# Block Diagram Reduction

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Appendix W Block Diagram Reduction W.3 4Mason s Rule and the Signal-Flow Graph A compact alternative notation to the block diagram is given by the signal- ow graph introduced Signal- ow by S. J. Mason (1953, 1956). As with the block diagram, the signal- ow graph o ers a visual tool for graph representing the causal relationships between the components of the system. The method consists of characterizing the system by a network of directed branches and associated gains (transfer functions) connected at nodes. Several block diagrams and their corresponding signal- ow graphs are shown in Fig. W.1. The two ways of depicting a system are equivalent, and you can use either diagram to apply Mason s rule (to be de ned shortly). In a signal- ow graph the internal signals in the diagram, such as the common input to several blocks or the output of a summing junction, are called nodes. The system input point and the system output point are also nodes; the input node has outgoing branches only, and the output node has incoming branches only. Mason de ned a path through a block diagram as a sequence of connected blocks, the route passing from one node to another in the direction of signal ow of the blocks without including any block more than once. A forward path is a path from the input to output such that no node is included more than once. If the nodes are numbered in a convenient order, then a forward path can be identi ed by the numbers that are included. Any closed path that returns to its starting node without passing through any node more than once is a loop, and a path that leads from a given variable back to the same variable is a loop path. The path gain is the product of component gains (transfer functions) making up the path. Similarly, the loop gain is the path gain associated with a loop that is, the product of gains in a loop. If two paths have a common component, they are said to touch. Notice particularly in this connection that the input and the output of a summing junction are not the same and that the summing junction is a one-way device from its inputs to its output. Mason s rule relates the graph to the algebra of the simultaneous equations it represents. 1 Consider Fig. W.1(c), where the signal at each node has been given a name and the gains are marked. Then the block diagram (or the signal- ow graph) represents the following system of equations: X 1 (s) = X 3 (s) + U(s); X 2 (s) = G 1 (s)x 1 (s) + G 2 (s)x 2 (s) + G 4 (s)x 3 (s); Y (s) = 1X 3 (s): Mason s rule states that the input output transfer function associated with a signal- ow graph is given by U(s) = 1 X G i i ; i Mason s rule 1 The derivation is based on Cramer s rule for solving linear equations by determinants and is described in Mason s papers. 1

2 2 APPENDIX W. BLOCK DIAGRAM REDUCTION Figure W.1: Block diagrams and corresponding signal ow graphs

3 W.3. 4MASON S RULE AND THE SIGNAL-FLOW GRAPH 3 where G i = path gain of the ith forward path, = the system determinant 1 P (all individual loop gains) + P P (gain products of all possible = two loops that do not touch) (gain products of all possible three loops that do not touch) + : : :, i = ith forward path determinant = value of for that part of the block diagram that does not touch the ith forward path. We will now illustrate the use of Mason s rule by several examples. Example W.1 Mason s Rule in a Simple System Find the transfer function for the block diagram in Fig. W.2. SOLUTION From the block diagram shown in Fig. W.2 we have Forward Path Path Gain 1236 G 1 = 1 1s (b 1 )(1) G 2 = 1 1s 1s (b 2 )(1) G 3 = 1 1s 1s 1s (b 3 )(1) Loop Path Gain 232 l 1 = a 1 =s 2342 l 2 = a 2 =s l 3 = a 3 =s 3 and the determinants are = 1 1 = = = 1 0: a1 s a 2 s 2 a 3 s Applying Mason s rule, we nd the transfer function to be U(s) = (b 1=s) + (b 2 =s 2 ) + (b 3 =s 3 ) 1 + (a 1 =s) + (a 2 =s 2 ) + (a 3 =s 3 ) = b 1 s 2 + b 2 s + b 3 s 3 + a 1 s 2 + a 2 s + a 3 : Mason s rule is particularly useful for more complex systems where there are several loops, some of which do not sum into the same point. Example W.2 Mason s Rule in a Complex System Find the transfer function for the system shown in Fig. W.3. SOLUTION From the block diagram, we nd that Forward Path Path Gain G 1 = H 1 H 2 H G 2 = H 4 Loop Path Gain 242 l 1 = H 1 H 5 (does not touch l 3 )

