CIRCUIT ANALYSIS TECHNIQUES


 Lorin Jacobs
 2 years ago
 Views:
Transcription
1 APPENDI B CIRCUIT ANALSIS TECHNIQUES The following methods can be used to combine impedances to simplify the topology of an electric circuit. Also, formulae are given for voltage and current division across/through impedances. SERIES IMPEDANCES + PARALLEL IMPEDANCES + DELTATOWE TRANSFORMATION Z C + + Z C Z C + + ZC Z Z Z C + + ZC Industrial Power Distribution, Second Edition. Ralph E. Fehr, III The Institute of Electrical and Electronics Engineers, Inc. Published 2016 by John Wiley & Sons, Inc. 361
2 362 APPENDI B CIRCUIT ANALSIS TECHNIQUES WETODELTA TRANSFORMATION Z Z + Z Z + Z Z A B B C C A Z Z + Z Z + Z Z A B B C C A Z C Z C Z Z Z Z + Z Z + Z Z A B B C C A (a) (b) VOLTAGE DIVIDER V V B V B = V + CURRENT DIVIDER I I B I B = I + MESHCURRENT ANALSIS A mesh is defined as any closed path through a planar circuit that contains no other closed paths. A planar circuit is one where no conductors cross over each other. To apply meshcurrent analysis to a nonplanar circuit, it must first be redrawn as a planar circuit. Example of a mesh
3 MESHCURRENT ANALSIS 363 Closed path Closed path Not a mesh Per Kirchhoff s voltage law (KVL), the sum of the voltage rises around any mesh must equal the sum of the voltage drops around the same mesh. This is an extension of the law of conservation of energy. Meshcurrent analysis leads to a system of n equations with n unknowns, where n is the number of meshes in the circuit. Matrix methods, such as Cramer s rule, are helpful to solve these systems of equations. Variables are assigned to each of the mesh currents. While the direction of the mesh current is arbitrary, clockwise currents are often assumed for uniformity. Meshcurrent analysis can be applied to the following circuit to produce the equations shown _ + _ i 1 i 2 i Writing the equation for the mesh defined by mesh current i 1, = (12 78 )i 1 +(27 66 )(i 1 i 2 ) The algebraic sign of the voltage source is determined by the terminal from which one leaves the source while traversing the mesh. Traversing the i 1 mesh clockwise, one leaves the positive terminal of the source, so is positive. Distributing the and simplifying algebraically by combining like terms, Similarly the second mesh gives which simplifies to ( )i 1 +( )i 2 = = (15 84 )i 2 + (33 73 )(i 2 i 3 ) + (27 66 )(i 2 i 1 ) The third mesh yields which simplifies to ( )i 1 + (75 73 )i 2 + ( )i 3 = = (14 62 )i 3 + (33 73 )(i 3 i 2 ) ( )i 2 + (47 70 )i 3 =
4 364 APPENDI B CIRCUIT ANALSIS TECHNIQUES Note that while traversing the third mesh clockwise, the negative terminal of the source is exited, so is used in the equation. These three equations can be written as a single matrix equation to facilitate implementation of a linear algebra solution method such as Cramer s Rule: ( )i 1 + ( )i 2 = ( )i 1 + (75 73 )i 2 + ( )i 3 = 0 ( )i 2 + (47 70 )i 3 = i 1 i 2 i = When a current source is present, an expression for the voltage across the current source cannot be written. Defining a supermesh that avoids the current source avoids the problem of not being able to express the voltage across the current source. A supermesh is not a mesh since it contains at least one closed path, but KVL applies to all closed paths through a circuit, not just meshes. An additional equation must be written addressing the current source (and any other circuit elements bypassed by the supermesh). The following circuit illustrates the use of a supermesh _ + _ i 1 i 2 i The equation for the first mesh can be written as usual: = (12 78 )i 1 + (27 66 )(i 1 i 2 ) ( )i 1 + ( )i 2 = Writing mesh equations for the second and third meshes would involve the current source, so the second two meshes are combined into a supermesh as follows: = (15 84 )i 2 + (14 62 )i 3 + (27 66 )(i 2 i 1 ) ( )i 1 + (42 72 )i 2 + (14 62 )i 3 = The third equation is developed from the branch avoided by the supermesh the branch containing the current source. By examining that branch, we can see by inspection that i 3 i 2 =
