ON THE RESISTIVE FUNCTION MEASURED BETWEEN TWO POINTS ON A GRID OR A LATTICE OF SIMILAR NON-LINEAR RESISTORS
|
|
- Annice Jones
- 5 years ago
- Views:
Transcription
1 INTERNATIONAL JOURNAL OF CIRCUIT THEORY AND APPLICATIONS LETTER TO THE EDITOR ON THE RESISTIVE FUNCTION MEASURED BETWEEN TWO POINTS ON A GRID OR A LATTICE OF SIMILAR NON-LINEAR RESISTORS EMANUEL GLUSKIN* Department of Electrical and Computer Engineering, Ben-Gurion University of the Negev, Beer-Sheva, Israel SUMMARY The known problem of the resistance measured between two closest nodes on a grid of similar resistors, often considered in a linear version in physics and electrical engineering textbooks, is considered in a non-linear version. An estimation, based on a simple polynomial model, shows that with increase in the number of the dimensions of the grid (or a lattice), the non-linearity of the characteristic of a single resistor is expressed more and more weakly in the input resistive function, and that the input resistive function has the same (in the sense of the degrees involved and polarities of the coefficients) polynomial structure as the individual characteristic of the single resistor John Wiley & Sons, Ltd. 1. INTRODUCTION In References 1 5 the well-known problem of the input resistance of an infinite square grid of similar linear resistors is considered. The input resistance is measured between a pair of close nodes which are denoted as P and Q. The input resistance is found in References 1 and 2 (see also References 3 6 for some generalizations of the linear problem) by means of an ingenious use of a superposition procedure, in which in each step the input current is supplied to only one of the points, P or Q, (and a point at infinity is involved), and thus the local electrical current distribution is symmetrical in each step, so that the computational problem of the infinity of the grid is overcome. The superposition procedure was possible because of the linearity of the system presented by the grid. Our intention here is to consider this problem in the context of a non-linear resistive individual characteristic, which is both mathematically and methodologically interesting, and may be helpful in, e.g. using the relatively easily measurable non-linear input resistive function of the grid for obtaining information about the distribution of the currents in the grid. Not seeking a complete solution, which, in a constructive form, would be very cumbersome in the non-linear case, we shall see in a simple approximation associated with a cut grid, how a non-linear term added to the voltage current (v i) characteristic of the resistors from which the grid is composed qualitatively influences the input voltage current characteristic of the grid. Some estimations and conclusions will be obtained also for the infinite non-linear grid. * Correspondence to: E. Gluskin, Department of Electrical and Computer Engineering, Ben-Gurion University of the Negev, Beer-Sheva, Israel. A more detailed work by the author on this topic is currently in preparation. CCC /98/ $17.50 Received 20 May John Wiley & Sons, Ltd. Revised 19 February 1997
2 208 LETTER TO THE EDITOR It will be somewhat more suitable to consider at the start not the resistivity v i, but the conductivity i v characteristic of the single resistor: i"f (v) (v and i relate to a single resistor) (1) Because of the non-linearity of the resistors, the input (between P and Q) conductivity characteristic of the grid is also a non-linear function of the input voltage i"ψ(v) (v and i relate to the input of the grid) We shall see that there is an analytical self-similarity of the grid which means that for a polynomial model of f (v), ψ(v) includes the same powers of v asf(v). The coefficients before the powers are generally different to the corresponding ones in f (v), but are of the same polarity. As a simple model we shall consider the individual characteristic f (v)"gv#εv (1a) If ε is small, the inverse resistivity characteristic v"f (i) will be approximately v+ri!εri with R"g. Introducing the notation α"!εr, we write this as v"ri#αi. Apart from the inversion of f (v), and apart from the fact that approximations of the type (1a) are usually physically good only for a small ε, there is no formal reason in what follows for ε to be small. Dealing with cut grids/lattices below, we shall obtain in our calculations expressions for the input characteristic in the form ψ(v)"a(gv#bεv) (2) with precisely calculated constants A and B which are, generally, different for different cuttings of the grid. The form (2) is correct also for the infinite grid, and we can find A precisely for the infinite grid, by noting that for any certain (here infinite) grid, setting ε"0 leads to a linear grid which can be easily computed. As for B, we shall be able to give for the infinite case, only an estimation for this parameter, based on a half-empirical assumption. Our main focus here is B, as associated with the non-linearity. We therefore shall concentrate our attention first of all on the relation ψ(v)&(gv#bεv) finding in some cases B to be much smaller than the value B"1 in the individual characteristic f (v)"gv#εv, which means a relative linearity of ψ(v), compared to f (v), which is an important point for any application Calculation for a cut grid 2. AN ESTIMATION OF ψ(v) FOR A NON-LINEAR 2D-GRID For an estimation, we shall reduce (see Figure 1) the infinite 2D-grid to the resistor, to which the terminals of the voltage source are connected, and the two meshes which are closest to this resistor. That is, we consider a finite circuit which includes three parallel branches, connected to the nodes P and Q. Two of the branches include, each, 3 resistors, and the central branch includes one resistor. Applying the voltage v to the points P and Q, and noting that in the branches which include 3 resistors the voltage fall is equally distributed between the resistors, we obtain the currents in the branches, respectively, as f v 3, f (v), f v John Wiley & Sons, Ltd.