4 4 APPENDIX W. BLOCK DIAGRAM REDUCTION Figure W.2: Block diagram for Example W.1 Figure W.3: Block diagram for Example W l 2 = H 2 H l 3 = H 3 H 7 (does not touch l 1 ) l 4 = H 4 H 7 H 6 H 5 and the determinants are Therefore, = 1 (H 1 H 5 + H 2 H 6 + H 3 H 7 + H 4 H 7 H 6 H 5 ) + (H 1 H 5 H 3 H 7 ) 1 = = 1 H 2 H 6 : U(s) = H 1 H 2 H 3 + H 4 H 4 H 2 H 6 1 H 1 H 5 H 2 H 6 H 3 H 7 H 4 H 7 H 6 H 5 + H 1 H 5 H 3 H 7 : Mason s rule is useful for solving relatively complicated block diagrams by hand. It yields the solution in the sense that it provides an explicit input output relationship for the system represented by the diagram. The advantage compared with path-by-path block-diagram reduction is that it is systematic and algorithmic rather than problem dependent. MATLAB and other control systems computer-aided software allow you to specify a system in terms of individual blocks in an overall

5 W.3. 4MASON S RULE AND THE SIGNAL-FLOW GRAPH 5 Figure W.4: Block diagram for Problem 2 system, and the software algorithms perform the required block-diagram reduction; therefore, Mason s rule is less important today than in the past. However, there are some derivations that rely on the concepts embodied by the rule, so it still has a role in the control designer s toolbox. Problems: Mason s Rule and the Signal-Flow Graph 1. 4 Find the transfer functions for the block diagrams in Fig. 3.53, using Mason s rule Use block-diagram algebra or Mason s rule to determine the transfer function between R(s) and Y(s) in Fig. W.4.

### SIGNAL FLOW GRAPHS. Prof. S. C.Pilli, Principal, Session: XI, 19/09/06 K.L.E.S. College of Engineering and Technology, Belgaum-590008.

SIGNA FOW GAPHS An alternate to block diagram is the signal flow graph due to S. J. Mason. A signal flow graph is a diagram that represents a set of simultaneous linear algebraic equations. Each signal

### (Refer Slide Time: 2:03)

Control Engineering Prof. Madan Gopal Department of Electrical Engineering Indian Institute of Technology, Delhi Lecture - 11 Models of Industrial Control Devices and Systems (Contd.) Last time we were

### How can we deal with a network branch which is part of two networks each with a source? R3 is carrying current supplied by each battery

Network nalysis ims: Consolidate use of KCL in circuit analysis. Use Principle of Superposition. Learn basics of Node Voltage nalysis (uses KCL) Learn basics of Mesh Current nalysis (uses KVL) Lecture

### G(s) = Y (s)/u(s) In this representation, the output is always the Transfer function times the input. Y (s) = G(s)U(s).

Transfer Functions The transfer function of a linear system is the ratio of the Laplace Transform of the output to the Laplace Transform of the input, i.e., Y (s)/u(s). Denoting this ratio by G(s), i.e.,

### http://users.ece.gatech.edu/~mleach/ece3050/notes/feedback/fbexamples.pdf

c Copyright 2009. W. Marshall Leach, Jr., Professor, Georgia Institute of Technology, School of Electrical and Computer Engineering. Feedback Amplifiers CollectionofSolvedProblems A collection of solved

### Background. Linear Convolutional Encoders (2) Output as Function of Input (1) Linear Convolutional Encoders (1)

S-723410 Convolutional Codes (1) 1 S-723410 Convolutional Codes (1) 3 Background Convolutional codes do not segment the data stream, but convert the entire data stream into another stream (codeword) Some

. 1/ CHAPTER- 4 SIMULATION RESULTS & DISCUSSION CHAPTER 4 SIMULATION RESULTS & DISCUSSION 4.1: ANT COLONY OPTIMIZATION BASED ON ESTIMATION OF DISTRIBUTION ACS possesses

### Kirchhoff s Laws. Kirchhoff's Law #1 - The sum of the currents entering a node must equal the sum of the currents exiting a node.

