# Analysis of Dynamic Circuits in MATLAB

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Transactions on Electrical Engineering, Vol. 4 (2015), No Analysis of Dynamic Circuits in MATLAB Iveta Tomčíková 1) 1) Technical University in Košice/Department of Theoretical and Industrial Electrical Engineering, Košice, Slovakia, Abstract The paper deals with the proposal for the analysis of dynamic linear circuits in the MATLAB environment. A very powerful tool for the analysis of the circuits is to transform the circuits directly into the complex frequency domain using the Laplace transform and then apply the circuit analysis techniques to solve them. Applying above-mentioned methods together with the symbolic computation of MATLAB, there is no difficulty in finding the complete response of second and higher-order dynamic circuits. Keywords Dynamic circuit, circuit analysis, complete response, Laplace transform, symbolic computation, MATLAB. I. INTRODUCTION The circuit analysis is very important part of electrical engineering education at the university level. Traditional didactic methods and highly specialized software package such as PSPICE that offers flexible approach to circuit analysis and design are nowadays usually used for teaching of the circuit analysis. PSPICE is very helpful tool for the numerical circuit analysis, but it hides to students a considerable fact that a symbolic model of the circuit is behind the numerical results. Better way to teach students the circuit analysis is to use software product that provides symbolic manipulation and can be used for symbolic circuit analysis, e.g. MATLAB, Maple, Mathematica, etc. Especially MATLAB is widespread software that is a very appropriate for lots of different university courses. The MATLAB environment is very user-friendly with powerful built-in routines that enable to solve a very wide variety of computations. Its great advantage is the fact that the user has not to spend time in learning the software but he can spend time in learning the principles of a given problem. The analysis of dynamic circuits implies the process of finding the complete response of dynamic circuits that involves solving of first, second and higher-order circuits in the time domain or in the complex frequency domain using the Laplace transform. In the time domain, the dynamic circuits are represented by the equations in form of differential equations, and in the complex frequency domain, the dynamic circuits are represented by a set of simultaneous algebraic equations, thus the circuit analysis techniques for resistive circuits can be used. However, in the both cases the analysis of dynamic circuits is sometimes a time-consuming activity. Finding the complete response of dynamic circuits in the complex frequency domain using MATLAB greatly reduces the effort demanded to solve the set of algebraic equations and to take the inverse Laplace transform made by hand, thus there is no difficulty in solving dynamic circuits. II. FINDING THE COMPLETE RESPONSE OF DYNAMIC CIRCUIT IN COMPLEX FREQUENCY DOMAIN A. The Laplace Transform and the Inverse Laplace Transform The solution of dynamic circuits in the complex frequency domain proposes two procedures. The first procedure consists of the following steps: the representation of the dynamic circuit in the time domain by differential equations and using the Laplace transform to transform the differential equations into algebraic equations. The second procedure involves the representation of the dynamic circuit in the complex frequency domain directly using the Laplace transformation and then the circuit analysis, thus in this case it will be eliminated the need to write differential equations representing the circuit. The Laplace transform for a time-domain function f ( t ) (it is zero for t < 0 ) of a real variable t is [1], [2] = { } = F( s) L f ( t) f ( t) e d t (1) 0 st where s is a complex variable (meaning the complex frequency, s = σ + jω ), and F( s ) is a complex frequency function. For every function f ( t ) a unique function F( s ) exists. This can be written as 1 { F s } f ( t) =L ( ) (2) meaning that f ( t ) is the inverse Laplace transform of the complex frequency function F( s ) and exists for t 0. The inverse Laplace transform is defined by the integral [1], [2] σ + j 1 st f ( t) = F( s) e d s 2π j. (3) σ j B. Complex Frequency-Domain Representation of Circuit Elements The complex frequency-domain (s-domain) representation of circuit elements is obtained using the Laplace transform of their voltage-current relationships.

