Power Factor in Electrical Power Systems with Non-Linear Loads

Save this PDF as:

Size: px
Start display at page:

Download "Power Factor in Electrical Power Systems with Non-Linear Loads"

Transcription

1 Power Factor i Electrical Power Systems with No-Liear Loads By: Gozalo Sadoval, ARTECHE / INELAP S.A. de C.V. Abstract. Traditioal methods of Power Factor Correctio typically focus o displacemet power factor ad therefore do ot achieve the total eergy savigs available i facilities havig both liear ad o liear loads. Oly through Total Power Factor Correctio ca the savigs ad power quality be maximized. This paper provides a simplified explaatio of both Power Factor (PF) ad Total Power Factor (TPF) for various electrical systems havig both liear ad o liear loads. Electrical diagrams ad waveforms are combied with represetative methods for aalysis ad help to demostrate the applicatio of mathematical calculatios. The cocept of Total Power Factor is frequetly misuderstood yet it is a importat cosideratio for electrical istallatios which employ power electroics equipmet. A uderstadig of Total Power Factor, alog with the potetial problems, is a importat cosideratio whe applyig o liear loads such as VFD s, DC motor drives, Uiterruptible Power Supplies, ad other power or frequecy coverters icludig SCR cotrollers. The article explais the differece betwee Power Factor ad Total Power Factor, ad the ifluece of o liear, harmoic producig loads o overall power system quality, reliability ad eergy efficiecy. Power factor with liear loads Whe the loads coected to the system are liear ad the voltage is siusoidal, the power factor is calculated with the followig equatio: ( ϕ) pf = cos () Ufortuately, this formula has led to a misuderstadig of the power factor cocept. Power factor is the proportioal relatio of the active power (or workig power) to the apparet power (total power delivered by the utility or cosumed by the load). Usig this defiitio, the power factor must be calculated as: P pf = () S Whe the loads are liear ad the voltage is siusoidal, the active, reactive ad apparet power are calculated with the followig equatios: ( ϕ) P = VI cos (3)

2 ( ϕ) Q = VI si (4) S = VI (5) ad we ca easily see that the power factor is calculated accordig to the equatio (), which is the resultat of the vector relatioship betwee the active, reactive ad apparet power show i figure. Figure Figure A low power factor meas a that a low amout of the total power delivered or cosumed (S) is used as workig power (P) ad a cosiderable amout is reactive power (Q). the purpose of power factor correctio is to reduce the reactive compoet of the total power. This achieves a more efficiet use of the eergy because whe the power factor is improved the workig power is equal (or early equal) to the total power, ad reactive power is zero or egligible. The most commo way to correct power factor is by addig a capacitor bak coected i parallel with the power system. The capacitor bak (Q C ) supplies most of the reactive power eeded by the load ad a small amout is supplied by the utility (Q ), as show i figure. The origial agle ( ) reduced to a smaller value ( ) cos( ϕ ) > ( ϕ ). cos ϕ betwee the apparet ad active power is ϕ ad the power factor is improved because It is very importat to ote that the reductio i the agle obtaied by the power factor improvemet is a result of the vector relatioship betwee the active, reactive ad apparet power, but what we are really doig is reducig the reactive power, cosequetly the apparet power is also reduced ad the power factor is icreased. Power factor with o liear loads ad siusoidal voltage Whe the loads are o liear but the voltage is siusoidal, the curret has harmoics ad the active, reactive ad apparet power should ot be calculated usig the traditioal methods as demostrated by equatios (3), (4) ad (5). This meas that the equatio () ca ot be used to calculate the power factor whe o liear loads are cocered. The active power is the mea (or average) value of the istataeous power, so it ca be calculated as:

3 P = VI cos ( ϕ ) (6) The rms value of curret is a fuctio of the total harmoic curret distortio ad the rms value of the fudametal compoet of curret: I = I + THD (7) I The power factor, for o liear loads, ca be calculated usig equatios (5), (6) ad (7): pf = cos( ϕ ) (8) + THD I There are two terms ivolved i the calculatio of the power factor: cos( ϕ ) ad + THD. The term ( ) pf because it I cos ϕ is called displacemet power factor ( ) depeds of the phase agle betwee the voltage ad the fudametal compoet of the curret, ad it is similar to the power factor calculated with liear loads ad siusoidal voltage. The term + THDI is called distortio power factor ( pf dist ) because it depeds of the curret harmoic distortio. The power factor calculated as the product of the displacemet power factor ad the pf : distortio power factor is kow as Total Power Factor ( ) T Total Power Factor ( ) T pf, where pf T = fpdisp fpdist (9) If the reactive power of the loads icreases, the displacemet agle betwee the voltage ad the fudametal compoet of the curret also icreases ad the total power factor decreases. Likewise, if the total harmoic curret distortio icreases, the total power factor decreases. Oe ca see by equatio (8) that Total Power Factor will always be lower tha the displacemet power factor wheever harmoic distortio is preset. Total power factor correctio ca oly be achieved whe both displacemet power factor ad distortio power factor are corrected. This requires a two step process:. Reduce the displacemet agle betwee voltage ad curret.. Reduce the total harmoic curret distortio. If either of these steps is take without the other the total power factor will be icreased but it may ot be high eough to reach the miimum value required by the utility. Additioally, if oe step is take without the other, the Total Power Factor Correctio ad the correspodig efficiecies will ot be achieved. A vector relatioship, as show i figure 3, ca be obtaied from the active power (P), the fudametal reactive power ( Q ) ad the fudametal apparet power ( S ) F prior to F displacemet power factor improvemet. This relatioship allows us to visualize the effect disp