5 NODEVOLTAGE ANALSIS 365 These three equations can be written as a single matrix equation as follows: ( )i 1 + ( )i 2 = ( )i 1 + (42 72 )i 2 + (14 62 )i 3 = i 2 + i 3 = i 1 i 2 i NODEVOLTAGE ANALSIS A node is defined as any closed path enclosing part of a circuit. When the size of a node approaches zero, it becomes a single point. Nodes Examples of nodes Per Kirchhoff s current law (KCL), the sum of the currents entering a node must equal the sum of the currents exiting that node. This is an extension of the law of conservation of charge. Nodevoltage analysis leads to a system of n equations with n unknowns, where n + 1 is the number of nodes in the circuit. The equation for the last node (the reference node) is not linearly independent of the first n equations, so it is not necessary. Matrix methods, such as Cramer s rule, are helpful to solve these systems of equations. Variables are assigned to represent the node voltages. Then, KCL is applied to each labeled node, assuming a zero voltage reference at the reference node at the bottom of the circuit v
6 366 APPENDI B CIRCUIT ANALSIS TECHNIQUES At the node labeled v 1, a current of 3 0 enters from the left. Assuming the other two currents exit the node, the following node equation can be written: 3 0 = v v Combining the v 1 terms and taking the reciprocals of the denominators, v = 3 0. A current of enters node from the right. Assuming the other two currents exit the node, the following node equation can be written: = v Combining the terms and taking the reciprocals of the denominators, v = These two equations can be written as a single matrix equation: v = v = [ ] [ ] [ ] v1 3 0 = When a voltage source is encountered during nodevoltage analysis, the current through the voltage source cannot be expressed v 1 _ Creating a supernode that encompasses the voltage source will allow the nodevoltage process to be used v 1 _
7 EVALUATING DETERMINANTS 367 Now, a single node equation can be written for the supernode, assuming the currents through the and impedances leave the supernode v = Simplifying the equation above, v = Since the supernode equation contains two unknowns, a second equation must be written. By examining the supernode, it can be seen that v 1 = These two equations can be written as a single matrix equation: v = v 1 + = [ ][ ] [ v = ]. EVALUATING DETERMINANTS The determinant of any square matrix can be evaluated in a number of ways. The determinant of a 2 2 matrix is defined as [ ] a b det = a b c d c d = ad bc. For a 3 3 matrix, the pattern that defines the 2 2 determinant can be expanded as follows: a b c d e f = (aek + bfg + cdh) (ceg + bdk + afh). g h k An alternative method is to decompose the larger determinant to 2 2 determinants using a process called minoring. Minoring must be used to evaluate determinants larger than 3 3. One way to apply minoring to find a 3 3 determinant is a b c d e f g h k = a e f h k b d f g k + c d e g h. Minoring can be done on any row or column. In the example above, minoring was done on the first row. It is advantageous to minor on a row or column that contains zeroes, since a zero entry will eliminate a term in the expansion. A minor has a dimension one less than the original determinant. Minoring can be repeated until every determinant is a 2 2 determinant. In the example above, when the a element is minored, the resulting minor is the 2 2 determinant remaining
8 368 APPENDI B CIRCUIT ANALSIS TECHNIQUES when the row and column containing the a element is eliminated. The minor must be multiplied by ( 1) i+j, where i and j are the row and column of the minored element in the original determinant. This term causes the algebraic sign of each term in the minor expansion to alternate. CRAMER S RULE This powerful linear algebra technique is useful for solving systems of equations, particularly when the coefficients are complex. For example, a 3 3 system of equations can be written as a matrix equation: ax + by + cz = m a b c x m dx + ey + fz = n d e f y = n gx + hy + kz = p g h k x p Four determinants can be defined: the first being the determinant of the coefficient matrix, and the next three being the determinant of the coefficient matrix with the constant vector substituted for one of the columns: a b c m b c a m c a b m D = d e f A = n e f B = d n f C = d e n g h k p h k g p k g h p The solution for the system of equations is x = A D, y = B D, z = C D. Cramer s rule can be extended to handle any size system of equations, and evaluation of the determinants can be done easily with software.