2 Transactions on Electrical Engineering, Vol. 4 (2015), No. x 65 The voltage-current relationship for a linear resistor having resistance R in the s-domain [1], [2] is U ( s) = R I ( s) (4) across the resistor, and I ( s ) is the Laplace transform of current through the resistor. The voltage-current relationship for a linear inductor having inductance L in the s-domain [1], [2] is U ( s) = sl I( s) Li(0) (5) across the inductor, I ( s ) is the Laplace transform of current through the inductor and i (0) is the initial condition. The equation (5) is used to represent the linear inductor in the s-domain as the series connection of two circuit elements that corresponds to the sum of the voltages on the right-hand side of (5). The voltage-current relationship for a linear capacitor having capacitance C in the s-domain [1], [2] is 1 u(0) U ( s) = I( s) sc + s (6) across the capacitor, I ( s ) is the Laplace transform of current through the capacitor and u (0) is the initial condition. The equation (6) is used to represent the linear capacitor in the s-domain as the series connection of two circuit elements that corresponds to the sum of the voltages on the right-hand side of (6). The voltage-current relationship for the coupled inductors having self-inductances L1, L 2 and mutual inductance M in the s-domain [1], [2] is U ( s) sl ± sm I ( s) L ± M i (0) U2( s) = ± sm sl 2 I2( s) ± M L 2 i2 (0) (7) where U1( s), U2( s ) is the Laplace transform of voltages u1 ( t), u2 ( t ) across the inductors L1, L 2 respectively, I1( s), I2( s ) is the Laplace transform of currents through the inductors L1, L 2 respectively, and i 1 (0), i 2 (0) are the initial conditions. Independent voltage and current sources are represented by the Laplace transform of their voltages and currents. The dependent voltage and current sources are represented by the Laplace transform of their controlled voltages and currents, and controlling voltages and currents. The ideal operational amplifier operates in the same way in the s-domain as it does in the time domain, thus the Laplace transform of its operating conditions is applied in the s-domain. C. Circuit Analysis Using the Laplace Transform After replacing all the circuit elements by their s-domain representation, the s-domain representation for the dynamic circuit with its initial conditions is obtained, and the circuit analysis techniques to solve the voltages and/or currents can be used. Usually two systematic circuit analysis techniques are used: the mesh analysis and the node analysis. Applying one of these methods to the circuit, a set of mesh or node equations is constituted by inspection. The mesh or node equations form a square system (the number of unknown is equal to the number of equations) of simultaneous algebraic equations containing the complex frequency s, thus it can be expressed in a matrix-vector form A( s) x( s) = b ( s) (8) where A ( s) is the coefficient matrix, x ( s) is a single column vector of unknowns (the mesh currents or node voltages), and b( s) is the single right-hand side column vector. Solving the set of equations (8) 1 x( s) = A ( s) b ( s) (9) the mesh currents or node voltages, described by a ratio of polynomials in the complex variable s, are obtained. If the mesh analysis is applied to the circuit, the s-domain element currents i ( s) are found as linear combination of the s-domain mesh currents x ( s) i( s) = K x ( s) (10) where K c is a matrix containing elements 0, +1, and 1. The s-domain element voltages can be expressed using the Laplace transform of element voltage-current relationships. If the node analysis is applied to the circuit, the s-domain element voltages u ( s) are found as linear combination of the s-domain node voltages c u( s) = K x ( s) (11) where K v is a matrix containing elements 0, +1, and 1. The s-domain element currents can be expressed using the Laplace transform of the element voltage-current relationship. To get the complete response of the dynamic circuit in the time domain, it is necessary to take the inverse Laplace transform of the obtained s-domain element voltages and currents. III. EXAMPLES OF CIRCUIT ANALYSIS USING THE LAPLACE TRANSFORM IN MATLAB Most of the steps for finding the complete response of a dynamic circuit given above can be easily executed using the MATLAB environment. The following text contains two representative examples of the circuit analysis using the Laplace transform in MATLAB. The analytical results of the solution together with the analysis in MATLAB and the graphical representation of the obtained results are given. v

3 Transactions on Electrical Engineering, Vol. 4 (2015), No. x 66 A. Analysis of Second-Order Circuit for the Overdamped, Critically Damped and Underdamped Case The circuit shown in Fig. 1 is at steady state before the switch is changed from position 1 to position 2 at time t = 0. Suppose that u 1 = 12 V, and u 2 = 24 V. Determine the element currents and voltages for t > 0 for three different sets of values of R1, R2, L and C: a) R 1 = 10 Ω, R 2 = 2 Ω, L = 2 H, C = 1 4 F, b) R 1 = 12 Ω, R 2 = 4 Ω, L = 2 H, C = 1 8 F, c) R 1 = 2 Ω, R 2 = 8 Ω, L = 1 H, C = 1 16 F. Fig. 2. The MATLAB program composed for analysis of the circuit shown in Fig. 1 using the Laplace transform. The branch currents for the critically damped case (for values of element parameters given in b) expressed in the analytical form are shown in Fig. 3. Fig. 1. Second-order circuit for the circuit analysis using the Laplace transform in MATLAB Example A. Since the circuit, driven by the DC voltage source, u 1, is supposed to be at steady state before the switch is changed from position 1 to position 2, the initial conditions, the current i (0) through inductor and the voltage u (0) across capacitor, can be found to be: u1 R2 i(0) =, u(0) = u1. (12) R + R R + R The s-domain representation of the circuit with the initial conditions is solved using the mesh analysis. The mesh equations with unknowns (mesh currents) I ( s), I ( s ) can be written in a matrix-vector form: 1 2 Fig. 3. The t-domain branch currents of the circuit shown in Fig. 1. MATLAB provides many functions to represent data graphically, thus there is no problem to plot the t-domain branch currents versus time (Fig. 4). 1 1 u u(0) 2 R1 + sl +, + Li(0) sc sc I1 s 1 1 I = 2 u(0), R 2 + sc sc s (13) All the remaining computations for solving of the circuit (with given parameters) were done by running the MATLAB program (Fig. 2). The MATLAB program is a set of MATLAB built-in instructions that are saved in M-file with extension.m (a dot and suffix m). Fig. 4. The t-domain currents of the circuit shown in Fig. 1 versus time.

