Chapter 5: Circuit Theorems


 Allison Hunt
 3 years ago
 Views:
Transcription
1 Chapter 5: Circuit Theorems This chapter provides a new powerful technique of solving complicated circuits that are more conceptual in nature than node/mesh analysis. Conceptually, the method is fairly straightforward to write down, however, they are complicated to apply in practice. One will find that the amount of work increases dramatically using these circuit theorems to solve a circuit. There are four theorems we will look at in this chapter: Source Transformations, Superposition Principle and Thévenin & Norton Equivalents, and Maimum Power Theorem. There are two useful ideas to take away from this chapter.. ThéveninNorton source transformations. In certain situations, there is a way to convert a voltage source into a current source and viceversa. Source transformations are going to change how you solve circuits and in some cases, a series of source transformations will have a dramatic affect in making the circuit calculation quite straight forward. Source transformations will become an additional technique in your arsenal in solving circuits regardless of the techniques one uses (T, Node, Mesh ).. ThéveninNorton Equivalents. The techniques of Chapter 4 (Mesh and Node) are very powerful and fairly straight forward to use. When we start studying nonlinear circuits (circuits with capacitors and inductors), mesh/node techniques will not be applicable. The way we will solve these nonlinear circuits are similar to solving irreducible dependent source circuits with KVL and KCL equations. The ThéveninNorton Equivalents will help us to simplify circuits as they get more and more challenging. Picture wise, we will breakup the circuit into two subcircuits: the (i) load circuit and (ii) subcircuit to be simplified via the ThéveninNorton Equivalent. That is, we will rewrite it into an equivalent form that is either a simple series circuit or a simple parallel circuit. Thévenin Norton Source Transformation There are transformations that convert current sources into voltage sources (and vice versa). That is, there is an equivalent circuit transformation between a series resistor and voltage source into a parallel resistor and current source: In order for these two circuits to be equivalent, the two separate subcircuits must have the same terminal current i and the terminal voltage v. In order to have the same terminal current and voltage, certain conditions must be met. These are derived by applying KVL to the series circuit and KCL to the parallel circuit. Here is what we get: KVL: v = v ir v v v i = i = R = R and v = i R KCL: i = i v/r R R R Th Th solving for i Th N Th N Th N Th N N N Th Th necessary conditions for equivalance That is, a ThéveninNorton Source Transformation converts a voltage source with a resistor in series into a current source and the same resistor in parallel: 5.
2 Eample 5. Use source transformations to solve for the current through the 3Ωresistor. Solution There are several ways one can solve for the current in this circuit: simple circuits or mesh/node. With the help of source transformations, you will see how useful these can be in reducing the circuit down to one loop before you can change a baby s diaper. Important Point: do not apply a source transformation to the 3Ω element. If a source transformation changes the 3Ω resistor to a parallel one, that specific information on the current through that 3Ω resistor is lost. Leave it alone! Let s convert this circuit using three different source transformations: Applying KVL/Mesh to the last loop, (3 4) i 6 = 0 i =.9A 3Ω 3 3Ω Superposition Theorem In Chapter, we briefly addressed the issue of a linear circuit. We stated then that for a circuit to be linear it meant that () the elements of the circuit are linear themselves and () the total response of an element (the voltages or currents) can be determined by the sum of the individual responses from each independent source in the circuit. This last statement is the Superposition theorem. Analogy: Newton s nd law (F net = F F F n) Superposition Principle The current i R through an element is equal to the algebraic sum of the currents (i, i i n) produced independently by each individual source. That is, i = = R i in isource all sources current produce by source current produce by sourcen Solving Strategies Step : Isolate one source and deactivate all other independent sources. To deactivate an independent current and voltage sources replace a voltage source with a short circuit (a short has v short = 0) current source with an open circuit (an open has i open = 0) With only one active source in the circuit, calculate the current/voltage of the element of interest. Repeat this process for each independent source in the circuit. Step : If there are nsources, then there are ncircuits to solve using superposition. Eample 5. Use superposition to find the current i through the 0kΩresistor. Solution Superposition states that to calculate the current i 0kΩ, this current is the sum of all of the individual currents produced by the V, 3mA and 9mAsources: i = i i i i i i 0k V 3mA 9mA 3 5.
3 Response of the Vsource Deactivate both current sources (replace them with opens), and calculate the current i produced by the V source using a KVL/Mesh loop: 36 i = i = ma 3 Response of the 3mAsource Deactivate the other sources (short V and open 9 ma sources), and calculate the current i produced by the 3mA source using CDR: 6 i 4 = (3mA) = ma = i Response of the 9mAsource Deactivate the other sources (short V and open 3 ma sources), and calculate the current i 3 produced by the 9mA source using CDR: i 3 = (9A) = 3 ma = i3 4 According to Superposition, the current i 0kΩ is the sum of all of these individual currents: i 4 0kΩ = i i i3 = 3 = ma = i 3 3 0kΩ Thévenin s Theorem History: this theorem was independently derived in 853 by Helmholtz and in 883 by Léon Charles Thévenin. Léon Charles Thévenin was a French telegraph engineer who etended Ohm's law to the analysis of comple electrical circuits. The goal of Thévenin s theorem is to identify and separate a portion of a comple circuit (called the subcircuit) and replace it with a Thévenin equivalent (series voltage source and resistor). In doing so, the solving of the circuit problem is greatly simplified. A typical application of a Thévenin theorem goes something like this: Step. Focus on a load and separate it out from the subcircuit to be replaced with a Thévenin equivalent. Step. Perform the necessary calculations to rewrite the subcircuit as a Thévenin equivalent circuit. Step 3. Reconnect the Thévenin equivalent circuit back onto the load circuit and now solve the simplified circuit. To accomplish this, one applies the Thévenin s Rules. Thévenin s Rules (Notation: v Th = v OC, R Th = R N, and i N = i Sc) a. Determine v oc with all sources activated If the open circuit contains sources about the open, apply KVL around the open circuit to determine v OC. 5.3