Module 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 informationElectric Circuits I. Nodal Analysis. Dr. Firas Obeidat
Electric Circuits I Nodal Analysis Dr. Firas Obeidat 1 Nodal Analysis Without Voltage Source Nodal analysis, which is based on a systematic application of Kirchhoff s current law (KCL). A node is defined
More informationCURRENT SOURCES EXAMPLE 1 Find the source voltage Vs and the current I1 for the circuit shown below SOURCE CONVERSIONS
CURRENT SOURCES EXAMPLE 1 Find the source voltage Vs and the current I1 for the circuit shown below EXAMPLE 2 Find the source voltage Vs and the current I1 for the circuit shown below SOURCE CONVERSIONS
More informationEE40 KVL KCL. Prof. Nathan Cheung 09/01/2009. Reading: Hambley Chapter 1
EE40 KVL KCL Prof. Nathan Cheung 09/01/2009 Reading: Hambley Chapter 1 Slide 1 Terminology: Nodes and Branches Node: A point where two or more circuit elements are connected Branch: A path that connects
More information6. MESH ANALYSIS 6.1 INTRODUCTION
6. MESH ANALYSIS INTRODUCTION PASSIVE SIGN CONVENTION PLANAR CIRCUITS FORMATION OF MESHES ANALYSIS OF A SIMPLE CIRCUIT DETERMINANT OF A MATRIX CRAMER S RULE GAUSSIAN ELIMINATION METHOD EXAMPLES FOR MESH
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 informationUnit 1 Matrices Notes Packet Period: Matrices
Algebra 2/Trig Unit 1 Matrices Notes Packet Name: Period: # Matrices (1) Page 203 204 #11 35 Odd (2) Page 203 204 #12 36 Even (3) Page 211 212 #4 6, 17 33 Odd (4) Page 211 212 #12 34 Even (5) Page 218
More informationmywbut.com Mesh Analysis
Mesh Analysis 1 Objectives Meaning of circuit analysis; distinguish between the terms mesh and loop. To provide more general and powerful circuit analysis tool based on Kirchhoff s voltage law (KVL) only.
More informationChapter 3 Methods of Analysis: 1) Nodal Analysis
Chapter 3 Methods of Analysis: 1) Nodal Analysis Dr. Waleed AlHanafy waleed alhanafy@yahoo.com Faculty of Electronic Engineering, Menoufia Univ., Egypt MSA Summer Course: Electric Circuit Analysis I (ESE
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 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 informationSystematic Circuit Analysis (T&R Chap 3)
Systematic Circuit Analysis (T&R Chap 3) Nodevoltage analysis Using the voltages of the each node relative to a ground node, write down a set of consistent linear equations for these voltages Solve this
More informationA2H Assignment #8 Cramer s Rule Unit 2: Matrices and Systems. DUE Date: Friday 12/2 as a Packet. 3x 2y = 10 5x + 3y = 4. determinant.
A2H Assignment #8 Cramer s Rule Unit 2: Matrices and Systems Name: DUE Date: Friday 12/2 as a Packet What is the Cramer s Rule used for? à Another method to solve systems that uses matrices and determinants.
More informationModule 2. DC Circuit. Version 2 EE IIT, Kharagpur
Module DC Circuit Lesson 4 Loop Analysis of resistive circuit in the context of dc voltages and currents Objectives Meaning of circuit analysis; distinguish between the terms mesh and loop. To provide
More informationEE292: Fundamentals of ECE
EE292: Fundamentals of ECE Fall 2012 TTh 10:0011:15 SEB 1242 Lecture 4 120906 http://www.ee.unlv.edu/~b1morris/ee292/ 2 Outline Review Voltage Divider Current Divider NodeVoltage Analysis 3 Network Analysis
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 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 informationLecture 3 BRANCHES AND NODES
Lecture 3 Definitions: Circuits, Nodes, Branches Kirchoff s Voltage Law (KVL) Kirchoff s Current Law (KCL) Examples and generalizations RC Circuit Solution 1 Branch: BRANCHES AND NODES elements connected
More informationELECTRICAL THEORY. Ideal Basic Circuit Element
ELECTRICAL THEORY PROF. SIRIPONG POTISUK ELEC 106 Ideal Basic Circuit Element Has only two terminals which are points of connection to other circuit components Can be described mathematically in terms
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 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 informationANNOUNCEMENT ANNOUNCEMENT
ANNOUNCEMENT Exam : Tuesday September 25, 208, 8 PM  0 PM Location: Elliott Hall of Music (see seating chart) Covers all readings, lectures, homework from Chapters 2 through 23 Multiple choice (58 questions)
More informationElectrical Technology (EE101F)
Electrical Technology (EE101F) Contents Series & Parallel Combinations KVL & KCL Introduction to Loop & Mesh Analysis Frequently Asked Questions NPTEL Link SeriesParallel esistances 1 V 3 2 There are
More informationMTH 306 Spring Term 2007
MTH 306 Spring Term 2007 Lesson 3 John Lee Oregon State University (Oregon State University) 1 / 27 Lesson 3 Goals: Be able to solve 2 2 and 3 3 linear systems by systematic elimination of unknowns without
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 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 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 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 informationNodal and Loop Analysis Techniques
IRW3652.I ALL 522 3:53 Page 65 Nodal and Loop Analysis Techniques LEARNING Goals In Chapter 2 we analyzed the simplest possible circuits, those containing only a singlenode pair or a single loop.