4 that a capacitor bak ( Q C F ) has o correctig the displacemet power factor (figure 4). Wheever capacitors are used, care should be take to avoid a resoat coditio betwee the capacitor bak ad the mai trasformer. Figure 3 Figure 4 The total power factor ca be improved by decreasig the harmoic curret distortio, which is accomplished by usig a filter istead of a capacitor bak. The capacitive part of the filter improves the displacemet power factor, while the combiatio of the reactor ad the capacitor bak decrease the total harmoic distortio of the curret. A twofold result is achieved, that is improvemet of the distortio power factor ad improvemet of the displacemet power factor. Impedace [ohms] Capacitive behavior Resistive behavior Iductive behavior Harmoic order Zreact Zcap Zfilter Figure 5 The Figure 5 shows the behavior of a harmoic passive filter as well as the behavior of its iductive ad capacitive parts. Below the tuig harmoic the behavior of the filter is like a capacitor ad above the tuig harmoic the behavior is like a iductor. At the tuig harmoic the behavior of the filter is like a resistor. We ca see that at the fudametal frequecy, the filter acts like a capacitor bak because its reactace is basically capacitive, so the filter improves the displacemet power factor.

5 At the tuig harmoic the filter is a very low impedace ad a great amout of curret at the tuig harmoic flows through it, decreasig the total harmoic curret distortio ad improvig the distortio power factor. The improvemet of both power factors (displacemet ad distortio) improves the total power factor. If a capacitor bak were used istead of a filter, the displacemet power factor would have bee improved. If there is o resoace, the distortio power factor does ot chage ad the total power factor icreases oly because the displacemet power factor also icreases, but the total power factor may ot be high eough to reach the miimum value required by the utility. If a resoat coditio is created betwee the capacitor bak ad the mai trasformer, the total harmoic curret distortio icreases so the distortio power factor degrades ad the fial result is a low total power factor eve with a capacitor bak ad a high displacemet power factor. The figure 6 shows the behavior of the total power factor for differet values of displacemet power factor ad total harmoic curret distortio. Total power factor behavior.00 Total power factor Displacemet power factor THD-I=0% THD-I=0% THD-I=0% THD-I=30% THD-I=40% THD-I=50% THD-I=60% THD-I=70% THD-I=80% THD-I=90% THD-I=00% Figure 6 Power factor with o liear loads ad voltage distortio Whe the loads are o liear ad the voltage is distorted the active, reactive ad apparet power ca ot be calculated usig traditioal methods such as the equatios (3), (4) ad (5). The commo equatio () ca ot be used to calculate the power factor either. The active power is the mea (or average) value of the istataeous power. If the phase agles of the voltage harmoics are eglected, the active power ca be calculated as:

6 P = N = ( ) Now, the power factor ca be calculated usig equatio (): V I cos ϕ (0) pf N ( ) VI cos ϕ = = VI () but, the voltage rms value is a fuctio of the total harmoic voltage distortio ad the rms value of the fudametal compoet of voltage: V = V + THD () V Usig equatios (7), () ad (), the power factor ca be calculated as follows: pf P = (3) S + THD I + THD V There are two terms ivolved i the calculatio of the power factor. The term P S is the relatioship betwee the total active power (icludig harmoics) ad the apparet fudametal power. This term should ot be called displacemet power factor because it ivolves the active power caused by the fudametal compoets ad harmoics. The term ( THD I + THD ) V dist, which depeds o the distortio of voltage ad curret. The power factor calculated as the product of the distortio power factor ad the proportio of the total active power to the fudametal pf : + is the distortio power factor ( pf ) apparet power is the total power factor ( ) T The term P S ca be expressed as: where I cos( ) S P S pf V I cos = S P = (4) T pf dist S ( ϕ ) + N = V I S cos ( ϕ ) V ϕ is the displacemet power factor ( pf ) be calculated as follows: disp (5), so the total power factor ca

