.Radio
Transmitter and Receiver 7.TV Receiver MATEEN YAQOOB DEPARTMENT OF
COMPUTER SCIENCE
Slide 25
Categories of Solids There are three categories of solids,
based on their conducting properties: conductors semiconductors
insulators MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE
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Electrical Resistivity and Conductivity of Selected Materials
at 293 K MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE
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Reviewing the previous table reveals that: The electrical
conductivity at room temperature is quite different for each of
these three kinds of solids Metals and alloys have the highest
conductivities followed by semiconductors and then by insulators
MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE
Slide 28
Band Theory of Solids In order to account for decreasing
resistivity with increasing temperature as well as other properties
of semiconductors, a new theory known as the band theory is
introduced. The essential feature of the band theory is that the
allowed energy states for electrons are nearly continuous over
certain ranges, called energy bands, with forbidden energy gaps
between the bands. MATEEN YAQOOB DEPARTMENT OF COMPUTER
SCIENCE
Slide 29
Valence band: Band occupied by the outermost electrons
Conduction: Lowest band with unoccupied states Conductor: Valence
band partially filled (half full) Cu. or Conduction band overlaps
the valence band MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE
Slide 30
Resistivity vs. Temperature Figure: (a) Resistivity versus
temperature for a typical conductor. Notice the linear rise in
resistivity with increasing temperature at all but very low
temperatures. (b) Resistivity versus temperature for a typical
conductor at very low temperatures. Notice that the curve flattens
and approaches a nonzero resistance as T 0. (c) Resistivity versus
temperature for a typical semiconductor. The resistivity increases
dramatically as T 0. MATEEN YAQOOB DEPARTMENT OF COMPUTER
SCIENCE
Slide 31
Band Theory and Conductivity Band theory helps us understand
what makes a conductor, insulator, or semiconductor. 1) Good
conductors like copper can be understood using the free electron 2)
It is also possible to make a conductor using a material with its
highest band filled, in which case no electron in that band can be
considered free. 3) If this filled band overlaps with the next
higher band, however (so that effectively there is no gap between
these two bands) then an applied electric field can make an
electron from the filled band jump to the higher level. This allows
conduction to take place, although typically with slightly higher
resistance than in normal metals. Such materials are known as
semimetals. MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE
Slide 32
Valence and Conduction Bands The band structures of insulators
and semiconductors resemble each other qualitatively. Normally
there exists in both insulators and semiconductors a filled energy
band (referred to as the valence band) separated from the next
higher band (referred to as the conduction band) by an energy gap.
If this gap is at least several electron volts, the material is an
insulator. It is too difficult for an applied field to overcome
that large an energy gap, and thermal excitations lack the energy
to promote sufficient numbers of electrons to the conduction band.
MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE
Slide 33
For energy gaps smaller than about 1 electron volt, it is
possible for enough electrons to be excited thermally into the
conduction band, so that an applied electric field can produce a
modest current. The result is a semiconductor. Smaller energy gaps
create semiconductors MATEEN YAQOOB DEPARTMENT OF COMPUTER
SCIENCE
Slide 34
Temperature and Resistivity When the temperature is increased
from T = 0, more and more atoms are found in excited states. The
increased number of electrons in excited states explains the
temperature dependence of the resistivity of semiconductors. Only
those electrons that have jumped from the valence band to the
conduction band are available to participate in the conduction
process in a semiconductor. More and more electrons are found in
the conduction band as the temperature is increased, and the
resistivity of the semiconductor therefore decreases. MATEEN YAQOOB
DEPARTMENT OF COMPUTER SCIENCE
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Holes and Intrinsic Semiconductors When electrons move into the
conduction band, they leave behind vacancies in the valence band.