More informationMAE140  Linear Circuits  Winter 09 Midterm, February 5
Instructions MAE40  Linear ircuits  Winter 09 Midterm, February 5 (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 informationQUIZ 1 SOLUTION. One way of labeling voltages and currents is shown below.
F 14 1250 QUIZ 1 SOLUTION EX: Find the numerical value of v 2 in the circuit below. Show all work. SOL'N: One method of solution is to use Kirchhoff's and Ohm's laws. The first step in this approach is
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 informationChapter 3. Loop and Cutset Analysis
Chapter 3. Loop and Cutset Analysis By: FARHAD FARADJI, Ph.D. Assistant Professor, Electrical Engineering, K.N. Toosi University of Technology http://wp.kntu.ac.ir/faradji/electriccircuits2.htm References:
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 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 informationChapter 2 Resistive Circuits
1. Sole circuits (i.e., find currents and oltages of interest) by combining resistances in series and parallel. 2. Apply the oltagediision and currentdiision principles. 3. Sole circuits by the nodeoltage
More informationMassachusetts Institute of Technology Department of Electrical Engineering and Computer Science : Circuits & Electronics Problem Set #1 Solution
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.2: Circuits & Electronics Problem Set # Solution Exercise. The three resistors form a series connection.
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 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 informationA Primer on Solving Systems of Linear Equations
A Primer on Solving Systems of Linear Equations In Signals and Systems, as well as other subjects in Unified, it will often be necessary to solve systems of linear equations, such as x + 2y + z = 2x +
More informationECE 2100 Circuit Analysis
ECE 2100 Circuit Analysis Lesson 3 Chapter 2 Ohm s Law Network Topology: nodes, branches, and loops Daniel M. Litynski, Ph.D. http://homepages.wmich.edu/~dlitynsk/ esistance ESISTANCE = Physical property
More informationDC CIRCUIT ANALYSIS. Loop Equations
All of the rules governing DC circuits that have been discussed so far can now be applied to analyze complex DC circuits. To apply these rules effectively, loop equations, node equations, and equivalent
More informationSystems of Equations
Prerequisites: Solving simultaneous equations in 2 variables; equation and graph of a straight line. Maths Applications: Finding circle equations; finding matrix inverses; intersections of lines and planes.
More informationProject 4: Introduction to Circuits The attached Project was prepared by Professor YihFang Huang from the department of Electrical Engineering.
Project 4: Introduction to Circuits The attached Project was prepared by Professor YihFang Huang from the department of Electrical Engineering. The examples given are example of basic problems that you
More informationNetwork Graphs and Tellegen s Theorem
Networ Graphs and Tellegen s Theorem The concepts of a graph Cut sets and Kirchhoff s current laws Loops and Kirchhoff s voltage laws Tellegen s Theorem The concepts of a graph The analysis of a complex
More informationWriting Circuit Equations
2 C H A P T E R Writing Circuit Equations Objectives By the end of this chapter, you should be able to do the following: 1. Find the complete solution of a circuit using the exhaustive, node, and mesh
More informationD.C.CIRCUITS. charged negatively if it has excess of electrons. The charge is measured in Coulombs and
D.C.CRCUTS Electrical /Quantities Definitions, Symbols and / Units Charge: A body is said to be changed positively, if it has deficit of electrons. t is said to be charged negatively if it has excess of
More informationELEC4612 Power System Analysis Power Flow Analysis
ELEC462 Power Sstem Analsis Power Flow Analsis Dr Jaashri Ravishankar jaashri.ravishankar@unsw.edu.au Busbars The meeting point of various components of a PS is called bus. The bus or busbar is a conductor
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 informationLinear Algebra and Vector Analysis MATH 1120
Faculty of Engineering Mechanical Engineering Department Linear Algebra and Vector Analysis MATH 1120 : Instructor Dr. O. Philips Agboola Determinants and Cramer s Rule Determinants If a matrix is square
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 informationChapter 2 Resistive Circuits
Chapter esistie Circuits Goal. Sole circuits by combining resistances in Series and Parallel.. Apply the VoltageDiision and CurrentDiision Principles.. Sole circuits by the NodeVoltage Technique.. Sole
More informationLinear algebra and differential equations (Math 54): Lecture 8
Linear algebra and differential equations (Math 54): Lecture 8 Vivek Shende February 11, 2016 Hello and welcome to class! Last time We studied the formal properties of determinants, and how to compute
More informationNetwork Analysis V. Mesh Equations Three Loops
Network Analysis V Mesh Equations Three Loops Circuit overview A B V1 12 V R1 R3 C R2 R4 I A I B I C D R6 E F R5 R7 R8 G V2 8 V H Using the method of mesh currents, solve for all the unknown values of
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 informationIntroductory Circuit Analysis
Introductory Circuit Analysis CHAPTER 6 Parallel dc Circuits OBJECTIVES Become familiar with the characteristics of a parallel network and how to solve for the voltage, current, and power to each element.
