Supplementary Course Notes Adding and Subtracting AC Voltages and Currents

Size: px
Start display at page:

Download "Supplementary Course Notes Adding and Subtracting AC Voltages and Currents"

Transcription

1 Supplementary Curse Ntes Adding and Subtracting AC Vltages and Currents As mentined previusly, when cmbining DC vltages r currents, we nly need t knw the plarity (vltage) and directin (current). In the case f alternating vltages and currents, the plarity and directin are peridically changing. There are mathematical and graphical ways t cmbine these, but these can be quite cumbersme and time cnsuming, especially when there are a number f cases t determine. Frtunately, Gerge Prteus Steinmetz develped a methd f cmbing AC signals which greatly simplifies this task; we knw it as Phasr Ntatin. It is based n the mre cmplicated general, mathematical trignmetric methd f cmbinatin, but is greatly simplified when applied t sinusidal vltages r currents f exactly the same frequency. Remembering the general parametric equatin fr a sinusidal signal, we ntice that if the frequency f all the signals cncerned is exactly the same, they will differ nly in magnitude and phase angle. It is these tw characteristics that are used t d a transfrmatin f the signal int a tw-dimensinal, cmplex valued number which can be mathematically cmbined quite simply by ding cmplex additin r subtractin t cmbine the signals. Once cmbined, they can be cnverted back in t a time functin representatin shuld that be required t determine time-related parameters. General transfrmatin f a sinusidal signal frm the time dmain t the phasr dmain: Table 1: Time t Phasr Cnversin Time Dmain t Phasr Dmain Cnversin Nte: phasrs are cnventinally shwn as RS values, but the cncept wrks equally well fr maximum (i.e. peak) values. Reverse transfrmatin frm the phasr dmain back int the time dmain: Table 2: Phasr t Time Cnversin Phasr Dmain t Time Dmain Cnversin Denard Lynch Page 1 f 10

2 Supplementary Curse Ntes Of curse in the reverse case, there is knw way f determining ω withut additinal infrmatin! The Phasr Diagram: Phasrs are just vectrs in a 2-D cmplex plane. A Phasr Diagram can be used t illustrate their relatinship and additin. An example: Figure 1: Phasr Diagram Final nte: Phasrs are transfrmatins f time-varying signals nly. Other AC circuit elements, as we will discuss next, will als be represented as cmplex-valued numbers, but are nt cnsidered phasrs. Denard Lynch Page 2 f 10

3 Supplementary Curse Ntes Oppsitin t Flw in AC Circuits In a steady-state DC circuit, resistrs, capacitrs and inductrs behave relatively simply: resistrs bey Ohm s Law, capacitr lk like pen circuits, and inductrs lk (almst) like shrt circuits. If the AC wrld, these elements can react a little differently. Resistrs still bey Ohm s Law, accrding t the instantaneus vltage r current at any given time. Thus, a resistr cnnected t an AC vltage surce will allw a current t flw accrding t Ω s Law at any instant, resulting in a sinusidal current f exactly the same frequency and phase as the vltage, and a magnitude predicted by Ω s Law at any instant. Fr inductrs that are cnnected t an AC surce: v L ( I sinωt) = ωli csωt = V csωt = V sin( ωt + ), wherev = ωli di d = L = L 90 dt dt Nte that the vltage leads the current by 90 0 (r, current lags vltage: eli). Define the term X L = ωl, as the Inductive Reactance, which is the Alternating Current wrld's ppsitin t flw fr an inductr, and is measured in hms, Ω. (L is the inductance in Henries, and ω is in Radians/secnd). Nte als that the reactance is directly prprtinal t bth the frequency and the Inductance. X L als = V I Similarly fr Capacitrs: which illustrates its equivalence t resistance, r ppsitin t flw. i C ( V sinωt) = ωcv csωt = I csωt = I sin( ωt + ), where I = ωcv dv d = C = C 90 dt dt Nte that here the current leads the vltage (vltage lags current, ice) by The quantity, V I 1 ωc =, where C is in Farads (F) is called the Capacitive Reactance, X C and is the Alternating Current wrld's equivalent f Resistance, R, fr a capacitr and is measured in hms, Ω. Nte that in this case, the reactance is inversely prprtinal t frequency and capacitance. Nte that R, X L and X C are all scalar quantities representing ppsitin t flw, and will all bey Ω s Law at any instant in time. If we have cnverted ur vltages and currents t phasrs, we need t d a cnversin f R and X s that the ppsitin t flw will bey Ω s Law perfectly and simply in the Phasr Dmain. Denard Lynch Page 3 f 10

4 Supplementary Curse Ntes Impedance, Z Define ne mre new term: Impedance, Z, which represents the ppsitin t flw in the phasr dmain. It is als measured in Ohms (Ω), and is defined as: (Nte: since Z is just a rati, it can be calculated frm either RS r ax values.) Since the phasr vltage and phasr current in this definitin are cmplex valued, the impedance, Z, is als cmplex valued. It is als measure in Ohms (Ω) and can represent the ppsitin t flw f any element r cmbinatin f elements. Using ur previus determinatins f the reactance f the basic elements (R, L, C) and the phase relatinship f the vltage and current when they are cnnected t an AC surce, we can develp an expressin fr the impedance f each type. Fr resistrs, since vltage and current are in phase, this is straightfrward: Z R = V 0 I 0 = Z 0 Ω = X R 0 Ω = R 0 Ω Fr inductrs, since current is 90 0 behind the vltage (eli) Z L V 0 = = Z 90 Ω = X L 90 Ω = ω L 90 Ω = I 90 jωl Fr capacitrs, since current is 90 0 ahead f the vltage (ice) Z C V 0 = I + 90 = Z 90 Ω = X C Ω = 90 Ω = j ωc ωc Fr any series cmbinatin f R-L-C in AC circuits, the impedance, Z = R +j(x L X C ), a linear cmbinatin f the individual elements impedance. Denard Lynch Page 4 f 10