Kirchhoff s Laws There are two laws necessary for solving circuit problems. For simple circuits, you have been applying these equations almost instinctively. Kirchhoff's Law #1 - The sum of the currents

### Collection of Solved Feedback Amplifier Problems

c Copyright 2009. W. Marshall Leach, Jr., Professor, Georgia Institute of Technology, School of Electrical and Computer Engineering. Collection of Solved Feedback Amplifier Problems This document contains

### Circuits 1 M H Miller

Introduction to Graph Theory Introduction These notes are primarily a digression to provide general background remarks. The subject is an efficient procedure for the determination of voltages and currents

### 1.3 Models & Applications

GOALS: 1. Translate word problems to algebra. 2. Solve word problems using a systematic approach. 3. For equations with more than one variable, solve for one of the variables in terms of the other. Study

### Graphs and Network Flows IE411 Lecture 1

Graphs and Network Flows IE411 Lecture 1 Dr. Ted Ralphs IE411 Lecture 1 1 References for Today s Lecture Required reading Sections 17.1, 19.1 References AMO Chapter 1 and Section 2.1 and 2.2 IE411 Lecture

### ECEN 5682 Theory and Practice of Error Control Codes

ECEN 5682 Theory and Practice of Error Control Codes Convolutional Codes University of Colorado Spring 2007 Linear (n, k) block codes take k data symbols at a time and encode them into n code symbols.

### Control System Definition

Control System Definition A control system consist of subsytems and processes (or plants) assembled for the purpose of controlling the outputs of the process. For example, a furnace produces heat as a

### Unit I Control System Modeling

Unit I Control System Modeling Introduction System An interconnection of elements and devices for a desired purpose. Control System An interconnection of components forming a system configuration that

### NODAL ANALYSIS. Circuits Nodal Analysis 1 M H Miller

NODAL ANALYSIS A branch of an electric circuit is a connection between two points in the circuit. In general a simple wire connection, i.e., a 'short-circuit', is not considered a branch since it is known

### 2 SYSTEM DESCRIPTION TECHNIQUES

2 SYSTEM DESCRIPTION TECHNIQUES 2.1 INTRODUCTION Graphical representation of any process is always better and more meaningful than its representation in words. Moreover, it is very difficult to arrange

### Direct-Current Circuits

Chapter 13 Direct-Current Circuits In This Chapter: Resistors in Series Resistors in Parallel EMF and Internal Resistance Kirchhoff s Rules Resistors in Series The equivalent resistance of a set of resistors

### 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

### Series and Parallel Resistive Circuits

Series and Parallel Resistive Circuits The configuration of circuit elements clearly affects the behaviour of a circuit. Resistors connected in series or in parallel are very common in a circuit and act

### Final Review Ch. 1 #2

Final Review Ch. 1 #2 9th Grade Algebra 1A / Algebra 1 Beta (Ms. Dalton) Student Name/ID: Instructor Note: Show all of your work! 1. Translate the phrase into an algebraic expression. The sum of and 2.

### 8.2. Solution by Inverse Matrix Method. Introduction. Prerequisites. Learning Outcomes

Solution by Inverse Matrix Method 8.2 Introduction The power of matrix algebra is seen in the representation of a system of simultaneous linear equations as a matrix equation. Matrix algebra allows us

### Modeling Guidelines Manual

Modeling Guidelines Manual [Insert company name here] July 2014 Author: John Doe john.doe@johnydoe.com Page 1 of 22 Table of Contents 1. Introduction... 3 2. Business Process Management (BPM)... 4 2.1.

### 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

### Gates, Circuits, and Boolean Algebra

Gates, Circuits, and Boolean Algebra Computers and Electricity A gate is a device that performs a basic operation on electrical signals Gates are combined into circuits to perform more complicated tasks

### The Solution of Linear Simultaneous Equations

Appendix A The Solution of Linear Simultaneous Equations Circuit analysis frequently involves the solution of linear simultaneous equations. Our purpose here is to review the use of determinants to solve

### 3LEARNING GOALS. Analysis Techniques

IRWI3_8232hr 9/3/4 8:54 AM Page 82 3 Nodal 3LEARNING GOALS and Loop Analysis Techniques 3. Nodal Analysis An analysis technique in which one node in an Nnode network is selected as the reference node and

### State-Space Representation of LTI Systems

214 Analysis and Design of Feedback Control Systems State-Space Representation of LTI Systems 1 Introduction Derek Rowell October 2002 The classical control theory and methods (such as root locus) that

### Algorithm & Flowchart & Pseudo code. Staff Incharge: S.Sasirekha

Algorithm & Flowchart & Pseudo code Staff Incharge: S.Sasirekha Computer Programming and Languages Computers work on a set of instructions called computer program, which clearly specify the ways to carry

