# LOOP ANALYSIS. The second systematic technique to determine all currents and voltages in a circuit

Save this PDF as:

Size: px
Start display at page:

Download "LOOP ANALYSIS. The second systematic technique to determine all currents and voltages in a circuit"

## Transcription

1 LOOP ANALYSS The second systematic technique to determine all currents and voltages in a circuit T S DUAL TO NODE ANALYSS - T FRST DETERMNES ALL CURRENTS N A CRCUT AND THEN T USES OHM S LAW TO COMPUTE NECESSARY OLTAGES THERE ARE STUATON WHERE NODE ANALYSS S NOT AN EFFCENT TECHNQUE AND WHERE THE NUMBER OF EQUATONS REQURED BY THS NEW METHOD S SGNFCANTLY SMALLER

2 Apply node analysis to this circuit + - R R R R R R There are non reference nodes There is one super node There is one node connected to the reference through a voltage source We need three equations to compute all node voltages BUT THERE S ONLY ONE CURRENT FLOWNG THROUGH ALL COMPONENTS AND F THAT CURRENT S DETERMNED ALL OLTAGES CAN BE COMPUTED WTH OHM S LAW STRATEGY:. Apply KL (sum of voltage drops =). Use Ohm s Law to express voltages in terms of the loop current. [ ] 8[ ] R R R [ ] R R 8[ ] R RESULT S ONE EQUATON N THE LOOP CURRENT!!! SHORTCUT Skip this equation Write this one directly

3 LOOPS, MESHES AND LOOP CURRENTS a b 7 c f 6 e 5 d A BASC CRCUT EACH COMPONENT S CHARACTERZED BY TS OLTAGE ACROSS AND TS CURRENT THROUGH A LOOP S A CLOSED PATH THAT DOES NOT GO TWCE OER ANY NODE. THS CRCUT HAS THREE LOOPS fabef ebcde fabcdef A MESH S A LOOP THAT DOES NOT ENCLOSE ANY OTHER LOOP. fabef, ebcde ARE MESHES A LOOP CURRENT S A (FCTCOUS) CURRENT THAT S ASSUMED TO FLOW AROUND A LOOP,, ARE LOOP CURRENTS A MESH CURRENT S A LOOP CURRENT ASSOCATED TO A MESH., ARE MESH CURRENTS CLAM: N A CRCUT, THE CURRENT THROUGH ANY COMPONENT CAN BE EXPRESSED N TERMS OF THE LOOP CURRENTS EXAMPLES FACT: NOT EERY LOOP CURRENT S REQURED TO COMPUTE ALL THE CURRENTS THROUGH COMPONENTS a a f be bc b 7 c f 6 e 5 d A BASC CRCUT THE DRECTON OF THE LOOP CURRENTS S SGNFCANT USNG LOOP a f b e b c TWO CURRENTS FOR EERY CRCUT THERE S A MNMUM NUMBER OF LOOP CURRENTS THAT ARE NECESSARY TO COMPUTE EERY CURRENT N THE CRCUT. SUCH A COLLECTON S CALLED A MNMAL SET (OF LOOP CURRENTS).

4 FOR A GEN CRCUT LET B NUMBER OF BRANCHES N NUMBER OF NODES THE MNMUM REQURED NUMBER OF LOOP CURRENTS S L B ( N ) MESH CURRENTS ARE ALWAYS NDEPENDENT AN EXAMPLE DETERMNATON OF LOOP CURRENTS KL ON LEFT MESH KL ON RGHT MESH v v v v S 5 USNG OHM S LAW v i R, v i R, v ( i i ) R v i R, v i R 5 5 REPLACNG AND REARRANGNG B 7 N 6 L 7 (6 ) TWO LOOP CURRENTS ARE REQURED. THE CURRENTS SHOWN ARE MESH CURRENTS. HENCE THEY ARE NDEPENDENT AND FORM A MNMAL SET N MATRX FORM R R R R i v S R R R R i v 5 S THESE ARE LOOP EQUATONS FOR THE CRCUT

5 WRTE THE MESH EQUATONS v R BOOKKEEPNG BRANCHES = 8 NODES = 7 LOOP CURRENTS NEEDED = i R AND WE ARE TOLD TO USE MESH CURRENTS! THS DEFNES THE LOOP CURRENTS TO BE USED DENTFY ALL OLTAGE DROPS v R i R v R ( i i R v R ) v R5 i R i R 5 WRTE KL ON EACH MESH TOP MESH: v BOTTOM: v USE OHM S LAW S vr vs vr R vr5 vr vs vr

6 DEELOPNG A SHORTCUT WRTE THE MESH EQUATONS + - R + - R R WHENEER AN ELEMENT HAS MORE THAN ONE LOOP CURRENT FLOWNG THROUGH T WE COMPUTE NET CURRENT N THE DRECTON OF TRAEL R 5 R DRAW THE MESH CURRENTS. ORENTATON CAN BE ARBTRARY. BUT BY CONENTON THEY ARE DEFNED CLOCKWSE NOW WRTE KL FOR EACH MESH AND APPLY OHM S LAW TO EERY RESSTOR. AT EACH LOOP FOLLOW THE PASSE SGN CONENTON USNG LOOP CURRENT REFERENCE DRECTON R ( ) R R5 R R ( ) R