4 Use any method to solve for v OC b. Determine R Th. Method: Independent Sources Only (Conceptual Approach) Deactivate all sources and determine R Th relative to the terminal points ab. Method: Independent & Dependent Sources (Calculational Approach) i. Place a short at the terminal points ab and calculate i sc using any method. ii. Determine R Th by using R Th = v OC/i. c. Redraw the equivalent Thévenin circuit. If the Thévenin resistance is negative, subtract it from the load resistance. Dependent sources only the book does not give any problems related to only dependent sources. I will not cover them; however, if you are transferring to San Jose State and plan on majoring in EE, at the end of the semester I can teach you these in about 05 minutes. Independent Sources Only Because this eample circuit only has an independent source, I will not only demonstrate how a Thévenin equivalent is determine between points ab, but also show that Method and Method are equivalent. Suppose the following circuit has its load identified and removed, so that we can determine the Thévenin equivalent between points ab. Both Method and Method determine v oc eactly the same. Step : Determine v oc What does the voltage v oc mean? It is the effective voltage that the load sees between the points a and b with respect to the rest of the circuit. Identifying nodes and applying node analysis because one of the nodes is v oc, we get Node v oc : ( ) v 0 oc ( ) v 0 ( ) 8 = 0 v oc = vth = V Node v : ( ) v ( ) v ( ) 8 = oc 6 What differentiates Method and Method is in the way that the Thévenin resistance is determined. Step : Determine R Th Method : Deactivating Sources There is only an independent voltage source in the circuit, and because of this, we can use Method. Method is NOT applicable if there are dependent sources in the circuit. We start by deactivating the voltage source by shorting it out (replace the 8Vsource with a wire) and place an Ohmmeter at the terminal points. Effectively, one way to envision how an Ohmmeter measures resistance is by sending out a calibrated current which is then compared to the returning current, and reads the resistance. The R Th resistance is (0 3 6) R Th = (0 3 6) = = 6Ω= What does this resistance physically mean? It is the effective resistance that the load sees between the points a and b with respect to the rest of the circuit. The Thévenin equivalent circuit has replaced a comple circuit with a Thévenin equivalent circuit with values of v Th = V and R Th = 6Ω: 5.4
5 Method : Determine i to calculate R Th = v OC/i Method is applicable for both independent and dependent sources in the circuit. Place a short across the open and calculate current i. There are three loops in the circuit and will use mesh analysis to determine i. Loop  i : ( 0 6) i (6) i 0 i = i 0 matri = form i = i = 0 Loop  i : (6 3) i (6) i 3 i 8 Loop  i : (0 3) i (0) i (3) i 0 solution i = However, I think that the most efficient way is using node analysis since there is only one node plus a KCL to get i. Applying node to the v y, we get y y = 6 y Node v : ( ) v ( ) 8 0 v = 5V Now apply KCL at node a: Calculating R Th, we get : = 8Ω 0Ω KCL i i i = = A = i = v = V = Ω OC 6 i A This agrees with our previous calculation of R Th and therefore, both methods are equivalent. A Eample 5.3 Determine the current through the 4Ωresistor using a Thévenin equivalent circuit (Methods and ). Solution First, remove the 4Ωresistor and convert the circuit into a Thévenin equivalent. Step : Determine voc = v Th. Focus on the open circuit and see how voc is defined. Since the 4Ω resistor is connected to an open loop, there is no current flowing through it and therefore, has no voltage drop. Applying a KVL loop around the open circuit loop, the 0Vsource will still contribute to voc and we have v = 0 v v = v 0 0 oc oc 0 5.5
6 So to determine voc, my real task is first to determine the voltage across the 0Ω resistor. Question: what is the most efficient way to determine v 0Ω? For Carlos, I d say a source transformation on the series 8V and 9Ω, converting them into a parallel current source and resistor, then add the current sources together. The circuit I get is Applying CDR to get the current of the 0Ω resistor and Ohm s law to get the voltage v 0Ω, we get 9 0 Ohm's CDR : i 0Ω = A = A v 5 law 0Ω = R0Ωi = 0 5 = 4V Using our KVL relation from above, we determine voc: v = v 0 = 4 0 = 4V = v = v oc 0Ω oc Th Step : Find R Th. Method : Deactivating Independent Sources Since the sources are all independent sources, we can use Method and deactivate all of sources to determine the Thévenin resistance R Th at ab. It is important to realize that in the previous voccircuit, there was no 4Ω resistor; however, in the R Thcircuit the 4Ω resistor reappears and must be accounted for. 5 0 = 4 (6 9) 0 = 4 = 0Ω= 5 0 Now that we have both the Thévenin voltage and resistance, I can draw the equivalent Thévenin circuit, add back in the load 4Ωresistor, and solve for the current i 4Ω. Using Ohm s law to determine the current i 4Ω, we see that the current flows from b to a, not a to b and therefore, I assign a negative value to this current: 4V i 4 Ω = = A = i Ω Interpretation of results. Effectively, when we calculate the Thévenin equivalent, the 4Ω resistor sees an effective voltage of 4V and an effective resistance of 6Ω from the view point of terminal points ab. That is, regardless of the load (whether it s a 4Ω resistor or some complicated load circuit), this load feels 4V and 6Ω from the rest of the circuit. It does not see the individual 0V, 8V and Asources but a combination of these that effectively reads the Thévenin voltage of 4V between ab. It does not see some combination of all of the resistors in the circuit, but the effective Thévenin resistance of 6Ω between ab. 5.6
7 . It clearly is a lot more work to calculate the current i 4Ω using Thévenin Equivalent circuits, whereas using Node or Mesh would have given us this current more efficiently. So what s all the fuss about learning these Thévenin equivalent techniques then if there are other techniques more efficient? To put this into prospective, resistive circuits are only half of this course, with the other have being capacitive and inductive circuits. Node and Mesh only apply to linear resistive circuits (as well as to ac phasor circuits). When we start capacitive and inductive circuits (nonlinear circuits), we will not be able to solve these circuits using Node or Mesh and we will be forced to use Simple Circuit Techniques (KVL & KCL) along with Thévenin equivalents to guide us. Thévenin equivalents are a powerful ally in solving these considerably tougher circuits. Method : Finding i to determine R Th = v OC/i When a short is placed across ab to calculate i, note that i = i 4Ω. I epect that you are most likely thinking of calculating i using mesh analysis (3 loops = loop equations plus a ). Even though this is a different circuit, I already have reduced the lefthand portion circuit with a source transformation that combined the 8Vsource with the Asource. I want to continue using this simplified circuit and make one more source transformation on the 0/9Asource to convert it into a voltage source. This is what I get: I immediately see that there is only one node voltage to solve for and a solver equation for i Node v : ( ) v ( ) ( 0) ( ) (0) = 0 v = 0.4V 0 v Solver : i = =.4A = i 4 4Ω Using a combination of voc and i, the Thévenin resistance is determined to be v 4V = = = Ω OC 0 i.4a Interpretation of results. Both methods ( & ) determine the Thévenin resistance, however, Method is only valid if there are ONLY independent sources. If there are any dependent sources, Method is NOT valid and ONLY Method can be used.. The combination voc/i, is a measure of the effective resistance (R Th) that the load sees relative to the terminal points ab. Norton Equivalent Circuits 5.7