### 8.3. Gauss elimination. Introduction. Prerequisites. Learning Outcomes

Gauss elimination 8.3 Introduction Engineers often need to solve large systems of linear equations; for example in determining the forces in a large framework or finding currents in a complicated electrical

### 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

### Application Report. Mixed Signal Products SLAA067

Application Report July 1999 Mixed Signal Products SLAA067 IMPORTANT NOTICE Texas Instruments and its subsidiaries (TI) reserve the right to make changes to their products or to discontinue any product

### Optimizations. Optimization Safety. Optimization Safety. Control Flow Graphs. Code transformations to improve program

Optimizations Code transformations to improve program Mainly: improve execution time Also: reduce program size Control low Graphs Can be done at high level or low level E.g., constant folding Optimizations

### Process Modelling from Insurance Event Log

Process Modelling from Insurance Event Log P.V. Kumaraguru Research scholar, Dr.M.G.R Educational and Research Institute University Chennai- 600 095 India Dr. S.P. Rajagopalan Professor Emeritus, Dr. M.G.R

### ( % . This matrix consists of \$ 4 5 " 5' the coefficients of the variables as they appear in the original system. The augmented 3 " 2 2 # 2 " 3 4&

Matrices define matrix We will use matrices to help us solve systems of equations. A matrix is a rectangular array of numbers enclosed in parentheses or brackets. In linear algebra, matrices are important

### Electric Circuits. Overview. Hani Mehrpouyan,

Electric Circuits Hani Mehrpouyan, Department of Electrical and Computer Engineering, Lecture 5 (Mesh Analysis) Sep 8 th, 205 Hani Mehrpouyan (hani.mehr@ieee.org) Boise State c 205 Overview With Ohm s

### 3.5. Solving Inequalities. Introduction. Prerequisites. Learning Outcomes

Solving Inequalities 3.5 Introduction An inequality is an expression involving one of the symbols,, > or

### ECE 331 Digital System Design

ECE 331 Digital System Design Introduction to and Analysis of Sequential Logic Circuits (Lecture #21) The slides included herein were taken from the materials accompanying Fundamentals of Logic Design,

### The Transportation Problem: LP Formulations

The Transportation Problem: LP Formulations An LP Formulation Suppose a company has m warehouses and n retail outlets A single product is to be shipped from the warehouses to the outlets Each warehouse

### 3.4. Solving Simultaneous Linear Equations. Introduction. Prerequisites. Learning Outcomes

Solving Simultaneous Linear Equations 3.4 Introduction Equations often arise in which there is more than one unknown quantity. When this is the case there will usually be more than one equation involved.

### International Islamic University Chittagong Department of Electrical & Electronics Engineering

International Islamic University Chittagong Department of Electrical & Electronics Engineering Course No: EEE 1102 Course Title: Electrical Circuit I Sessional Experiment No : 01 Experiment Name: Introduction

### 3.4. Solving simultaneous linear equations. Introduction. Prerequisites. Learning Outcomes

Solving simultaneous linear equations 3.4 Introduction Equations often arise in which there is more than one unknown quantity. When this is the case there will usually be more than one equation involved.

### Solving for Voltage and Current

Chapter 3 Solving for Voltage and Current Nodal Analysis If you know Ohm s Law, you can solve for all the voltages and currents in simple resistor circuits, like the one shown below. In this chapter, we

### CSEE 3827: Problem Set 4

CSEE 3827: Problem Set 4 Solutions. Gray codes have the useful property that consecutive numbers differ in only a single bit position. Design a 3-bit Gray code counter FSM with no inputs and three outputs.

### Minimum Spanning Trees

Minimum Spanning Trees Algorithms and 18.304 Presentation Outline 1 Graph Terminology Minimum Spanning Trees 2 3 Outline Graph Terminology Minimum Spanning Trees 1 Graph Terminology Minimum Spanning Trees

### A SYSTEMATIC METHOD FOR CALCULATING TRANSISTOR CIRCUIT

A SYSTEMATIC METHOD FOR CALCULATING TRANSISTOR CIRCUIT BIASING VOLTAGES December 22, 2012 J.L. 1 A BASIC COMMON-EMITTER BJT AMPLIFIER 1 In this context a few practical examples are given to demonstrate

### Operational Amplifiers

Operational Amplifiers Introduction The operational amplifier (op-amp) is a voltage controlled voltage source with very high gain. It is a five terminal four port active element. The symbol of the op-amp

