PHY202 PDF - Course Content

COURSE CONTENT Direct current (DC) is the unidirectional flow of electric charge. DC flows through a conductor such as a wire, it can also flow through semiconductors, even through a vacuum as in electron or ion beams. Direct current may be converted from an alternating current supply by use of a r...

COURSE CONTENT Direct current (DC) is the unidirectional flow of electric charge. DC flows through a conductor such as a wire, it can also flow through semiconductors, even through a vacuum as in electron or ion beams. Direct current may be converted from an alternating current supply by use of a rectifier. Direct current may be converted into alternating current via an inverter. Direct current has many uses such as charging of batteries, production of large power supplies for electronic systems such as motors, etc. An electrical device is represented by a circuit diagram or network constructed from series and parallel arrangements of two-terminal elements. The analysis of the circuit diagram predicts the performance of the actual device. A two-terminal element in general form is shown in Fig.1, with a single device represented by the rectangular symbol and two perfectly conducting leads ending at connecting points A and B. Active elements are voltage or current sources which are able to supply energy to the network. Resistors, inductors, and capacitors are passive elements which take energy from the sources and either convert it to another form or store it in an electric or magnetic field. Figure 1: Two-Terminal Element Generally, network analysis is any structured technique used to mathematically analyze a circuit (a “network” of interconnected components). Quite often the technician or engineer will encounter circuits containing multiple sources of power or component configurations that defy simplification by series/parallel analysis techniques. In those cases, the engineer will be forced to use other means such as circuital laws in analyzing such complex circuits. The equivalent resistance of a circuit (or network) between its any two points (or terminals) is given by that single resistance which can replace the entire given circuit between these two points. Serial Arrangement of Resistance The Fig. 2a shows two resistors connected in series between point A and point B. Part (b) shows three resistors in series, and part (c) shows four in series. Therefore, there is no limit of number of resistors that can make up the resistors in series. Figure 2: series connection of (a) two resistors (b) three resistors (c) four resistors When a voltage source is connected from point A to point B, the only way for current to get from one point to the other in any of the connections of Figure 2 is to go through each of the resistors. Therefore, a series circuit is defined as circuit that provides only one path for current between two points so that the current is the same through each series resistor. Likewise, serial arrangement can be in a form as represented in fig. 3a-e. Figure 3a-e: further serial arrangements of resistors. For any number of individual resistors connected in series, the total resistance is the sum of each of the individual values. 𝑅𝑇 = 𝑅1 + 𝑅2 + 𝑅3 + ⋯ + 𝑅𝑛 Each parallel path in a circuit is called a BRANCH. Two resistors connected in parallel are shown in Figure 4(a). The equivalent current distribution as shown in Fig(4b), the current out of the source (𝐼𝑇 ) divides when it gets to point B. 𝐼1 passes through R1 and 𝐼2 passes through R2. The two currents come back together at point A. Figure 4 (a) Parallel arrangement of resistors (b) current distribution in the circuit. A parallel circuit are identified if there is more than one current path (branch) between two points, and if the voltage between those two points also appears across each of the branches, then there is a parallel circuit between those two points. Furthermore, parallel resistors can be drawn in different ways between two points labeled A and B, as shown in the Figure 5. Figure 5: Examples of circuits with 2 parallel paths. The equivalent resistance (𝑅 𝑇 ) 1 1 1 1 1 = + + +⋯+ 𝑅𝑇 𝑅1 𝑅2 𝑅3 𝑅𝑛 For the case of two resistors, the equation of for calculating the equivalent resistors are represented as 𝑅1 𝑅2 𝑅𝑇 = 𝑅1 + 𝑅2 The equation simply indicates that the total resistance for two resistors in parallel is equal to the product of the two resistors divided by the sum of the two resistors. This equation can be summarized as the “product over the sum” formula. Certain theorems are used which when applied to the solutions of electric networks, wither simplify the network itself or render their analytical solution very easy. These theorems can also be applied to an A.C. system, with the only difference that impedances replace the ohmic resistance of D.C. system. Different electric circuits (according to their properties) are defined below We will now discuss the various network theorems which are of great help in solving complicated networks. Incidentally, a network is said to be completely solved or analyzed when all voltages and all currents in its different elements are determined. There are two general approaches to network analysis: (i) Direct Method Here, the network is left in its original form while determining its different voltages and currents. Such methods are usually restricted to fairly simple circuits and include Kirchhoff’s laws, Loop analysis, Nodal analysis, superposition theorem, Compensation theorem and Reciprocity theorem etc. (ii) Network Reduction Method Here, the original network is converted into a much simpler equivalent circuit for rapid calculation of different quantities. This method can be applied to simple as well as complicated networks. Examples of this method are: Delta/Star and Star/Delta conversions, Thevenin’s theorem and Norton’s Theorem etc. Kirchhoff’s Laws Gustav Robert Kirchoff (1824 - 1887) a German physicist, published the first systematic description of the laws of circuit analysis. These laws are known as Kirchhoff ’s current law (KCL) and Kirchoff ’s voltage law (KVL). 1) Current law: or point law, it States that in any network of wires carrying currents the algebraic sum of the currents meeting at Junction (or point) is zero. It is also called as point law. It can be written as the total current leaving a junction is equal to the total current entering that junction. It is obviously true because there is no accumulation of charge at the junction of the network. Figure 6: Five conductors carrying current and meeting at a point. From the Fig. 6, the current is entering the node A through 𝐼4 and leaving the node through 𝐼1 , 𝐼2 , 𝐼3 and 𝐼5 𝑀𝑎𝑡ℎ𝑒𝑚𝑎𝑡𝑖𝑐𝑎𝑙𝑙𝑦; + 𝐼1 − 𝐼2 − 𝐼3 + 𝐼4 − 𝐼5 = 0 KCL 𝐼4 + 𝐼1 = 𝐼2 + 𝐼3 + 𝐼5 Using this notation, KCL can be written using a summation; ∑𝑛𝑖=1 𝐼𝑖 = 0 Terms used in explanation of DC Networks Circuit: A circuit is a closed conducting path through which an electric current either flows or is intended flow. Parameters: The various elements of an electric circuit are called its parameters like resistance, inductance and capacitance. These parameters may be lumped or distributed. Linear Circuit: A linear circuit is one whose parameters are constant i.e. they do not change with voltage or current. Non-linear Circuit is that circuit whose parameters change with voltage or current. Bilateral Circuit: A bilateral circuit is one whose properties or characteristics are the same in either direction. The usual transmission line is bilateral, because it can be made to perform its function equally well in either direction. Unilateral Circuit: It is that circuit whose properties or characteristics change with the direction of its operation. A diode rectifier is a unilateral circuit, because it cannot perform rectification in both directions. Passive Network: is one which contains no source of e.m.f. in it. Active Network is one which contains one or more than one source of e.m.f. Node is a junction in a circuit where two or more circuit elements are connected together. Branch is that part of a network which lies between two junctions. Loop. It is a close path in a circuit in which no element or node is encountered more than once. Mesh. It is a loop that contains no other loop within it. Determine the number of loops, branches, nodes and the meshes in the circuit as shown in Figure 7. Figure 7: The circuit showing 6 nodes, 7 branches, 3 Loops and 2 Meshes. Kirchhoff’s Mesh Law or Voltage Law (KVL) It states that the algebraic sum of the products of currents and resistances (IR) in each of the conductors in any closed path (or mesh) in a network plus the algebraic sum of the emfs. in that path is zero. Likewise, it can be stated as in any closed electric circuit, the sum of potential drops (I.R) is equal to the sum of the impressed emfs. This law represents the statement for the conservation of Energy. In other words, Σ IR + Σ e.m.f. = 0...round a mesh Figure 8: Demonstrating (a) node, loop and branch (b) KCL (c) KVL In applying KVL, attention is needed in determination of voltage drop/rise in a loop, otherwise results will come out to be wrong as shown in Fig. 9 Figure 9: Showing the Voltage drop and Voltage rise Sign of Battery E.M.F. A rise in voltage should be given a + ve sign and a fall in voltage a - ve sign (Figure 9). Keeping this in mind, it is clear that as we go from the − ve terminal of a battery to its + ve terminal (Fig. 9), there is a rise in potential, hence this voltage should be given a + ve sign. If, on the other hand, we go from + ve terminal to − ve terminal, then there is a fall in potential, hence this voltage should be preceded by − ve sign. It is important to note that the sign of the battery e.m.f. is independent of the direction of the current through that branch. Sign of IR Drop Let’s consider the case of a resistor (Figure 9), if we go through a resistor in the same direction as the current, then there is a fall in potential because current flows from a higher to a lower potential. Hence, this voltage fall should be taken as −ve. However, if we go in a direction opposite to that of the current, then there is a rise in voltage. Hence, this voltage rise should be given a positive sign (+ve) It is clear that the sign of voltage drop across a resistor depends on the direction of current through that resistor but is independent of the polarity of any other source of e.m.f. in the circuit under consideration. In applying Kirchhoff’s laws to electrical networks, the question of assuming proper direction of current usually arises. The direction of current flow may be assumed either clockwise or anticlockwise. If the assumed direction of current is not the actual direction, then on solving the electrical networks, this current will be found to have a minus sign. If the answer is positive, then assumed direction is the same as actual direction However, the important point is that once a particular direction has been assumed, the same should be used throughout the solution of the question. Lesson Classwork 1. What is DC Networks? 2. Differentiate between series and parallel combination. 3. Briefly explain the Kirchhoff’s Current Law (KCL) 2.0 Introduction Electric circuit analysis with the help of Kirchhoff’s laws usually involves solution of two or three simultaneous equations. These equations can be solved by a systematic elimination of the variables but the procedure is often lengthy and laborious and hence more liable to error. Determinants and Cramer’s rule provide a simple and straight method for solving network equations through manipulation of their coefficients. Of course, if the number of simultaneous equations happens to be very large, use of a digital computer can make the task easy. Determinants The symbol 𝑎 𝑏 | | is called a determinant of the second order (or 2 × 2 determinant) because it contains two 𝑐 𝑑 rows (ab and cd) and two columns (ac and bd). The numbers a, b, c and d are called the elements or constituents of the determinant. Their number in the present case is 22 = 4. The above result for a second order determinant can be expressed as “upper left times lower right minus upper right times lower left” Example: Solve the following two simultaneous equations by the method of determinants Maxwell’s Loop Curent Method This method which is particularly well-suited to coupled circuit solutions employs a system of loop or mesh currents instead of branch currents (as in Kirchhoff’s laws). Here, the currents in different meshes are assigned continuous paths so that they do not split at a junction into branch currents. This method eliminates a great deal of tedious work involved in the branch-current method and is best suited when energy sources are voltage sources rather than current sources. Basically, this method consists of writing loop voltage equations by Kirchhoff’s voltage law in terms of unknown loop currents. The Figure shows two batteries E1 and E2 connected in a network consisting of five resistors. Let the loop currents for the three meshes be I1, I2 and I3. It is obvious that current through 𝑅4 is(𝐼1 − 𝐼2 ) and that through 𝑅5 is(𝐼2 − 𝐼3 ). However, when 𝑅4 is considered part of second loop, current through it is (𝐼2 − 𝐼1 ). Similarly, when 𝑅5 is considered part of third loop, current through is (𝐼3 − 𝐼2 ). Applying KVL to the three loops; 𝐸1 − 𝐼1 𝑅1 − 𝑅4 (𝐼1 − 𝐼2 ) = 0 𝐼1 (𝑅1 + 𝑅4 ) − 𝐼2 𝑅4 − 𝐸1 = 0…………..Loop 1 −𝐼2 𝑅2 − 𝑅5 (𝐼2 − 𝐼3 ) − 𝑅4 (𝐼2 − 𝐼1 ) = 0 𝐼2 𝑅4 − 𝐼2 (𝑅2 + 𝑅4 + 𝑅5 ) + 𝐼3 𝑅5 = 0………Loop 2 −𝐼3 𝑅3 − 𝐸2 − 𝑅5 (𝐼3 − 𝐼2 ) = 0 𝐼2 𝑅5 − 𝐼3 (𝑅3 + 𝑅5 ) − 𝐸2 = 0……..Loop 3 Mesh Analysis Using Matrix Form Consider the network of Figure below, which contains resistances and independent voltage sources and has three meshes. Let the three mesh currents be designated as I1, I2 and I3 and all the three may be assumed to flow in the clockwise direction for obtaining symmetry in mesh equations. EXAMPLE: Write the impedance matrix of the network and find the value of current I3 Lesson Classwork 1. Briefly explain the procedures of using Determinant and Cramer’s rule in solving DC networks 2. Differentiate between Self resistance and Mutual resistance in mesh analysis 3. Briefly explain the Kirchhoff’s Current Law (KCL) 3.0 Introduction The Thevenin’s equivalent form of any two-terminal resistive circuit consists of an equivalent voltage source 𝑉𝑇𝐻 and an equivalent resistance 𝑅𝑇𝐻. The values of the equivalent voltage and resistance depend on the values in the original circuit. Any two-terminal resistive circuit can be simplified to a Thevenin’s equivalent regardless of its complexity. The Thevenin’s voltage is the open circuit (no-load) voltage between two specified output terminals in a circuit. The Thevenin’s equivalent resistance is the total resistance appearing between two specified output terminals in a circuit with all sources replaced by their internal resistances (which for an ideal voltage source is zero). The current flowing through a load resistance RL connected across any two terminals A and B of a linear, active bilateral network is given by V oc || (Ri + RL) where Voc is the open-circuit voltage (i.e. voltage across the two terminals when RL is removed) and Ri is the internal resistance of the network as viewed back into the open-circuited network from terminals A and B with all voltage sources replaced by their internal resistance (if any) and current sources by infinite resistance. Thevenin’s theorem provides a mathematical technique for replacing a given network, as viewed from two output terminals, by a single voltage source with a series resistance. It makes the solving process of complicated networks to be easy and achieve quickly. The application of this extremely useful theorem will be explained with the help of the following simple example. Although a Thevenin’s equivalent circuit is not of the same form as the original circuit, but, it acts the same in terms of the output voltage and current. How to Thevenize a Given Circuit 1. Temporarily remove the resistance (called load resistance R L) whose current is required. 2. Find the open-circuit voltage Voc which appears across the two terminals from where resistance has been removed. It is also called Thevenin’s voltage Vth. 3. Compute the resistance of the whose network as looked into from these two terminals after all voltage sources have been removed leaving behind their internal resistances (if any) and current sources have been replaced by open-circuit i.e. infinite resistance. It is also called Thevenin’s resistance Rth or Ti. 4. Replace the entire network by a single Thevenin’s source, whose voltage is Vth or Voc and whose internal resistance is Rth or Ri. 5. Connect RL back to its terminals from where it was previously removed. 6. Finally, calculate the current flowing through RL by using the equation, I = Vth/(Rth + RL) or I = Voc/(Ri + RL) NORTON’S THEOREM Norton’s Theorem This theorem is an alternative to the Thevenin’s theorem. It is a twin theorem to Thevenin’s theorem. Like Thevenin’s theorem, Norton’s theorem provides a method of reducing a more complex circuit to a simpler form. As Thevenin’s theorem reduces a two-terminal active network of linear resistances and generators to an equivalent constant-voltage source and series resistance, Norton’s theorem replaces the network by an equivalent constant-current source and a parallel resistance. The form of Norton’s equivalent circuit is shown in Figure 1. Regardless of how complex the original circuit is, it can always be reduced to this equivalent form. The equivalent current source is designated IN, and the equivalent resistance is designated RN. Therefore Norton’s theorem states that any two-terminal active network containing voltage sources and resistance when viewed from its output terminals, is equivalent to a constant-current source and a parallel resistance. The constant current is equal to the current which would flow in a short- circuit placed across the terminals and parallel resistance is the resistance of the network when viewed from these open-circuited terminals after all voltage and current sources have been removed and replaced by their internal resistances. The voltage between any two points in a network is equal to ISC. Ri where ISC is the short-circuit current between the two points and Ri is the resistance of the network as viewed from these points with all voltage sources being replaced by their internal resistances (if any) and current sources replaced by open-circuits. We RN define in the same way as Rth: it is the total resistance appearing between two terminals in a given circuit with all sources replaced by their internal resistances. In summary, any load resistor connected between the terminals of a Norton’s equivalent circuit will have the same current through it and the same voltage across it as if it were connected to the terminals of the original circuit. A summary of steps for theoretically applying Norton’s theorem is as follows: 1. Short the two terminals between which you want to find the Norton equivalent circuit. 2. Determine the current (IN) through the shorted terminals. 3. Determine the resistance (RN) between the two terminals (opened) with all voltage sources shorted and all current sources opened (ISC). 4. Connect and in parallel to produce the complete Norton equivalent for the original circuit. Norton’s equivalent circuit can also be derived from Thevenin’s equivalent circuit by use of the source conversion method. Lesson Classwork 1. Briefly explain the procedures of using Determinant and Cramer’s rule in solving DC networks 2. Differentiate between Self resistance and Mutual resistance in mesh analysis 3. Briefly explain the Kirchhoff’s Current Law (KCL) Study Session IV: DC Networks- Maximum Power Transfer Theorem 4.0 Introduction Maximum Power Transfer Theorem The theorem is applicable to all branch of electrical engineering particularly useful for analyzing communication networks. The overall efficiency of a network supplying maximum power to any branch is 50 per cent. For this reason, the application of this theorem to power transmission and distribution networks is limited because, in their case, the goal is high efficiency and not maximum power transfer. However, in the case of electronic and communication networks, very often, the goal is either to receive or transmit maximum power (through at reduced efficiency) especially when power involved is only a few milliwatts or microwatts. Frequently, the problem of maximum power transfer is of crucial significance in the operation of transmission lines and antennas. Maximum Power Transfer Theorem states that a resistive load will abstract maximum power from a network when the load resistance is equal to the resistance of the network as viewed from the output terminals, with all energy sources removed leaving behind their internal resistances In Fig above, a load resistance of RL is connected across the terminals A and B of a network which consists of a generator of e.m.f. E and internal resistance Rg and a series resistance R which, in fact, represents the lumped resistance of the connecting wires. Let Ri = Rg + R = internal resistance of the network as viewed from A and B. According MPT, RL will abstract maximum power from the network when RL = Ri. Power Transfer Efficiency if PL is the power supplied to the load and PT is the total power supplied by the voltage source then power transfer efficiency is given by 𝑃𝐿 𝜂= 𝑃𝑇 𝑃𝑇 = 𝑃𝐿 + 𝑃𝑖 = 𝐼 2 𝑅𝐿 + 𝐼 2 𝑅𝑖 𝑃𝐿 𝐼 2 𝑅𝐿 𝑅𝐿 𝜂= = 2 = 𝑃𝑇 𝐼 𝑅𝐿 + 𝐼 2 𝑅𝑖 𝑅𝐿 + 𝑅𝑖 1 𝑅𝑖 1 + ( ⁄𝑅 ) 𝐿 The maximum value of 𝜂 is unity when 𝑅𝐿 = ∞ 𝜂 = 1; 𝑤ℎ𝑒𝑛 𝑅𝐿 = ∞ 𝜂 = 0.5; 𝑤ℎ𝑒𝑛 𝑅𝐿 = 𝑅𝑖 This means that, under maximum power transfer condition, the power transfer efficiency is only 50%. Lesson Classwork 1. Briefly explain the matching condition for maximum power transfer theorem. 2. State power transfer theorem. 3. A voltage source delivers 4 A when the load connected to it is 5 Ω and 2 A when the load becomes 20Ω. Calculate a. Maximum power which source can supply b. Power transfer efficiency of the source with R L of 20Ω. c. the power transfer efficiency when the source delivers 60 W