5 Supplementary Curse Ntes Impedances in series r in parallel ad exactly like resistrs in series r in parallel, except that we must use the cmplex-valued numbers. Using this cmplex impedance, Z, alng with phasr vltage and current representatins, Ohm's Law can be applied in a straightfrward fashin t A.C. circuits made up f cmbinatins f resistive, inductive and capacitive cmpnents. Nte: that althugh Z is cmplex, it is nt a phasr as it is nt a sinusidally varying quantity. Nte: that reactance(x) and resistance(r) are scalars and nt cmplex valued. While impedance is a cmplex vectr which can be represented n a 2-D plane, Z des nt represent a time-varying quantity (like a phasr des). Impedance, Z, can als be represented and added n a 2-D plane called an Impedance Diagram (similar t, but nt a Phasr Diagram!): Figure 2: Impedance Diagram Denard Lynch Page 5 f 10

6 Supplementary Curse Ntes Finally ur favurite laws. These are virtually identical t thse fr the DC wrld, except we use (cmplex-valued) phasrs: Ohms Law fr AC Circuits: V = IZ, and Z = V I Kirchhff s Laws fr AC circuits: KVL: The Σ phasr vltages arund a lp = 0 KCL: The Σ phasr currents int a nde = 0 Nrtn and Thévèn equivalents and current/vltage surce cnversin als wrk exactly the same as the DC cunterparts, except that phasr and cmplex impedance values are used. With phasr vltages and currents and [cmplex] impedances, we can apply almst all f the same rules and Laws we used in DC circuits in the AC wrld. Rectangular Crdinates Table 3: Cmplex-number ath Summary Plar Crdinates + (A + jb) + (C + jd) = (A+C) + j(b+d) Add like vectrs; usually easiest t cnvert t rectangular crdinates then add. If θ is the same fr bth, can add magnitudes (A + jb) - (C + jd) = (A-C) + j(b-d) (A + jb) (C + jd) = AB + j(bc+ad) + j 2 (BD) = (AC-BD) + j(bc+ad) (A + jb) (C + jd), need Cmplex Cnjugate Cmplex Cnjugate: (A + jb)* = (A - jb), then N = N* NN* = (cmplex prduct) (real number) and... (A + jb) (C + jd) = {(AC+BD) (C 2 +D 2 )} + j{(bc-ad) (C 2 +D 2 )} As abve, subtract like vectrs, but easiest t cnvert t rectangular crdinates first. If θ is the same fr bth, can subtract magnitudes (A θ 1 ) (Β θ 2 ) = (A B) (θ 1 +θ 2 ) (A θ 1 ) (Β θ 2 ) = (A B) (θ 1 θ 2 ) NOTE: quantities in the phasr dmain can be, and usually are, given in effective r RS terms. Generally assume that any Phasr quantities are RS unless therwise stated. Denard Lynch Page 6 f 10

7 Supplementary Curse Ntes Table 4: Phasr Representatin Summary Time Dmain Phasr Dmain (Plar) Phasr Dmain (Rectangular) Asin(ωt+θ) A +θ Acsθ + jasinθ Asin(ωt-θ) A θ Acsθ - jasinθ Acs(ωt+ θ )=Asinω(t+θ+90 0 ) Asinω(t+ θ )=Acsω(t+θ-90 0 ) -Asin(ωt+ θ )= Asinω(ωt+θ±180 0 ) -A ±θ = Α (±θ±180 0 ) If A +θ = (Acsθ + jasinθ) then A θ = (Acsθ - jasinθ) Denard Lynch Page 7 f 10

8 EE204 Basic Electrnics and Electric Pwer Curse Ntes Eg. 1: An R-L-C series AC circuit: Given the current magnitude f 2.48A, and the reactance values: V R = 116.6V V L = 93.5V V C = 65.7V Nte: these are scalar magnitudes! I= A + 47Ω X L = 37.7Ω X C =26.5Ω a) Determine the impedance f each element, and their ttal impedance as seem by the surce. b) Determine the phasr vltage acrss each element and the phasr vltage f the surce (Anther Nte: the default assumptin is that any values fr V r I are RS, and cmpnent values are fr reactance an in Ωs unless specified therwise.) Denard Lynch 2012 Page 8 f 10 Sep 23, 2012

9 EE204 Basic Electrnics and Electric Pwer Curse Ntes Eg. 2: A series - parallel AC circuit I S V R Given: E S = V = (32.2 +j13.68)v, f = 30kHz R = 10Ω, C =.22µF, L = 100µH E S I C (a) V C I L V L Find: I S, I C, I L, V R, V L, V C, Z T. Denard Lynch 2012 Page 9 f 10 Sep 23, 2012

10 EE204 Basic Electrnics and Electric Pwer Curse Ntes Eg. 3: Sme mre practice with AC circuits Find the unknwn vltages fr the fllwing circuits and express yu answer in plar ntatin. (Nte: there are tw pssible answers fr (b); prvide bth.) V C =80V V C =200V E S = V V R =? E S = V V L =? (b) Denard Lynch 2012 Page 10 f 10 Sep 23, 2012

Supplementary Course Notes Adding and Subtracting AC Voltages and Currents

Supplementary Course Notes Adding and Subtracting AC Voltages and Currents Supplementary Curse Ntes Adding and Subtracting AC Vltages and Currents As mentined previusly, when cmbining DC vltages r currents, we nly need t knw the plarity (vltage) and directin (current). In the

More information

Revision: August 19, E Main Suite D Pullman, WA (509) Voice and Fax

Revision: August 19, E Main Suite D Pullman, WA (509) Voice and Fax .7.4: Direct frequency dmain circuit analysis Revisin: August 9, 00 5 E Main Suite D Pullman, WA 9963 (509) 334 6306 ice and Fax Overview n chapter.7., we determined the steadystate respnse f electrical

More information

ECE 2100 Circuit Analysis

ECE 2100 Circuit Analysis ECE 2100 Circuit Analysis Lessn 25 Chapter 9 & App B: Passive circuit elements in the phasr representatin Daniel M. Litynski, Ph.D. http://hmepages.wmich.edu/~dlitynsk/ ECE 2100 Circuit Analysis Lessn

More information

Sections 15.1 to 15.12, 16.1 and 16.2 of the textbook (Robbins-Miller) cover the materials required for this topic.