8 Independent & Dependent Sources There are two important changes when dealing with independent and dependent sources in circuit where Thévenin s theorem is going to be applied:. Without dependent sources, zeroing all the independent sources leaves a resistive circuit so that the Thévenin resistance can be calculated directly. However, dependent sources typically alter the Thévenin resistance so those can't be zeroed. That is, Method is NOT VALID whenever dependent sources are in the circuit and the only way to determine R Th is by using Medthod; determine the ratio v OC/i = R Th.. Thévenin circuit with dependent sources can have positive and negative Thévenin resistance. It is not like there is an actual negative resistor that one can actual purchase at the store or something like that, but has implications that circuits have a impedance matching and can be resonated. Eample 5.4 Obtain the Thévenin equivalent for the following circuit. Solution The fact that there is a dependent source in the circuit immediately tells us that Method cannot be applied to this circuit. Only Method is valid. Step : Determine voc Focusing on the open circuit where voc is defined, the open immediately tells us that the kω resistor is a dead resistor since there is no current flowing through it and has no voltage drop. So the voltage across the 3kΩ resistor is equivalent to voc: v = v = 3i oc Clearly, to determine voc we shift our attention to first determining i. Thinking in terms of source transformations first, converting the.5ma source into a voltage source allows us to reduce the circuit to one loop. 3kΩ X : X X X oc X KVL (6 4 3)i = 5 i i = ma v = 3i = 3V Step : Determine i sc such that R Th = voc /isc. Since I ve already modified the left side of the circuit, I will continue using it. Placing the short across ab, i sc is the current through the kω resistor. To determine i sc we can either use (i) mesh analysis which will produce 3 equations and 3 unknowns (i sc, i, i z), (ii) node analysis which also has 3 equations and 3 unknowns (i sc, i, v y) or (ii) simple circuits that has equations and unknowns (i sc, i ). Let s do simple circuits! 5.8
9 I will use VDR to determine the voltage 3i (and current i ) and use Ohm s law to get i sc: 3 VDR: v3kω = vkω = 3i = (5 i ) 3 0 solving for i i = 3 ma solving vkω for i i = = ma = i kω Applying R Th = v OC/i, we find that voc 3V R = Th 3kΩ i = ma = Step 3 Redraw the equivalent circuit: sc Eample 5.5 Determine the current through the Ωresistor at ab using a Thévenin equivalent circuit. Solution Step : Determine v oc The simplest way to determine v oc is by applying KVL around the top loop. Why? Because I immediately know the dependent current i since it is in series with the 4Asource. i= 4A voc 8 i 6i = 0 voc = 4V Step : Determine i sc such that R Th = v oc/i sc. Placing the short at ab sets up three loops (i, i sc, i ). However, this short really sets up an unusual situation where the loop currenti (which also has a dependent source controlled by i) is directly connected to i sc via a constant 4Asource. Note that I have completely ignored i. The trick is to see that doing a mesh loop for loopi and applying KCL between ab, there are equations and unknowns (i sc, i) that quickly determines i sc: solving for i Loop  i : 6i i = 8 i = A KCL at ab: i = 4 i = 6A Applying R Th = v OC/i, we find that v 4 V = = = Ω oc 4 isc 6 A Step 3: Redraw the equivalent circuit and solve the circuit problem. 5.9
10 Using Ohm s law, the current through the resistor is voc 4 iω = = = A = iω Ω 4 Because this current is positive, the current runs positive from terminal a to terminal b. Interpretation of Negative Thévenin Resistance Suppose now that the load resistor was replaced with a variable resistor (i.e., decade resistor bo) such that we adjusted the resistance. How does the circuit behave? One would think that if one sets R ab = 0Ω, then maimum current would flow through the load resistance R ab. Let s look at this: voc 4 iload = = = 6 A Rload 4 0 If I now adjusted the decade resistor bo to R load = Ω, the current now reads A, double the value of the short circuit current: 4 iload = = A 4 And as you can see, something very special happens as R load 4Ω, the current goes to infinity: 4 lim iload = lim Rload 4Ω Rload 4Ω 4 R load Physical interpretation The key phase is impedance matching. Negative resistance cannot physically occur in the case where the circuit is linear and contains only passive components (resistors, capacitors and inductors). But for active circuits, where dependent sources (amplifiers) are applied, a virtual negative resistance can be realized. The physical meaning of negative resistance is that power is absorbed by the circuit with zero phase shift rather than dissipated. Maimum Power Delivered to a Circuit The Maimum Power Transfer Theorem is not so much a means of analysis as it is an aid to system design. Simply stated, the maimum amount of power will be dissipated by a load resistance when that load resistance is equal to the Thévenin resistance of the circuit supplying the power. If the load resistance is lower or higher than the Thévenin resistance, its dissipated power will be less than maimum. This is essentially what is aimed for in radio transmitter design where the antenna or transmission line impedance is matched to final power amplifier impedance for maimum radio frequency power output (impedance matching). Impedance, the overall opposition to AC and DC current, is very similar to resistance, and must be equal between source and load for the greatest amount of power to be transferred to the load. A load impedance that is too high will result in low power output. A load impedance that is too low will not only result in low power output, but possibly overheating of the amplifier due to the power dissipated in its internal Thévenin impedance. Mathematically, one can determine the maimum power delivered to the load by calculating the critical point where dp load/dr load = 0: 5.0
11 Rload vload dpload d R load vload = vth Pload = = v Th = 0 Rload Rload drload dr load ( Rload ) After doing some algebra, we arrive d R load Rload Rload = = 0 R 4 Th = R dr load ( Rload ) ( Rload ) Eample 5.6 What is the maimum power that can be delivered to the load resistor R L? load Solution Step : Find v OC What is the most efficient way to solve for v OC? I first note that the controlling voltage v y is known since the Ωresistor is in series with the 9Asource: v y = 8V. So I can replace the VCCS as a 36Asource instead. So if I do the numbers, this is what I get: 4 nodes sources = node eqs [ node (v ) & SN eq (v,v ) (v,v ) DC (i ) OC eqs, unknowns (v OC,v,v,i ) Writing out the equations: node v : v v 0 = 9 OC OC SN: ( )v 3i v v 3i = 7 : v v = OC DC: v 3i = i In matri form and solution: 38 v = = 79.3V OC v 9 OC 90 v 7 v = = 63.3V Step : Find i. Let s do the numbers: 3 9 solutions = v 84 i v = = 6.3V 38 i = =.7V 3 5.