7 LEARNNG EXAMPLE: FND o USNG LOOP ANALYS AN ALTERNATE SELECTON OF LOOP CURRENTS SHORTCUT: POLARTES ARE NOT NEEDED. APPLY OHM S LAW TO EACH ELEMENT AS KL S BENG WRTTEN REARRANGE k 6k 6k 9k */ k 6. 5mA k 6k and add 5 ma EXPRESS ARABLE OF NTEREST AS FUNCTON OF LOOP CURRENTS O NOW REARRANGE O THS SELECTON S MORE EFFCENT k 6k */ 6k 9k 9 */ and substract k 8 ma

8 F THE CRCUT CONTANS ONLY NDEPENDENT SOURCE THE MESH EQUATONS CAN BE WRTTEN BY NSPECTON A PRACTCE EXAMPLE MUST HAE ALL MESH CURRENTS WTH THE SAME ORENTATON N LOOP K THE COEFFCENT OF k S THE SUM OF RESSTANCES AROUND THE LOOP. LOOP coefficient of coefficien t of coefficient of k 6k 6k RHS 6[ ] THE RGHT HAND SDE S THE ALGEBRAC SUM OF OLTAGE SOURCES AROUND THE LOOP (OLTAGE RSES - OLTAGE DROPS) THE COEFFCENT OF j S THE SUM OF RESSTANCES COMMON TO BOTH k AND j AND WTH A NEGATE SGN. LOOP k 6k LOOP k 9k 6 Loop LOOP coefficient of coefficient of 9k k coefficient of k RHS 6[ ] (6 k) ( k) (k 6k k)

9 LEARNNG EXTENSON. DRAW THE MESH CURRENTS. WRTE MESH EQUATONS MESH k k k) k [ ] ( k (k 6k) (6 MESH ) DDE BY k. GET NUMBERS FOR COEFFCENTS ON THE LEFT AND ma ON THE RHS. SOLE EQUATONS 8 [ ma] 8 9[ ma] * / [ ma] O and add 6k [ ] 5

10 WRTE THE MESH EQUATONS k k k k. DRAW MESH CURRENTS 6k 9 BOOKKEEPNG: B = 7, N =. WRTE MESH EQUATONS. USE KL MESH : k 6k( ) MESH : k( ) k( ) MESH : 9 6k( ) k( ) MESH : k( ) k 9 CHOOSE YOUR FAORTE TECHNQUE TO SOLE THE SYSTEM OF EQUATONS EQUATONS BY NSPECTON 8k 6k 8k k k 6k k k 9 k 6k 9

11 CRCUTS WTH NDEPENDENT CURRENT SOURCES KL THERE S NO RELATONSHP BETWEEN AND THE SOURCE CURRENT! HOWEER... MESH CURRENT S CONSTRANED MESH EQUATON MESH ma BY NSPECTON k 8k k (ma) 9 ma O 6k [ 8k ] CURRENT SOURCES THAT ARE NOT SHARED BY OTHER MESHES (OR LOOPS) SERE TO DEFNE A MESH (LOOP) CURRENT AND REDUCE THE NUMBER OF REQURED EQUATONS TO OBTAN APPLY KL TO ANY CLOSED PATH THAT NCLUDES

12 LEARNNG EXAMPLE COMPUTE O USNG MESH ANALYSS KL FOR o TWO MESH CURRENTS ARE DEFNED BY CURRENT SOURCES ma ma MESH USE KL TO COMPUTE o BY NSPECTON k(ma) k( ma) k k k k ma

13 LEARNNG EXTENSONS WE ACTUALLY NEED THE CURRENT ON THE RGHT MESH. HENCE, USE MESH ANALYSS MESH : ma MESH : [ ] k( ) 6k 5 5mA ma ma O 6k [ ] 5 MESH : ma MESH : k k O 6 ma 6k 8[ ]

14 Problem.6 (6th Ed). Write loop equations. S S + - Determine O k k k S = ma, S = 6. Select loop currents. 6k SELECTNG THE SOLUTON METHOD + O non-reference nodes. meshes One current source, one super node BOTH APPROACHES SEEM COMPARABLE. CHOOSE LOOP ANALYSS n this case we use meshes. We note that the current source could define one mesh. _ Loop S Loop S k( ) k( ) Loop k( ) 6k k( ) Since we need to compute o it is efficient to solve for only. HNT: Divide the loop equations by k. Coefficients become numbers and voltage source becomes ma. Loop Loop We use the fact that = s S (6 )[ ma] * / k */ and add eqs 6 S ma O 6k 7 ma

15 CURRENT SOURCES SHARED BY LOOPS - THE SUPERMESH APPROACH. WRTE CONSTRANT EQUATON DUE TO MESH CURRENTS SHARNG CURRENT SOURCES ma. WRTE EQUATONS FOR THE OTHER MESHES ma. DEFNE A SUPERMESH BY (MENTALLY) REMONG THE SHARED CURRENT SOURCE. SELECT MESH CURRENTS 5. WRTE KL FOR THE SUPERMESH 6 k k k( ) k ( ) SUPERMESH NOW WE HAE THREE EQUATONS N THREE UNKNOWNS. THE MODEL S COMPLETE

16 CURRENT SOURCES SHARED BY MESHES - THE GENERAL LOOP APPROACH THE STRATEGY S TO DEFNE LOOP CURRENTS THAT DO NOT SHARE CURRENT SOURCES - EEN F T MEANS ABANDONNG MESHES FOR CONENENCE START USNG MESH CURRENTS UNTL REACHNG A SHARED SOURCE. AT THAT PONT DEFNE A NEW LOOP. N ORDER TO GUARANTEE THAT F GES AN NDEPENDENT EQUATON ONE MUST MAKE SURE THAT THE LOOP NCLUDES COMPONENTS THAT ARE NOT PART OF PREOUSLY DEFNED LOOPS A POSSBLE STRATEGY S TO CREATE A LOOP BY OPENNG THE CURRENT SOURCE THE LOOP EQUATONS FOR THE LOOPS WTH CURRENT SOURCES ARE ma ma THE LOOP EQUATON FOR THE THRD LOOP S 6 [ ] k k( ) k( ) k ( ) THE MESH CURRENTS OBTANED WTH THS METHOD ARE DFFERENT FROM THE ONES OBTANED WTH A SUPERMESH. EEN FOR THOSE DEFNED USNG MESHES.