7 pf T = pf disp + N = VI cos( ϕ ) pf S dist (6) I a similar way to the case of the o liear loads ad siusoidal voltage, if the reactive power of the loads icreases, the displacemet agle betwee the fudametal compoets of voltage ad curret also icreases ad the total power factor decreases. If the distortio of curret ad voltage icreases the distortio power factor decreases ad the total power factor decreases as well. If we wat to improve the power factor the it is ecessary to:. Reduce the displacemet agle betwee the fudametal compoets of voltage ad curret.. Reduce the total harmoic distortio of both the curret ad the voltage. A vector relatioship, show i figure 7, ca be obtaied from the fudametal active ( P ), reactive ( Q ) ad apparet power ( ) F F improvemet. This vector relatioship allows for the use of a capacitor bak ( ) S before the displacemet power factor Q to correct the displacemet power factor (show i figure 8), but agai, a resoat coditio betwee the capacitor bak ad the mai trasformer must be avoided. I a similar way to the case of power factor with o liear loads ad siusoidal voltage, the total power factor ca be improved by decreasig the harmoic curret distortio, ad this ca be achieved by usig a harmoic filter. The capacitive part of the filter improves the displacemet power factor while the combiatio of the reactor ad the capacitor bak decreases the total harmoic distortio of the curret, thus improvig the distortio power factor, as well as the total power factor. C F Figure 7 Figure 8 The figure 9 shows the behavior of the distortio power factor for differet values of total harmoic curret ad voltage distortio.

8 Distortio power factor behavior.0000 Distortio power factor % 3% 6% 9% % 5% 8% % 4% 7% 30% THD-I THD-V=% THD-V=3% THD-V=5% THD-V=7% THD-V=9% THD-V=% THD-V=3% THD-V=5% Figure 9 Example: Cosider a load of 850 kw, 000 kva with 38% total harmoic curret distortio. The displacemet power factor is calculated as = The rms value of curret is: Therefore the total power factor is: I = I = I pf = cos ϕ so I ( ) = = + THD I Coclusio: While traditioal methods of improvig power factor may help to reduce some utility power factor charges, simple power factor correctio does ot assure adequate improvemet of total power factor or achieve the total beefits of performig total power factor correctio. Oly whe measures are take to correct both the displacemet ad distortio power factors ca eergy efficiecy be achieved. The implemetatio of total power factor improvemet ca achieve sigificat cost savigs such as elimiatio of utility power factor charges, reduced eergy demad, reduced electrical equipmet operatig temperatures ad exteder electrical equipmet life.

CHAPTER 3 DIGITAL CODING OF SIGNALS

CHAPTER 3 DIGITAL CODING OF SIGNALS Computers are ofte used to automate the recordig of measuremets. The trasducers ad sigal coditioig circuits produce a voltage sigal that is proportioal to a quatity

More information

*The most important feature of MRP as compared with ordinary inventory control analysis is its time phasing feature.

Itegrated Productio ad Ivetory Cotrol System MRP ad MRP II Framework of Maufacturig System Ivetory cotrol, productio schedulig, capacity plaig ad fiacial ad busiess decisios i a productio system are iterrelated.

More information

Soving Recurrence Relations

Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree

More information

where: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return

EVALUATING ALTERNATIVE CAPITAL INVESTMENT PROGRAMS By Ke D. Duft, Extesio Ecoomist I the March 98 issue of this publicatio we reviewed the procedure by which a capital ivestmet project was assessed. The

More information

Measuring Magneto Energy Output and Inductance Revision 1

Measurig Mageto Eergy Output ad Iductace evisio Itroductio A mageto is fudametally a iductor that is mechaically charged with a iitial curret value. That iitial curret is produced by movemet of the rotor

More information

9.8: THE POWER OF A TEST

9.8: The Power of a Test CD9-1 9.8: THE POWER OF A TEST I the iitial discussio of statistical hypothesis testig, the two types of risks that are take whe decisios are made about populatio parameters based

More information

Capacitor banks, range. Capacitor banks, STD range. Three-phase measurement

N e w g e e r a t i o o f C I R C U T O R c a p a c i t o r b a k s Capacitor baks, rage Capacitor baks, STD rage Itelliget capacitor baks Three-phase measuremet Built-i etwork aalzer The qualit at the

More information

CHAPTER 3 THE TIME VALUE OF MONEY

CHAPTER 3 THE TIME VALUE OF MONEY OVERVIEW A dollar i the had today is worth more tha a dollar to be received i the future because, if you had it ow, you could ivest that dollar ad ear iterest. Of all

More information

The Euler Totient, the Möbius and the Divisor Functions

The Euler Totiet, the Möbius ad the Divisor Fuctios Rosica Dieva July 29, 2005 Mout Holyoke College South Hadley, MA 01075 1 Ackowledgemets This work was supported by the Mout Holyoke College fellowship

More information

Improvement of Energy Efficiency in Power Electronics at Partial Load

Improvemet of ergy fficiecy i Power lectroics at Partial Load Klaus Muehlbauer, ieter Gerlig Istitute for lectrical rives ad Actuators, iversitaet der Budeswehr Mueche, Germay klaus.muehlbauer@uibw.de,

More information

.04. This means \$1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth

Questio 1: What is a ordiary auity? Let s look at a ordiary auity that is certai ad simple. By this, we mea a auity over a fixed term whose paymet period matches the iterest coversio period. Additioally,

More information

1 Correlation and Regression Analysis

1 Correlatio ad Regressio Aalysis I this sectio we will be ivestigatig the relatioship betwee two cotiuous variable, such as height ad weight, the cocetratio of a ijected drug ad heart rate, or the cosumptio

More information

How to use what you OWN to reduce what you OWE

How to use what you OWN to reduce what you OWE Maulife Oe A Overview Most Caadias maage their fiaces by doig two thigs: 1. Depositig their icome ad other short-term assets ito chequig ad savigs accouts.