12 4 loops sources = loops eqs ( regular (i ) & SL (i,i,i ) DC (i,v ) y Solver (i ) 3 6 eqs, 6 unknowns (i,i,i,i,v,i ) Writing out the equations, Loop : i i = 3i ( ) 3 y SL: i i 0 i = 3i : loop loop i 3 3 y y 3 i i = 9, i i = v DC : v = i i Solver : i = i i The matri and solutions are i 0 i = 3.58A i i =.6A i i A solutions = = i i 0 = 0.9A v v y 0 y =.65V i 0 i = 7.67A Step 3: Find R TH and solve the circuit problem Applying R Th = v OC/i, we find that v 79.3V = = = Ω oc 0.3 isc 7.67A R = 0.3 Ω v = 79.3V Th Th maimum power theorem v L (79.3 / ) R ma Th = RL = 0.3Ω P = = = 53 W = P R 0.3 L ma 5.
Circuit Theorems Overview Linearity Superposition Source Transformation Thévenin and Norton Equivalents Maximum Power Transfer
Circuit Theorems Overview Linearity Superposition Source Transformation Thévenin and Norton Equivalents Maximum Power Transfer J. McNames Portland State University ECE 221 Circuit Theorems Ver. 1.36 1
More informationChapter 5. Department of Mechanical Engineering
Source Transformation By KVL: V s =ir s + v By KCL: i s =i + v/r p is=v s /R s R s =R p V s /R s =i + v/r s i s =i + v/r p Two circuits have the same terminal voltage and current Source Transformation
More informationUNIT 4 DC EQUIVALENT CIRCUIT AND NETWORK THEOREMS
UNIT 4 DC EQUIVALENT CIRCUIT AND NETWORK THEOREMS 1.0 Kirchoff s Law Kirchoff s Current Law (KCL) states at any junction in an electric circuit the total current flowing towards that junction is equal
More informationECE2262 Electric Circuits
ECE2262 Electric Circuits Equivalence Chapter 5: Circuit Theorems Linearity Superposition Thevenin s and Norton s Theorems Maximum Power Transfer Analysis of Circuits Using Circuit Theorems 1 5. 1 Equivalence
More informationNotes for course EE1.1 Circuit Analysis TOPIC 3 CIRCUIT ANALYSIS USING SUBCIRCUITS
Notes for course EE1.1 Circuit Analysis 200405 TOPIC 3 CIRCUIT ANALYSIS USING SUBCIRCUITS OBJECTIVES 1) To introduce the Source Transformation 2) To consider the concepts of Linearity and Superposition
More informationChapter 10 AC Analysis Using Phasors
Chapter 10 AC Analysis Using Phasors 10.1 Introduction We would like to use our linear circuit theorems (Nodal analysis, Mesh analysis, Thevenin and Norton equivalent circuits, Superposition, etc.) to
More informationECE2262 Electric Circuits. Chapter 5: Circuit Theorems
ECE2262 Electric Circuits Chapter 5: Circuit Theorems 1 Equivalence Linearity Superposition Thevenin s and Norton s Theorems Maximum Power Transfer Analysis of Circuits Using Circuit Theorems 2 5. 1 Equivalence
More informationCHAPTER FOUR CIRCUIT THEOREMS
4.1 INTRODUCTION CHAPTER FOUR CIRCUIT THEOREMS The growth in areas of application of electric circuits has led to an evolution from simple to complex circuits. To handle the complexity, engineers over
More informationLecture #3. Review: Power
Lecture #3 OUTLINE Power calculations Circuit elements Voltage and current sources Electrical resistance (Ohm s law) Kirchhoff s laws Reading Chapter 2 Lecture 3, Slide 1 Review: Power If an element is
More informationSinusoidal Steady State Analysis (AC Analysis) Part I
Sinusoidal Steady State Analysis (AC Analysis) Part I Amin Electronics and Electrical Communications Engineering Department (EECE) Cairo University elc.n102.eng@gmail.com http://scholar.cu.edu.eg/refky/
More informationChapter 10: Sinusoidal SteadyState Analysis
Chapter 10: Sinusoidal SteadyState Analysis 10.1 10.2 10.3 10.4 10.5 10.6 10.9 Basic Approach Nodal Analysis Mesh Analysis Superposition Theorem Source Transformation Thevenin & Norton Equivalent Circuits
More informationEE 3120 Electric Energy Systems Study Guide for Prerequisite Test Wednesday, Jan 18, pm, Room TBA
EE 3120 Electric Energy Systems Study Guide for Prerequisite Test Wednesday, Jan 18, 2006 67 pm, Room TBA First retrieve your EE2110 final and other course papers and notes! The test will be closed book
More informationCHAPTER 4. Circuit Theorems
CHAPTER 4 Circuit Theorems The growth in areas of application of electrical circuits has led to an evolution from simple to complex circuits. To handle such complexity, engineers over the years have developed
More information3.1 Superposition theorem
Many electric circuits are complex, but it is an engineer s goal to reduce their complexity to analyze them easily. In the previous chapters, we have mastered the ability to solve networks containing independent
More informationMidterm Exam (closed book/notes) Tuesday, February 23, 2010
University of California, Berkeley Spring 2010 EE 42/100 Prof. A. Niknejad Midterm Exam (closed book/notes) Tuesday, February 23, 2010 Guidelines: Closed book. You may use a calculator. Do not unstaple
More informationNotes for course EE1.1 Circuit Analysis TOPIC 10 2PORT CIRCUITS
Objectives: Introduction Notes for course EE1.1 Circuit Analysis 45 Reexamination of 1port subcircuits Admittance parameters for port circuits TOPIC 1 PORT CIRCUITS Gain and port impedance from port
More informationChapter 2. Engr228 Circuit Analysis. Dr Curtis Nelson
Chapter 2 Engr228 Circuit Analysis Dr Curtis Nelson Chapter 2 Objectives Understand symbols and behavior of the following circuit elements: Independent voltage and current sources; Dependent voltage and
More informationElectric Circuits II Sinusoidal Steady State Analysis. Dr. Firas Obeidat
Electric Circuits II Sinusoidal Steady State Analysis Dr. Firas Obeidat 1 Table of Contents 1 2 3 4 5 Nodal Analysis Mesh Analysis Superposition Theorem Source Transformation Thevenin and Norton Equivalent
More informationThevenin Norton Equivalencies  GATE Study Material in PDF
Thevenin Norton Equivalencies  GATE Study Material in PDF In these GATE 2018 Notes, we explain the Thevenin Norton Equivalencies. Thevenin s and Norton s Theorems are two equally valid methods of reducing
More informationSeries & Parallel Resistors 3/17/2015 1
Series & Parallel Resistors 3/17/2015 1 Series Resistors & Voltage Division Consider the singleloop circuit as shown in figure. The two resistors are in series, since the same current i flows in both
More informationENGG 225. David Ng. Winter January 9, Circuits, Currents, and Voltages... 5
ENGG 225 David Ng Winter 2017 Contents 1 January 9, 2017 5 1.1 Circuits, Currents, and Voltages.................... 5 2 January 11, 2017 6 2.1 Ideal Basic Circuit Elements....................... 6 3 January
More information4/27 Friday. I have all the old homework if you need to collect them.