17 S ( FND OLTAGES R R S S R ACROSS S R RESSTORS - + S Now we need a loop current that does not go over any current source and passes through all unused components. HNT: F ALL CURRENT SOURCES ARE REMOED THERE S ONLY ONE LOOP LEFT MESH EQUATONS FOR LOOPS WTH CURRENT SOURCES s S S KL OF REMANNG LOOP For loop analysis we notice... Three independent current sources. Four meshes. One current source shared by two meshes. Careful choice of loop currents should make only one loop equation necessary. Three loop currents can be chosen using meshes and not sharing any source. SOLE FOR THE CURRENT. USE OHM S LAW TO CMPUTE REQURED OLTAGES R R ( ) R ( ) R ( ) ( ) ) ) R ( R ) ( R

18 - + R R R R S S S S A COMMENT ON METHOD SELECTON The same problem can be solved by node analysis but it requires equations R R R R R R S S S S S

19 CRCUTS WTH DEPENDENT SOURCES Treat the dependent source as though it were independent. Add one equation for the controlling variable k ma X BY SOURCES DETERMNED CURRENTS MESH ) ( ) ( k k k x MESH : ) ( ) ( k k MESH : ) ( k x x CONTROLLN G ARABLES COMBNE EQUATONS. DDE BY k 8

20 SOLE USNG MATLAB 8 PUT N MATRX FORM Since we divided by k the RHS is ma and all the coefficients are numbers 8 >> is the MATLAB prompt. What follows is the command entered DEFNE THE MATRX» R=[,,,; %FRST ROW,, -, ; %SECOND ROW,,,-; %THRD ROW,-,-,] %FOURTH ROW R = DEFNE THE RGHT HAND SDE ECTOR» =[;;8;] = 8 - SOLE AND GET THE ANSWER» =R\ The answers are in ma =

21 LEARNNG EXTENSON: Dependent Sources Find o USNG MESH CURRENTS USNG LOOP CURRENTS We treat the dependent source as one more voltage source x LOOP k( ) k MESH k k( ) 6 k( ) MESH x k LOOP k( ) 6k x NOW WE EXPRESS THE CONTROLLNG ARABLE N TERMS OF THE LOOP CURRENTS x k k k( ) and solve... x k k k ma,. 5mA REPLACE AND REARRANGE 6k 6k O SOLUTONS 6k 9[ ] 6k 8k.5mA,. 5mA NOTCE THE DFFERENCE BETWEEN MESH CURRENT AND LOOP CURRENT EEN THOUGH THEY ARE ASSOCATED TO THE SAME PATH The selection of loop currents simplifies expression for x and computation of o.

22 DEPENDENT CURRENT SOURCE. CURRENT SOURCES NOT SHARED BY MESHES WE ARE ASKED FOR o. WE ONLY NEED TO SOLE FOR REPLACE AND REARRANGE x x k k( ) 8k k ma ma 8 We treat the dependent source as a conventional source Equations for meshes with current sources O 6k [ ] Then KL on the remaining loop(s) And express the controlling variable, x, in terms of loop currents

23 DRAW MESH CURRENTS WRTE MESH EQUATONS. MESH MESH : k k( ) CONTROLLNG ARABLE N TERMS OF LOOP CURRENTS x : k x k k( ) REPLACE AND REARRANGE 6k 6k k 6k SOLE FOR k 6mA O k [ ]

24 n the following we shall solve using loop analysis two circuits that had previously been solved using node analysis This is one circuit. we recap first the node analysis approach and then we solve using loop analysis

25 LEARNNG EXAMPLE FND THE OLTAGE o RECAP OF NODE : k AT SUPER NODE X ma k k k k k k : ma CONTROLLNG ARABLE X SOLE EQUATONS NOW X 6 X X ARABLE OF NTEREST O

26 DETERMNE o USNG LOOP ANALYSS Write loop equations Loop : ma Loop : ma Loop : k k( ) X Loop : k( ) k X Controlling variable: k( ) X k k 6 ma, ma k 8 ariable of nterest O k START SELECTON USNG MESHES SELECT A GENERAL LOOP TO AOD SHARNG A CURRENT SOURCE

27 LEARNNG EXAMPLE Find the current o RECAP OF NODE : node: 6 (constraint eq.) 5 X k k k k k 5 : 5 X k k CONTROLLNG ARABLES X X k 7 eqs in 7 variables ARABLE OF NTEREST O 5 k

28 Find the current o using mesh analysis Write loop/mesh equations Select mesh currents Loop: k k( ) k( ) Loop : k( ) 6 k( ) 5 Loop : X Loop : k( ) X Loop 5: k( ) k( ) X Loop 6: k( ) k k( ) Controlling variables X X k 8 eqs in 8 unknowns 5 6 ariable of interest: O 6

### + + + - - This circuit than can be reduced to a planar circuit

MeshCurrent Method The meshcurrent s analog of the nodeoltage method. We sole for a new set of arables, mesh currents, that automatcally satsfy KCLs. As such, meshcurrent method reduces crcut soluton to