These vacancies are called holes. Because holes represent the
absence of negative charges, it is useful to think of them as
positive charges. Whereas the electrons move in a direction
opposite to the applied electric field, the holes move in the
direction of the electric field. A semiconductor in which there is
a balance between the number of electrons in the conduction band
and the number of holes in the valence band is called an intrinsic
semiconductor. Examples of intrinsic semiconductors include pure
carbon and germanium. MATEEN YAQOOB DEPARTMENT OF COMPUTER
SCIENCE
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Impurity Semiconductor It is possible to fine-tune a
semiconductors properties by adding a small amount of another
material, called a dopant, to the semiconductor creating what is a
called an impurity semiconductor. As an example, silicon has four
electrons in its outermost shell (this corresponds to the valence
band) and arsenic has five. Thus while four of arsenics outer-shell
electrons participate in covalent bonding with its nearest
neighbors (just as another silicon atom would), the fifth electron
is very weakly bound. It takes only about 0.05 eV to move this
extra electron into the conduction band. The effect is that adding
only a small amount of arsenic to silicon greatly increases the
electrical conductivity. MATEEN YAQOOB DEPARTMENT OF COMPUTER
SCIENCE
Slide 39
Extra weakly bound valence electron from As lies in an energy
level close to the empty conduction band. These levels donate
electrons to the conduction band. MATEEN YAQOOB DEPARTMENT OF
COMPUTER SCIENCE
Slide 40
n-type Semiconductor The addition of arsenic to silicon creates
what is known as an n-type semiconductor (n for negative), because
it is the electrons close to the conduction band that will
eventually carry electrical current. The new arsenic energy levels
just below the conduction band are called donor levels because an
electron there is easily donated to the conduction band. MATEEN
YAQOOB DEPARTMENT OF COMPUTER SCIENCE
Slide 41
Ga has only three electrons and creates a hole in one of the
bonds. As electrons move into the hole the hole moves driving
electric current Impurity creates empty energy levels just above
the filled valence band MATEEN YAQOOB DEPARTMENT OF COMPUTER
SCIENCE
Slide 42
Acceptor Levels Consider what happens when indium is added to
silicon. Indium has one less electron in its outer shell than
silicon. The result is one extra hole per indium atom. The
existence of these holes creates extra energy levels just above the
valence band, because it takes relatively little energy to move
another electron into a hole Those new indium levels are called
acceptor levels because they can easily accept an electron from the
valence band. Again, the result is an increased flow of current
(or, equivalently, lower electrical resistance) as the electrons
move to fill holes under an applied electric field It is always
easier to think in terms of the flow of positive charges (holes) in
the direction of the applied field, so we call this a p-type
semiconductor (p for positive). acceptor levels p-Type
semiconductors In addition to intrinsic and impurity
semiconductors, there are many compound semiconductors, which
consist of equal numbers of two kinds of atoms. MATEEN YAQOOB
DEPARTMENT OF COMPUTER SCIENCE
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At a pn junction holes diffuse from the p side MATEEN YAQOOB
DEPARTMENT OF COMPUTER SCIENCE
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pn junction Region depleted from mobile carriers Potential
barrier prevents further diffusion of holes and electrons. Zero
current for no external E field MATEEN YAQOOB DEPARTMENT OF
COMPUTER SCIENCE
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Intrinsic Semiconductor Elemental or pure semiconductors have
equal numbers of holes and electrons Depends on temperature, type,
and size. Compound Semiconductors can be formed from two (or more)
elements (e.g., GaAs) MATEEN YAQOOB DEPARTMENT OF COMPUTER
SCIENCE
Slide 46
Extrinsic Semiconductors A pure semiconductors where a small
amount of another element is added to replace atoms in the lattice
(doping). The aim is to produce an excess of either electrons
(n-type) or holes (p-type) Typical doping concentrations are one
part in ten million Doping must be uniform throughout the lattice
so that charges do not accumulate MATEEN YAQOOB DEPARTMENT OF
COMPUTER SCIENCE
Slide 47
N-Type and P-Type One valence electron too many (n-type)
Arsenic, antimony, bismuth, phosphorus One valence electron too few
(p-type) Aluminum, indium, gallium, boron MATEEN YAQOOB DEPARTMENT
OF COMPUTER SCIENCE
Slide 48
The PN Junction Diode Start with a P and N type material. Note
that there are excess negatives in the n-type and excess positives
in the p-type Merge the two some of the negatives migrate over to
the p-type, filling in the holes. The yellow region is called the
