Electromagnetism Lecture#12-13 MUHAMMAD MATEEN YAQOOB THE UNIVERSITY OF LAHORE SARGODHA CAMPUS

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Transcript of Electromagnetism Lecture#12-13 MUHAMMAD MATEEN YAQOOB THE UNIVERSITY OF LAHORE SARGODHA CAMPUS

  • Slide 1
  • Electromagnetism Lecture#12-13 MUHAMMAD MATEEN YAQOOB THE UNIVERSITY OF LAHORE SARGODHA CAMPUS
  • Slide 2
  • 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
  • Slide 3
  • 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
  • Slide 4
  • 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
  • Slide 5
  • 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
  • Slide 6
  • 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
  • Slide 10
  • THE SOURCE-FREE RC CIRCUIT MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE
  • Slide 11
  • Slide 12
  • 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
  • Slide 15
  • THE SOURCE FREE RL CIRCUIT MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE
  • Slide 16
  • Slide 17
  • 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.
  • Slide 20
  • 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
  • Slide 21
  • 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
  • Slide 26
  • Electrical Resistivity and Conductivity of Selected Materials at 293 K MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE
  • Slide 27
  • 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) Re