4 Transactions on Electrical Engineering, Vol. 4 (2015), No. x 67 The graphical representation of the t-domain element voltages versus time for the critically damped case is in Fig. 5. The time domain representation of the voltage u out of the circuit for time t > 0 was determined in analytical form (Fig. 8). Fig. 7. The MATLAB program composed for the circuit analysis shown in Fig. 6 using the Laplace transform. Fig. 5. The t-domain element voltages of the circuit shown in Fig. 1 versus time. B. The Complete Response of Second-Order Circuit Containing Operational Amplifier The circuit shown in Fig. 6 is at steady state before the input voltage u in = 1 V at time t = 0 is applied. Find the output voltage u out ( t ) for t > 0. Suppose that: R 1 = 1 kω, R 2 = 200 Ω, C 1 = mf, C 2 = 50 µf, and all the initial conditions are zero. Fig. 8. The t-domain output voltage (analytical form) of the circuit shown in Fig. 6. In Fig. 9 the graphical representation of the voltage u out of the circuit and the input voltage for time t > 0 is shown. Fig. 6. Second-order circuit for the circuit analysis using the Laplace transform in MATLAB Example B. Since all the initial conditions are assumed zero, the complete response of the circuit is the transient response to the input unity step function applied at time t = 0. The s-domain representation of the circuit is solved using the node analysis. The node equations with voltages U1( s), U2( s), U0( s ) can be written in a matrix-vector form U1( s) 1 s sc 1 R + sc 1 R U ( s) = R2 sc2 U 0( s) 0 (14) All remaining computations for analysis of the circuit (with given parameters) were done by running the MATLAB program (Fig. 7) for the circuit analysis shown in Fig. 6 using the Laplace transform. Fig. 9. The t-domain output voltage (graphical form) of the circuit shown in Fig. 6. IV. ADVANTAGES OF PROPOSED PROCEDURE FOR ANALYSIS OF DYNAMIC CIRCUITS Using MATLAB environment for analysis of dynamic circuits has lots of advantages but the most significant ones are the effort, time, and the analytical form of the voltages and currents in the solved circuits. It is evident that the effort demanded to solve such type of problems is greatly reduced because it is not necessary

5 Transactions on Electrical Engineering, Vol. 4 (2015), No. x 68 to solve the set of algebraic equations and to take the inverse Laplace transform. The elapsed time that represents the processing and execution of the instructions in the composed MATLAB programs for given representative examples of the circuit analysis takes very short time. For the MATLAB program of Example A (Fig. 2) the elapsed time is seconds, and for the MATLAB program of Example B the elapsed time is seconds. The complete MATLAB program (all three cases with t-domain and graphical outputs) of Example A takes up seconds only, and of Example B takes up seconds. The last valuable advantage of MATLAB environment is the fact that the analytical form of the voltages and currents in the t-domain are available after circuit analysis in contrast with some highly specialized software packages. V. CONCLUSIONS The procedure for analysis of a dynamic circuit using the Laplace transform in the MATLAB environment was proposed. It consists of the steps that must be done by hand: finding the initial conditions for given circuit, drawing the s-domain representation of the given circuit, applying systematic circuit analysis techniques to s-domain representation of the given circuit and writing a set of simultaneous algebraic equations containing the terms of the complex variable s; rearranging the obtained set of equations in a matrixvector form, and the steps that will be a result of running the composed MATLAB program: assigning the values to the parameters of given circuit, defining the coefficient matrix and the right-hand side column vector, solving the set of algebraic equations, taking the inverse Laplace transform, printing the symbolic (analytic) expression of sought voltages and/or currents in a format that resembles type-set mathematics, plotting the graph of sought voltages and/or currents versus time. There exists also a possibility to compose the MATLAB program that can do all the steps for the dynamic circuit analysis. In this case, the analysis techniques, which enable to formulate the circuit equations in systematic and automatic way, have to be used. Using MATLAB for solving these problems saves a lot of time and effort and everybody who needs to analyse the circuit can concentrate on gaining an insight into properties of the circuit, and making conclusions. REFERENCES [1] R.C.Dorf, J.A.Svoboda: Introduction to Electric Circuits. John Wiley &Sons, Inc., 2007, ISBN [2] D. Mayer: Introduction in Theory of Electric Circuits. SNTL Prague, 1984 (in Czech). [3] MATLAB User s Guide (Mathematics, Programming, Graphics), MathWorks 2009.