4/27 Friday Last HW: do not need to turn it. Solution will be posted on the web. I have all the old homework if you need to collect them. Final exam: 79pm, Monday, 4/30 at Lambert Fieldhouse F101 Calculator
More informationD C Circuit Analysis and Network Theorems:
UNIT1 D C Circuit Analysis and Network Theorems: Circuit Concepts: Concepts of network, Active and passive elements, voltage and current sources, source transformation, unilateral and bilateral elements,
More informationChapter 4 Circuit Theorems
Chapter 4 Circuit Theorems 1. Linearity and Proportionality. Source Transformation 3. Superposition Theorem 4. Thevenin s Theorem and Norton s Theorem 5. Maximum Power Transfer Theorem Mazita Sem 1 111
More informationChapter 10: Sinusoidal SteadyState Analysis
Chapter 0: Sinusoidal SteadyState Analysis Sinusoidal Sources If a circuit is driven by a sinusoidal source, after 5 tie constants, the circuit reaches a steadystate (reeber the RC lab with t = τ). Consequently,
More informationEECE208 Intro to Electrical Engineering Lab. 5. Circuit Theorems  Thevenin Theorem, Maximum Power Transfer, and Superposition
EECE208 Intro to Electrical Engineering Lab Dr. Charles Kim 5. Circuit Theorems  Thevenin Theorem, Maximum Power Transfer, and Superposition Objectives: This experiment emphasizes e following ree circuit
More informationDEPARTMENT OF COMPUTER ENGINEERING UNIVERSITY OF LAHORE
DEPARTMENT OF COMPUTER ENGINEERING UNIVERSITY OF LAHORE NAME. Section 1 2 3 UNIVERSITY OF LAHORE Department of Computer engineering Linear Circuit Analysis Laboratory Manual 2 Compiled by Engr. Ahmad Bilal
More informationChapter 5 Solution P5.22, 3, 6 P5.33, 5, 8, 15 P5.43, 6, 8, 16 P5.52, 4, 6, 11 P5.62, 4, 9
Chapter 5 Solution P5.22, 3, 6 P5.33, 5, 8, 15 P5.43, 6, 8, 16 P5.52, 4, 6, 11 P5.62, 4, 9 P 5.22 Consider the circuit of Figure P 5.22. Find i a by simplifying the circuit (using source transformations)
More informationEECE251 Circuit Analysis I Lecture Integrated Program Set 3: Circuit Theorems
EECE251 Circuit Analysis I Lecture Integrated Program Set 3: Circuit Theorems Shahriar Mirabbasi Department of Electrical and Computer Engineering University of British Columbia shahriar@ece.ubc.ca 1 Linearity
More informationV x 4 V x. 2k = 5
Review Problem: d) Dependent sources R3 V V R Vx  R2 Vx V2 ) Determine the voltage V5 when VV Need to find voltage Vx then multiply by dependent source multiplier () Node analysis 2 V x V x R R 2 V x
More informationBFF1303: ELECTRICAL / ELECTRONICS ENGINEERING. Alternating Current Circuits : Basic Law
BFF1303: ELECTRICAL / ELECTRONICS ENGINEERING Alternating Current Circuits : Basic Law Ismail Mohd Khairuddin, Zulkifil Md Yusof Faculty of Manufacturing Engineering Universiti Malaysia Pahang Alternating
More informationExperiment #6. Thevenin Equivalent Circuits and Power Transfer
Experiment #6 Thevenin Equivalent Circuits and Power Transfer Objective: In this lab you will confirm the equivalence between a complicated resistor circuit and its Thevenin equivalent. You will also learn
More informationLecture Notes on DC Network Theory
Federal University, NdufuAlike, Ikwo Department of Electrical/Electronics and Computer Engineering (ECE) Faculty of Engineering and Technology Lecture Notes on DC Network Theory Harmattan Semester by
More informationSinusoidal Steady State Analysis (AC Analysis) Part II
Sinusoidal Steady State Analysis (AC Analysis) Part II Amin Electronics and Electrical Communications Engineering Department (EECE) Cairo University elc.n102.eng@gmail.com http://scholar.cu.edu.eg/refky/
More informationChapter 5 Objectives
Chapter 5 Engr228 Circuit Analysis Dr Curtis Nelson Chapter 5 Objectives State and apply the property of linearity State and apply the property of superposition Investigate source transformations Define
More informationPOLYTECHNIC UNIVERSITY Electrical Engineering Department. EE SOPHOMORE LABORATORY Experiment 2 DC circuits and network theorems
POLYTECHNIC UNIVERSITY Electrical Engineering Department EE SOPHOMORE LABORATORY Experiment 2 DC circuits and network theorems Modified for Physics 18, Brooklyn College I. Overview of Experiment In this
More informationReview of Circuit Analysis
Review of Circuit Analysis Fundamental elements Wire Resistor Voltage Source Current Source Kirchhoff s Voltage and Current Laws Resistors in Series Voltage Division EE 42 Lecture 2 1 Voltage and Current
More informationELEC 250: LINEAR CIRCUITS I COURSE OVERHEADS. These overheads are adapted from the Elec 250 Course Pack developed by Dr. Fayez Guibaly.
