Advanced Circuits Topics - Part 1 by Dr. Colton (Fall 2017)
|
|
- Erick Grant
- 5 years ago
- Views:
Transcription
1 Advanced rcuts Topcs - Part by Dr. olton (Fall 07) Part : Some thngs you should already know from Physcs 0 and 45 These are all thngs that you should have learned n Physcs 0 and/or 45. Ths secton s organzed lke ths so you can see the parallels between resstor problems and complex mpedance problems: Outlne:. Voltage and urrent. esstors and Ohm s aw 3. Seres and Parallel esstors 4. Solvng rcut Problems Wth Equvalent esstances 5. Interlude: A Quck Summary of omplex Numbers 6. omplex Voltage and urrent 7. omplex Impedances and Ohm s aw 8. Seres and Parallel Impedances 9. Solvng rcut Problems Wth Equvalent Impedances. Voltage and urrent The basc water analogy works well for me: f electrcty s lke water flowng n ppes, then the current s the rate at whch the water s flowng n the ppe (kg/s) and the voltage s the heght of the ppe (potental energy of the water). Batteres whch rase the potental of the charges are lke pumps pumpng the water up to a hgher elevaton. The potental dfference of a crcut element s the dfference n heght from one sde of the element to the other.. esstors and Ohm s aw esstors and conductors are the ppes themselves. A conductor s lke a very large dameter ppe that allows for a lot of current flow. A resstor s a lke a small dameter ppe whch lmts the flow: the larger the resstor, the smaller the dameter and the smaller the flow. The resstor ppe s angled downwards, so that the end at whch the water comes out s lower than the end at whch the water goes n. The potental dfference of a resstor s gven by Ohm s law: Δ 3. Seres and Parallel esstors A combnaton of resstors can be labeled ether as seres or parallel, dependng on how they are connected: ppes can ether be one after the other (seres) or sde-by-sde (parallel). Seres resstors share the same current; parallel resstors share the same potental dfference. SEIES PAAE / / Shortcut notaton: // / /
2 4. Solvng rcut Problems Wth Equvalent esstances Networks of resstors can often be decomposed nto combnatons of resstors that are n seres and parallel wth each other, called equvalent resstances of the resstor combnatons. Example Problem : What s the total resstance of the 4 resstors? 3 4 Soluton: // // Suppose the four resstors were hooked up to a battery wth voltage V B. The total current flowng from the battery would be gven by Ohm s law for the crcut as a whole: / /. The current and voltage of any of the resstors n ths problem can be obtaned by applcaton of seres/parallel resstor formulas along wth Ohm s law. The same s true for nearly any crcut problem nvolvng a sngle voltage source. For more complcated stuatons, Krchhoff s two crcut laws can be used to obtan smultaneous equatons, but that s beyond the scope of ths document. 5. Interlude: A Quck Summary of omplex Numbers We wll be usng complex numbers n Physcs 44 and 44 as a tool for descrbng oscllatng electromagnetc phenomena: A crcuts n Physcs 44, and electromagnetc waves n Physcs 44. A complex number can be wrtten ether n rectangular form, or polar form,. The polar form can equvalently be expressed as a complex exponental. The complex exponental form follows drectly from Euler s formula: cossn, and by lookng at the x- and y-components of the polar coordnates, cos, sn. o s called the ampltude, magntude, or modulus of the complex number. In Mathematca t s obtaned va the command Abs. o s called the phase angle, phase factor, phase, angle, argument, or ampltude of the complex number. In Mathematca t s obtaned va the command Arg. For example, consder the complex number 3 + 4: = (3, 4) n rectangular form, = (5, 53.3º) n polar form, and = 5 e or 5e n complex exponental form, snce 53.3º = rad. I often wrte the polar form usng ths shortcut notaton: A. The symbol can be read as, at an angle of. By the rules of exponents, when you multply two complex numbers n polar form, the ampltudes multply and the angles add. Smlarly, when you dvde n polar form, the ampltudes dvde and the angles subtract. Thus you can wrte: (3 + 4) (5 + ) = = (3 + 4) / (5 + ) = / = Yes, unfortunately the word ampltude can at tmes refer to ether A or ϕ. I myself wll reserve ampltude for A, though.