### Systematic Circuit Analysis (T&R Chap 3)

Systematc Crcut Analyss TR Chap ) Nodeoltage analyss Usng the oltages of the each node relate to a ground node, wrte down a set of consstent lnear equatons for these oltages Sole ths set of equatons usng,

### EE201 Circuit Theory I 2015 Spring. Dr. Yılmaz KALKAN

EE201 Crcut Theory I 2015 Sprng Dr. Yılmaz KALKAN 1. Basc Concepts (Chapter 1 of Nlsson - 3 Hrs.) Introducton, Current and Voltage, Power and Energy 2. Basc Laws (Chapter 2&3 of Nlsson - 6 Hrs.) Voltage

### Linear Circuits Analysis. Superposition, Thevenin /Norton Equivalent circuits

Lnear Crcuts Analyss. Superposton, Theenn /Norton Equalent crcuts So far we hae explored tmendependent (resste) elements that are also lnear. A tmendependent elements s one for whch we can plot an / cure.

### (6)(2) (-6)(-4) (-4)(6) + (-2)(-3) + (4)(3) + (2)(-3) = -12-24 + 24 + 6 + 12 6 = 0

Chapter 3 Homework Soluton P3.-, 4, 6, 0, 3, 7, P3.3-, 4, 6, P3.4-, 3, 6, 9, P3.5- P3.6-, 4, 9, 4,, 3, 40 ---------------------------------------------------- P 3.- Determne the alues of, 4,, 3, and 6

### CIRCUIT ELEMENTS AND CIRCUIT ANALYSIS

EECS 4 SPING 00 Lecture 9 Copyrght egents of Unversty of Calforna CICUIT ELEMENTS AND CICUIT ANALYSIS Lecture 5 revew: Termnology: Nodes and branches Introduce the mplct reference (common) node defnes

### ( ) Homework Solutions Physics 8B Spring 09 Chpt. 32 5,18,25,27,36,42,51,57,61,76

Homework Solutons Physcs 8B Sprng 09 Chpt. 32 5,8,25,27,3,42,5,57,,7 32.5. Model: Assume deal connectng wres and an deal battery for whch V bat = E. Please refer to Fgure EX32.5. We wll choose a clockwse

### Homework Solutions Physics 8B Spring 2012 Chpt. 32 5,18,25,27,36,42,51,57,61,76

Homework Solutons Physcs 8B Sprng 202 Chpt. 32 5,8,25,27,3,42,5,57,,7 32.5. Model: Assume deal connectng wres and an deal battery for whch V bat =. Please refer to Fgure EX32.5. We wll choose a clockwse

### Mesh-Current Method (Loop Analysis)

Mesh-Current Method (Loop Analysis) Nodal analysis was developed by applying KCL at each non-reference node. Mesh-Current method is developed by applying KVL around meshes in the circuit. A mesh is a loop

### Electric circuit components. Direct Current (DC) circuits

Electrc crcut components Capactor stores charge and potental energy, measured n Farads (F) Battery generates a constant electrcal potental dfference ( ) across t. Measured n olts (). Resstor ressts flow

### NODE ANALYSIS. One of the systematic ways to determine every voltage and current in a circuit

NODE ANALYSIS One of the systematic ways to determine eery oltage and current in a circuit The ariables used to describe the circuit will be Node oltages -- The oltages of each node with respect to a pre-selected

### Introduction: Analysis of Electronic Circuits

/30/008 ntroducton / ntroducton: Analyss of Electronc Crcuts Readng Assgnment: KVL and KCL text from EECS Just lke EECS, the majorty of problems (hw and exam) n EECS 3 wll be crcut analyss problems. Thus,

### Circuit Reduction Techniques

Crcut Reducton Technques Comnaton of KVLs, KCLs, and characterstcs equatons result n a set of lnear equatons for the crcut arales. Whle the aoe set of equaton s complete and contans all necessary nformaton,

### Peak Inverse Voltage

9/13/2005 Peak Inerse Voltage.doc 1/6 Peak Inerse Voltage Q: I m so confused! The brdge rectfer and the fullwae rectfer both prode full-wae rectfcaton. Yet, the brdge rectfer use 4 juncton dodes, whereas

### 120 CHAPTER 3 NODAL AND LOOP ANALYSIS TECHNIQUES SUMMARY PROBLEMS SECTION 3.1

IRWI03_082132v3 8/26/04 9:41 AM Page 120 120 CHAPTER 3 NODAL AND LOOP ANALYSIS TECHNIQUES SUMMARY Nodal analysis for an Nnode circuit Select one node in the Nnode circuit as the reference node. Assume

### 8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by

6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng

### Analysis of Small-signal Transistor Amplifiers

Analyss of Small-sgnal Transstor Amplfers On completon of ths chapter you should be able to predct the behaour of gen transstor amplfer crcuts by usng equatons and/or equalent crcuts that represent the

### 4. Basic Nodal and Mesh Analysis

1 4. Basic Nodal and Mesh Analysis This chapter introduces two basic circuit analysis techniques named nodal analysis and mesh analysis 4.1 Nodal Analysis For a simple circuit with two nodes, we often

### 2. Introduction and Chapter Objectives

eal Analog Crcuts Chapter : Crcut educton. Introducton and Chapter Objectes In Chapter, we presented Krchoff s laws (whch goern the nteractons between crcut elements) and Ohm s law (whch goerns the oltagecurrent