Elec 250: Linear Circuits I 5/4/08 ELEC 250: LINEAR CIRCUITS I COURSE OVERHEADS These overheads are adapted from the Elec 250 Course Pack developed by Dr. Fayez Guibaly. S.W. Neville Elec 250: Linear Circuits
More informationChapter 10 Sinusoidal Steady State Analysis Chapter Objectives:
Chapter 10 Sinusoidal Steady State Analysis Chapter Objectives: Apply previously learn circuit techniques to sinusoidal steadystate analysis. Learn how to apply nodal and mesh analysis in the frequency
More informationLecture # 2 Basic Circuit Laws
CPEN 206 Linear Circuits Lecture # 2 Basic Circuit Laws Dr. Godfrey A. Mills Email: gmills@ug.edu.gh Phone: 026907363 February 5, 206 Course TA David S. Tamakloe CPEN 206 Lecture 2 205_206 What is Electrical
More informationNotes on Electric Circuits (Dr. Ramakant Srivastava)
Notes on Electric ircuits (Dr. Ramakant Srivastava) Passive Sign onvention (PS) Passive sign convention deals with the designation of the polarity of the voltage and the direction of the current arrow
More informationChapter 4. Techniques of Circuit Analysis
Chapter 4. Techniques of Circuit Analysis By: FARHAD FARADJI, Ph.D. Assistant Professor, Electrical Engineering, K.N. Toosi University of Technology http://wp.kntu.ac.ir/faradji/electriccircuits1.htm Reference:
More informationThevenin equivalent circuits
Thevenin equivalent circuits We have seen the idea of equivalency used in several instances already. 1 2 1 2 same as 1 2 same as 1 2 R 3 same as = 0 V same as 0 A same as same as = EE 201 Thevenin 1 The
More informationElectric Current. Note: Current has polarity. EECS 42, Spring 2005 Week 2a 1
Electric Current Definition: rate of positive charge flow Symbol: i Units: Coulombs per second Amperes (A) i = dq/dt where q = charge (in Coulombs), t = time (in seconds) Note: Current has polarity. EECS
More informationModule 2. DC Circuit. Version 2 EE IIT, Kharagpur
Module 2 DC Circuit esson 8 evenin s and Norton s theorems in the context of dc voltage and current sources acting in a resistive network Objectives To understand the basic philosophy behind the evenin
More informationBasic. Theory. ircuit. Charles A. Desoer. Ernest S. Kuh. and. McGrawHill Book Company
Basic C m ш ircuit Theory Charles A. Desoer and Ernest S. Kuh Department of Electrical Engineering and Computer Sciences University of California, Berkeley McGrawHill Book Company New York St. Louis San
More informationEIT Review. Electrical Circuits DC Circuits. Lecturer: Russ Tatro. Presented by Tau Beta Pi The Engineering Honor Society 10/3/2006 1
EIT Review Electrical Circuits DC Circuits Lecturer: Russ Tatro Presented by Tau Beta Pi The Engineering Honor Society 10/3/2006 1 Session Outline Basic Concepts Basic Laws Methods of Analysis Circuit
More informationIntroduction to AC Circuits (Capacitors and Inductors)
Introduction to AC Circuits (Capacitors and Inductors) Amin Electronics and Electrical Communications Engineering Department (EECE) Cairo University elc.n102.eng@gmail.com http://scholar.cu.edu.eg/refky/
More informationDOWNLOAD PDF AC CIRCUIT ANALYSIS PROBLEMS AND SOLUTIONS
Chapter 1 : Resistors in Circuits  Practice â The Physics Hypertextbook In AC circuit analysis, if the circuit has sources operating at different frequencies, Superposition theorem can be used to solve
More information09Circuit Theorems Text: , 4.8. ECEGR 210 Electric Circuits I
09Circuit Theorems Text: 4.1 4.3, 4.8 ECEGR 210 Electric Circuits I Overview Introduction Linearity Superposition Maximum Power Transfer Dr. Louie 2 Introduction Nodal and mesh analysis can be tedious
More informationLABORATORY MODULE ELECTRIC CIRCUIT
LABORATORY MODULE ELECTRIC CIRCUIT HIGH VOLTAGE AND ELECTRICAL MEASUREMENT LAB ELECTRICAL ENGINEERING DEPARTMENT FACULTY OF ENGINEERING UNIVERSITAS INDONESIA DEPOK 2018 MODULE 1 LABORATORY BRIEFING All
More informationCOOKBOOK KVL AND KCL A COMPLETE GUIDE
1250 COOKBOOK KVL AND KCL A COMPLETE GUIDE Example circuit: 1) Label all source and component values with a voltage drop measurement (+, ) and a current flow measurement (arrow): By the passive sign convention,
More informationResonant Matching Networks
Chapter 1 Resonant Matching Networks 1.1 Introduction Frequently power from a linear source has to be transferred into a load. If the load impedance may be adjusted, the maximum power theorem states that
More informationSolution: Based on the slope of q(t): 20 A for 0 t 1 s dt = 0 for 3 t 4 s. 20 A for 4 t 5 s 0 for t 5 s 20 C. t (s) 20 C. i (A) Fig. P1.