3 6. omplex Voltage and urrent Suppose you have an oscllaton such as an A voltage or current that you want to represent va complex numbers. The equaton for a generc oscllaton would be ths, wth arbtrary ampltude V 0 and phase ϕ: cos One represents the oscllatory functon va complex numbers lke ths: cos e e e Startng e e wth ths step, there s an mpled take the real part of everythng on the rght hand sde There s often an mpled e as well, so ths can become: ϕ Thus a voltage oscllatng n tme can be represented as a sngle complex number. Ths trck wll make the math much easer for many calculatons. Therefore, f for example you see 3.V30, what t means s that a voltage s oscllatng n tme as 3.V cos. Presumably the oscllaton frequency ω would be gven elsewhere n the problem, or maybe you d be lookng for thngs as a functon of ω. Smlarly, f you see.5a0 t means that a current s oscllatng as.5a cos. In summary: cos cos 7. omplex Impedances and Ohm s aw If you have a crcut wth a snusodal drvng functon (e.g. voltage) and f you are only concerned about the steady-state (long term) response, then the concept of complex mpedances smplfes matters consderably. Actually, any perodc drvng functons can be handled ths way snce wth Fourer analyss you can decompose any perodc sgnal nto a sum of snusodal ones. If you are supplyng a snusodal voltage at frequency ω, after a long tme all currents and potental dfferences of ndvdual crcut elements wll also be snusodal and oscllatng at the same frequency ω. As dscussed n the last secton, we wll use complex numbers to represent these voltages and currents, and we wll restrct ourselves to crcuts made of resstors, capactors, and nductors. As you wll see, complex numbers wll allow us to easly handle phase shfts between crcut elements. To solve such systems we want to defne an Ohm s aw -lke quantty for capactors and nductors; then we can use the seres and parallel resstance formulas whch were derved va Ohm s aw to descrbe complcated crcuts. That quantty s called the mpedance, and wll be a complex number Z: Δ s lke Δ for a resstor. 3
4 We wll set the zero of tme such that the drvng battery voltage s a cosne functon wth no phase shft,.e. t reaches ts maxmum at t = 0. et s call the ampltude V 0 : cos 0 The total current suppled by the battery wll also be snusodal but t may have a dfferent phase than the voltage. et s call the current s ampltude, and say that t lags the voltage by a phase (f the current s actually leadng the voltage, then wll be negatve): cos Keep n mnd that the voltages and currents are real quanttes, not complex; the pont of the complex exponentals s just to smplfy the mathematcs of snusodal sgnals wth phase shfts. At the end of a problem, f you want to know the actual voltages or currents just convert back from the complex numbers usng the ϕ cos types of correspondences. Impedance of esstor. esstors are easy: Z =. Ohm s aw apples drectly wth no werd modfcatons. Impedance of apactor.δ, and f, then when you ntegrate I you get a factor of / from the exponental, tmes the current I tself. Thus the capactor equaton becomes: Δ, and the resstance-lke quantty s (also often wrtten as ). Impedance of Inductor. As s later n hapter 7 of Grffths, and you learned n Phys 0, Δ. When you take the dervatve of I, you get a factor of from the exponental, tmes the current I tself. Thus the nductor equaton becomes: Δ, and the resstance-lke quantty s. In summary: If you use these complex mpedances (and have snusodal responses), then Ohm s law works for capactors and nductors and for the crcut as a whole just lke t does for resstors: Δ 8. Seres and Parallel Impedances Because the seres and parallel resstor formulas were derved va Ohm s law, they carry over completely wth complex mpedances. Seres: Parallel: / / // These equatons are very powerful because they can be used wth combnatons of any types of mpedances. 4