### Chapter 08. Methods of Analysis

Chapter 08 Methods of Analysis Source: Circuit Analysis: Theory and Practice Delmar Cengage Learning C-C Tsai Outline Source Conversion Mesh Analysis Nodal Analysis Delta-Wye ( -Y) Conversion Bridge Networks

### Math 2280 Section 002 [SPRING 2013] 1

Math 2280 Section 002 [SPRING 2013] 1 Today well learn about a method for solving systems of differential equations, the method of elimination, that is very similar to the elimination methods we learned

### Module 6. Channel Coding. Version 2 ECE IIT, Kharagpur

Module 6 Channel Coding Lesson 35 Convolutional Codes After reading this lesson, you will learn about Basic concepts of Convolutional Codes; State Diagram Representation; Tree Diagram Representation; Trellis

### 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

### Principles of Dat Da a t Mining Pham Tho Hoan hoanpt@hnue.edu.v hoanpt@hnue.edu. n

Principles of Data Mining Pham Tho Hoan hoanpt@hnue.edu.vn References [1] David Hand, Heikki Mannila and Padhraic Smyth, Principles of Data Mining, MIT press, 2002 [2] Jiawei Han and Micheline Kamber,

### NETWORK STRUCTURES FOR IIR SYSTEMS. Solution 12.1

NETWORK STRUCTURES FOR IIR SYSTEMS Solution 12.1 (a) Direct Form I (text figure 6.10) corresponds to first implementing the right-hand side of the difference equation (i.e. the eros) followed by the left-hand

### Week_11: Network Models

Week_11: Network Models 1 1.Introduction The network models include the traditional applications of finding the most efficient way to link a number of locations directly or indirectly, finding the shortest

### 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

### Lecture 20: Transmission (ABCD) Matrix.

Whites, EE 48/58 Lecture 0 Page of 7 Lecture 0: Transmission (ABC) Matrix. Concerning the equivalent port representations of networks we ve seen in this course:. Z parameters are useful for series connected

### = [a ij ] 2 3. Square matrix A square matrix is one that has equal number of rows and columns, that is n = m. Some examples of square matrices are

This document deals with the fundamentals of matrix algebra and is adapted from B.C. Kuo, Linear Networks and Systems, McGraw Hill, 1967. It is presented here for educational purposes. 1 Introduction In

### CHAPTER 13 ANALYSIS OF CLOCKED SEQUENTIAL CIRCUITS

CHAPTER 13 ANALYSIS OF CLOCKED SEQUENTIAL CIRCUITS This chapter in the book includes: Objectives Study Guide 13.1 A Sequential Parity Checker 13.2 Analysis by Signal Tracing and Timing Charts 13.3 State

### Chapter 7 Direct-Current Circuits

Chapter 7 Direct-Current Circuits 7. Introduction...7-7. Electromotive Force...7-3 7.3 Resistors in Series and in Parallel...7-5 7.4 Kirchhoff s Circuit Rules...7-7 7.5 Voltage-Current Measurements...7-9

### Scheduling of Flexible Manufacturing Systems Using Fuzzy Logic

Scheduling of Flexible Manufacturing Systems Using Fuzzy Logic by Pramot Srinoi A/Prof. Ebrahim Shayan Dr. Fatemeh Ghotb, School of Mathematical Sciences Abstract This paper presents a research project

### An out-put signal of 3 dbmw indicates a power twice that of the 0 db signal. Hence, 3 dbmw = 2 milli Watts.

CATV FIBER OPTIC NETWORK SYSTEM DESIGN PART III - A Practical Guide To Fiber Optic System Design The third part of this series of articles takes a closer look at the practical aspects of Fiber Optic System

### Chapter 4 DECISION ANALYSIS

ASW/QMB-Ch.04 3/8/01 10:35 AM Page 96 Chapter 4 DECISION ANALYSIS CONTENTS 4.1 PROBLEM FORMULATION Influence Diagrams Payoff Tables Decision Trees 4.2 DECISION MAKING WITHOUT PROBABILITIES Optimistic Approach

### ALGORITHMS AND FLOWCHARTS

ALGORITHMS AND FLOWCHARTS ALGORITHMS AND FLOWCHARTS A typical programming task can be divided into two phases: Problem solving phase produce an ordered sequence of steps that describe solution of problem