More information

Output Analysis (2, Chapters 10 &11 Law)

B. Maddah ENMG 6 Simulatio 05/0/07 Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio should

More information

The analysis of the Cournot oligopoly model considering the subjective motive in the strategy selection

The aalysis of the Courot oligopoly model cosiderig the subjective motive i the strategy selectio Shigehito Furuyama Teruhisa Nakai Departmet of Systems Maagemet Egieerig Faculty of Egieerig Kasai Uiversity

More information

An optimization approach to calculation of passive filter parameters based on particle swarm optimization

Europea Associatio for the Developmet of eewable Eergies, Eviromet ad Power Quality (EA4EPQ) Iteratioal Coferece o eewable Eergies ad Power Quality (ICEPQ ) Satiago de Compostela (Spai), 8th to 30th March,

More information

In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008

I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces

More information

A probabilistic proof of a binomial identity

A probabilistic proof of a biomial idetity Joatho Peterso Abstract We give a elemetary probabilistic proof of a biomial idetity. The proof is obtaied by computig the probability of a certai evet i two

More information

(VCP-310) 1-800-418-6789

Maual VMware Lesso 1: Uderstadig the VMware Product Lie I this lesso, you will first lear what virtualizatio is. Next, you ll explore the products offered by VMware that provide virtualizatio services.

More information

Ekkehart Schlicht: Economic Surplus and Derived Demand

Ekkehart Schlicht: Ecoomic Surplus ad Derived Demad Muich Discussio Paper No. 2006-17 Departmet of Ecoomics Uiversity of Muich Volkswirtschaftliche Fakultät Ludwig-Maximilias-Uiversität Müche Olie at http://epub.ub.ui-mueche.de/940/

More information

3. Greatest Common Divisor - Least Common Multiple

3 Greatest Commo Divisor - Least Commo Multiple Defiitio 31: The greatest commo divisor of two atural umbers a ad b is the largest atural umber c which divides both a ad b We deote the greatest commo gcd

More information

Partial Di erential Equations

Partial Di eretial Equatios Partial Di eretial Equatios Much of moder sciece, egieerig, ad mathematics is based o the study of partial di eretial equatios, where a partial di eretial equatio is a equatio

More information

PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM

PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical ad Mathematical Scieces 2015, 1, p. 15 19 M a t h e m a t i c s AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM A. G. GULYAN Chair of Actuarial Mathematics

More information

Learning outcomes. Algorithms and Data Structures. Time Complexity Analysis. Time Complexity Analysis How fast is the algorithm? Prof. Dr.

Algorithms ad Data Structures Algorithm efficiecy Learig outcomes Able to carry out simple asymptotic aalysisof algorithms Prof. Dr. Qi Xi 2 Time Complexity Aalysis How fast is the algorithm? Code the

More information

Lecture 5: Span, linear independence, bases, and dimension

Lecture 5: Spa, liear idepedece, bases, ad dimesio Travis Schedler Thurs, Sep 23, 2010 (versio: 9/21 9:55 PM) 1 Motivatio Motivatio To uderstad what it meas that R has dimesio oe, R 2 dimesio 2, etc.;

More information

7. Sample Covariance and Correlation

1 of 8 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 7. Sample Covariace ad Correlatio The Bivariate Model Suppose agai that we have a basic radom experimet, ad that X ad Y

More information

Incremental calculation of weighted mean and variance

Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically

More information

Throughput of Ideally Routed Wireless Ad Hoc Networks

Throughput of Ideally Routed Wireless Ad Hoc Networks Gábor Németh, Zoltá Richárd Turáyi, 2 ad Adrás Valkó 2 Commuicatio Networks Laboratory 2 Traffic Lab, Ericsso Research, Hugary I. INTRODUCTION At the

More information

A GUIDE TO BUILDING SMART BUSINESS CREDIT

A GUIDE TO BUILDING SMART BUSINESS CREDIT Establishig busiess credit ca be the key to growig your compay DID YOU KNOW? Busiess Credit ca help grow your busiess Soud paymet practices are key to a solid

More information

Basic Elements of Arithmetic Sequences and Series

MA40S PRE-CALCULUS UNIT G GEOMETRIC SEQUENCES CLASS NOTES (COMPLETED NO NEED TO COPY NOTES FROM OVERHEAD) Basic Elemets of Arithmetic Sequeces ad Series Objective: To establish basic elemets of arithmetic

More information

GENERAL INFORMATION FOR PROXIMITY SWITCHES

78 C wire proximity switch The devices operate exactly like mechaical switches, with the coected load beig switched i series. They ca be used ito PLC iputs like relays. Notice should be take o the ifluece

More information

0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5

Sectio 13 Kolmogorov-Smirov test. Suppose that we have a i.i.d. sample X 1,..., X with some ukow distributio P ad we would like to test the hypothesis that P is equal to a particular distributio P 0, i.e.