5 9. Solvng rcut Problems Wth Equvalent Impedances Usng these mpedances to determne the ampltude and phase of the current from the power supply s straghtforward: Step. Fnd the total mpedance of the crcut,, usng seres/parallel rules. It s a complex number. Step. Wrte n polar form: ϕ Step 3. Use Ohm s law to get the total current, also a complex number: Step 4. onvert back to a cosne oscllaton to get your real answer f desred: cos, usng the same phase conventon as wrtten n secton 7 above (I lags V when s postve). Note that f the phase angle of the complex mpedance s negatve, then the phase angle of the complex current wll be postve, and vce versa. Once the total current s known n complex form, other quanttes of nterest can generally also be found usng Phys 0 technques nvolvng Ohm s law and seres/parallel mpedances. Example Problem What current s suppled here? Soluton: Step : Zeq Z Z Step : Now wrte 3 4 n polar form: Step 3: length=5, angle = tan (4/3) = 0.97 rad Step 4: If desred, take the real part to get the actual formula for the current: 0.A cos 0.97 Example Problem 3 For ths crcut, what s (a) V and (b) V? 5
6 6 Step : Z eq Z eq Step : Put n polar form, ϕ: Z eq tan (I assumed that ω > /ω so the vector shown above s ndeed n the frst quadrant. auton: f not n the frst or fourth quadrants the tan functon wll gve wrong answers. As mentoned above Mathematca s Arg command s helpful, and much better than the regular tan functon.) Steps 3 & 4: I = V/Z, and convert to a real oscllaton: t V I tan cos 0 (a) Everythng s n seres, so ths same current goes through the resstor. Use V = I by Ohm s aw, and the answer s: t V V tan cos 0 (b) Use the complex Ohm s law on the nductor: Δ Δ tan Δ tan That last step was done by wrtng tself n polar form. From there t s just complex number multplcaton and convertng to real numbers f you want: Z /
7 Δ Δ tan cos tan Example Problem 4 What s the current comng from the power supply, n terms of V 0, ω, t,,,, and? Frst fnd the total mpedance of the crcut: Z Z Z // Z Z eq I ll use Mathematca to help wth the algebra: It needs to be put nto polar form n order to fnd ts magntude and phase. The polar coordnates can easly be found n Mathematca wth these commands: and Abs Arg Then the current s gven by cos for that Z and ϕ. It s probably not worth the space to wrte out the full-blown answer. 7
Physics 4B. A positive value is obtained, so the current is counterclockwise around the circuit.
Physcs 4B Solutons to Chapter 7 HW Chapter 7: Questons:, 8, 0 Problems:,,, 45, 48,,, 7, 9 Queston 7- (a) no (b) yes (c) all te Queston 7-8 0 μc Queston 7-0, c;, a;, d; 4, b Problem 7- (a) Let be the current
More informationSection 8.3 Polar Form of Complex Numbers
80 Chapter 8 Secton 8 Polar Form of Complex Numbers From prevous classes, you may have encountered magnary numbers the square roots of negatve numbers and, more generally, complex numbers whch are the
More informationDesigning Information Devices and Systems II Spring 2018 J. Roychowdhury and M. Maharbiz Discussion 3A
EECS 16B Desgnng Informaton Devces and Systems II Sprng 018 J. Roychowdhury and M. Maharbz Dscusson 3A 1 Phasors We consder snusodal voltages and currents of a specfc form: where, Voltage vt) = V 0 cosωt
More informationComplex Numbers, Signals, and Circuits
Complex Numbers, Sgnals, and Crcuts 3 August, 009 Complex Numbers: a Revew Suppose we have a complex number z = x jy. To convert to polar form, we need to know the magntude of z and the phase of z. z =
More informationPhysics 1202: Lecture 11 Today s Agenda
Physcs 122: Lecture 11 Today s Agenda Announcements: Team problems start ths Thursday Team 1: Hend Ouda, Mke Glnsk, Stephane Auger Team 2: Analese Bruder, Krsten Dean, Alson Smth Offce hours: Monday 2:3-3:3
More informationElectrical Circuits 2.1 INTRODUCTION CHAPTER
CHAPTE Electrcal Crcuts. INTODUCTION In ths chapter, we brefly revew the three types of basc passve electrcal elements: resstor, nductor and capactor. esstance Elements: Ohm s Law: The voltage drop across
More informationSections begin this week. Cancelled Sections: Th Labs begin this week. Attend your only second lab slot this week.