3 LETTER TO THE EDITOR 209 Figure 1. The cut grid for the 2D-case. We connect the source of voltage v to the points P and Q. The circuit is composed of non-linear similar resistors (conductors) which are described by the function i" f (v) and the total input current of the circuit is i"ψ(v)"f (v)#2f v 3 "gv#εv#2gv 3 #2ε v 3 " 5 3 gv#11 9 εv"5 3 gv#11 15 εv Thus in the 2D-case, for the cut grid, ψ(v)&(gv# εv) with B"11/15+0)733, i.e. with the relative role of the nonlinear term weaker than in the individual characteristic (1a). The precision of this ψ(v), associated with cutting the grid, (we consider first A in (2)) is seen if we transfer to the linear case, setting ε"0. Then we obtain i" gv+(1)67/r)v, instead of i"2gv"(2/r)v, asin References 1 5 for an infinite linear grid. Certainly, the cutting causes also some imprecision in B. This imprecision cannot be precisely estimated in the approach taken, but we can see that successive (iterating) cutting influences A more strongly than B, and we can find a half-empirical correction to the value of B which corresponds to a cut grid, which leads to an estimation of B for the infinite grid. We are essentially using that ψ(v) can include only the powers of v which appear in f (v). This follows from the fact that using Kirchhoff s equations, we perform only linear operations with the voltages and currents, and thus the transformation f (v)pψ(v) may be associated only with scaling changes, as we have seen for the 2D cut grid, and as we shall also see in Section 3 for a 3D cut grid John Wiley & Sons, Ltd.
4 210 LETTER TO THE EDITOR 2.2 Estimation of ψ(v) for infinite grid Taking larger pieces of the grid we always have, as is easy to understand, an increase in A (more paths for the current) and a decrease in B (less current in each resistor, and thus a more weakly revealed non-linearity). Passing from the most cut grid which includes only one resistor between P and Q to the cut grid considered above which includes 7 resistors, we transfer from A"1 and B"1 toa"and B". The value of A is changed more strongly than the value of B: compared to i.e. A A " 5 3!1 :1"2 3 B B " 11 15!1 :1"4 15 A A B B "2)5. Now using the fact that the transfer from the grid with 7 resistors to the infinite grid is associated with the change in A from to 2, and assuming that the ratio ( A/A)/ B/B is less changed, with the iterative approximations of the infinite lattice by increasing cut grids, than the relative changes in B or A by themselves, let us estimate the value of B for the infinite grid, B, using the value 2)5 for this ratio, and the value B" for 7-resistor grid for finding the addition ΔB (ΔB(0): B R "B# B+B! B"B1! B B +B 1! 1 A 2)5 A " !2 5 2!5 3 : 5 3 " "0)675 Thus, for the infinite 2D grid our estimation is or For ε small the inverse resistivity characteristic ψ (v)+2(gv#0)675εv) v"ψ (i)+r 2 i!0)675εr R 2 i + R i!0)169εri, 2 v"ψ (i)"r 2 i#0)169αi where we returned to the notations R"g and α"!εr, for which the inverse characteristic of the individual resistor is (for α small) v"ri#αi John Wiley & Sons, Ltd.
5 LETTER TO THE EDITOR THE ESTIMATION FOR A 3D-LATTICE Somewhat extending the traditional borders of the problem, let us also consider a 3D- grid, which is an orthogonal lattice with one non-linear resistor connected between each of the two nodes which are closest to each other. Six resistors are now connected to each node. The solution for the linear version of the infinite circuit is R "R/3, or g "3/R, which is easily obtained by the superposition procedure, quite similarly to that in References 1 5 for the 2D case Calculation of ψ(v) for the cut lattice and a conclusion about the role of the dimension Consider the cutting, which here also leaves only two different kinds of branches, we have for the 3D case (see Figure 2) four similar branches, each with 3 resistors, which are closest to the resistor connected in parallel with the source of the voltage v. For the cut lattice the input current is i"f (v)#4 f v 3 "gv#εv#4g v 3 #4ε v 3 " 7 3 gv#13 9 εv"7 3 gv#13 21 εv For the linear case we set ε"0, obtaining i" gv+(2)33/r)v for the cut lattice instead of 3gv"(3/R)v for the infinite linear lattice, which here too shows an indifferent precision in A, associated with the cutting of the lattice, but, nevertheless, since, now ψ(v)&(gv#εv), there is an important difference from the 2D case: in the 3D case the role of the non-linearity of a single resistor is more weakly expressed in ψ(v), B"13/21+0)619 instead of the value 0)733 in the 2D case. Noting, regarding this point, that for a 1D case, when the grid is precisely an infinite chain of the resistors, the input characteristic measured between two closest connections ( nodes ) of the chain is, clearly, the individual characteristic, ψ(v)"f (v) (i.e. B"1), and considering all of the values for B obtained for the cut grids of different dimensions: 1, 0)733, 0)619, we come to the conclusion that for the model chosen (and, actually, for any f (v) with a monotonic non-linearity), with an increase in the number of the dimensions of the grid, the role of the non-linearity of a single element from which the grid is composed is monotonically decreased as seen from the input. For the model (1a) this decrease is not as strong as Figure 2. The cut lattice for the 3-D case 1998 John Wiley & Sons, Ltd.