Problem 1.24 The plot in Fig. P1.24 displays the cumulative charge q(t) that has entered a certain device up to time t. Sketch a plot of the corresponding current i(t). q 20 C 0 1 2 3 4 5 t (s) 20 C Figure
More informationOUTCOME 3  TUTORIAL 2
Unit : Unit code: QCF evel: 4 Credit value: 15 SYABUS Engineering Science /601/1404 OUTCOME 3  TUTORIA Be able to apply DC theory to solve electrical and electronic engineering problems DC electrical
More informationINTRODUCTION TO ELECTRONICS
INTRODUCTION TO ELECTRONICS Basic Quantities Voltage (symbol V) is the measure of electrical potential difference. It is measured in units of Volts, abbreviated V. The example below shows several ways
More informationChapter 4: Methods of Analysis
Chapter 4: Methods of Analysis When SCT are not applicable, it s because the circuit is neither in series or parallel. There exist extremely powerful mathematical methods that use KVL & KCL as its basis
More informationUNIVERSITY F P RTLAND Sch l f Engineering
UNIVERSITY F P RTLAND Sch l f Engineering EE271Electrical Circuits Laboratory Spring 2004 Dr. Aziz S. Inan & Dr. Joseph P. Hoffbeck Lab Experiment #4: Electrical Circuit Theorems  p. 1 of 5  Electrical
More informationDC STEADY STATE CIRCUIT ANALYSIS
DC STEADY STATE CIRCUIT ANALYSIS 1. Introduction The basic quantities in electric circuits are current, voltage and resistance. They are related with Ohm s law. For a passive branch the current is: I=
More informationLecture 6: Impedance (frequency dependent. resistance in the s world), Admittance (frequency. dependent conductance in the s world), and
Lecture 6: Impedance (frequency dependent resistance in the s world), Admittance (frequency dependent conductance in the s world), and Consequences Thereof. Professor Ray, what s an impedance? Answers:
More informationDesigning Information Devices and Systems I Fall 2018 Lecture Notes Note Introduction: Opamps in Negative Feedback
EECS 16A Designing Information Devices and Systems I Fall 2018 Lecture Notes Note 18 18.1 Introduction: Opamps in Negative Feedback In the last note, we saw that can use an opamp as a comparator. However,
More informationChapter 4: Techniques of Circuit Analysis
Chapter 4: Techniques of Circuit Analysis This chapter gies us many useful tools for soling and simplifying circuits. We saw a few simple tools in the last chapter (reduction of circuits ia series and
More informationVoltage Dividers, Nodal, and Mesh Analysis
Engr228 Lab #2 Voltage Dividers, Nodal, and Mesh Analysis Name Partner(s) Grade /10 Introduction This lab exercise is designed to further your understanding of the use of the lab equipment and to verify
More informationNetworks and Systems Prof. V. G. K. Murti Department of Electrical Engineering Indian Institute of Technology, Madras
Networks and Systems Prof. V. G. K. Murti Department of Electrical Engineering Indian Institute of Technology, Madras Lecture  34 Network Theorems (1) Superposition Theorem Substitution Theorem The next
More informationECE2262 Electric Circuit
ECE2262 Electric Circuit Chapter 7: FIRST AND SECONDORDER RL AND RC CIRCUITS Response to FirstOrder RL and RC Circuits Response to SecondOrder RL and RC Circuits 1 2 7.1. Introduction 3 4 In dc steady
More informationUniversity of Alabama Department of Physics and Astronomy. Problem Set 4
University of Alabama Department of Physics and Astronomy PH 26 LeClair Fall 20 Problem Set 4. A battery has an ideal voltage V and an internal resistance r. A variable load resistance R is connected to
More informationBasic Laws. Bởi: Sy Hien Dinh
Basic Laws Bởi: Sy Hien Dinh INTRODUCTION Chapter 1 introduced basic concepts such as current, voltage, and power in an electric circuit. To actually determine the values of this variable in a given circuit
More informationElectronics. Basics & Applications. group talk Daniel Biesinger
Electronics Basics & Applications group talk 23.7.2010 by Daniel Biesinger 1 2 Contents Contents Basics Simple applications Equivalent circuit Impedance & Reactance More advanced applications  RC circuits
More informationCircuit Analysis. by John M. Santiago, Jr., PhD FOR. Professor of Electrical and Systems Engineering, Colonel (Ret) USAF. A Wiley Brand FOR
Circuit Analysis FOR A Wiley Brand by John M. Santiago, Jr., PhD Professor of Electrical and Systems Engineering, Colonel (Ret) USAF FOR A Wiley Brand Table of Contents. ' : '" '! " ' ' '... ',. 1 Introduction
More informationEE201 Review Exam I. 1. The voltage Vx in the circuit below is: (1) 3V (2) 2V (3) 2V (4) 1V (5) 1V (6) None of above
EE201, Review Probs Test 1 page1 Spring 98 EE201 Review Exam I Multiple Choice (5 points each, no partial credit.) 1. The voltage Vx in the circuit below is: (1) 3V (2) 2V (3) 2V (4) 1V (5) 1V (6)
More informationNetwork Topology2 & Dual and Duality Choice of independent branch currents and voltages: The solution of a network involves solving of all branch currents and voltages. We know that the branch current
More informationLCR Series Circuits. AC Theory. Introduction to LCR Series Circuits. Module. What you'll learn in Module 9. Module 9 Introduction
Module 9 AC Theory LCR Series Circuits Introduction to LCR Series Circuits What you'll learn in Module 9. Module 9 Introduction Introduction to LCR Series Circuits. Section 9.1 LCR Series Circuits. Amazing
More informationKirchhoff's Laws and Circuit Analysis (EC 2)
Kirchhoff's Laws and Circuit Analysis (EC ) Circuit analysis: solving for I and V at each element Linear circuits: involve resistors, capacitors, inductors Initial analysis uses only resistors Power sources,
More informationChapter 10: Sinusoidal SteadyState Analysis
Chapter 0: Sinusoidal SteadyState Analysis Sinusoidal Sources If a circuit is driven by a sinusoidal source, after 5 tie constants, the circuit reaches a steadystate (reeber the RC lab with t τ). Consequently,
More informationQUESTION BANK SUBJECT: NETWORK ANALYSIS (10ES34)
QUESTION BANK SUBJECT: NETWORK ANALYSIS (10ES34) NOTE: FOR NUMERICAL PROBLEMS FOR ALL UNITS EXCEPT UNIT 5 REFER THE EBOOK ENGINEERING CIRCUIT ANALYSIS, 7 th EDITION HAYT AND KIMMERLY. PAGE NUMBERS OF
More informationES250: Electrical Science. HW1: Electric Circuit Variables, Elements and Kirchhoff s Laws
ES250: Electrical Science HW1: Electric Circuit Variables, Elements and Kirchhoff s Laws Introduction Engineers use electric circuits to solve problems that are important to modern society, such as: 1.
More informationECE 1311: Electric Circuits. Chapter 2: Basic laws
ECE 1311: Electric Circuits Chapter 2: Basic laws Basic Law Overview Ideal sources series and parallel Ohm s law Definitions open circuits, short circuits, conductance, nodes, branches, loops Kirchhoff's
More informationNotes for course EE1.1 Circuit Analysis TOPIC 4 NODAL ANALYSIS
Notes for course EE1.1 Circuit Analysis 200405 TOPIC 4 NODAL ANALYSIS OBJECTIVES 1) To develop Nodal Analysis of Circuits without Voltage Sources 2) To develop Nodal Analysis of Circuits with Voltage
More informationHomework 3 Solution. Due Friday (5pm), Feb. 14, 2013
University of California, Berkeley Spring 2013 EE 42/100 Prof. K. Pister Homework 3 Solution Due Friday (5pm), Feb. 14, 2013 Please turn the homework in to the drop box located next to 125 Cory Hall (labeled
More informationE40M Review  Part 1
E40M Review Part 1 Topics in Part 1 (Today): KCL, KVL, Power Devices: V and I sources, R Nodal Analysis. Superposition Devices: Diodes, C, L Time Domain Diode, C, L Circuits Topics in Part 2 (Wed): MOSFETs,
More informationElectrical Circuits I Lecture 8
Electrical Circuits I Lecture 8 Thevenin and Norton theorems Thevenin theorem tells us that we can replace the entire network, exclusive of the load resistor, by an equivalent circuit
More informationNETWORK ANALYSIS AND NETWORK THEORMS
ASSIGNMENT NETWORK ANALYSIS AND NETWORK THEORMS VAIBHAV SHARMA Adm.no.2012MS0044 Network Analysis and Theorems Generally speaking, network analysis is any structured technique used to mathematically analyze
More informationSinusoids and Phasors
CHAPTER 9 Sinusoids and Phasors We now begins the analysis of circuits in which the voltage or current sources are timevarying. In this chapter, we are particularly interested in sinusoidally timevarying
More informationModule 2. DC Circuit. Version 2 EE IIT, Kharagpur
Module 2 DC Circuit Lesson 5 Nodevoltage analysis of resistive circuit in the context of dc voltages and currents Objectives To provide a powerful but simple circuit analysis tool based on Kirchhoff s
More informationSeries Parallel Analysis of AC Circuits
HAE 9 eries arallel Analysis of A ircuits hapter Outline 9. A eries ircuits 9.2 A arallel ircuits 9.3 A eries arallel ircuits 9.4 Analysis of Multipleource A ircuits Using uperposition 9. A EIE IUI In
More informationELEC273 Lecture Notes Set 11 AC Circuit Theorems
ELEC273 Lecture Notes Set C Circuit Theorems The course web site is: http://users.encs.concordia.ca/~trueman/web_page_273.htm Final Exam (confirmed): Friday December 5, 207 from 9:00 to 2:00 (confirmed)
More informationTransient Analysis of FirstOrder Circuits: Approaches and Recommendations
Transient Analysis of FirstOrder Circuits: Approaches and Recommendations Khalid AlOlimat Heath LeBlanc ECCS Department ECCS Department Ohio Northern University Ohio Northern University Ada, OH 45810
More informationENGR 2405 Class No Electric Circuits I
ENGR 2405 Class No. 48056 Electric Circuits I Dr. R. Williams Ph.D. rube.williams@hccs.edu Electric Circuit An electric circuit is an interconnec9on of electrical elements Charge Charge is an electrical
More informationDiscussion Question 6A
Discussion Question 6 P212, Week 6 Two Methods for Circuit nalysis Method 1: Progressive collapsing of circuit elements In last week s discussion, we learned how to analyse circuits involving batteries
More informationE1.1 Analysis of Circuits ( ) Revision Lecture 1 1 / 13
RevisionLecture 1: E1.1 Analysis of Circuits (20144530) Revision Lecture 1 1 / 13 Format Question 1 (40%): eight short parts covering the whole syllabus. Questions 2 and 3: single topic questions (answer
More informationHomework 2. Due Friday (5pm), Feb. 8, 2013
University of California, Berkeley Spring 2013 EE 42/100 Prof. K. Pister Homework 2 Due Friday (5pm), Feb. 8, 2013 Please turn the homework in to the drop box located next to 125 Cory Hall (labeled EE
More informationElectronics II. Final Examination
The University of Toledo f17fs_elct27.fm 1 Electronics II Final Examination Problems Points 1. 11 2. 14 3. 15 Total 40 Was the exam fair? yes no The University of Toledo f17fs_elct27.fm 2 Problem 1 11
More informationStudy Notes on Network Theorems for GATE 2017
Study Notes on Network Theorems for GATE 2017 Network Theorems is a highly important and scoring topic in GATE. This topic carries a substantial weight age in GATE. Although the Theorems might appear to
More informationMAE140  Linear Circuits  Fall 14 Midterm, November 6
MAE140  Linear Circuits  Fall 14 Midterm, November 6 Instructions (i) This exam is open book. You may use whatever written materials you choose, including your class notes and textbook. You may use a
More informationRC, RL, and LCR Circuits
RC, RL, and LCR Circuits EK307 Lab Note: This is a two week lab. Most students complete part A in week one and part B in week two. Introduction: Inductors and capacitors are energy storage devices. They
More informationBasic Electrical Circuits Analysis ECE 221
Basic Electrical Circuits Analysis ECE 221 PhD. Khodr Saaifan http://trsys.faculty.jacobsuniversity.de k.saaifan@jacobsuniversity.de 1 2 Reference: Electric Circuits, 8th Edition James W. Nilsson, and
More informationSchedule. ECEN 301 Discussion #20 Exam 2 Review 1. Lab Due date. Title Chapters HW Due date. Date Day Class No. 10 Nov Mon 20 Exam Review.
Schedule Date Day lass No. 0 Nov Mon 0 Exam Review Nov Tue Title hapters HW Due date Nov Wed Boolean Algebra 3. 3.3 ab Due date AB 7 Exam EXAM 3 Nov Thu 4 Nov Fri Recitation 5 Nov Sat 6 Nov Sun 7 Nov Mon
More information(amperes) = (coulombs) (3.1) (seconds) Time varying current. (volts) =
3 Electrical Circuits 3. Basic Concepts Electric charge coulomb of negative change contains 624 0 8 electrons. Current ampere is a steady flow of coulomb of change pass a given point in a conductor in
More informationElectric Circuits I FINAL EXAMINATION
EECS:300, Electric Circuits I s6fs_elci7.fm  Electric Circuits I FINAL EXAMINATION Problems Points.. 3. 0 Total 34 Was the exam fair? yes no 5//6 EECS:300, Electric Circuits I s6fs_elci7.fm  Problem
More informationElectrical Engineering Technology
Electrical Engineering Technology 1 ECET 17700  DAQ & Control Systems Lecture # 9 Loading, Thévenin Model & Norton Model Professors Robert Herrick & J. Michael Jacob Module 1 Circuit Loading Lecture 9
More information