More information

Stat 104 Lecture 2. Variables and their distributions. DJIA: monthly % change, 2000 to Finding the center of a distribution. Median.

Stat 04 Lecture Statistics 04 Lecture (IPS. &.) Outlie for today Variables ad their distributios Fidig the ceter Measurig the spread Effects of a liear trasformatio Variables ad their distributios Variable:

More information

Sequences and Series

CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their

More information

The second difference is the sequence of differences of the first difference sequence, 2

Differece Equatios I differetial equatios, you look for a fuctio that satisfies ad equatio ivolvig derivatives. I differece equatios, istead of a fuctio of a cotiuous variable (such as time), we look for

More information

Confidence Intervals for One Mean

Chapter 420 Cofidece Itervals for Oe Mea Itroductio This routie calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) at a stated cofidece level for a

More information

Your organization has a Class B IP address of 166.144.0.0 Before you implement subnetting, the Network ID and Host ID are divided as follows:

Subettig Subettig is used to subdivide a sigle class of etwork i to multiple smaller etworks. Example: Your orgaizatio has a Class B IP address of 166.144.0.0 Before you implemet subettig, the Network

More information

Phase Shift Transformers Modelling

Phase Shift Trasformers Modellig ersio:.. CGMES v.. 8 May ENTSO-E ASBL Aveue de Cortebergh Brussels Belgium Tel + 7 9 5 Fax + 7 9 5 ifo@etsoe.eu www. etsoe.eu Phase Shift Trasformers Modellig Cotets Cotets....

More information

WindWise Education. 2 nd. T ransforming the Energy of Wind into Powerful Minds. editi. A Curriculum for Grades 6 12

WidWise Educatio T rasformig the Eergy of Wid ito Powerful Mids A Curriculum for Grades 6 12 Notice Except for educatioal use by a idividual teacher i a classroom settig this work may ot be reproduced

More information

AQA STATISTICS 1 REVISION NOTES

AQA STATISTICS 1 REVISION NOTES AVERAGES AND MEASURES OF SPREAD www.mathsbox.org.uk Mode : the most commo or most popular data value the oly average that ca be used for qualitative data ot suitable if

More information

PENSION ANNUITY. Policy Conditions Document reference: PPAS1(7) This is an important document. Please keep it in a safe place.

PENSION ANNUITY Policy Coditios Documet referece: PPAS1(7) This is a importat documet. Please keep it i a safe place. Pesio Auity Policy Coditios Welcome to LV=, ad thak you for choosig our Pesio Auity.

More information

Vladimir N. Burkov, Dmitri A. Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT

Keywords: project maagemet, resource allocatio, etwork plaig Vladimir N Burkov, Dmitri A Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT The paper deals with the problems of resource allocatio betwee

More information

Recursion and Recurrences

Chapter 5 Recursio ad Recurreces 5.1 Growth Rates of Solutios to Recurreces Divide ad Coquer Algorithms Oe of the most basic ad powerful algorithmic techiques is divide ad coquer. Cosider, for example,

More information

Domain 1: Designing a SQL Server Instance and a Database Solution

Maual SQL Server 2008 Desig, Optimize ad Maitai (70-450) 1-800-418-6789 Domai 1: Desigig a SQL Server Istace ad a Database Solutio Desigig for CPU, Memory ad Storage Capacity Requiremets Whe desigig a

More information

CHAPTER 11 Financial mathematics

CHAPTER 11 Fiacial mathematics I this chapter you will: Calculate iterest usig the simple iterest formula ( ) Use the simple iterest formula to calculate the pricipal (P) Use the simple iterest formula

More information

Linear Algebra II. 4 Determinants. Notes 4 1st November Definition of determinant

MTH6140 Liear Algebra II Notes 4 1st November 2010 4 Determiats The determiat is a fuctio defied o square matrices; its value is a scalar. It has some very importat properties: perhaps most importat is

More information

Chatpun Khamyat Department of Industrial Engineering, Kasetsart University, Bangkok, Thailand ocpky@hotmail.com

SOLVING THE OIL DELIVERY TRUCKS ROUTING PROBLEM WITH MODIFY MULTI-TRAVELING SALESMAN PROBLEM APPROACH CASE STUDY: THE SME'S OIL LOGISTIC COMPANY IN BANGKOK THAILAND Chatpu Khamyat Departmet of Idustrial

More information

CHAPTER 7: Central Limit Theorem: CLT for Averages (Means)

CHAPTER 7: Cetral Limit Theorem: CLT for Averages (Meas) X = the umber obtaied whe rollig oe six sided die oce. If we roll a six sided die oce, the mea of the probability distributio is X P(X = x) Simulatio:

More information

Lesson 17 Pearson s Correlation Coefficient

Outlie Measures of Relatioships Pearso s Correlatio Coefficiet (r) -types of data -scatter plots -measure of directio -measure of stregth Computatio -covariatio of X ad Y -uique variatio i X ad Y -measurig

More information

Tagore Engineering College Department of Electrical and Electronics Engineering EC 2314 Digital Signal Processing University Question Paper Part-A

Tagore Egieerig College Departmet of Electrical ad Electroics Egieerig EC 34 Digital Sigal Processig Uiversity Questio Paper Part-A Uit-I. Defie samplig theorem?. What is kow as Aliasig? 3. What is LTI

More information

THE ARITHMETIC OF INTEGERS. - multiplication, exponentiation, division, addition, and subtraction

THE ARITHMETIC OF INTEGERS - multiplicatio, expoetiatio, divisio, additio, ad subtractio What to do ad what ot to do. THE INTEGERS Recall that a iteger is oe of the whole umbers, which may be either positive,

More information

Semiconductor Devices

emicoductor evices Prof. Zbigiew Lisik epartmet of emicoductor ad Optoelectroics evices room: 116 e-mail: zbigiew.lisik@p.lodz.pl Uipolar devices IFE T&C JFET Trasistor Uipolar evices - Trasistors asic

More information

Unit 20 Hypotheses Testing

Uit 2 Hypotheses Testig Objectives: To uderstad how to formulate a ull hypothesis ad a alterative hypothesis about a populatio proportio, ad how to choose a sigificace level To uderstad how to collect

More information

Domain 1 - Describe Cisco VoIP Implementations

Maual ONT (642-8) 1-800-418-6789 Domai 1 - Describe Cisco VoIP Implemetatios Advatages of VoIP Over Traditioal Switches Voice over IP etworks have may advatages over traditioal circuit switched voice etworks.

More information

Baan Service Master Data Management

Baa Service Master Data Maagemet Module Procedure UP069A US Documetiformatio Documet Documet code : UP069A US Documet group : User Documetatio Documet title : Master Data Maagemet Applicatio/Package :

More information

Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.

This documet was writte ad copyrighted by Paul Dawkis. Use of this documet ad its olie versio is govered by the Terms ad Coditios of Use located at http://tutorial.math.lamar.edu/terms.asp. The olie versio

More information

Lecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009)

18.409 A Algorithmist s Toolkit October 27, 2009 Lecture 13 Lecturer: Joatha Keler Scribe: Joatha Pies (2009) 1 Outlie Last time, we proved the Bru-Mikowski iequality for boxes. Today we ll go over the

More information

CALCULATION OF THE FAULT LEVEL CONTRIBUTION OF DISTRIBUTED GENERATION TO DISTRIBUTION NETWORK

51707-C--004-1-C-ERASMS-PC- CALCLAON OF HE FAL LEVEL CONRBON OF DSRBED ENERAON O DSRBON NEWORK ABSRAC Mariá Mešter Paper deals with the calculatio of the fault level cotributio of distributed geeratio

More information

Lesson 15 ANOVA (analysis of variance)

Outlie Variability -betwee group variability -withi group variability -total variability -F-ratio Computatio -sums of squares (betwee/withi/total -degrees of freedom (betwee/withi/total -mea square (betwee/withi

More information

Gibbs Distribution in Quantum Statistics

Gibbs Distributio i Quatum Statistics Quatum Mechaics is much more complicated tha the Classical oe. To fully characterize a state of oe particle i Classical Mechaics we just eed to specify its radius

More information

Finding the circle that best fits a set of points

Fidig the circle that best fits a set of poits L. MAISONOBE October 5 th 007 Cotets 1 Itroductio Solvig the problem.1 Priciples............................... Iitializatio.............................

More information

Basic Measurement Issues. Sampling Theory and Analog-to-Digital Conversion

Theory ad Aalog-to-Digital Coversio Itroductio/Defiitios Aalog-to-digital coversio Rate Frequecy Aalysis Basic Measuremet Issues Reliability the extet to which a measuremet procedure yields the same results

More information

Standard Errors and Confidence Intervals

Stadard Errors ad Cofidece Itervals Itroductio I the documet Data Descriptio, Populatios ad the Normal Distributio a sample had bee obtaied from the populatio of heights of 5-year-old boys. If we assume

More information

PSYCHOLOGICAL STATISTICS

UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc. Cousellig Psychology (0 Adm.) IV SEMESTER COMPLEMENTARY COURSE PSYCHOLOGICAL STATISTICS QUESTION BANK. Iferetial statistics is the brach of statistics

More information

Is there employment discrimination against the disabled? Melanie K Jones i. University of Wales, Swansea

Is there employmet discrimiatio agaist the disabled? Melaie K Joes i Uiversity of Wales, Swasea Abstract Whilst cotrollig for uobserved productivity differeces, the gap i employmet probabilities betwee

More information

FM4 CREDIT AND BORROWING

FM4 CREDIT AND BORROWING Whe you purchase big ticket items such as cars, boats, televisios ad the like, retailers ad fiacial istitutios have various terms ad coditios that are implemeted for the cosumer

More information

FOUNDATIONS OF MATHEMATICS AND PRE-CALCULUS GRADE 10

FOUNDATIONS OF MATHEMATICS AND PRE-CALCULUS GRADE 10 [C] Commuicatio Measuremet A1. Solve problems that ivolve liear measuremet, usig: SI ad imperial uits of measure estimatio strategies measuremet strategies.