6 212 LETTER TO THE EDITOR the increase in the dimension. Similar conclusions relate to the comparison of the estimated input conductivity characteristics of infinite grids/lattices. 3.2 Estimation of ψ(v) for the infinite lattice Following the approach of Section 2.2, we estimate ψ(v) for the infinite 3D-grid. We consider that for the 3D-grid the transfer from the most cut lattice with one resistor to the cut 3D-lattice with 13 resistors, analysed in the above, is associated with changes in A from 1 to, and in B from 1to, i.e. the relative changes of the absolute values relate as A A B B "7/3!1 1 : 13/21!1 " "7 2 "3)5 This ratio is assumed to be changed weakly with transfer to larger pieces of lattice. Since the precise value A "3 yields for the transfer to the infinite lattice ΔA/A"(3!)/", we thus estimate B R " ! " ! 49 4 " )569 or Thus, for the 3D infinite grid our estimation for ψ(v) is ψ (v)+3(gv#0)569εv) For a small ε the inverse resistivity characteristic is written, using the notations R"g and α"!εr,as v"ψ "R 3 i!0)569 εr R 3 i v+ R 3 i#0)063αi 4. CONCLUSIONS There is an interesting analytical problem of calculating the grid/lattice of non-linear resistors, which is a generalization of the linear problem of References 1 6. This problem seems to be relevant to both topics of circuit theory and physical measurements, deserving attention of both physicists and circuit specialists. The analysis, devoted, first of all, to the mapping of the individual characteristic ( f ) of the resistor to the input characteristic (ψ) of the grid/lattice, fpψ was focused on the relative weights of the non-linear terms in f (v) and ψ(v) in models with cut grids, and on the possibility to simply estimate ψ(v) for the infinite grid/lattice. We summarized some of the results of the investigation in the Table I which includes the resistivity characteristics related to the case when the nonlinear term is much smaller than the linear one. We see from the table that the corresponding decreasing of the relative weight of the non-linear term with increases in the dimension of the grids are 1D: 1 : 1"1; 2D: 0)169 : 0)5"0 338; 3D: 0)063 : 0)3333" John Wiley & Sons, Ltd.
7 LETTER TO THE EDITOR 213 Table I The input resistivity characteristic of the infinite grid The individual resistivity 1D-case 2D-case 3D-case characteristic of resistors ν"ri#αi Ri#αi R 2 i#0)169αi R 3 i#0)063αi (precisely equals the individual (approximately) (approximately) characteristic) These weight values are analogies of the parameter B in equation (2), now applied to the resistive characteristic. Contrary to the case of conductivity characteristic, for the resistivity characteristic the decrease in this parameter is larger than the corresponding increase in the dimension of the grid. Though some of the derived conclusions clearly relate to any monotonic non-linear characteristic, the analysis was, in general, limited by (a) certain analytical model of f (v); (b) the use of a cut grid/lattice in precise calculations; (c) the assumption (which is especially important for the transfer to an infinite grid), based on (see Section 2.2) transfer from one particular cut grid to another, of a relatively weak change in the ratio of the relative changes in the parameters A and B with increase in the cut grid; and (d) consideration of only time-independent (algebraic) problem, ignoring inductive and capacitive features of realistic elements, which would be revealed for the input voltage dependent on time. Regarding the use of the finite polynomial conductivity characteristic f (v), it needs to be noted that the finite polynomial is turned, in principle, into an infinite series of powers for the inverse individual resistivity function. Corresponding to this, for v not small, the input resistivity function of the grid/lattice must also be given by an infinite series of powers. If we were, on the contrary, to postulate the individual resistivity characteristic as a finite polynomial, for v not small then the inverse individual conductivity characteristic would be given by an infinite series, and the input conductivity function of the grid/lattice also would be given by an infinite series. This injustice with respect to the analytical form of one of the characteristics, either conductive or resistive, gives some support to the usual use of analytical models in realistic physical problems only in the cases of weak non-linearity, when we can ignore the high powers of the argument in each characteristic. This is in contrast to the case of a singular non-linearity which may be taken as a strong one in a realistic problem, not causing significant analytical difficulties. (See e.g. the examples in References 7, 8 and 9). REFERENCES 1. E. M. Purcell, Electricity and Magnetism, McGraw-Hill, New York, 1965, p L. O. Chua, C. A. Desoer, and E. S. Kuh, inear and Nonlinear Circuits, McGraw-Hill, New York, 1987, pp A. H. Zemanian, A classical puzzle: the driving point resistances of infinite grids, IEEE Circuits Systems Mag., 7 9 (1984). 4. A. H. Zemanian, Infinite electrical networks, Proc. IEEE, 64, 6 17 (1976). 5. B. van der Pol and H. Bremmer, Operational Calculus Based on the ¹wo-Sided aplace ¹ransform, Cambridge Univ. Press, Cambridge, G. Venezian, On the resistance between two points on a grid, Am. J. Phys. 62, (1994). 7. L. O. Chua, Introduction to Nonlinear Network ¹heory, McGraw-Hill, New York, E. Gluskin, On the theory of an integral equation, Adv. in App. Math., 15, (1994). 9. E. Gluskin, Nonlinear systems: between a law and a definition, Reports on Progress in Physics, Vol. 60, no. 10 (Oct. 1997) John Wiley & Sons, Ltd.