Sections 15.1 to 15.12, 16.1 and 16.2 of the textbook (Robbins-Miller) cover the materials required for this topic. Tpic : AC Fundamentals, Sinusidal Wavefrm, and Phasrs Sectins 5. t 5., 6. and 6. f the textbk (Rbbins-Miller) cver the materials required fr this tpic.. Wavefrms in electrical systems are current r vltage

More information

ECE 2100 Circuit Analysis

ECE 2100 Circuit Analysis ECE 00 Circuit Analysis Lessn 6 Chapter 4 Sec 4., 4.5, 4.7 Series LC Circuit C Lw Pass Filter Daniel M. Litynski, Ph.D. http://hmepages.wmich.edu/~dlitynsk/ ECE 00 Circuit Analysis Lessn 5 Chapter 9 &

More information

Fields and Waves I. Lecture 3

Fields and Waves I. Lecture 3 Fields and Waves I ecture 3 Input Impedance n Transmissin ines K. A. Cnnr Electrical, Cmputer, and Systems Engineering Department Rensselaer Plytechnic Institute, Try, NY These Slides Were Prepared by

More information

NUMBERS, MATHEMATICS AND EQUATIONS

NUMBERS, MATHEMATICS AND EQUATIONS AUSTRALIAN CURRICULUM PHYSICS GETTING STARTED WITH PHYSICS NUMBERS, MATHEMATICS AND EQUATIONS An integral part t the understanding f ur physical wrld is the use f mathematical mdels which can be used t

More information

Module 4: General Formulation of Electric Circuit Theory

Module 4: General Formulation of Electric Circuit Theory Mdule 4: General Frmulatin f Electric Circuit Thery 4. General Frmulatin f Electric Circuit Thery All electrmagnetic phenmena are described at a fundamental level by Maxwell's equatins and the assciated

More information

Physics 2B Chapter 23 Notes - Faraday s Law & Inductors Spring 2018

Physics 2B Chapter 23 Notes - Faraday s Law & Inductors Spring 2018 Michael Faraday lived in the Lndn area frm 1791 t 1867. He was 29 years ld when Hand Oersted, in 1820, accidentally discvered that electric current creates magnetic field. Thrugh empirical bservatin and

More information

Copyright Paul Tobin 63

Copyright Paul Tobin 63 DT, Kevin t. lectric Circuit Thery DT87/ Tw-Prt netwrk parameters ummary We have seen previusly that a tw-prt netwrk has a pair f input terminals and a pair f utput terminals figure. These circuits were

More information

Lab 11 LRC Circuits, Damped Forced Harmonic Motion

Lab 11 LRC Circuits, Damped Forced Harmonic Motion Physics 6 ab ab 11 ircuits, Damped Frced Harmnic Mtin What Yu Need T Knw: The Physics OK this is basically a recap f what yu ve dne s far with circuits and circuits. Nw we get t put everything tgether

More information

Chapter 30. Inductance

Chapter 30. Inductance Chapter 30 nductance 30. Self-nductance Cnsider a lp f wire at rest. f we establish a current arund the lp, it will prduce a magnetic field. Sme f the magnetic field lines pass thrugh the lp. et! be the

More information

Building to Transformations on Coordinate Axis Grade 5: Geometry Graph points on the coordinate plane to solve real-world and mathematical problems.

Building to Transformations on Coordinate Axis Grade 5: Geometry Graph points on the coordinate plane to solve real-world and mathematical problems. Building t Transfrmatins n Crdinate Axis Grade 5: Gemetry Graph pints n the crdinate plane t slve real-wrld and mathematical prblems. 5.G.1. Use a pair f perpendicular number lines, called axes, t define

More information

BASIC DIRECT-CURRENT MEASUREMENTS

BASIC DIRECT-CURRENT MEASUREMENTS Brwn University Physics 0040 Intrductin BASIC DIRECT-CURRENT MEASUREMENTS The measurements described here illustrate the peratin f resistrs and capacitrs in electric circuits, and the use f sme standard

More information

Coupled Inductors and Transformers

Coupled Inductors and Transformers Cupled nductrs and Transfrmers Self-nductance When current i flws thrugh the cil, a magnetic flux is prduced arund it. d d di di v= = = dt di dt dt nductance: = d di This inductance is cmmnly called self-inductance,

More information

Lecture 6: Phase Space and Damped Oscillations

Lecture 6: Phase Space and Damped Oscillations Lecture 6: Phase Space and Damped Oscillatins Oscillatins in Multiple Dimensins The preius discussin was fine fr scillatin in a single dimensin In general, thugh, we want t deal with the situatin where:

More information

CHAPTER 5. Solutions for Exercises

CHAPTER 5. Solutions for Exercises HAPTE 5 Slutins fr Exercises E5. (a We are given v ( t 50 cs(00π t 30. The angular frequency is the cefficient f t s we have ω 00π radian/s. Then f ω / π 00 Hz T / f 0 ms m / 50 / 06. Furthermre, v(t attains

More information

ZVS Boost Converter. (a) (b) Fig 6.29 (a) Quasi-resonant boost converter with M-type switch. (b) Equivalent circuit.

ZVS Boost Converter. (a) (b) Fig 6.29 (a) Quasi-resonant boost converter with M-type switch. (b) Equivalent circuit. EEL6246 Pwer Electrnics II Chapter 6 Lecture 6 Dr. Sam Abdel-Rahman ZVS Bst Cnverter The quasi-resnant bst cnverter by using the M-type switch as shwn in Fig. 6.29(a) with its simplified circuit shwn in

More information

PHYS College Physics II Final Examination Review

PHYS College Physics II Final Examination Review PHYS 1402- Cllege Physics II Final Examinatin Review The final examinatin will be based n the fllwing Chapters/Sectins and will cnsist f tw parts. Part 1, cnsisting f Multiple Chice questins, will accunt

More information

2. Find i, v, and the power dissipated in the 6-Ω resistor in the following figure.

2. Find i, v, and the power dissipated in the 6-Ω resistor in the following figure. CSC Class exercise DC Circuit analysis. Fr the ladder netwrk in the fllwing figure, find I and R eq. Slutin Req 4 ( 6 ) 5Ω 0 0 I Re q 5 A. Find i, v, and the pwer dissipated in the 6-Ω resistr in the fllwing