depletion zone. More positive than rest of N More negative than
rest of P MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE
Slide 49
Biasing the Junction Apply a voltage as indicated. The free
charge carriers (negative charges in the N material and positive
charges in the P material) are attracted to the ends of the
crystal. No charge flows across the junction and the depletion zone
grows. This is called reverse bias. Switch polarity. Now the
negative charges are driven toward the junction in the N material
and the positive charges also are driven toward the junction in the
P material. The depletion zone shrinks and will disappear if the
voltage exceeds a threshold. This is called forward bias. MATEEN
YAQOOB DEPARTMENT OF COMPUTER SCIENCE
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Diode Circuit Symbols Reverse BiasForward Bias N material
(cathode) P material (anode) MATEEN YAQOOB DEPARTMENT OF COMPUTER
SCIENCE
Slide 51
Types of Diodes Rectifier Diode Used in power supplies Signal
Diode Used in switches, detectors, mixers, etc. Zener Diode Voltage
regulation operated reverse bias in the avalanche region Reference
Diode Used like zener for voltage regulation MATEEN YAQOOB
DEPARTMENT OF COMPUTER SCIENCE
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Rectification Conversion of ac to dc. Many devices
(transistors) are unidirectional current devices DC required for
proper operation. MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE
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Half Wave Rectifier MATEEN YAQOOB DEPARTMENT OF COMPUTER
SCIENCE
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Full Wave Rectifier MATEEN YAQOOB DEPARTMENT OF COMPUTER
SCIENCE
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Bridge Rectifier MATEEN YAQOOB DEPARTMENT OF COMPUTER
SCIENCE
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LEDs When forward biased, electrons from the N-type material
may recombine with holes in the P-type material. System energy is
decreased Excess energy emitted as light Indium gallium nitride
(InGaN) semiconductors have been used to make colored LEDs Stop
lights Progress toward white LEDs is promising MATEEN YAQOOB
DEPARTMENT OF COMPUTER SCIENCE
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Laser Diodes N material P material Highly Reflective Mirror
Partially Reflective Mirror Used in forward bias. Electrons move
into depletion zone and recombine with holes, producing light (like
an LED). More electron-hole recombinations can be stimulated by
this photon, producing more photons at the same wavelength. The
mirrors reflect the photons back and forth through the depletion
zone, stimulating more photon at each pass. Eventually, the beam
passes out of the right hand mirror. MATEEN YAQOOB DEPARTMENT OF
COMPUTER SCIENCE
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Laser Diode Application Used as CD and DVD detector Laser Diode
Photodiode Also used as bar code readers, laser pointers, fiber
optics, etc. MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE
Slide 59
LCD Liquid Crystal Display State between solid and liquid
Requires only a little heat to change material into a liquid Used
as thermometers or mood rings Electric currents are used to orient
the crystals in predictable ways The liquid crystals polarize the
light (either internal light source or external) to give light and
dark areas on the display MATEEN YAQOOB DEPARTMENT OF COMPUTER
SCIENCE
Slide 60
Field Effect Transistors (FET) The three terminals of the FET
are known as the drain, source, and gate, and these correspond to
the collector, emitter, and base, respectively, of a bipolar
transistor. Figure: (a) A schematic of a FET. The two gate regions
are connected internally. (b) The circuit symbol for the FET,
assuming the source-to-drain channel is of n-type material and the
gate is p-type. If the channel is p-type and the gate n-type, then
the arrow is reversed. (c) An amplifier circuit containing a
FET.
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Integrated Circuits The most important use of all these
semiconductor devices today is not in discrete components, but
rather in integrated circuits called chips. Some integrated
circuits contain a million or more components such as resistors,
capacitors, and transistors. Two benefits: miniaturization and
processing speed.
Slide 62
Moores Law and Computing Power Figure: Moores law, showing the
progress in computing power over a 30-year span, illustrated here
with Intel chip names. The Pentium 4 contains over 50 million
transistors.
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Nanotechnology Nanotechnology is generally defined as the
scientific study and manufacture of materials on a submicron scale.
These scales range from single atoms (on the order of.1 nm up to 1
micron (1000 nm). This technology has applications in engineering,
chemistry, and the life sciences and, as such, is
interdisciplinary.