### 1. LINEAR EQUATIONS. A linear equation in n unknowns x 1, x 2,, x n is an equation of the form

1. LINEAR EQUATIONS A linear equation in n unknowns x 1, x 2,, x n is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b, where a 1, a 2,..., a n, b are given real numbers. For example, with x and

### Regression Analysis Prof. Soumen Maity Department of Mathematics Indian Institute of Technology, Kharagpur. Lecture - 2 Simple Linear Regression

Regression Analysis Prof. Soumen Maity Department of Mathematics Indian Institute of Technology, Kharagpur Lecture - 2 Simple Linear Regression Hi, this is my second lecture in module one and on simple

### Predictive Dynamix Inc

Predictive Modeling Technology Predictive modeling is concerned with analyzing patterns and trends in historical and operational data in order to transform data into actionable decisions. This is accomplished

Introduction to Simulink MEEN 364 Simulink is a software package for modeling, simulating, and analyzing dynamical systems. It supports linear and nonlinear systems, modeled in continuous time, sampled

### A Programme Implementation of Several Inventory Control Algorithms

BULGARIAN ACADEMY OF SCIENCES CYBERNETICS AND INFORMATION TECHNOLOGIES Volume, No Sofia 20 A Programme Implementation of Several Inventory Control Algorithms Vladimir Monov, Tasho Tashev Institute of Information

### Solving Simultaneous Equations and Matrices

Solving Simultaneous Equations and Matrices The following represents a systematic investigation for the steps used to solve two simultaneous linear equations in two unknowns. The motivation for considering

### EE 1202 Experiment #2 Resistor Circuits

EE 1202 Experiment #2 Resistor Circuits 1. ntroduction and Goals: Demonstrates the voltage-current relationships in DC and AC resistor circuits. Providing experience in using DC power supply, digital multimeter,

### Terminology and Symbols in Control Engineering

Technical Information Terminology and Symbols in Control Engineering 1 Part 1 Fundamentals Technical Information Part 1: Fundamentals Part 2: Self-operated Regulators Part 3: Control Valves Part 4: Communication

### Module 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

### GameTime: A Toolkit for Timing Analysis of Software

GameTime: A Toolkit for Timing Analysis of Software Sanjit A. Seshia and Jonathan Kotker EECS Department, UC Berkeley {sseshia,jamhoot}@eecs.berkeley.edu Abstract. Timing analysis is a key step in the

### BASIC ELECTRONICS PROF. T.S. NATARAJAN DEPT OF PHYSICS IIT MADRAS LECTURE-4 SOME USEFUL LAWS IN BASIC ELECTRONICS

BASIC ELECTRONICS PROF. T.S. NATARAJAN DEPT OF PHYSICS IIT MADRAS LECTURE-4 SOME USEFUL LAWS IN BASIC ELECTRONICS Hello everybody! In a series of lecture on basic electronics, learning by doing, we now

### Circuit Analysis using the Node and Mesh Methods

Circuit Analysis using the Node and Mesh Methods We have seen that using Kirchhoff s laws and Ohm s law we can analyze any circuit to determine the operating conditions (the currents and voltages). The

### SYNCHRONOUS COUNTERS

SYNCHRONOUS COUNTERS Synchronous digital counters have a common clock signal that controls all flip-flop stages. Since a common clock controls all flip-flops simultaneously, there are no cumulative delays

### PROPERTIES AND EXAMPLES OF Z-TRANSFORMS

PROPERTIES AD EXAMPLES OF Z-TRASFORMS I. Introduction In the last lecture we reviewed the basic properties of the z-transform and the corresponding region of convergence. In this lecture we will cover

### Firewall Design: Consistency, Completeness, Compactness

Firewall Design: Consistency, Completeness, Compactness Alex X. Liu alex@cs.utexas.edu Department of Computer Sciences The University of Texas at Austin Austin, Texas 78712-1188, U.S.A. March, 2004 Co-author:

### Module 2 Communication Switching. Version 1 ECE, IIT Kharagpur

Module 2 Communication Switching Lesson 2 Circuit Switching INSTRUCTIONAL OBJECTIVES GENERAL This lesson is aimed at developing the concept and application of circuit switching which is a very important

### Parallel Computing using MATLAB Distributed Compute Server ZORRO HPC

Parallel Computing using MATLAB Distributed Compute Server ZORRO HPC Goals of the session Overview of parallel MATLAB Why parallel MATLAB? Multiprocessing in MATLAB Parallel MATLAB using the Parallel Computing