On the symmetry features of some electrical circuits
INTERNATIONAL JOURNAL OF CIRCUIT THEORY AND APPLICATIONS Int. J. Circ. Theor. Appl. 2006; 34:637 644 Published online 26 September 2006 in Wiley InterScience (www.interscience.wiley.com)..377 On the symmetry
More informationCalculation of the general impedance between adjacent nodes of infinite uniform N-dimensional resistive, inductive, or capacitive lattices
A 2009-2517: ALULATON OF THE GENERAL MPEDANE BETWEEN ADJAENT NODES OF NFNTE UNFORM N-DMENSONAL RESSTVE, NDUTVE, OR APATVE LATTES Peter Osterberg, University of Portland Dr. Osterberg is an associate professor
More informationSOME MOTIVATING ARGUMENTS FOR TEACHING ELECTRICAL ENGINEERING STUDENTS
Far East Journal of Mathematical Education Volume 6, Number 1, 2011, Pages 65-80 This paper is available online at http://pphmj.com/journals/fjme.htm 2011 Pushpa Publishing House SOME MOTIVATING ARGUMENTS
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 informationTHE topic under discussion is very important in practice.
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 46, NO. 5, MAY 1999 529 Circuits and Systems Expositions The Fluorescent Lamp Circuit Emanuel Gluskin Abstract The
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 informationOutline. Week 5: Circuits. Course Notes: 3.5. Goals: Use linear algebra to determine voltage drops and branch currents.
Outline Week 5: Circuits Course Notes: 3.5 Goals: Use linear algebra to determine voltage drops and branch currents. Components in Resistor Networks voltage source current source resistor Components in
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 informationNONLINEAR DC ANALYSIS
ECE 552 Numerical Circuit Analysis Chapter Six NONLINEAR DC ANALYSIS OR: Solution of Nonlinear Algebraic Equations I. Hajj 2017 Nonlinear Algebraic Equations A system of linear equations Ax = b has a
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 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 informationBasic. Theory. ircuit. Charles A. Desoer. Ernest S. Kuh. and. McGraw-Hill 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 McGraw-Hill Book Company New York St. Louis San
More informationREUNotes08-CircuitBasics May 28, 2008
Chapter One Circuits (... introduction here... ) 1.1 CIRCUIT BASICS Objects may possess a property known as electric charge. By convention, an electron has one negative charge ( 1) and a proton has one
More informationAN INDEPENDENT LOOPS SEARCH ALGORITHM FOR SOLVING INDUCTIVE PEEC LARGE PROBLEMS
Progress In Electromagnetics Research M, Vol. 23, 53 63, 2012 AN INDEPENDENT LOOPS SEARCH ALGORITHM FOR SOLVING INDUCTIVE PEEC LARGE PROBLEMS T.-S. Nguyen *, J.-M. Guichon, O. Chadebec, G. Meunier, and
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 informationChapter 4 Circuit Theorems: Linearity & Superposition
Chapter 4 Circuit Theorems: Linearity & Superposition Dr. Waleed Al-Hanafy waleed alhanafy@yahoo.com Faculty of Electronic Engineering, Menoufia Univ., Egypt MSA Summer Course: Electric Circuit Analysis
More informationE246 Electronics & Instrumentation. Lecture 1: Introduction and Review of Basic Electronics
E246 Electronics & Instrumentation Lecture 1: Introduction and Review of Basic Electronics Course Personnel Instructor: Yi Guo Office: Burchard 207 Office Hours: Tuesday & Thursday 2-3pm Ph: (201) 216-5658
More informationDirect Current Circuits. February 18, 2014 Physics for Scientists & Engineers 2, Chapter 26 1
Direct Current Circuits February 18, 2014 Physics for Scientists & Engineers 2, Chapter 26 1 Kirchhoff s Junction Rule! The sum of the currents entering a junction must equal the sum of the currents leaving
More informationParallel Circuits. Chapter
Chapter 5 Parallel Circuits Topics Covered in Chapter 5 5-1: The Applied Voltage V A Is the Same Across Parallel Branches 5-2: Each Branch I Equals V A / R 5-3: Kirchhoff s Current Law (KCL) 5-4: Resistance
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 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 informationMansfield Independent School District AP Physics C: Electricity and Magnetism Year at a Glance
Mansfield Independent School District AP Physics C: Electricity and Magnetism Year at a Glance First Six-Weeks Second Six-Weeks Third Six-Weeks Lab safety Lab practices and ethical practices Math and Calculus
More information1. Review of Circuit Theory Concepts