More informationResistor. l A. Factors affecting the resistance are 1. Crosssectional area, A 2. Length, l 3. Resistivity, ρ
Chapter 2 Basic Laws. Ohm s Law 2. Branches, loops and nodes definition 3. Kirchhoff s Law 4. Series resistors circuit and voltage division. 5. Equivalent parallel circuit and current division. 6. WyeDelta
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 informationdet(ka) = k n det A.
Properties of determinants Theorem. If A is n n, then for any k, det(ka) = k n det A. Multiplying one row of A by k multiplies the determinant by k. But ka has every row multiplied by k, so the determinant
More informationvtusolution.in Initial conditions Necessity and advantages: Initial conditions assist
Necessity and advantages: Initial conditions assist Initial conditions To evaluate the arbitrary constants of differential equations Knowledge of the behavior of the elements at the time of switching Knowledge
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 informationPreamble. Circuit Analysis II. Mesh Analysis. When circuits get really complex methods learned so far will still work,
Preamble Circuit Analysis II Physics, 8 th Edition Custom Edition Cutnell & Johnson When circuits get really complex methods learned so far will still work, but they can take a long time to do. A particularly
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 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 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 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 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 information6.4 Determinants and Cramer s Rule
6.4 Determinants and Cramer s Rule Objectives Determinant of a 2 x 2 Matrix Determinant of an 3 x 3 Matrix Determinant of a n x n Matrix Cramer s Rule If a matrix is square (that is, if it has the same
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 informationSome Notes on Linear Algebra
Some Notes on Linear Algebra prepared for a first course in differential equations Thomas L Scofield Department of Mathematics and Statistics Calvin College 1998 1 The purpose of these notes is to present
More informationDesigning Information Devices and Systems II Fall 2016 Murat Arcak and Michel Maharbiz Homework 0. This homework is due August 29th, 2016, at Noon.