### Bode Diagrams of Transfer Functions and Impedances ECEN 2260 Supplementary Notes R. W. Erickson

Bode Diagrams of Transfer Functions and Impedances ECEN 2260 Supplementary Notes. W. Erickson In the design of a signal processing network, control system, or other analog system, it is usually necessary

### ES250: Electrical Science. HW7: Energy Storage Elements

ES250: Electrical Science HW7: Energy Storage Elements Introduction This chapter introduces two more circuit elements, the capacitor and the inductor whose elements laws involve integration or differentiation;

### Electrical engineering education in the field of electric circuits theory at AGH University of Science and Technology in Kraków

st World Conference on Technology and Engineering Education WIETE Kraków, Poland, 4-7 September Electrical engineering education in the field of electric circuits theory at AGH University of Science and

### Mutual Inductance and Transformers F3 3. r L = ω o

utual Inductance and Transformers F3 1 utual Inductance & Transformers If a current, i 1, flows in a coil or circuit then it produces a magnetic field. Some of the magnetic flux may link a second coil

### EE 221 Circuits II. Chapter 13 Magnetically Coupled Circuits

EE Circuits II Chapter 3 Magnetically Coupled Circuits Magnetically Coupled Circuits 3. What is a transformer? 3. Mutual Inductance 3.3 Energy in a Coupled Circuit 3.4 inear Transformers 3.5 Ideal Transformers

### Lab 1: Introduction to PSpice

Lab 1: Introduction to PSpice Objectives A primary purpose of this lab is for you to become familiar with the use of PSpice and to learn to use it to assist you in the analysis of circuits. The software

### W03 Analysis of DC Circuits. Yrd. Doç. Dr. Aytaç Gören

W03 Analysis of DC Circuits Yrd. Doç. Dr. Aytaç Gören ELK 2018 - Contents W01 Basic Concepts in Electronics W02 AC to DC Conversion W03 Analysis of DC Circuits (self and condenser) W04 Transistors and

### Lecture 24. Inductance and Switching Power Supplies (how your solar charger voltage converter works)

Lecture 24 Inductance and Switching Power Supplies (how your solar charger voltage converter works) Copyright 2014 by Mark Horowitz 1 Roadmap: How Does This Work? 2 Processor Board 3 More Detailed Roadmap

### Eðlisfræði 2, vor 2007

[ Assignment View ] [ Print ] Eðlisfræði 2, vor 2007 30. Inductance Assignment is due at 2:00am on Wednesday, March 14, 2007 Credit for problems submitted late will decrease to 0% after the deadline has

### 2.1 Introduction. 2.2 Terms and definitions

.1 Introduction An important step in the procedure for solving any circuit problem consists first in selecting a number of independent branch currents as (known as loop currents or mesh currents) variables,

### Engineering: Electrical Engineering

Engineering: Electrical Engineering CRAFTY Curriculum Foundations Project Clemson University, May 4 7, 2000 Ben Oni, Report Editor Kenneth Roby and Susan Ganter, Workshop Organizers Summary This report

### Outline. Systems and Signals 214 / 244 & Energy Systems 244 / 344. Ideal Inductor. Ideal Inductor (cont... )

Outline Systems and Signals 214 / 244 & Energy Systems 244 / 344 Inductance, Leakage Inductance, Mutual Inductance & Transformers 1 Inductor revision Ideal Inductor Non-Ideal Inductor Dr. P.J. Randewijk

### Fast analytical techniques for electrical and electronic circuits. Jet Propulsion Laboratory California Institute of Technology

Fast analytical techniques for electrical and electronic circuits Vatché Vorpérian Jet Propulsion Laboratory California Institute of Technology PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE

### EE301 Lesson 14 Reading: 10.1-10.4, 10.11-10.12, 11.1-11.4 and 11.11-11.13

CAPACITORS AND INDUCTORS Learning Objectives EE301 Lesson 14 a. Define capacitance and state its symbol and unit of measurement. b. Predict the capacitance of a parallel plate capacitor. c. Analyze how

### Slide 1 / 26. Inductance. 2011 by Bryan Pflueger

Slide 1 / 26 Inductance 2011 by Bryan Pflueger Slide 2 / 26 Mutual Inductance If two coils of wire are placed near each other and have a current passing through them, they will each induce an emf on one

### PIN CONFIGURATION FEATURES ORDERING INFORMATION ABSOLUTE MAXIMUM RATINGS. D, F, N Packages

DESCRIPTION The µa71 is a high performance operational amplifier with high open-loop gain, internal compensation, high common mode range and exceptional temperature stability. The µa71 is short-circuit-protected

### Laboratory 4: Feedback and Compensation

Laboratory 4: Feedback and Compensation To be performed during Week 9 (Oct. 20-24) and Week 10 (Oct. 27-31) Due Week 11 (Nov. 3-7) 1 Pre-Lab This Pre-Lab should be completed before attending your regular

### J.L. Kirtley Jr. Electric network theory deals with two primitive quantities, which we will refer to as: 1. Potential (or voltage), and

Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.061 Introduction to Power Systems Class Notes Chapter 1: eiew of Network Theory J.L. Kirtley Jr. 1 Introduction

### RLC Resonant Circuits

C esonant Circuits Andrew McHutchon April 20, 203 Capacitors and Inductors There is a lot of inconsistency when it comes to dealing with reactances of complex components. The format followed in this document

ECSE 4440 Control System Engineering Fall 2001 Project 3 Controller Design in Frequency Domain TA 1. Abstract 2. Introduction 3. Controller design in Frequency domain 4. Experiment 5. Colclusion 1. Abstract