1. Review of Circuit Theory Concepts Lecture notes: Section 1 ECE 65, Winter 2013, F. Najmabadi Circuit Theory is an pproximation to Maxwell s Electromagnetic Equations circuit is made of a bunch of elements
More informationChapter 10 Sinusoidal Steady State Analysis Chapter Objectives:
Chapter 10 Sinusoidal Steady State Analysis Chapter Objectives: Apply previously learn circuit techniques to sinusoidal steady-state analysis. Learn how to apply nodal and mesh analysis in the frequency
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 informationModule 2. DC Circuit. Version 2 EE IIT, Kharagpur
Module 2 DC Circuit Lesson 5 Node-voltage 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 informationChapter 21 Electric Current and Direct- Current Circuits
Chapter 21 Electric Current and Direct- Current Circuits Units of Chapter 21 Electric Current Resistance and Ohm s Law Energy and Power in Electric Circuits Resistors in Series and Parallel Kirchhoff s
More informationIntroduction. HFSS 3D EM Analysis S-parameter. Q3D R/L/C/G Extraction Model. magnitude [db] Frequency [GHz] S11 S21 -30
ANSOFT Q3D TRANING Introduction HFSS 3D EM Analysis S-parameter Q3D R/L/C/G Extraction Model 0-5 -10 magnitude [db] -15-20 -25-30 S11 S21-35 0 1 2 3 4 5 6 7 8 9 10 Frequency [GHz] Quasi-static or full-wave
More informationLusin sequences under CH and under Martin s Axiom
F U N D A M E N T A MATHEMATICAE 169 (2001) Lusin sequences under CH and under Martin s Axiom by Uri Abraham (Beer-Sheva) and Saharon Shelah (Jerusalem) Abstract. Assuming the continuum hypothesis there
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 informationarxiv: v1 [cs.oh] 18 Jan 2016
SUPERPOSITION PRINCIPLE IN LINEAR NETWORKS WITH CONTROLLED SOURCES arxiv:64563v [csoh] 8 Jan 26 CIRO VISONE Abstract The manuscript discusses a well-known issue that, despite its fundamental role in basic
More informationChapter 26 Direct-Current and Circuits. - Resistors in Series and Parallel - Kirchhoff s Rules - Electric Measuring Instruments - R-C Circuits
Chapter 26 Direct-Current and Circuits - esistors in Series and Parallel - Kirchhoff s ules - Electric Measuring Instruments - -C Circuits . esistors in Series and Parallel esistors in Series: V ax I V
More informationHomework 1 solutions
Electric Circuits 1 Homework 1 solutions (Due date: 2014/3/3) This assignment covers Ch1 and Ch2 of the textbook. The full credit is 100 points. For each question, detailed derivation processes and accurate
More informationPhysics 1214 Chapter 19: Current, Resistance, and Direct-Current Circuits
Physics 1214 Chapter 19: Current, Resistance, and Direct-Current Circuits 1 Current current: (also called electric current) is an motion of charge from one region of a conductor to another. Current When
More informationSeries & Parallel Resistors 3/17/2015 1
Series & Parallel Resistors 3/17/2015 1 Series Resistors & Voltage Division Consider the single-loop circuit as shown in figure. The two resistors are in series, since the same current i flows in both
More informationDo not fill out the information below until instructed to do so! Name: Signature: Section Number:
Do not fill out the information below until instructed to do so! Name: Signature: E-mail: Section Number: No calculators are allowed in the test. Be sure to put a box around your final answers and clearly
More informationA REFORMULATION OF THE RADON-NIKODYM THEOREM
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 47, Number 2, February 1975 A REFORMULATION OF THE RADON-NIKODYM THEOREM JONATHAN LEWIN AND MIRIT LEWIN ABSTRACT. The Radon-Nikodym theorems of Segal
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 informationAP Physics C. Electric Circuits III.C
AP Physics C Electric Circuits III.C III.C.1 Current, Resistance and Power The direction of conventional current Suppose the cross-sectional area of the conductor changes. If a conductor has no current,
More informationPh February, Kirchhoff's Rules Author: John Adams, I. Theory
Ph 122 23 February, 2006 I. Theory Kirchhoff's Rules Author: John Adams, 1996 quark%/~bland/docs/manuals/ph122/elstat/elstat.doc This experiment seeks to determine if the currents and voltage drops in
More informationPHY102 Electricity Course Summary
TOPIC 1 ELECTOSTTICS PHY1 Electricity Course Summary Coulomb s Law The magnitude of the force between two point charges is directly proportional to the product of the charges and inversely proportional