### Node and Mesh Analysis

Node and Mesh Analysis 1 Copyright ODL Jan 2005 Open University Malaysia Circuit Terminology Name Definition Node Essential node Path Branch Essential Branch Loop Mesh A point where two ore more branches

### The circuit shown on Figure 1 is called the common emitter amplifier circuit. The important subsystems of this circuit are:

polar Juncton Transstor rcuts Voltage and Power Amplfer rcuts ommon mtter Amplfer The crcut shown on Fgure 1 s called the common emtter amplfer crcut. The mportant subsystems of ths crcut are: 1. The basng

### Michał Tadeusiewicz. Electric Circuits. Technical University of Łódź International Faculty of Engineering. Łódź 2009

Mchał Tadeusewcz Electrc Crcuts Techncal Unersty of Łódź Internatonal Faculty of Engneerng Łódź 9 Contents Preface.. 5. Fundamental laws of electrcal crcuts 7.. Introducton... 7.. Krchhoff s oltage Law

### Graph Theory and Cayley s Formula

Graph Theory and Cayley s Formula Chad Casarotto August 10, 2006 Contents 1 Introducton 1 2 Bascs and Defntons 1 Cayley s Formula 4 4 Prüfer Encodng A Forest of Trees 7 1 Introducton In ths paper, I wll

### Chapter 6 Inductance, Capacitance, and Mutual Inductance

Chapter 6 Inductance Capactance and Mutual Inductance 6. The nductor 6. The capactor 6.3 Seres-parallel combnatons of nductance and capactance 6.4 Mutual nductance 6.5 Closer look at mutual nductance Oerew

### EXPLORATION 2.5A Exploring the motion diagram of a dropped object

-5 Acceleraton Let s turn now to moton that s not at constant elocty. An example s the moton of an object you release from rest from some dstance aboe the floor. EXPLORATION.5A Explorng the moton dagram

### State function: eigenfunctions of hermitian operators-> normalization, orthogonality completeness

Schroednger equaton Basc postulates of quantum mechancs. Operators: Hermtan operators, commutators State functon: egenfunctons of hermtan operators-> normalzaton, orthogonalty completeness egenvalues and

### 07-Nodal Analysis Text: ECEGR 210 Electric Circuits I

07Nodal Analysis Text: 3.1 3.4 ECEGR 210 Electric Circuits I Overview Introduction Nodal Analysis Nodal Analysis with Voltage Sources Dr. Louie 2 Basic Circuit Laws Ohm s Law Introduction Kirchhoff s Voltage

### Semiconductor sensors of temperature

Semconductor sensors of temperature he measurement objectve 1. Identfy the unknown bead type thermstor. Desgn the crcutry for lnearzaton of ts transfer curve.. Fnd the dependence of forward voltage drop

### Example: Determine the power supplied by each of the sources, independent and dependent, in this circuit:

Example: Determine the power supplied by each of the sources, independent and dependent, in this circuit: Solution: We ll begin by choosing the bottom node to be the reference node. Next we ll label the

### Questions that we may have about the variables

Antono Olmos, 01 Multple Regresson Problem: we want to determne the effect of Desre for control, Famly support, Number of frends, and Score on the BDI test on Perceved Support of Latno women. Dependent

Introducton Faraday's Law o Inducton In ths lab, you wll study Faraday's Law o nducton usng a wand wth col whch swngs through a magnetc eld. You wll also examne converson o mechanc energy nto electrc energy

### Problem set #5 EE 221, 09/26/ /03/2002 1

Chapter 3, Problem 42. Problem set #5 EE 221, 09/26/2002 10/03/2002 1 In the circuit of Fig. 3.75, choose v 1 to obtain a current i x of 2 A. Chapter 3, Solution 42. We first simplify as shown, making

### Causal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting

Causal, Explanatory Forecastng Assumes cause-and-effect relatonshp between system nputs and ts output Forecastng wth Regresson Analyss Rchard S. Barr Inputs System Cause + Effect Relatonshp The job of

### PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 12

14 The Ch-squared dstrbuton PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 1 If a normal varable X, havng mean µ and varance σ, s standardsed, the new varable Z has a mean 0 and varance 1. When ths standardsed

### Solution : (a) FALSE. Let C be a binary one-error correcting code of length 9. Then it follows from the Sphere packing bound that.

MATH 29T Exam : Part I Solutons. TRUE/FALSE? Prove your answer! (a) (5 pts) There exsts a bnary one-error correctng code of length 9 wth 52 codewords. (b) (5 pts) There exsts a ternary one-error correctng

### Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Alexander-Sadiku Fundamentals of Electric Circuits Chapter 3 Methods of Analysis Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1 Methods of Analysis - Chapter

### THE METHOD OF LEAST SQUARES THE METHOD OF LEAST SQUARES

The goal: to measure (determne) an unknown quantty x (the value of a RV X) Realsaton: n results: y 1, y 2,..., y j,..., y n, (the measured values of Y 1, Y 2,..., Y j,..., Y n ) every result s encumbered

### Passive Filters. References: Barbow (pp 265-275), Hayes & Horowitz (pp 32-60), Rizzoni (Chap. 6)

Passve Flters eferences: Barbow (pp 6575), Hayes & Horowtz (pp 360), zzon (Chap. 6) Frequencyselectve or flter crcuts pass to the output only those nput sgnals that are n a desred range of frequences (called

### Section B9: Zener Diodes

Secton B9: Zener Dodes When we frst talked about practcal dodes, t was mentoned that a parameter assocated wth the dode n the reverse bas regon was the breakdown voltage, BR, also known as the peak-nverse