### Impedance Matching. Using transformers Using matching networks

Impedance Matching The plasma industry uses process power over a wide range of frequencies: from DC to several gigahertz. A variety of methods are used to couple the process power into the plasma load,

### Lecture Notes: ECS 203 Basic Electrical Engineering Semester 1/2010. Dr.Prapun Suksompong 1 June 16, 2010

Sirindhorn International Institute of Technology Thammasat University School of Information, Computer and Communication Technology Lecture Notes: ECS 203 Basic Electrical Engineering Semester 1/2010 Dr.Prapun

### SERIES-PARALLEL DC CIRCUITS

Name: Date: Course and Section: Instructor: EXPERIMENT 1 SERIES-PARALLEL DC CIRCUITS OBJECTIVES 1. Test the theoretical analysis of series-parallel networks through direct measurements. 2. Improve skills

### 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

### System Modeling and Control for Mechanical Engineers

Session 1655 System Modeling and Control for Mechanical Engineers Hugh Jack, Associate Professor Padnos School of Engineering Grand Valley State University Grand Rapids, MI email: jackh@gvsu.edu Abstract

### 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

### 3.2 Sources, Sinks, Saddles, and Spirals

3.2. Sources, Sinks, Saddles, and Spirals 6 3.2 Sources, Sinks, Saddles, and Spirals The pictures in this section show solutions to Ay 00 C By 0 C Cy D 0. These are linear equations with constant coefficients

### Lab #9: AC Steady State Analysis

Theory & Introduction Lab #9: AC Steady State Analysis Goals for Lab #9 The main goal for lab 9 is to make the students familar with AC steady state analysis, db scale and the NI ELVIS frequency analyzer.

### DCMS DC MOTOR SYSTEM User Manual

DCMS DC MOTOR SYSTEM User Manual release 1.3 March 3, 2011 Disclaimer The developers of the DC Motor System (hardware and software) have used their best efforts in the development. The developers make

### Dynamic Process Modeling. Process Dynamics and Control

Dynamic Process Modeling Process Dynamics and Control 1 Description of process dynamics Classes of models What do we need for control? Modeling for control Mechanical Systems Modeling Electrical circuits

### Power supplies. EE328 Power Electronics Assoc. Prof. Dr. Mutlu BOZTEPE Ege University, Dept. of E&E

Power supplies EE328 Power Electronics Assoc. Prof. Dr. Mutlu BOZTEPE Ege University, Dept. of E&E EE328 POWER ELECTRONICS Outline of lecture Introduction to power supplies Modelling a power transformer

### Electricity and Magnetism, Georgia, GEOSTM (Georgian National Agency for Standards and Metrology)

single references 1 1.018 V Temperature (23 ± 3) C 6 µv 2 95% No 1 single references 10 10 V Temperature (23 ± 3) C 25 µv 2 95% No 2 low ranges DMM > 0.01 10 V Temperature (23 ± 3) C 2E-06U + 3E-06 to

### Designing Stable Compensation Networks for Single Phase Voltage Mode Buck Regulators

Designing Stable Compensation Networks for Single Phase Voltage Mode Buck Regulators Technical Brief December 3 TB47. Author: Doug Mattingly Assumptions This Technical Brief makes the following assumptions:.

### Homework #11 203-1-1721 Physics 2 for Students of Mechanical Engineering

Homework #11 203-1-1721 Physics 2 for Students of Mechanical Engineering 2. A circular coil has a 10.3 cm radius and consists of 34 closely wound turns of wire. An externally produced magnetic field of

### Department of Electrical and Electronic Engineering, California State University, Sacramento

Department of Electrical and Electronic Engineering, California State University, Sacramento Engr 17 Introductory Circuit Analysis, graded, 3 units Instructor: Tatro - Spring 2016 Section 2, Call No. 30289,

### Creating a Usable Power Supply from a Solar Panel

Creating a Usable Power Supply from a Solar Panel An exploration in DC- DC converters By Kathleen Ellis Advised by Dr. Derin Sherman Department of Physics, Cornell College November 21, 2012 Introduction

### University of Rochester Department of Electrical and Computer Engineering ECE113 Lab. #7 Higher-order filter & amplifier designs March, 2012

University of Rochester Department of Electrical and Computer Engineering ECE113 Lab. #7 Higherorder filter & amplifier designs March, 2012 Writeups, due one week after the lab is performed, should provide

### Basic Op Amp Circuits

Basic Op Amp ircuits Manuel Toledo INEL 5205 Instrumentation August 3, 2008 Introduction The operational amplifier (op amp or OA for short) is perhaps the most important building block for the design of

### Understanding Power Impedance Supply for Optimum Decoupling

Introduction Noise in power supplies is not only caused by the power supply itself, but also the load s interaction with the power supply (i.e. dynamic loads, switching, etc.). To lower load induced noise,

### Application of network analyzer in measuring the performance functions of power supply

J Indian Inst Sci, July Aug 2006, 86, 315 325 Indian Institute of Science Application of network analyzer in measuring the performance functions of power supply B SWAMINATHAN* AND V RAMANARAYANAN Power