### Fixed Point Theorems

Fixed Point Theorems Definition: Let X be a set and let T : X X be a function that maps X into itself. (Such a function is often called an operator, a transformation, or a transform on X, and the notation

### Chapter 2 Objectives

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

### Solving simultaneous equations. Jackie Nicholas

Mathematics Learning Centre Solving simultaneous equations Jackie Nicholas c 2005 University of Sydney Mathematics Learning Centre, University of Sydney 1 1 Simultaneous linear equations We will introduce

### Introduction Number Systems and Conversion

UNIT 1 Introduction Number Systems and Conversion Objectives 1. Introduction The first part of this unit introduces the material to be studied later. In addition to getting an overview of the material

### Basic Laws Circuit Theorems Methods of Network Analysis Non-Linear Devices and Simulation Models

EE Modul 1: Electric Circuits Theory Basic Laws Circuit Theorems Methods of Network Analysis Non-Linear Devices and Simulation Models EE Modul 1: Electric Circuits Theory Current, Voltage, Impedance Ohm

### Operation Count; Numerical Linear Algebra

10 Operation Count; Numerical Linear Algebra 10.1 Introduction Many computations are limited simply by the sheer number of required additions, multiplications, or function evaluations. If floating-point

### An Introduction to Applied Mathematics: An Iterative Process

An Introduction to Applied Mathematics: An Iterative Process Applied mathematics seeks to make predictions about some topic such as weather prediction, future value of an investment, the speed of a falling

### Presentation Outline

Feedback-based Congestion Control Gateway Router in Home M2M Network Lee Chin KHO Yasuo TAN Azman Osman LIM Japan Advanced Institute of Science and Technology School of Information Science Ishikawa 2 Presentation

### Control Flow Testing

Control low esting Structural esting In structural testing, the software is viewed as a white box and test cases are determined from the implementation of the software. Structural testing techniques include

### An Introduction to the Mofied Nodal Analysis

An Introduction to the Mofied Nodal Analysis Michael Hanke May 30, 2006 1 Introduction Gilbert Strang provides an introduction to the analysis of electrical circuits in his book Introduction to Applied

### The scenario: Initial state Target (Goal) states A set of intermediate states A set of operations that move the agent from one state to another.

CmSc310 Artificial Intelligence Solving Problems by Searching - Uninformed Search 1. Introduction All AI tasks involve searching. General idea: You know where you are (initial state) and where you want

### Chapter 28. Direct Current Circuits

Chapter 28 Direct Current Circuits Direct Current When the current in a circuit has a constant direction, the current is called direct current Most of the circuits analyzed will be assumed to be in steady

### Network Model APPENDIXD. D.1 Basic Concepts

APPENDIXD Network Model In the relational model, the data and the relationships among data are represented by a collection of tables. The network model differs from the relational model in that data are

### Some sequences have a fixed length and have a last term, while others go on forever.

Sequences and series Sequences A sequence is a list of numbers (actually, they don t have to be numbers). Here is a sequence: 1, 4, 9, 16 The order makes a difference, so 16, 9, 4, 1 is a different sequence.

### Analysis of Algorithms, I

Analysis of Algorithms, I CSOR W4231.002 Eleni Drinea Computer Science Department Columbia University Thursday, February 26, 2015 Outline 1 Recap 2 Representing graphs 3 Breadth-first search (BFS) 4 Applications

### 1. A Sequential Parity Checker

Chapter 13: Analysis of Clocked Sequential Circuits 1. A Sequential Parity Checker When binary data is transmitted or stored, an extra bit (called a parity bit) is frequently added for purposes of error

### PHYSICS 176. Experiment 3. Kirchhoff's Laws. Three resistors (Nominally: 1 Kilohm, 2 Kilohm, 3 Kilohm).

PHYSICS 176 Experiment 3 Kirchhoff's Laws Equipment: Supplies: Digital Multimeter, Power Supply (0-20 V.). Three resistors (Nominally: 1 Kilohm, 2 Kilohm, 3 Kilohm). A. Kirchhoff's Loop Law Suppose that

### Chemical Process Simulation

Chemical Process Simulation The objective of this course is to provide the background needed by the chemical engineers to carry out computer-aided analyses of large-scale chemical processes. Major concern