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 informationAPPENDIX: TRANSMISSION LINE MODELLING AND PORT-BASED CIRCUITS
APPENDIX: TRANSMISSION LINE MODELLING AND PORT-BASED CIRCUITS A. MODELLING TRANSMISSION LINES THROUGH CIRCUITS In Chapter 5 we introduced the so-called basic rule for modelling circuital systems through
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 informationErrors in Electrical Measurements
1 Errors in Electrical Measurements Systematic error every times you measure e.g. loading or insertion of the measurement instrument Meter error scaling (inaccurate marking), pointer bending, friction,
More informationSTATE UNIVERSITY OF NEW YORK COLLEGE OF TECHNOLOGY CANTON, NEW YORK
STATE UNIVERSITY OF NEW YORK COLLEGE OF TECHNOLOGY CANTON, NEW YORK COURSE OUTLINE ELEC 261 ELECTRICITY Prepared By: Dr. Rashid Aidun CANINO SCHOOL OF ENGINEERING TECHNOLOGY ENGINEERING SCIENCE & ELECTRICAL
More informationNEW CONCEPT FOR ANGULAR POSITION MEASUREMENTS. I.A. Premaratne, S.A.D.A.N. Dissanayake and D.S. Wickramasinghe
NEW CONCEPT FOR ANGULAR POSITION MEASUREMENTS I.A. Premaratne, S.A.D.A.N. Dissanayake and D.S. Wickramasinghe Department of Electrical and Computer Engineering, Open University of Sri Lanka INTRODUCTION
More informationPhysics for Scientists and Engineers 4th Edition 2017
A Correlation and Narrative Summary of Physics for Scientists and Engineers 4th Edition 2017 To the AP Physics C: Electricity and Magnetism Course Description AP is a trademark registered and/or owned
More informationSinusoidal Response of RLC Circuits
Sinusoidal Response of RLC Circuits Series RL circuit Series RC circuit Series RLC circuit Parallel RL circuit Parallel RC circuit R-L Series Circuit R-L Series Circuit R-L Series Circuit Instantaneous
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 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 informationEDEXCEL NATIONALS UNIT 5 - ELECTRICAL AND ELECTRONIC PRINCIPLES. ASSIGNMENT No. 3 - ELECTRO MAGNETIC INDUCTION
EDEXCEL NATIONALS UNIT 5 - ELECTRICAL AND ELECTRONIC PRINCIPLES ASSIGNMENT No. 3 - ELECTRO MAGNETIC INDUCTION NAME: I agree to the assessment as contained in this assignment. I confirm that the work submitted
More informationConvex Invertible Cones, Nevalinna-Pick Interpolation. and the Set of Lyapunov Solutions
Convex Invertible Cones, Nevalinna-Pick Interpolation and the Set of Lyapunov Solutions Nir Cohen Department of Applied Mathematics, Cmpinas State University, CP 6065, Campinas, SP 13081-970, Brazil. E-mail:
More informationUNIVERSITY OF TECHNOLOGY, JAMAICA Faculty of Engineering and Computing School of Engineering
UNIVERSITY OF TECHNOLOGY, JAMAICA Faculty of Engineering and Computing School of Engineering SYLLABUS OUTLINE FACULTY: SCHOOL/DEPT: COURSE OF STUDY: Engineering and Computing Engineering Diploma in Electrical
More informationToday in Physics 217: circuits
Today in Physics 217: circuits! Review of DC circuits: Kirchhoff s rules! Solving equations from Kirchhoff s rules for simple DC circuits 2 December 2002 Physics 217, Fall 2002 1 Lumped circuit elements:
More informationLecture Notes on DC Network Theory
Federal University, Ndufu-Alike, Ikwo Department of Electrical/Electronics and Computer Engineering (ECE) Faculty of Engineering and Technology Lecture Notes on DC Network Theory Harmattan Semester by
More informationChapter 28. Direct Current Circuits
Chapter 28 Direct Current Circuits Circuit Analysis Simple electric circuits may contain batteries, resistors, and capacitors in various combinations. For some circuits, analysis may consist of combining
More informationAP Physics C - E & M
AP Physics C - E & M Current and Circuits 2017-07-12 www.njctl.org Electric Current Resistance and Resistivity Electromotive Force (EMF) Energy and Power Resistors in Series and in Parallel Kirchoff's
More informationChapter 21 Electric Current and Direct- Current Circuits
Chapter 21 Electric Current and Direct- Current Circuits 1 Overview of Chapter 21 Electric Current and Resistance Energy and Power in Electric Circuits Resistors in Series and Parallel Kirchhoff s Rules
More information2.004 Dynamics and Control II Spring 2008
MIT OpenCourseWare http://ocwmitedu 00 Dynamics and Control II Spring 00 For information about citing these materials or our Terms of Use, visit: http://ocwmitedu/terms Massachusetts Institute of Technology
More informationResidual resistance simulation of an air spark gap switch.