Copyright. All rights reserved. LTspice IV Getting Started Guide 2 Benefits of Using LTspice IV Stable SPICE circuit simulation with Unlimited number of nodes Schematic/symbol editor Waveform viewer Library

### Department of Chemical Engineering ChE-101: Approaches to Chemical Engineering Problem Solving MATLAB Tutorial VI

Department of Chemical Engineering ChE-101: Approaches to Chemical Engineering Problem Solving MATLAB Tutorial VI Solving a System of Linear Algebraic Equations (last updated 5/19/05 by GGB) Objectives:

### Capacitors in Circuits

apacitors in ircuits apacitors store energy in the electric field E field created by the stored charge In circuit apacitor may be absorbing energy Thus causes circuit current to be reduced Effectively

### South Carolina College- and Career-Ready (SCCCR) Pre-Calculus

South Carolina College- and Career-Ready (SCCCR) Pre-Calculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know

### AC 2009-2024: STUDENT OUTLOOK TOWARD MEDIA-BASED MODULES IN ELECTRONICS AND NETWORK ANALYSIS

AC 2009-2024: STUDENT OUTLOOK TOWARD MEDIA-BASED MODULES IN ELECTRONICS AND NETWORK ANALYSIS Jean-Claude Thomassian, State University of New York, Maritime College Dr. Jean-Claude Thomassian received his

### Elementary Functions

Chapter Three Elementary Functions 31 Introduction Complex functions are, of course, quite easy to come by they are simply ordered pairs of real-valued functions of two variables We have, however, already

### EXAMPLE 8: An Electrical System (Mechanical-Electrical Analogy)

EXAMPLE 8: An Electrical System (Mechanical-Electrical Analogy) A completely analogous procedure can be used to find the state equations of electrical systems (and, ultimately, electro-mechanical systems

### ISSCC 2003 / SESSION 10 / HIGH SPEED BUILDING BLOCKS / PAPER 10.5

ISSCC 2003 / SESSION 10 / HIGH SPEED BUILDING BLOCKS / PAPER 10.5 10.5 Broadband ESD Protection Circuits in CMOS Technology Sherif Galal, Behzad Razavi Electrical Engineering Department, University of

### SCHWEITZER ENGINEERING LABORATORIES, COMERCIAL LTDA.

Pocket book of Electrical Engineering Formulas Content 1. Elementary Algebra and Geometry 1. Fundamental Properties (real numbers) 1 2. Exponents 2 3. Fractional Exponents 2 4. Irrational Exponents 2 5.

### S. Boyd EE102. Lecture 1 Signals. notation and meaning. common signals. size of a signal. qualitative properties of signals.

S. Boyd EE102 Lecture 1 Signals notation and meaning common signals size of a signal qualitative properties of signals impulsive signals 1 1 Signals a signal is a function of time, e.g., f is the force

### Assessment Plan for Learning Outcomes for BA/BS in Physics

Department of Physics and Astronomy Goals and Learning Outcomes 1. Students know basic physics principles [BS, BA, MS] 1.1 Students can demonstrate an understanding of Newton s laws 1.2 Students can demonstrate

### What are the place values to the left of the decimal point and their associated powers of ten?

The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything

### Further Mathematics for Engineering Technicians

Unit 28: Further Mathematics for Engineering Technicians Unit code: QCF Level 3: Credit value: 10 Guided learning hours: 60 Aim and purpose H/600/0280 BTEC Nationals This unit aims to enhance learners

### METHODOLOGICAL CONSIDERATIONS OF DRIVE SYSTEM SIMULATION, WHEN COUPLING FINITE ELEMENT MACHINE MODELS WITH THE CIRCUIT SIMULATOR MODELS OF CONVERTERS.

SEDM 24 June 16th - 18th, CPRI (Italy) METHODOLOGICL CONSIDERTIONS OF DRIVE SYSTEM SIMULTION, WHEN COUPLING FINITE ELEMENT MCHINE MODELS WITH THE CIRCUIT SIMULTOR MODELS OF CONVERTERS. Áron Szûcs BB Electrical

### Power Electronics. Prof. K. Gopakumar. Centre for Electronics Design and Technology. Indian Institute of Science, Bangalore.

Power Electronics Prof. K. Gopakumar Centre for Electronics Design and Technology Indian Institute of Science, Bangalore Lecture - 1 Electric Drive Today, we will start with the topic on industrial drive

### LM833,LMF100,MF10. Application Note 779 A Basic Introduction to Filters - Active, Passive,and. Switched Capacitor. Literature Number: SNOA224A

LM833,LMF100,MF10 Application Note 779 A Basic Introduction to Filters - Active, Passive,and Switched Capacitor Literature Number: SNOA224A A Basic Introduction to Filters Active, Passive, and Switched-Capacitor

### Abstract. Cycle Domain Simulator for Phase-Locked Loops

Abstract Cycle Domain Simulator for Phase-Locked Loops Norman James December 1999 As computers become faster and more complex, clock synthesis becomes critical. Due to the relatively slower bus clocks