### The complex inverse trigonometric and hyperbolic functions

Physcs 116A Wnter 010 The complex nerse trgonometrc and hyperbolc functons In these notes, we examne the nerse trgonometrc and hyperbolc functons, where the arguments of these functons can be complex numbers

### Bipolar Junction Transistor (BJT)

polar Juncton Transstor (JT) Lecture notes: Sec. 3 Sedra & Smth (6 th Ed): Sec. 6.1-6.4* Sedra & Smth (5 th Ed): Sec. 5.1-5.4* * Includes detals of JT dece operaton whch s not coered n ths course F. Najmabad,

### The Development of Web Log Mining Based on Improve-K-Means Clustering Analysis

The Development of Web Log Mnng Based on Improve-K-Means Clusterng Analyss TngZhong Wang * College of Informaton Technology, Luoyang Normal Unversty, Luoyang, 471022, Chna wangtngzhong2@sna.cn Abstract.

### The OC Curve of Attribute Acceptance Plans

The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4

### Chapter 2. Circuit Analysis Techniques

Chapter 2 Circuit Analysis Techniques 1 Objectives To formulate the node-voltage equations. To solve electric circuits using the node voltage method. To introduce the mesh current method. To formulate

### Section 5.4 Annuities, Present Value, and Amortization

Secton 5.4 Annutes, Present Value, and Amortzaton Present Value In Secton 5.2, we saw that the present value of A dollars at nterest rate per perod for n perods s the amount that must be deposted today

// Snusdal Steady-State Analyss ntrductn dal Analyss Mesh Analyss Superpstn Therem Surce Transfrmatn Thevenn and rtn Equvalent Crcuts OP-amp AC Crcuts Applcatns ntrductn Steps t Analyze ac Crcuts: The

### FORCED CONVECTION HEAT TRANSFER IN A DOUBLE PIPE HEAT EXCHANGER

FORCED CONVECION HEA RANSFER IN A DOUBLE PIPE HEA EXCHANGER Dr. J. Mchael Doster Department of Nuclear Engneerng Box 7909 North Carolna State Unversty Ralegh, NC 27695-7909 Introducton he convectve heat

### The Bridge Rectifier

9/4/004 The Brdge ectfer.doc 1/9 The Brdge ectfer Now consder ths juncton dode rectfer crcut: 1 Lne (t) - O (t) _ 4 3 We call ths crcut the brdge rectfer. Let s analyze t and see what t does! Frst, we

### Analysis and Modeling of Magnetic Coupling

Analyss and Modelng of Magnetc Couplng Bryce Hesterman Adanced Energy Industres Tuesday, Aprl 7 Dscoery earnng Center Unersty Of Colorado, Boulder, Colorado Dener Chapter, IEEE Power Electroncs Socety

### The example below solves a system in the unknowns α and β:

The Fnd Functon The functon Fnd returns a soluton to a system of equatons gven by a solve block. You can use Fnd to solve a lnear system, as wth lsolve, or to solve nonlnear systems. The example below

### SIMPLE LINEAR CORRELATION

SIMPLE LINEAR CORRELATION Smple lnear correlaton s a measure of the degree to whch two varables vary together, or a measure of the ntensty of the assocaton between two varables. Correlaton often s abused.

### 3. Bipolar Junction Transistor (BJT)

3. polar Juncton Transstor (JT) Lecture notes: Sec. 3 Sedra & Smth (6 th Ed): Sec. 6.1-6.4* Sedra & Smth (5 th Ed): Sec. 5.1-5.4* * Includes detals of JT dece operaton whch s not coered n ths course EE

### ME 563 HOMEWORK # 1 (Solutions) Fall 2010

ME 563 HOMEWORK # 1 (Solutons) Fall 2010 PROBLEM 1: (40%) Derve the equatons of moton for the three systems gven usng Newton-Euler technques (A, B, and C) and energy/power methods (A and B only). System

### BERNSTEIN POLYNOMIALS

On-Lne Geometrc Modelng Notes BERNSTEIN POLYNOMIALS Kenneth I. Joy Vsualzaton and Graphcs Research Group Department of Computer Scence Unversty of Calforna, Davs Overvew Polynomals are ncredbly useful

### Chapter 4: Methods of Analysis

Chapter 4: Methods of Analysis 4.1 Motivation 4.2 Nodal Voltage Analysis 4.3 Simultaneous Eqs. & Matrix Inversion 4.4 Nodal Voltage Analysis with Voltage Sources 4.5 Mesh Current Analysis 4.6 Mesh Current

### Project Networks With Mixed-Time Constraints

Project Networs Wth Mxed-Tme Constrants L Caccetta and B Wattananon Western Australan Centre of Excellence n Industral Optmsaton (WACEIO) Curtn Unversty of Technology GPO Box U1987 Perth Western Australa

### PRACTICE 1: MUTUAL FUNDS EVALUATION USING MATLAB.

PRACTICE 1: MUTUAL FUNDS EVALUATION USING MATLAB. INDEX 1. Load data usng the Edtor wndow and m-fle 2. Learnng to save results from the Edtor wndow. 3. Computng the Sharpe Rato 4. Obtanng the Treynor Rato

### Level Annuities with Payments Less Frequent than Each Interest Period

Level Annutes wth Payments Less Frequent than Each Interest Perod 1 Annuty-mmedate 2 Annuty-due Level Annutes wth Payments Less Frequent than Each Interest Perod 1 Annuty-mmedate 2 Annuty-due Symoblc approach