Residual resistance simulation of an air spark gap switch. V. V. Tikhomirov, S.E. Siahlo February 27, 2015 arxiv:1502.07499v1 [physics.acc-ph] 26 Feb 2015 Research Institute for Nuclear Problems, Belarusian
More informationShape Optimization of Impactors Against a Finite Width Shield Using a Modified Method of Local Variations #
Mechanics Based Design of Structures and Machines, 35: 113 125, 27 Shape Optimization of Impactors Against a Finite Width Shield Using a Modified Method of Local Variations # G. Ben-Dor, A. Dubinsky, and
More informationElectricity & Magnetism
Electricity & Magnetism D.C. Circuits Marline Kurishingal Note : This chapter includes only D.C. In AS syllabus A.C is not included. Recap... Electrical Circuit Symbols : Draw and interpret circuit diagrams
More informationChapter 18. Direct Current Circuits
Chapter 18 Direct Current Circuits Sources of emf The source that maintains the current in a closed circuit is called a source of emf Any devices that increase the potential energy of charges circulating
More informationE2.2 Analogue Electronics
E2.2 Analogue Electronics Instructor : Christos Papavassiliou Office, email : EE 915, c.papavas@imperial.ac.uk Lectures : Monday 2pm, room 408 (weeks 2-11) Thursday 3pm, room 509 (weeks 4-11) Problem,
More informationEDEXCEL NATIONALS UNIT 5 - ELECTRICAL AND ELECTRONIC PRINCIPLES. ASSIGNMENT No.2 - CAPACITOR NETWORK
EDEXCEL NATIONALS UNIT 5 - ELECTRICAL AND ELECTRONIC PRINCIPLES ASSIGNMENT No.2 - CAPACITOR NETWORK NAME: I agree to the assessment as contained in this assignment. I confirm that the work submitted is
More informationTransients on Integrated Power System
Chapter 3 Transients on Integrated Power System 3.1 Line Dropping and Load Rejection 3.1.1 Line Dropping In three phase circuit capacitance switching, the determination of the voltage trapped after switching
More informationUniversity Of Pennsylvania Department of Physics PHYS 141/151 Engineering Physics II (Course Outline)
University Of Pennsylvania Department of Physics PHYS 141/151 Engineering Physics II (Course Outline) Instructor: Dr. Michael A. Carchidi Textbooks: Sears & Zemansky s University Physics by Young and Freedman
More informationarxiv: v2 [math-ph] 23 Jun 2014
Note on homological modeling of the electric circuits Eugen Paal and Märt Umbleja arxiv:1406.3905v2 [math-ph] 23 Jun 2014 Abstract Based on a simple example, it is explained how the homological analysis
More informationPhysics 112. Study Notes for Exam II
Chapter 20 Electric Forces and Fields Physics 112 Study Notes for Exam II 4. Electric Field Fields of + and point charges 5. Both fields and forces obey (vector) superposition Example 20.5; Figure 20.29
More informationCome & Join Us at VUSTUDENTS.net
Come & Join Us at VUSTUDENTS.net For Assignment Solution, GDB, Online Quizzes, Helping Study material, Past Solved Papers, Solved MCQs, Current Papers, E-Books & more. Go to http://www.vustudents.net and
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 informationParallel Resistors (32.6)
Parallel Resistors (32.6) Resistors connected at both ends are called parallel resistors The important thing to note is that: the two left ends of the resistors are at the same potential. Also, the two
More informationTHE INDUCTANCE OF A SINGLE LAYER COIL DERIVED FROM CAPACITANCE
THE INDUCTANCE OF A SINGLE LAYER COIL DERIVED FROM CAPACITANCE The inductance of a coil can be derived from the magnetic reluctance to its flux, and for a single layer coil this reluctance can be derived
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 informationChapter 10: Sinusoidal Steady-State Analysis
Chapter 10: Sinusoidal Steady-State 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 informationParallel Resistors (32.6)
Parallel Resistors (32.6) Resistors connected at both ends are called parallel resistors Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring 2010 1 / 1 Parallel Resistors (32.6)
More informationExperiment 2: Analysis and Measurement of Resistive Circuit Parameters
Experiment 2: Analysis and Measurement of Resistive Circuit Parameters Report Due In-class on Wed., Mar. 28, 2018 Pre-lab must be completed prior to lab. 1.0 PURPOSE To (i) verify Kirchhoff's laws experimentally;
More informationM. C. Escher: Waterfall. 18/9/2015 [tsl425 1/29]
M. C. Escher: Waterfall 18/9/2015 [tsl425 1/29] Direct Current Circuit Consider a wire with resistance R = ρl/a connected to a battery. Resistor rule: In the direction of I across a resistor with resistance
More informationNetworks and Systems Prof. V. G. K. Murti Department of Electrical Engineering Indian Institution of Technology, Madras
Networks and Systems Prof. V. G. K. Murti Department of Electrical Engineering Indian Institution of Technology, Madras Lecture - 32 Network Function (3) 2-port networks: Symmetry Equivalent networks Examples
More informationChapter 3 Methods of Analysis: 1) Nodal Analysis
Chapter 3 Methods of Analysis: 1) Nodal Analysis Dr. Waleed Al-Hanafy waleed alhanafy@yahoo.com Faculty of Electronic Engineering, Menoufia Univ., Egypt MSA Summer Course: Electric Circuit Analysis I (ESE
More informationNote on homological modeling of the electric circuits
Journal of Physics: Conference Series OPEN ACCESS Note on homological modeling of the electric circuits To cite this article: E Paal and M Umbleja 2014 J. Phys.: Conf. Ser. 532 012022 Related content -
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 informationAppendix A Installing QUCS
Appendix A Installing QUCS In this appendix, we will discuss how to install QUCS [1]. Note that QUCS has a lot of components, many of which we will not use. Nevertheless, we will install all components
More informationNETWORK ANALYSIS WITH APPLICATIONS
NETWORK ANALYSIS WITH APPLICATIONS Third Edition William D. Stanley Old Dominion University Prentice Hall Upper Saddle River, New Jersey I Columbus, Ohio CONTENTS 1 BASIC CIRCUIT LAWS 1 1-1 General Plan
More informationFundamental of Electrical circuits
Fundamental of Electrical circuits 1 Course Description: Electrical units and definitions: Voltage, current, power, energy, circuit elements: resistors, capacitors, inductors, independent and dependent
More information2. The following diagram illustrates that voltage represents what physical dimension?