### Electrical Engineering 234

Electrical Engineering 234 Electrical Engineering Circuit Laboratory by Robert C. Maher, Assistant Professor with Duane T. Hickenbottom, Graduate Assistant University of Nebraska Lincoln Department of

### CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA

We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical

### Math Review. for the Quantitative Reasoning Measure of the GRE revised General Test

Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important

### 12.5: CHI-SQUARE GOODNESS OF FIT TESTS

125: Chi-Square Goodness of Fit Tests CD12-1 125: CHI-SQUARE GOODNESS OF FIT TESTS In this section, the χ 2 distribution is used for testing the goodness of fit of a set of data to a specific probability

### Unit/Standard Number. High School Graduation Years 2010, 2011 and 2012

1 Secondary Task List 100 SAFETY 101 Demonstrate an understanding of State and School safety regulations. 102 Practice safety techniques for electronics work. 103 Demonstrate an understanding of proper

### the points are called control points approximating curve

Chapter 4 Spline Curves A spline curve is a mathematical representation for which it is easy to build an interface that will allow a user to design and control the shape of complex curves and surfaces.

### Equipment: Power Supply, DAI, Variable resistance (8311), Variable inductance (8321)

Lab 4: 3-phase circuits. Objective: to study voltage-current relationships in 3-phase circuits; to learn to make delta and Y connections; to calculate and measure real, apparent, and reactive powers. Equipment:

### EE 242 EXPERIMENT 5: COMPUTER SIMULATION OF THREE-PHASE CIRCUITS USING PSPICE SCHEMATICS 1

EE 242 EXPERIMENT 5: COMPUTER SIMULATION OF THREE-PHASE CIRCUITS USING PSPICE SCHEMATICS 1 Objective: To build, simulate, and analyze three-phase circuits using OrCAD Capture Pspice Schematics under balanced

### Diodes have an arrow showing the direction of the flow.

The Big Idea Modern circuitry depends on much more than just resistors and capacitors. The circuits in your computer, cell phone, Ipod depend on circuit elements called diodes, inductors, transistors,

### Design of an Auxiliary Power Distribution Network for an Electric Vehicle

Design of an Auxiliary Power Distribution Network for an Electric Vehicle William Chen, Simon Round and Richard Duke Department of Electrical & Computer Engineering University of Canterbury, Christchurch,

### Reading: HH Sections 4.11 4.13, 4.19 4.20 (pgs. 189-212, 222 224)

6 OP AMPS II 6 Op Amps II In the previous lab, you explored several applications of op amps. In this exercise, you will look at some of their limitations. You will also examine the op amp integrator and

### Module 11: Conducted Emissions

Module 11: Conducted Emissions 11.1 Overview The term conducted emissions refers to the mechanism that enables electromagnetic energy to be created in an electronic device and coupled to its AC power cord.

### Discrete Convolution and the Discrete Fourier Transform

Discrete Convolution and the Discrete Fourier Transform Discrete Convolution First of all we need to introduce what we might call the wraparound convention Because the complex numbers w j e i πj N have

### Last time : energy storage elements capacitor.

Last time : energy storage elements capacitor. Charge on plates Energy stored in the form of electric field Passive sign convention Vlt Voltage drop across real capacitor can not change abruptly because

### CONCEPT-II. Overview of demo examples

CONCEPT-II CONCEPT-II is a frequency domain method of moment (MoM) code, under development at the Institute of Electromagnetic Theory at the Technische Universität Hamburg-Harburg (www.tet.tuhh.de). Overview

### Simulation and Analysis of Parameter Identification Techniques for Induction Motor Drive

International Journal of Electronic and Electrical Engineering. ISSN 0974-2174 Volume 7, Number 10 (2014), pp. 1027-1035 International Research Publication House http://www.irphouse.com Simulation and

### Experimental Identification an Interactive Online Course

Proceedings of the 17th World Congress The International Federation of Automatic Control Experimental Identification an Interactive Online Course L. Čirka, M. Fikar, M. Kvasnica, and M. Herceg Institute

### OPTIMAL DISPATCH OF POWER GENERATION SOFTWARE PACKAGE USING MATLAB

OPTIMAL DISPATCH OF POWER GENERATION SOFTWARE PACKAGE USING MATLAB MUHAMAD FIRDAUS BIN RAMLI UNIVERSITI MALAYSIA PAHANG v ABSTRACT In the reality practical power system, power plants are not at the same

### Simulation of Ungrounded Shipboard Power Systems in PSpice

Simulation of Ungrounded Shipboard Power Systems in PSpice Haibo Zhang IEEE Student Member Karen L.Butler IEEE Member Power System Automation Lab Electrical Engineering Department Texas A&M University

### Fundamentals of Electronic Circuit Design. By Hongshen Ma

Fundamentals of Electronic Circuit Design By Hongshen Ma Preface Why Study Electronics? Purely mechanical problems are often only a subset of larger multidomain problems faced by the designer. Particularly,