### 1E6 Electrical Engineering AC Circuit Analysis and Power Lecture 12: Parallel Resonant Circuits

E6 Electrcal Engneerng A rcut Analyss and Power ecture : Parallel esonant rcuts. Introducton There are equvalent crcuts to the seres combnatons examned whch exst n parallel confguratons. The ssues surroundng

### Solution of Algebraic and Transcendental Equations

CHAPTER Soluton of Algerac and Transcendental Equatons. INTRODUCTION One of the most common prolem encountered n engneerng analyss s that gven a functon f (, fnd the values of for whch f ( = 0. The soluton

### Using Series to Analyze Financial Situations: Present Value

2.8 Usng Seres to Analyze Fnancal Stuatons: Present Value In the prevous secton, you learned how to calculate the amount, or future value, of an ordnary smple annuty. The amount s the sum of the accumulated

### The Magnetic Field. Concepts and Principles. Moving Charges. Permanent Magnets

. The Magnetc Feld Concepts and Prncples Movng Charges All charged partcles create electrc felds, and these felds can be detected by other charged partcles resultng n electrc force. However, a completely

### Multiple stage amplifiers

Multple stage amplfers Ams: Examne a few common 2-transstor amplfers: -- Dfferental amplfers -- Cascode amplfers -- Darlngton pars -- current mrrors Introduce formal methods for exactly analysng multple

### Small-Signal Analysis of BJT Differential Pairs

5/11/011 Dfferental Moe Sall Sgnal Analyss of BJT Dff Par 1/1 SallSgnal Analyss of BJT Dfferental Pars Now lets conser the case where each nput of the fferental par conssts of an entcal D bas ter B, an

### Chapter 4: Techniques of Circuit Analysis

4.1 Terminology Example 4.1 a. Nodes: a, b, c, d, e, f, g b. Essential Nodes: b, c, e, g c. Branches: v 1, v 2, R 1, R 2, R 3, R 4, R 5, R 6, R 7, I d. Essential Branch: v 1 -R 1, R 2 -R 3, v 2 -R 4, R

### Ping Pong Fun - Video Analysis Project

Png Pong Fun - Vdeo Analyss Project Objectve In ths experment we are gong to nvestgate the projectle moton of png pong balls usng Verner s Logger Pro Software. Does the object travel n a straght lne? What

### Efficient Project Portfolio as a tool for Enterprise Risk Management

Effcent Proect Portfolo as a tool for Enterprse Rsk Management Valentn O. Nkonov Ural State Techncal Unversty Growth Traectory Consultng Company January 5, 27 Effcent Proect Portfolo as a tool for Enterprse

### 8 Algorithm for Binary Searching in Trees

8 Algorthm for Bnary Searchng n Trees In ths secton we present our algorthm for bnary searchng n trees. A crucal observaton employed by the algorthm s that ths problem can be effcently solved when the

### s-domain Circuit Analysis

S-Doman naly -Doman rcut naly Tme doman t doman near rcut aplace Tranform omplex frequency doman doman Tranformed rcut Dfferental equaton lacal technque epone waveform aplace Tranform nvere Tranform -

### 4. Bipolar Junction Transistors. 4. Bipolar Junction Transistors TLT-8016 Basic Analog Circuits 2005/2007 1

4. polar Juncton Transstors 4. polar Juncton Transstors TLT-806 asc Analog rcuts 2005/2007 4. asc Operaton of the npn polar Juncton Transstor npn JT conssts of thn p-type layer between two n-type layers;

### benefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).

REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or

### Introduction to Regression

Introducton to Regresson Regresson a means of predctng a dependent varable based one or more ndependent varables. -Ths s done by fttng a lne or surface to the data ponts that mnmzes the total error. -

### Harvard University Division of Engineering and Applied Sciences. Fall Lecture 3: The Systems Approach - Electrical Systems

Harvard Unversty Dvson of Engneerng and Appled Scences ES 45/25 - INTRODUCTION TO SYSTEMS ANALYSIS WITH PHYSIOLOGICAL APPLICATIONS Fall 2000 Lecture 3: The Systems Approach - Electrcal Systems In the last

### 2.4 Bivariate distributions

page 28 2.4 Bvarate dstrbutons 2.4.1 Defntons Let X and Y be dscrete r.v.s defned on the same probablty space (S, F, P). Instead of treatng them separately, t s often necessary to thnk of them actng together

### PROFIT RATIO AND MARKET STRUCTURE

POFIT ATIO AND MAKET STUCTUE By Yong Yun Introducton: Industral economsts followng from Mason and Ban have run nnumerable tests of the relaton between varous market structural varables and varous dmensons

### HÜCKEL MOLECULAR ORBITAL THEORY

1 HÜCKEL MOLECULAR ORBITAL THEORY In general, the vast maorty polyatomc molecules can be thought of as consstng of a collecton of two electron bonds between pars of atoms. So the qualtatve pcture of σ

### Chapter 4 Objectives

Chapter 4 Engr228 Circuit Analysis Dr Curtis Nelson Chapter 4 Objectives Understand and be able to use the node-voltage method to solve a circuit; Understand and be able to use the mesh-current method

### Binary Dependent Variables. In some cases the outcome of interest rather than one of the right hand side variables is discrete rather than continuous

Bnary Dependent Varables In some cases the outcome of nterest rather than one of the rght hand sde varables s dscrete rather than contnuous The smplest example of ths s when the Y varable s bnary so that

### 3.4 Operation in the Reverse Breakdown Region Zener Diodes

3/3/2008 secton_3_4_zener_odes 1/4 3.4 Operaton n the everse Breakdown egon Zener odes eadng Assgnment: pp. 167171 A Zener ode The 3 techncal dfferences between a juncton dode and a Zener dode: 1. 2. 3.