BioE 1310 - Exam 1 2/20/2018 Answer Sheet - Correct answer is A for all questions 1. A particular voltage divider with 10 V across it consists of two resistors in series. One resistor is 7 KΩ and the other
More informationElectric Charge and Electric field
Electric Charge and Electric field ConcepTest 16.1a Electric Charge I Two charged balls are repelling each other as they hang from the ceiling. What can you say about their charges? 1) one is positive,
More informationTopic 5.2 Heating Effect of Electric Currents
Topic 5.2 Heating Effect of Electric Currents Kari Eloranta 2017 Jyväskylän Lyseon lukio International Baccalaureate February 14, 2017 Topic 5.2 Heating Effect of Electric Currents In subtopic 5.2 we study
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 informationProblem info Geometry model Labelled Objects Results Nonlinear dependencies
Problem info Problem type: Transient Magnetics (integration time: 9.99999993922529E-09 s.) Geometry model class: Plane-Parallel Problem database file names: Problem: circuit.pbm Geometry: Circuit.mod Material
More informationPELLISSIPPI STATE TECHNICAL COMMUNITY COLLEGE MASTER SYLLABUS ELECTRICITY & MAGNETISM W/LAB PHY 2310
PELLISSIPPI STATE TECHNICAL COMMUNITY COLLEGE MASTER SYLLABUS ELECTRICITY & MAGNETISM W/LAB PHY 2310 Class Hours: 3.0 Credit Hours: 4.0 Laboratory Hours: 3.0 Date Revised: Spring 01 Catalog Course Description:
More informationExam 1--PHYS 202--Spring 2013
Name: Class: Date: Exam 1--PHYS 202--Spring 2013 Multiple Choice Identify the choice that best completes the statement or answers the question 1 A metallic object holds a charge of 38 10 6 C What total
More informationCONCEPTUAL TOOLS By: Neil E. Cotter CIRCUITS OHM'S LAW Physics
Physics DERIV: Ohm's law is almost always derived from basic physics with a starting assumption that the electric field inside a resistor is constant. We first investigate this assumption. The electric
More informationLETTER TO THE EDITOR
INTERNATIONAL JOURNAL OF CIRCUIT THEORY AND APPLICATIONS, VOL. 25, 43 49 (1997) LETTER TO THE EDITOR NEW TRANSMISSION LINE SYSTEMS FOR ACCUMULATING POWER FROM DISTRIBUTED RENEWABLE ENERGY MING YING KUO,
More informationDirect Current Circuits
Name: Date: PC1143 Physics III Direct Current Circuits 5 Laboratory Worksheet Part A: Single-Loop Circuits R 1 = I 0 = V 1 = R 2 = I 1 = V 2 = R 3 = I 2 = V 3 = R 12 = I 3 = V 12 = R 23 = V 23 = R 123
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 informationAP Physics C Electricity & Magnetism Mid Term Review
AP Physics C Electricity & Magnetism Mid Term Review 1984 37. When lighted, a 100-watt light bulb operating on a 110-volt household circuit has a resistance closest to (A) 10-2 Ω (B) 10-1 Ω (C) 1 Ω (D)
More informationNEEDS Thermoelectric Compact Model Documentation Version 1.0.0
NEEDS Thermoelectric Compact Model Documentation Version 1.0.0 Published on August 31, 2015 Introduction The NEEDS thermoelectric compact model (TEsegment.va) describes a homogeneous segment of thermoelectric
More informationRICH VARIETY OF BIFURCATIONS AND CHAOS IN A VARIANT OF MURALI LAKSHMANAN CHUA CIRCUIT
International Journal of Bifurcation and Chaos, Vol. 1, No. 7 (2) 1781 1785 c World Scientific Publishing Company RICH VARIETY O BIURCATIONS AND CHAOS IN A VARIANT O MURALI LAKSHMANAN CHUA CIRCUIT K. THAMILMARAN
More information