### 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

### The Membrane Equation

The Membrane Equation Professor David Heeger September 5, 2000 RC Circuits Figure 1A shows an RC (resistor, capacitor) equivalent circuit model for a patch of passive neural membrane. The capacitor represents

### 8 Speed control of Induction Machines

8 Speed control of Induction Machines We have seen the speed torque characteristic of the machine. In the stable region of operation in the motoring mode, the curve is rather steep and goes from zero torque

### Ver 3537 E1.1 Analysis of Circuits (2014) E1.1 Circuit Analysis. Problem Sheet 1 (Lectures 1 & 2)

Ver 3537 E. Analysis of Circuits () Key: [A]= easy... [E]=hard E. Circuit Analysis Problem Sheet (Lectures & ). [A] One of the following circuits is a series circuit and the other is a parallel circuit.

### 3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes

Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the general

### Keywords: input noise, output noise, step down converters, buck converters, MAX1653EVKit

Maxim > Design Support > Technical Documents > Tutorials > Power-Supply Circuits > APP 986 Keywords: input noise, output noise, step down converters, buck converters, MAX1653EVKit TUTORIAL 986 Input and

### Apprentice Telecommunications Technician Test (CTT) Study Guide

Apprentice Telecommunications Technician Test (CTT) Study Guide 1 05/2014 Study Guide for Pacific Gas & Electric Company Apprentice Telecommunications Technician Qualifying Test (CTT) About the Test The

### Mathematics Online Instructional Materials Correlation to the 2009 Algebra I Standards of Learning and Curriculum Framework

Provider York County School Division Course Syllabus URL http://yorkcountyschools.org/virtuallearning/coursecatalog.aspx Course Title Algebra I AB Last Updated 2010 - A.1 The student will represent verbal

### Analysis of Common-Collector Colpitts Oscillator

Analysis of Common-Collector Colpitts Oscillator H R Pota May 20, 2005 Introduction Murphy s rule when paraphrased for oscillators reads [], Amplifiers will oscillate but oscillators won t. As we all know,

### Algebra Practice Problems for Precalculus and Calculus

Algebra Practice Problems for Precalculus and Calculus Solve the following equations for the unknown x: 1. 5 = 7x 16 2. 2x 3 = 5 x 3. 4. 1 2 (x 3) + x = 17 + 3(4 x) 5 x = 2 x 3 Multiply the indicated polynomials

### ENERGY TRANSFER SYSTEMS AND THEIR DYNAMIC ANALYSIS

ENERGY TRANSFER SYSTEMS AND THEIR DYNAMIC ANALYSIS Many mechanical energy systems are devoted to transfer of energy between two points: the source or prime mover (input) and the load (output). For chemical

### CITY UNIVERSITY LONDON. BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION

No: CITY UNIVERSITY LONDON BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION ENGINEERING MATHEMATICS 2 (resit) EX2005 Date: August

### 3 Orthogonal Vectors and Matrices

3 Orthogonal Vectors and Matrices The linear algebra portion of this course focuses on three matrix factorizations: QR factorization, singular valued decomposition (SVD), and LU factorization The first

### Single-Stage High Power Factor Flyback for LED Lighting

Application Note Stockton Wu AN012 May 2014 Single-Stage High Power Factor Flyback for LED Lighting Abstract The application note illustrates how the single-stage high power factor flyback converter uses

### Impedance Matching and Matching Networks. Valentin Todorow, December, 2009

Impedance Matching and Matching Networks Valentin Todorow, December, 2009 RF for Plasma Processing - Definition of RF What is RF? The IEEE Standard Dictionary of Electrical and Electronics Terms defines

### Solution of Linear Systems

Chapter 3 Solution of Linear Systems In this chapter we study algorithms for possibly the most commonly occurring problem in scientific computing, the solution of linear systems of equations. We start

### CURRENT ELECTRICITY INTRODUCTION TO RESISTANCE, CAPACITANCE AND INDUCTANCE

CURRENT ELECTRICITY INTRODUCTION TO RESI STANCE, CAPACITANCE AND INDUCTANCE P R E A M B L E This problem is adapted from an on-line knowledge enhancement module for a PGCE programme. It is used to cover

### Indiana State Core Curriculum Standards updated 2009 Algebra I

Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and

### Design of Four Input Buck-Boost DC-DC Converter for Renewable Energy Application

Design of Four Input Buck-Boost DC-DC Converter for Renewable Energy Application A.Thiyagarajan Assistant Professor, Department of Electrical and Electronics Engineering Karpagam Institute of Technology

### Numerical Analysis. Professor Donna Calhoun. Fall 2013 Math 465/565. Office : MG241A Office Hours : Wednesday 10:00-12:00 and 1:00-3:00

Numerical Analysis Professor Donna Calhoun Office : MG241A Office Hours : Wednesday 10:00-12:00 and 1:00-3:00 Fall 2013 Math 465/565 http://math.boisestate.edu/~calhoun/teaching/math565_fall2013 What is

### Algebra I Vocabulary Cards

Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression