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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

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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

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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

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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

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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

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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.

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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.

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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).

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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.

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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.