More information

MATHEMATICS SYLLABUS SECONDARY 5th YEAR

MATHEMATICS SYLLABUS SECONDARY 5th YEAR Eurpean Schls Office f the Secretary-General Pedaggical Develpment Unit Ref. : 011-01-D-8-en- Orig. : EN MATHEMATICS SYLLABUS SECONDARY 5th YEAR 6 perid/week curse APPROVED BY THE JOINT TEACHING COMMITTEE

More information

Schedule. Time Varying electromagnetic fields (1 Week) 6.1 Overview 6.2 Faraday s law (6.2.1 only) 6.3 Maxwell s equations

Schedule. Time Varying electromagnetic fields (1 Week) 6.1 Overview 6.2 Faraday s law (6.2.1 only) 6.3 Maxwell s equations chedule Time Varying electrmagnetic fields (1 Week) 6.1 Overview 6.2 Faraday s law (6.2.1 nly) 6.3 Maxwell s equatins Wave quatin (3 Week) 6.5 Time-Harmnic fields 7.1 Overview 7.2 Plane Waves in Lssless

More information

Edexcel GCSE Physics

Edexcel GCSE Physics Edexcel GCSE Physics Tpic 10: Electricity and circuits Ntes (Cntent in bld is fr Higher Tier nly) www.pmt.educatin The Structure f the Atm Psitively charged nucleus surrunded by negatively charged electrns

More information

MODULE FOUR. This module addresses functions. SC Academic Elementary Algebra Standards:

MODULE FOUR. This module addresses functions. SC Academic Elementary Algebra Standards: MODULE FOUR This mdule addresses functins SC Academic Standards: EA-3.1 Classify a relatinship as being either a functin r nt a functin when given data as a table, set f rdered pairs, r graph. EA-3.2 Use

More information

[COLLEGE ALGEBRA EXAM I REVIEW TOPICS] ( u s e t h i s t o m a k e s u r e y o u a r e r e a d y )

[COLLEGE ALGEBRA EXAM I REVIEW TOPICS] ( u s e t h i s t o m a k e s u r e y o u a r e r e a d y ) (Abut the final) [COLLEGE ALGEBRA EXAM I REVIEW TOPICS] ( u s e t h i s t m a k e s u r e y u a r e r e a d y ) The department writes the final exam s I dn't really knw what's n it and I can't very well

More information

1. Transformer A transformer is used to obtain the approximate output voltage of the power supply. The output of the transformer is still AC.

1. Transformer A transformer is used to obtain the approximate output voltage of the power supply. The output of the transformer is still AC. PHYSIS 536 Experiment 4: D Pwer Supply I. Intrductin The prcess f changing A t D is investigated in this experiment. An integrated circuit regulatr makes it easy t cnstruct a high-perfrmance vltage surce

More information

LECTURES 4 AND 5 THREE-PHASE CONNECTIONS (1)

LECTURES 4 AND 5 THREE-PHASE CONNECTIONS (1) ECE 330 POWER CIRCUITS AND ELECTROMECHANICS LECTURES 4 AND 5 THREEPHASE CONNECTIONS (1) AcknwledgmentThese handuts and lecture ntes given in class are based n material frm Prf. Peter Sauer s ECE 330 lecture

More information

Flipping Physics Lecture Notes: Simple Harmonic Motion Introduction via a Horizontal Mass-Spring System

Flipping Physics Lecture Notes: Simple Harmonic Motion Introduction via a Horizontal Mass-Spring System Flipping Physics Lecture Ntes: Simple Harmnic Mtin Intrductin via a Hrizntal Mass-Spring System A Hrizntal Mass-Spring System is where a mass is attached t a spring, riented hrizntally, and then placed

More information

Series and Parallel Resonances

Series and Parallel Resonances Series and Parallel esnances Series esnance Cnsider the series circuit shwn in the frequency dmain. The input impedance is Z Vs jl jl I jc C H s esnance ccurs when the imaginary part f the transfer functin

More information

Relationships Between Frequency, Capacitance, Inductance and Reactance.

Relationships Between Frequency, Capacitance, Inductance and Reactance. P Physics Relatinships between f,, and. Relatinships Between Frequency, apacitance, nductance and Reactance. Purpse: T experimentally verify the relatinships between f, and. The data cllected will lead

More information

Section I5: Feedback in Operational Amplifiers

Section I5: Feedback in Operational Amplifiers Sectin I5: eedback in Operatinal mplifiers s discussed earlier, practical p-amps hae a high gain under dc (zer frequency) cnditins and the gain decreases as frequency increases. This frequency dependence

More information

Power Flow in Electromagnetic Waves. The time-dependent power flow density of an electromagnetic wave is given by the instantaneous Poynting vector

Power Flow in Electromagnetic Waves. The time-dependent power flow density of an electromagnetic wave is given by the instantaneous Poynting vector Pwer Flw in Electrmagnetic Waves Electrmagnetic Fields The time-dependent pwer flw density f an electrmagnetic wave is given by the instantaneus Pynting vectr P t E t H t ( ) = ( ) ( ) Fr time-varying

More information

Plan o o. I(t) Divide problem into sub-problems Modify schematic and coordinate system (if needed) Write general equations

Plan o o. I(t) Divide problem into sub-problems Modify schematic and coordinate system (if needed) Write general equations STAPLE Physics 201 Name Final Exam May 14, 2013 This is a clsed bk examinatin but during the exam yu may refer t a 5 x7 nte card with wrds f wisdm yu have written n it. There is extra scratch paper available.

More information

Chapter 3 Kinematics in Two Dimensions; Vectors

Chapter 3 Kinematics in Two Dimensions; Vectors Chapter 3 Kinematics in Tw Dimensins; Vectrs Vectrs and Scalars Additin f Vectrs Graphical Methds (One and Tw- Dimensin) Multiplicatin f a Vectr b a Scalar Subtractin f Vectrs Graphical Methds Adding Vectrs

More information

MODULE 1. e x + c. [You can t separate a demominator, but you can divide a single denominator into each numerator term] a + b a(a + b)+1 = a + b

MODULE 1. e x + c. [You can t separate a demominator, but you can divide a single denominator into each numerator term] a + b a(a + b)+1 = a + b . REVIEW OF SOME BASIC ALGEBRA MODULE () Slving Equatins Yu shuld be able t slve fr x: a + b = c a d + e x + c and get x = e(ba +) b(c a) d(ba +) c Cmmn mistakes and strategies:. a b + c a b + a c, but

More information

A Novel Isolated Buck-Boost Converter

A Novel Isolated Buck-Boost Converter vel slated uck-st Cnverter S-Sek Kim *,WOO-J JG,JOOG-HO SOG, Ok-K Kang, and Hee-Jn Kim ept. f Electrical Eng., Seul atinal University f Technlgy, Krea Schl f Electrical and Cmputer Eng., Hanyang University,

More information

ALE 21. Gibbs Free Energy. At what temperature does the spontaneity of a reaction change?