More information

NEW HIGH PERFORMANCE COMPUTATIONAL METHODS FOR MORTGAGES AND ANNUITIES. Yuri Shestopaloff,

NEW HIGH PERFORMNCE COMPUTTIONL METHODS FOR MORTGGES ND NNUITIES Yuri Shestopaloff, Geerally, mortgage ad auity equatios do ot have aalytical solutios for ukow iterest rate, which has to be foud usig umerical

More information

Thermodynamic Laws/Definition of Entropy

This is a review of Thermodyamics ad Statistical Mechaics. Thermodyamic Laws/Defiitio of Etropy st law of thermodyamics, the coservatio of eergy: du = dq dw = Q W, ( where dq is heat eterig the system

More information

Study on the application of the software phase-locked loop in tracking and filtering of pulse signal

Advaced Sciece ad Techology Letters, pp.31-35 http://dx.doi.org/10.14257/astl.2014.78.06 Study o the applicatio of the software phase-locked loop i trackig ad filterig of pulse sigal Sog Wei Xia 1 (College

More information

Descriptive statistics deals with the description or simple analysis of population or sample data.

Descriptive statistics Some basic cocepts A populatio is a fiite or ifiite collectio of idividuals or objects. Ofte it is impossible or impractical to get data o all the members of the populatio ad a small

More information

Convexity, Inequalities, and Norms

Covexity, Iequalities, ad Norms Covex Fuctios You are probably familiar with the otio of cocavity of fuctios. Give a twicedifferetiable fuctio ϕ: R R, We say that ϕ is covex (or cocave up) if ϕ (x) 0 for

More information

Problem Set 1 Oligopoly, market shares and concentration indexes

Advaced Idustrial Ecoomics Sprig 2016 Joha Steek 29 April 2016 Problem Set 1 Oligopoly, market shares ad cocetratio idexes 1 1 Price Competitio... 3 1.1 Courot Oligopoly with Homogeous Goods ad Differet

More information

Statistical inference: example 1. Inferential Statistics

Statistical iferece: example 1 Iferetial Statistics POPULATION SAMPLE A clothig store chai regularly buys from a supplier large quatities of a certai piece of clothig. Each item ca be classified either

More information

1.3 Binomial Coefficients

18 CHAPTER 1. COUNTING 1. Biomial Coefficiets I this sectio, we will explore various properties of biomial coefficiets. Pascal s Triagle Table 1 cotais the values of the biomial coefficiets ( ) for 0to

More information

The following example will help us understand The Sampling Distribution of the Mean. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles

The followig eample will help us uderstad The Samplig Distributio of the Mea Review: The populatio is the etire collectio of all idividuals or objects of iterest The sample is the portio of the populatio

More information

Confidence Intervals. CI for a population mean (σ is known and n > 30 or the variable is normally distributed in the.

Cofidece Itervals A cofidece iterval is a iterval whose purpose is to estimate a parameter (a umber that could, i theory, be calculated from the populatio, if measuremets were available for the whole populatio).

More information

Subject CT5 Contingencies Core Technical Syllabus

Subject CT5 Cotigecies Core Techical Syllabus for the 2015 exams 1 Jue 2014 Aim The aim of the Cotigecies subject is to provide a groudig i the mathematical techiques which ca be used to model ad value

More information

LOCATIONAL MARGINAL PRICING FRAMEWORK IN SECURED DISPATCH SCHEDULING UNDER CONTINGENCY CONDITION

IJRET: Iteratioal Joural of Research i Egieerig ad Techology eissn: 2319-1163 pissn: 2321-7308 LOCATIONAL MARGINAL PRICING FRAMEWORK IN SECURED DISPATCH SCHEDULING UNDER CONTINGENCY CONDITION R.Maiamda

More information

LEASE-PURCHASE DECISION

Public Procuremet Practice STANDARD The decisio to lease or purchase should be cosidered o a case-by case evaluatio of comparative costs ad other factors. 1 Procuremet should coduct a cost/ beefit aalysis

More information

INVESTMENT PERFORMANCE COUNCIL (IPC) Guidance Statement on Calculation Methodology

Adoptio Date: 4 March 2004 Effective Date: 1 Jue 2004 Retroactive Applicatio: No Public Commet Period: Aug Nov 2002 INVESTMENT PERFORMANCE COUNCIL (IPC) Preface Guidace Statemet o Calculatio Methodology

More information

Analyzing Longitudinal Data from Complex Surveys Using SUDAAN

Aalyzig Logitudial Data from Complex Surveys Usig SUDAAN Darryl Creel Statistics ad Epidemiology, RTI Iteratioal, 312 Trotter Farm Drive, Rockville, MD, 20850 Abstract SUDAAN: Software for the Statistical

More information

I. Why is there a time value to money (TVM)?

Itroductio to the Time Value of Moey Lecture Outlie I. Why is there the cocept of time value? II. Sigle cash flows over multiple periods III. Groups of cash flows IV. Warigs o doig time value calculatios

More information

CHAPTER 3 The Simple Surface Area Measurement Module

CHAPTER 3 The Simple Surface Area Measuremet Module I chapter 2, the quality of charcoal i each batch might chage due to traditioal operatio. The quality test shall be performed before usig it as a adsorbet.