### The node voltage method

The node voltage method Equivalent resistance Voltage / current dividers Source transformations Node voltages Mesh currents Superposition Not every circuit lends itself to short-cut methods. Sometimes

### Chapter 3 Group Theory p. 1 - Remark: This is only a brief summary of most important results of groups theory with respect

Chapter 3 Group Theory p. - 3. Compact Course: Groups Theory emark: Ths s only a bref summary of most mportant results of groups theory wth respect to the applcatons dscussed n the followng chapters. For

Newtonan modellng In ths actvty you wll see how Newton s Laws of Moton are used to connect the moton of an object wth the forces actng on t. You wll practse applyng Newton s three laws n some real contexts.

### 6. EIGENVALUES AND EIGENVECTORS 3 = 3 2

EIGENVALUES AND EIGENVECTORS The Characterstc Polynomal If A s a square matrx and v s a non-zero vector such that Av v we say that v s an egenvector of A and s the correspondng egenvalue Av v Example :

### 9 Arithmetic and Geometric Sequence

AAU - Busness Mathematcs I Lecture #5, Aprl 4, 010 9 Arthmetc and Geometrc Sequence Fnte sequence: 1, 5, 9, 13, 17 Fnte seres: 1 + 5 + 9 + 13 +17 Infnte sequence: 1,, 4, 8, 16,... Infnte seres: 1 + + 4

### 2 The TTL Inverter. (i) An input transistor, T 1, which performs a current steering function, can be thought of as a back-to-back diode arrangement.

The TTL Inverter.1 Crcut Structure The crcut dagram of the Transstor Transstor Logc nverter s shown n Fg..1. Ths crcut overcomes the lmtatons of the sngle transstor nverter crcut. Some of the notable features

### Logistic Regression. Lecture 4: More classifiers and classes. Logistic regression. Adaboost. Optimization. Multiple class classification

Lecture 4: More classfers and classes C4B Machne Learnng Hlary 20 A. Zsserman Logstc regresson Loss functons revsted Adaboost Loss functons revsted Optmzaton Multple class classfcaton Logstc Regresson

### The Greedy Method. Introduction. 0/1 Knapsack Problem

The Greedy Method Introducton We have completed data structures. We now are gong to look at algorthm desgn methods. Often we are lookng at optmzaton problems whose performance s exponental. For an optmzaton

### Nonlinear data mapping by neural networks

Nonlnear data mappng by neural networks R.P.W. Dun Delft Unversty of Technology, Netherlands Abstract A revew s gven of the use of neural networks for nonlnear mappng of hgh dmensonal data on lower dmensonal

### 1 Approximation Algorithms

CME 305: Dscrete Mathematcs and Algorthms 1 Approxmaton Algorthms In lght of the apparent ntractablty of the problems we beleve not to le n P, t makes sense to pursue deas other than complete solutons

### Aryabhata s Root Extraction Methods. Abhishek Parakh Louisiana State University Aug 31 st 2006

Aryabhata s Root Extracton Methods Abhshek Parakh Lousana State Unversty Aug 1 st 1 Introducton Ths artcle presents an analyss of the root extracton algorthms of Aryabhata gven n hs book Āryabhatīya [1,

### where the coordinates are related to those in the old frame as follows.

Chapter 2 - Cartesan Vectors and Tensors: Ther Algebra Defnton of a vector Examples of vectors Scalar multplcaton Addton of vectors coplanar vectors Unt vectors A bass of non-coplanar vectors Scalar product

### 2. Linear Algebraic Equations

2. Lnear Algebrac Equatons Many physcal systems yeld smultaneous algebrac equatons when mathematcal functons are requred to satsfy several condtons smultaneously. Each condton results n an equaton that

### Chapter 12 Inductors and AC Circuits

hapter Inductors and A rcuts awrence B. ees 6. You may make a sngle copy of ths document for personal use wthout wrtten permsson. Hstory oncepts from prevous physcs and math courses that you wll need for

### Section 5.3 Annuities, Future Value, and Sinking Funds

Secton 5.3 Annutes, Future Value, and Snkng Funds Ordnary Annutes A sequence of equal payments made at equal perods of tme s called an annuty. The tme between payments s the payment perod, and the tme

### 5.74 Introductory Quantum Mechanics II

MIT OpenCourseWare http://ocw.mt.edu 5.74 Introductory Quantum Mechancs II Sprng 9 For nformaton about ctng these materals or our Terms of Use, vst: http://ocw.mt.edu/terms. 4-1 4.1. INTERACTION OF LIGHT

### Forecasting the Direction and Strength of Stock Market Movement

Forecastng the Drecton and Strength of Stock Market Movement Jngwe Chen Mng Chen Nan Ye cjngwe@stanford.edu mchen5@stanford.edu nanye@stanford.edu Abstract - Stock market s one of the most complcated systems

### IT09 - Identity Management Policy

IT09 - Identty Management Polcy Introducton 1 The Unersty needs to manage dentty accounts for all users of the Unersty s electronc systems and ensure that users hae an approprate leel of access to these

### 21 Vectors: The Cross Product & Torque

21 Vectors: The Cross Product & Torque Do not use our left hand when applng ether the rght-hand rule for the cross product of two vectors dscussed n ths chapter or the rght-hand rule for somethng curl