ALE 21. Gibbs Free Energy. At what temperature does the spontaneity of a reaction change? Name Chem 163 Sectin: Team Number: ALE 21. Gibbs Free Energy (Reference: 20.3 Silberberg 5 th editin) At what temperature des the spntaneity f a reactin change? The Mdel: The Definitin f Free Energy S

More information

Q1. A) 48 m/s B) 17 m/s C) 22 m/s D) 66 m/s E) 53 m/s. Ans: = 84.0 Q2.

Q1. A) 48 m/s B) 17 m/s C) 22 m/s D) 66 m/s E) 53 m/s. Ans: = 84.0 Q2. Phys10 Final-133 Zer Versin Crdinatr: A.A.Naqvi Wednesday, August 13, 014 Page: 1 Q1. A string, f length 0.75 m and fixed at bth ends, is vibrating in its fundamental mde. The maximum transverse speed

More information

Three charges, all with a charge of 10 C are situated as shown (each grid line is separated by 1 meter).

Three charges, all with a charge of 10 C are situated as shown (each grid line is separated by 1 meter). Three charges, all with a charge f 0 are situated as shwn (each grid line is separated by meter). ) What is the net wrk needed t assemble this charge distributin? a) +0.5 J b) +0.8 J c) 0 J d) -0.8 J e)

More information

Electric and Electronic Engineering

Electric and Electronic Engineering Electric and Electrnic Engineering Intrductin Cmputer engineering is cncerned with the integratin f circuits and systems nt small pieces f silicn tday. A typical cmputer engineer has a wrking knwledge

More information

Q1. A string of length L is fixed at both ends. Which one of the following is NOT a possible wavelength for standing waves on this string?

Q1. A string of length L is fixed at both ends. Which one of the following is NOT a possible wavelength for standing waves on this string? Term: 111 Thursday, January 05, 2012 Page: 1 Q1. A string f length L is fixed at bth ends. Which ne f the fllwing is NOT a pssible wavelength fr standing waves n this string? Q2. λ n = 2L n = A) 4L B)

More information

Flipping Physics Lecture Notes: Simple Harmonic Motion Introduction via a Horizontal Mass-Spring System

Flipping Physics Lecture Notes: Simple Harmonic Motion Introduction via a Horizontal Mass-Spring System Flipping Physics Lecture Ntes: Simple Harmnic Mtin Intrductin via a Hrizntal Mass-Spring System A Hrizntal Mass-Spring System is where a mass is attached t a spring, riented hrizntally, and then placed

More information

AIP Logic Chapter 4 Notes

AIP Logic Chapter 4 Notes AIP Lgic Chapter 4 Ntes Sectin 4.1 Sectin 4.2 Sectin 4.3 Sectin 4.4 Sectin 4.5 Sectin 4.6 Sectin 4.7 4.1 The Cmpnents f Categrical Prpsitins There are fur types f categrical prpsitins. Prpsitin Letter

More information

Chapter - 7 ALTERNATING CURRENT

Chapter - 7 ALTERNATING CURRENT hapter - 7 AENANG UEN A simple type f ac is ne which varies with time is simple harmnic manner- epresented by sine curve. ac vltage = sin t - Where = NAB Amplitude and = NAB ac current - - sin Where, Amplitude

More information

Phys102 Final-061 Zero Version Coordinator: Nasser Wednesday, January 24, 2007 Page: 1

Phys102 Final-061 Zero Version Coordinator: Nasser Wednesday, January 24, 2007 Page: 1 Crdinatr: Nasser Wednesday, January 4, 007 Page: 1 Q1. Tw transmitters, S 1 and S shwn in the figure, emit identical sund waves f wavelength λ. The transmitters are separated by a distance λ /. Cnsider

More information

Corrections for the textbook answers: Sec 6.1 #8h)covert angle to a positive by adding period #9b) # rad/sec

Corrections for the textbook answers: Sec 6.1 #8h)covert angle to a positive by adding period #9b) # rad/sec U n i t 6 AdvF Date: Name: Trignmetric Functins Unit 6 Tentative TEST date Big idea/learning Gals In this unit yu will study trignmetric functins frm grade, hwever everything will be dne in radian measure.

More information

Competency Statements for Wm. E. Hay Mathematics for grades 7 through 12:

Competency Statements for Wm. E. Hay Mathematics for grades 7 through 12: Cmpetency Statements fr Wm. E. Hay Mathematics fr grades 7 thrugh 12: Upn cmpletin f grade 12 a student will have develped a cmbinatin f sme/all f the fllwing cmpetencies depending upn the stream f math

More information

5 th grade Common Core Standards

5 th grade Common Core Standards 5 th grade Cmmn Cre Standards In Grade 5, instructinal time shuld fcus n three critical areas: (1) develping fluency with additin and subtractin f fractins, and develping understanding f the multiplicatin

More information

Emphases in Common Core Standards for Mathematical Content Kindergarten High School

Emphases in Common Core Standards for Mathematical Content Kindergarten High School Emphases in Cmmn Cre Standards fr Mathematical Cntent Kindergarten High Schl Cntent Emphases by Cluster March 12, 2012 Describes cntent emphases in the standards at the cluster level fr each grade. These

More information

Lecture 13: Electrochemical Equilibria

Lecture 13: Electrochemical Equilibria 3.012 Fundamentals f Materials Science Fall 2005 Lecture 13: 10.21.05 Electrchemical Equilibria Tday: LAST TIME...2 An example calculatin...3 THE ELECTROCHEMICAL POTENTIAL...4 Electrstatic energy cntributins