EECS 16B Designing Information Devices and Systems II Fall 2016 Murat Arcak and Michel Maharbiz Homework 0 This homework is due August 29th, 2016, at Noon. 1. Homework process and study group (a) Who else
More informationR R V I R. Conventional Current. Ohms Law V = IR
DC Circuits opics EMF and erminal oltage esistors in Series and in Parallel Kirchhoff s ules EMFs in Series and in Parallel Capacitors in Series and in Parallel Ammeters and oltmeters Conventional Current
More information7.6 The Inverse of a Square Matrix
7.6 The Inverse of a Square Matrix Copyright Cengage Learning. All rights reserved. What You Should Learn Verify that two matrices are inverses of each other. Use GaussJordan elimination to find inverses
More informationMethods of Solution of Linear Simultaneous Equations
Methods of Solution of Linear Simultaneous Equations ET151 Circuits 1 Professor Fiore, jfiore@mvcc.edu Several circuit analysis methods such as branch current analysis, mesh analysis and nodal analysis,
More informationMTH 215: Introduction to Linear Algebra
MTH 215: Introduction to Linear Algebra Lecture 5 Jonathan A. Chávez Casillas 1 1 University of Rhode Island Department of Mathematics September 20, 2017 1 LU Factorization 2 3 4 Triangular Matrices Definition
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 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 informationarxiv: v2 [mathph] 23 Jun 2014
Note on homological modeling of the electric circuits Eugen Paal and Märt Umbleja arxiv:1406.3905v2 [mathph] 23 Jun 2014 Abstract Based on a simple example, it is explained how the homological analysis
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 informationChapter 2 Analysis Methods
Chapter Analysis Methods. Nodal Analysis Problem.. Two current sources with equal internal resistances feed a load as shown in Fig... I a ¼ 00 A; I b ¼ 00 A; R ¼ 00 X; R L ¼ 00 X: (a) Find the current
More informationMAE140 Linear Circuits Fall 2016 Final, December 6th Instructions
MAE40 Linear Circuits Fall 206 Final, December 6th Instructions. This exam is open book. You may use whatever written materials you choose, including your class notes and textbook. You may use a handheld
More informationEngineering Fundamentals and Problem Solving, 6e
Engineering Fundamentals and Problem Solving, 6e Chapter 17 Electrical Circuits Chapter Objectives Compute the equivalent resistance of resistors in series and in parallel Apply Ohm s law to a resistive
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 informationSolving Systems of Linear Equations. Classification by Number of Solutions
Solving Systems of Linear Equations Case 1: One Solution Case : No Solution Case 3: Infinite Solutions Independent System Inconsistent System Dependent System x = 4 y = Classification by Number of Solutions
More informationANALYTICAL MATHEMATICS FOR APPLICATIONS 2018 LECTURE NOTES 3
ANALYTICAL MATHEMATICS FOR APPLICATIONS 2018 LECTURE NOTES 3 ISSUED 24 FEBRUARY 2018 1 Gaussian elimination Let A be an (m n)matrix Consider the following row operations on A (1) Swap the positions any
More informationBasics of Network Theory (PartI)
Basics of Network Theory (PartI). A square waveform as shown in figure is applied across mh ideal inductor. The current through the inductor is a. wave of peak amplitude. V 0 0.5 t (m sec) [Gate 987: Marks]
More informationBASIC NETWORK ANALYSIS
SECTION 1 BASIC NETWORK ANALYSIS A. Wayne Galli, Ph.D. Project Engineer Newport News Shipbuilding SeriesParallel dc Network Analysis......................... 1.1 BranchCurrent Analysis of a dc Network......................
More informationDeterminants and Cramer's Rule
eterminants and ramer's Rule This section will deal with how to find the determinant of a square matrix. Every square matrix can be associated with a real number known as its determinant. The determinant
More informationDC NETWORK THEOREMS. Learning Objectives
C H A P T E R 2 Learning Objectives Electric Circuits and Network Theorems Kirchhoff s Laws Determination of Voltage Sign Assumed Direction of Current Solving Simultaneous Equations Determinants Solving
More information48520 Electronics & Circuits: Web Tutor
852 Electronics & Circuits: Web Tutor Topic : Resistive Circuits 2 Help for Exercise.: Nodal Analysis, circuits with I, R and controlled sources. The purpose of this exercise is to further extend Nodal
More informationChapter 2 Direct Current Circuits
Chapter 2 Direct Current Circuits 2.1 Introduction Nowadays, our lives are increasingly dependent upon the availability of devices that make extensive use of electric circuits. The knowledge of the electrical
More informationProceedings of the 34th Annual Conference on Information Sciences and Systems (CISS 2000), Princeton, New Jersey, March, 2000
2000 Conference on Information Sciences and Systems, Princeton University, March 1517, 2000 Proceedings of the 34th Annual Conference on Information Sciences and Systems (CISS 2000), Princeton, New Jersey,
More informationElectrical Eng. fundamental Lecture 1
Electrical Eng. fundamental Lecture 1 Contact details: helhelw@staffs.ac.uk Introduction Electrical systems pervade our lives; they are found in home, school, workplaces, factories,
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 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 informationIn this lecture, we will consider how to analyse an electrical circuit by applying KVL and KCL. As a result, we can predict the voltages and currents
In this lecture, we will consider how to analyse an electrical circuit by applying KVL and KCL. As a result, we can predict the voltages and currents around an electrical circuit. This is a short lecture,
More informationElectrical Circuits I
Electrical Circuits I This lecture discusses the mathematical modeling of simple electrical linear circuits. When modeling a circuit, one ends up with a set of implicitly formulated algebraic and differential
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 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 information