More information

Hypothesis testing. Null and alternative hypotheses

Hypothesis testig Aother importat use of samplig distributios is to test hypotheses about populatio parameters, e.g. mea, proportio, regressio coefficiets, etc. For example, it is possible to stipulate

More information

Definition. A variable X that takes on values X 1, X 2, X 3,...X k with respective frequencies f 1, f 2, f 3,...f k has mean

1 Social Studies 201 October 13, 2004 Note: The examples i these otes may be differet tha used i class. However, the examples are similar ad the methods used are idetical to what was preseted i class.

More information

Systems Design Project: Indoor Location of Wireless Devices

Systems Desig Project: Idoor Locatio of Wireless Devices Prepared By: Bria Murphy Seior Systems Sciece ad Egieerig Washigto Uiversity i St. Louis Phoe: (805) 698-5295 Email: bcm1@cec.wustl.edu Supervised

More information

I. Chi-squared Distributions

1 M 358K Supplemet to Chapter 23: CHI-SQUARED DISTRIBUTIONS, T-DISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad t-distributios, we first eed to look at aother family of distributios, the chi-squared distributios.

More information

Modified Line Search Method for Global Optimization

Modified Lie Search Method for Global Optimizatio Cria Grosa ad Ajith Abraham Ceter of Excellece for Quatifiable Quality of Service Norwegia Uiversity of Sciece ad Techology Trodheim, Norway {cria, ajith}@q2s.tu.o

More information

1 Computing the Standard Deviation of Sample Means

Computig the Stadard Deviatio of Sample Meas Quality cotrol charts are based o sample meas ot o idividual values withi a sample. A sample is a group of items, which are cosidered all together for our aalysis.

More information

ODBC. Getting Started With Sage Timberline Office ODBC

ODBC Gettig Started With Sage Timberlie Office ODBC NOTICE This documet ad the Sage Timberlie Office software may be used oly i accordace with the accompayig Sage Timberlie Office Ed User Licese Agreemet.

More information

Complex Numbers. where x represents a root of Equation 1. Note that the ± sign tells us that quadratic equations will have

Comple Numbers I spite of Calvi s discomfiture, imagiar umbers (a subset of the set of comple umbers) eist ad are ivaluable i mathematics, egieerig, ad sciece. I fact, i certai fields, such as electrical

More information

11.2 Nuclear Reactions: Fission

11.2 Nuclear Reactios: Fissio Followig Fermi s work i 1938, Otto Hah, Lise Meiter, ad Fritz Strassma discovered that whe eutros bombarded uraium atoms, the reactio produced smaller uclei that were about

More information

Non-life insurance mathematics. Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring

No-life isurace mathematics Nils F. Haavardsso, Uiversity of Oslo ad DNB Skadeforsikrig Mai issues so far Why does isurace work? How is risk premium defied ad why is it importat? How ca claim frequecy

More information

Building Blocks Problem Related to Harmonic Series

TMME, vol3, o, p.76 Buildig Blocks Problem Related to Harmoic Series Yutaka Nishiyama Osaka Uiversity of Ecoomics, Japa Abstract: I this discussio I give a eplaatio of the divergece ad covergece of ifiite

More information

APPLICATION NOTE 30 DFT or FFT? A Comparison of Fourier Transform Techniques

APPLICATION NOTE 30 DFT or FFT? A Compariso of Fourier Trasform Techiques This applicatio ote ivestigates differeces i performace betwee the DFT (Discrete Fourier Trasform) ad the FFT(Fast Fourier Trasform)

More information

Heat (or Diffusion) equation in 1D*

Heat (or Diffusio) equatio i D* Derivatio of the D heat equatio Separatio of variables (refresher) Worked eamples *Kreysig, 8 th Ed, Sectios.4b Physical assumptios We cosider temperature i a log thi wire

More information

Gregory Carey, 1998 Linear Transformations & Composites - 1. Linear Transformations and Linear Composites

Gregory Carey, 1998 Liear Trasformatios & Composites - 1 Liear Trasformatios ad Liear Composites I Liear Trasformatios of Variables Meas ad Stadard Deviatios of Liear Trasformatios A liear trasformatio

More information

Annuities Under Random Rates of Interest II By Abraham Zaks. Technion I.I.T. Haifa ISRAEL and Haifa University Haifa ISRAEL.

Auities Uder Radom Rates of Iterest II By Abraham Zas Techio I.I.T. Haifa ISRAEL ad Haifa Uiversity Haifa ISRAEL Departmet of Mathematics, Techio - Israel Istitute of Techology, 3000, Haifa, Israel I memory

More information

Properties of MLE: consistency, asymptotic normality. Fisher information.

Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout

More information