More information

Honors Physics Final Review Summary

Honors Physics Final Review Summary Hnrs Physics Final Review Summary Wrk Dne By A Cnstant Frce: Wrk describes a frce s tendency t change the speed f an bject. Wrk is dne nly when an bject mves in respnse t a frce, and a cmpnent f the frce

More information

Applying Kirchoff s law on the primary circuit. V = - e1 V+ e1 = 0 V.D. e.m.f. From the secondary circuit e2 = v2. K e. Equivalent circuit :

Applying Kirchoff s law on the primary circuit. V = - e1 V+ e1 = 0 V.D. e.m.f. From the secondary circuit e2 = v2. K e. Equivalent circuit : TRANSFORMERS Definitin : Transfrmers can be defined as a static electric machine which cnverts electric energy frm ne ptential t anther at the same frequency. It can als be defined as cnsists f tw electric

More information

B. Definition of an exponential

B. Definition of an exponential Expnents and Lgarithms Chapter IV - Expnents and Lgarithms A. Intrductin Starting with additin and defining the ntatins fr subtractin, multiplicatin and divisin, we discvered negative numbers and fractins.

More information

ENGI 4430 Parametric Vector Functions Page 2-01

ENGI 4430 Parametric Vector Functions Page 2-01 ENGI 4430 Parametric Vectr Functins Page -01. Parametric Vectr Functins (cntinued) Any nn-zer vectr r can be decmpsed int its magnitude r and its directin: r rrˆ, where r r 0 Tangent Vectr: dx dy dz dr

More information

Synchronous Motor V-Curves

Synchronous Motor V-Curves Synchrnus Mtr V-Curves 1 Synchrnus Mtr V-Curves Intrductin Synchrnus mtrs are used in applicatins such as textile mills where cnstant speed peratin is critical. Mst small synchrnus mtrs cntain squirrel

More information

Lecture 5: Equilibrium and Oscillations

Lecture 5: Equilibrium and Oscillations Lecture 5: Equilibrium and Oscillatins Energy and Mtin Last time, we fund that fr a system with energy cnserved, v = ± E U m ( ) ( ) One result we see immediately is that there is n slutin fr velcity if

More information

Chapter 16. Capacitance. Capacitance, cont. Parallel-Plate Capacitor, Example 1/20/2011. Electric Energy and Capacitance

Chapter 16. Capacitance. Capacitance, cont. Parallel-Plate Capacitor, Example 1/20/2011. Electric Energy and Capacitance summary C = ε A / d = πε L / ln( b / a ) ab C = 4πε 4πε a b a b >> a Chapter 16 Electric Energy and Capacitance Capacitance Q=CV Parallel plates, caxial cables, Earth Series and parallel 1 1 1 = + +..

More information

We can see from the graph above that the intersection is, i.e., [ ).

We can see from the graph above that the intersection is, i.e., [ ). MTH 111 Cllege Algebra Lecture Ntes July 2, 2014 Functin Arithmetic: With nt t much difficulty, we ntice that inputs f functins are numbers, and utputs f functins are numbers. S whatever we can d with

More information

Physics 102. Second Midterm Examination. Summer Term ( ) (Fundamental constants) (Coulomb constant)

Physics 102. Second Midterm Examination. Summer Term ( ) (Fundamental constants) (Coulomb constant) ε µ0 N mp T kg Kuwait University hysics Department hysics 0 Secnd Midterm Examinatin Summer Term (00-0) July 7, 0 Time: 6:00 7:0 M Name Student N Instructrs: Drs. bdel-karim, frusheh, Farhan, Kkaj, a,

More information

Lecture 7: Damped and Driven Oscillations

Lecture 7: Damped and Driven Oscillations Lecture 7: Damped and Driven Oscillatins Last time, we fund fr underdamped scillatrs: βt x t = e A1 + A csω1t + i A1 A sinω1t A 1 and A are cmplex numbers, but ur answer must be real Implies that A 1 and

More information

ECEN 4872/5827 Lecture Notes

ECEN 4872/5827 Lecture Notes ECEN 4872/5827 Lecture Ntes Lecture #5 Objectives fr lecture #5: 1. Analysis f precisin current reference 2. Appraches fr evaluating tlerances 3. Temperature Cefficients evaluatin technique 4. Fundamentals

More information

NWACC Dept of Mathematics Dept Final Exam Review for Trig - Part 3 Trigonometry, 9th Edition; Lial, Hornsby, Schneider Fall 2008

NWACC Dept of Mathematics Dept Final Exam Review for Trig - Part 3 Trigonometry, 9th Edition; Lial, Hornsby, Schneider Fall 2008 NWACC Dept f Mathematics Dept Final Exam Review fr Trig - Part Trignmetry, 9th Editin; Lial, Hrnsby, Schneider Fall 008 Departmental Objectives: Departmental Final Exam Review fr Trignmetry Part : Chapters

More information

Introduction to Electronic circuits.

Introduction to Electronic circuits. Intrductn t Electrnc crcuts. Passve and Actve crcut elements. Capactrs, esstrs and Inductrs n AC crcuts. Vltage and current dvders. Vltage and current surces. Amplfers, and ther transfer characterstc.

More information

Electric Current and Resistance

Electric Current and Resistance Electric Current and Resistance Electric Current Electric current is the rate f flw f charge thrugh sme regin f space The SI unit f current is the ampere (A) 1 A = 1 C / s The symbl fr electric current

More information

Electrochemistry. Reduction: the gaining of electrons. Reducing agent (reductant): species that donates electrons to reduce another reagent.

Electrochemistry. Reduction: the gaining of electrons. Reducing agent (reductant): species that donates electrons to reduce another reagent. Electrchemistry Review: Reductin: the gaining f electrns Oxidatin: the lss f electrns Reducing agent (reductant): species that dnates electrns t reduce anther reagent. Oxidizing agent (xidant): species

More information

Sinusoidal Steady State Analysis (AC Analysis) Part I

Sinusoidal Steady State Analysis (AC Analysis) Part I Sinusoidal Steady State Analysis (AC Analysis) Part I Amin Electronics and Electrical Communications Engineering Department (EECE) Cairo University elc.n102.eng@gmail.com http://scholar.cu.edu.eg/refky/

More information

Kinetics of Particles. Chapter 3

Kinetics of Particles. Chapter 3 Kinetics f Particles Chapter 3 1 Kinetics f Particles It is the study f the relatins existing between the frces acting n bdy, the mass f the bdy, and the mtin f the bdy. It is the study f the relatin between

More information

AP Physics. Summer Assignment 2012 Date. Name. F m = = + What is due the first day of school? a. T. b. = ( )( ) =

AP Physics. Summer Assignment 2012 Date. Name. F m = = + What is due the first day of school? a. T. b. = ( )( ) = P Physics Name Summer ssignment 0 Date I. The P curriculum is extensive!! This means we have t wrk at a fast pace. This summer hmewrk will allw us t start n new Physics subject matter immediately when

More information

CHM112 Lab Graphing with Excel Grading Rubric

CHM112 Lab Graphing with Excel Grading Rubric Name CHM112 Lab Graphing with Excel Grading Rubric Criteria Pints pssible Pints earned Graphs crrectly pltted and adhere t all guidelines (including descriptive title, prperly frmatted axes, trendline

More information

Experiment #3. Graphing with Excel

Experiment #3. Graphing with Excel Experiment #3. Graphing with Excel Study the "Graphing with Excel" instructins that have been prvided. Additinal help with learning t use Excel can be fund n several web sites, including http://www.ncsu.edu/labwrite/res/gt/gt-

More information

4) What is the magnitude of the net electric field at the center of the square?

4) What is the magnitude of the net electric field at the center of the square? Fur charges are n the fur crners f a square. Q = +5C, Q = -0C, Q 3 = +5C, Q 4 = -0C. The side length f each side f the square is 3 m. Q Q ) What is the directin f the frce n Q due t ONLY Q 4? (a) up (b)

More information

Chapter 32. Maxwell s Equations and Electromagnetic Waves

Chapter 32. Maxwell s Equations and Electromagnetic Waves Chapter 32 Maxwell s Equatins and Electrmagnetic Waves Maxwell s Equatins and EM Waves Maxwell s Displacement Current Maxwell s Equatins The EM Wave Equatin Electrmagnetic Radiatin MFMcGraw-PHY 2426 Chap32-Maxwell's

More information

Design and Simulation of Dc-Dc Voltage Converters Using Matlab/Simulink

Design and Simulation of Dc-Dc Voltage Converters Using Matlab/Simulink American Jurnal f Engineering Research (AJER) 016 American Jurnal f Engineering Research (AJER) e-issn: 30-0847 p-issn : 30-0936 Vlume-5, Issue-, pp-9-36 www.ajer.rg Research Paper Open Access Design and

More information

MICROWAVE COMMUNICATIONS AND RADAR

MICROWAVE COMMUNICATIONS AND RADAR MICROWAVE COMMUNICATIONS AND RADAR Generic Architecture: Signal Amplificatin Guide Antenna Prcessing Micrwave r ptical Signal Prcessing Detectin Guide Antenna tuning, resnance waveguides transitins cupling

More information

Differentiation Applications 1: Related Rates

Differentiation Applications 1: Related Rates Differentiatin Applicatins 1: Related Rates 151 Differentiatin Applicatins 1: Related Rates Mdel 1: Sliding Ladder 10 ladder y 10 ladder 10 ladder A 10 ft ladder is leaning against a wall when the bttm

More information

Physics 101 Math Review. Solutions

Physics 101 Math Review. Solutions Physics 0 Math eview Slutins . The fllwing are rdinary physics prblems. Place the answer in scientific ntatin when apprpriate and simplify the units (Scientific ntatin is used when it takes less time t

More information

Subject description processes

Subject description processes Subject representatin 6.1.2. Subject descriptin prcesses Overview Fur majr prcesses r areas f practice fr representing subjects are classificatin, subject catalging, indexing, and abstracting. The prcesses

More information

EE292: Fundamentals of ECE

EE292: Fundamentals of ECE EE292: Fundamentals of ECE Fall 2012 TTh 10:00-11:15 SEB 1242 Lecture 18 121025 http://www.ee.unlv.edu/~b1morris/ee292/ 2 Outline Review RMS Values Complex Numbers Phasors Complex Impedance Circuit Analysis

More information

Chemistry 20 Lesson 11 Electronegativity, Polarity and Shapes

Chemistry 20 Lesson 11 Electronegativity, Polarity and Shapes Chemistry 20 Lessn 11 Electrnegativity, Plarity and Shapes In ur previus wrk we learned why atms frm cvalent bnds and hw t draw the resulting rganizatin f atms. In this lessn we will learn (a) hw the cmbinatin

More information

Electric Circuit Theory

Electric Circuit Theory Electric Circuit Theory Nam Ki Min nkmin@korea.ac.kr 010-9419-2320 Chapter 11 Sinusoidal Steady-State Analysis Nam Ki Min nkmin@korea.ac.kr 010-9419-2320 Contents and Objectives 3 Chapter Contents 11.1

More information

ChE 471: LECTURE 4 Fall 2003

ChE 471: LECTURE 4 Fall 2003 ChE 47: LECTURE 4 Fall 003 IDEL RECTORS One f the key gals f chemical reactin engineering is t quantify the relatinship between prductin rate, reactr size, reactin kinetics and selected perating cnditins.

More information

Department of Electrical Engineering, University of Waterloo. Introduction

Department of Electrical Engineering, University of Waterloo. Introduction Sectin 4: Sequential Circuits Majr Tpics Types f sequential circuits Flip-flps Analysis f clcked sequential circuits Mre and Mealy machines Design f clcked sequential circuits State transitin design methd

More information

ANSWER KEY FOR MATH 10 SAMPLE EXAMINATION. Instructions: If asked to label the axes please use real world (contextual) labels

ANSWER KEY FOR MATH 10 SAMPLE EXAMINATION. Instructions: If asked to label the axes please use real world (contextual) labels ANSWER KEY FOR MATH 10 SAMPLE EXAMINATION Instructins: If asked t label the axes please use real wrld (cntextual) labels Multiple Chice Answers: 0 questins x 1.5 = 30 Pints ttal Questin Answer Number 1

More information

EEO 401 Digital Signal Processing Prof. Mark Fowler

EEO 401 Digital Signal Processing Prof. Mark Fowler EEO 401 Digital Signal Prcessing Prf. Mark Fwler Intrductin Nte Set #1 ading Assignment: Ch. 1 f Prakis & Manlakis 1/13 Mdern systems generally DSP Scenari get a cntinuus-time signal frm a sensr a cnt.-time

More information

, which yields. where z1. and z2

, which yields. where z1. and z2 The Gaussian r Nrmal PDF, Page 1 The Gaussian r Nrmal Prbability Density Functin Authr: Jhn M Cimbala, Penn State University Latest revisin: 11 September 13 The Gaussian r Nrmal Prbability Density Functin

More information

LHS Mathematics Department Honors Pre-Calculus Final Exam 2002 Answers

LHS Mathematics Department Honors Pre-Calculus Final Exam 2002 Answers LHS Mathematics Department Hnrs Pre-alculus Final Eam nswers Part Shrt Prblems The table at the right gives the ppulatin f Massachusetts ver the past several decades Using an epnential mdel, predict the

More information

Preparation work for A2 Mathematics [2017]

Preparation work for A2 Mathematics [2017] Preparatin wrk fr A2 Mathematics [2017] The wrk studied in Y12 after the return frm study leave is frm the Cre 3 mdule f the A2 Mathematics curse. This wrk will nly be reviewed during Year 13, it will

More information

Potential and Capacitance

Potential and Capacitance Ptential and apacitance Electric Ptential Electric ptential (V) = Electric ptential energy (U e ) per unit charge () Define: ptential energy U e = 0 at infinity (r = ) lim U 0 r e Nte the similarity f

More information

Computational modeling techniques

Computational modeling techniques Cmputatinal mdeling techniques Lecture 4: Mdel checing fr ODE mdels In Petre Department f IT, Åb Aademi http://www.users.ab.fi/ipetre/cmpmd/ Cntent Stichimetric matrix Calculating the mass cnservatin relatins

More information

THE FLUXOID QUANTUM AND ELECTROGRAVITATIONAL DYNAMICS. Chapter 8. This work extends chapter 6 titled, "Field Mass Generation and Control", while

THE FLUXOID QUANTUM AND ELECTROGRAVITATIONAL DYNAMICS. Chapter 8. This work extends chapter 6 titled, Field Mass Generation and Control, while 133 THE FLUXOID QUANTUM AND ELECTROGRAVITATIONAL DYNAMICS Chapter 8 This wrk extends chapter 6 titled, "Field Mass Generatin and Cntrl", while als develping a new cnceptual apprach t mass-field vehicle

More information

11. DUAL NATURE OF RADIATION AND MATTER

11. DUAL NATURE OF RADIATION AND MATTER 11. DUAL NATURE OF RADIATION AND MATTER Very shrt answer and shrt answer questins 1. Define wrk functin f a metal? The minimum energy required fr an electrn t escape frm the metal surface is called the

More information

INSTRUCTIONAL PLAN Day 2

INSTRUCTIONAL PLAN Day 2 INSTRUCTIONAL PLAN Day 2 Subject: Trignmetry Tpic: Other Trignmetric Ratis, Relatinships between Trignmetric Ratis, and Inverses Target Learners: Cllege Students Objectives: At the end f the lessn, students

More information

Oscillator. Introduction of Oscillator Linear Oscillator. Stability. Wien Bridge Oscillator RC Phase-Shift Oscillator LC Oscillator

Oscillator. Introduction of Oscillator Linear Oscillator. Stability. Wien Bridge Oscillator RC Phase-Shift Oscillator LC Oscillator Oscillatr Intrductin f Oscillatr Linear Oscillatr Wien Bridge Oscillatr Phase-Shift Oscillatr L Oscillatr Stability Oscillatrs Oscillatin: an effect that repeatedly and regularly fluctuates abut the mean

More information

I. Analytical Potential and Field of a Uniform Rod. V E d. The definition of electric potential difference is

I. Analytical Potential and Field of a Uniform Rod. V E d. The definition of electric potential difference is Length L>>a,b,c Phys 232 Lab 4 Ch 17 Electric Ptential Difference Materials: whitebards & pens, cmputers with VPythn, pwer supply & cables, multimeter, crkbard, thumbtacks, individual prbes and jined prbes,

More information

Rigid Body Dynamics (continued)

Rigid Body Dynamics (continued) Last time: Rigid dy Dynamics (cntinued) Discussin f pint mass, rigid bdy as useful abstractins f reality Many-particle apprach t rigid bdy mdeling: Newtn s Secnd Law, Euler s Law Cntinuus bdy apprach t

More information

Math Foundations 10 Work Plan

Math Foundations 10 Work Plan Math Fundatins 10 Wrk Plan Units / Tpics 10.1 Demnstrate understanding f factrs f whle numbers by: Prime factrs Greatest Cmmn Factrs (GCF) Least Cmmn Multiple (LCM) Principal square rt Cube rt Time Frame

More information

Phy 213: General Physics III 6/14/2007 Chapter 28 Worksheet 1

Phy 213: General Physics III 6/14/2007 Chapter 28 Worksheet 1 Ph 13: General Phsics III 6/14/007 Chapter 8 Wrksheet 1 Magnetic Fields & Frce 1. A pint charge, q= 510 C and m=110-3 m kg, travels with a velcit f: v = 30 ˆ s i then enters a magnetic field: = 110 T ˆj.

More information

Transduction Based on Changes in the Energy Stored in an Electrical Field

Transduction Based on Changes in the Energy Stored in an Electrical Field Lecture 6-3 Transductin Based n Changes in the Energy Stred in an Electrical ield Department f Mechanical Engineering Example:Capacitive Pressure Sensr Pressure sensitive capacitive device With separatin

More information

Preparation work for A2 Mathematics [2018]

Preparation work for A2 Mathematics [2018] Preparatin wrk fr A Mathematics [018] The wrk studied in Y1 will frm the fundatins n which will build upn in Year 13. It will nly be reviewed during Year 13, it will nt be retaught. This is t allw time

More information

Math 302 Learning Objectives

Math 302 Learning Objectives Multivariable Calculus (Part I) 13.1 Vectrs in Three-Dimensinal Space Math 302 Learning Objectives Plt pints in three-dimensinal space. Find the distance between tw pints in three-dimensinal space. Write

More information