Announcements Sectons begn ths week Cancelled Sectons: Th 122. Labs begn ths week. Attend your only second lab slot ths week. Cancelled labs: ThF 25. Please check your Lab secton. Homework #1 onlne Due
More informationC/CS/Phy191 Problem Set 3 Solutions Out: Oct 1, 2008., where ( 00. ), so the overall state of the system is ) ( ( ( ( 00 ± 11 ), Φ ± = 1
C/CS/Phy9 Problem Set 3 Solutons Out: Oct, 8 Suppose you have two qubts n some arbtrary entangled state ψ You apply the teleportaton protocol to each of the qubts separately What s the resultng state obtaned
More informationFE REVIEW OPERATIONAL AMPLIFIERS (OP-AMPS)( ) 8/25/2010
FE REVEW OPERATONAL AMPLFERS (OP-AMPS)( ) 1 The Op-amp 2 An op-amp has two nputs and one output. Note the op-amp below. The termnal labeled l wth the (-) sgn s the nvertng nput and the nput labeled wth
More informationIntroduction to circuit analysis. Classification of Materials
Introducton to crcut analyss OUTLINE Electrcal quanttes Charge Current Voltage Power The deal basc crcut element Sgn conventons Current versus voltage (I-V) graph Readng: 1.2, 1.3,1.6 Lecture 2, Slde 1
More informationI 2 V V. = 0 write 1 loop equation for each loop with a voltage not in the current set of equations. or I using Ohm s Law V 1 5.
Krchoff s Laws Drect: KL, KL, Ohm s Law G G Ohm s Law: 6 (always get equaton/esor) Ω 5 Ω 6Ω 4 KL: : 5 : 5 eq. are dependent (n general, get n ndep. for nodes) KL: 4 wrte loop equaton for each loop wth
More information1 Matrix representations of canonical matrices
1 Matrx representatons of canoncal matrces 2-d rotaton around the orgn: ( ) cos θ sn θ R 0 = sn θ cos θ 3-d rotaton around the x-axs: R x = 1 0 0 0 cos θ sn θ 0 sn θ cos θ 3-d rotaton around the y-axs:
More information( ) = ( ) + ( 0) ) ( )
EETOMAGNETI OMPATIBIITY HANDBOOK 1 hapter 9: Transent Behavor n the Tme Doman 9.1 Desgn a crcut usng reasonable values for the components that s capable of provdng a tme delay of 100 ms to a dgtal sgnal.
More informationPhysics 114 Exam 2 Spring Name:
Physcs 114 Exam Sprng 013 Name: For gradng purposes (do not wrte here): Queston 1. 1... 3. 3. Problem Answer each of the followng questons. Ponts for each queston are ndcated n red wth the amount beng
More informationINDUCTANCE. RC Cicuits vs LR Circuits
INDUTANE R cuts vs LR rcuts R rcut hargng (battery s connected): (1/ )q + (R)dq/ dt LR rcut = (R) + (L)d/ dt q = e -t/ R ) = / R(1 - e -(R/ L)t ) q ncreases from 0 to = dq/ dt decreases from / R to 0 Dschargng
More informationInner Product. Euclidean Space. Orthonormal Basis. Orthogonal
Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,
More informationKirchhoff second rule
Krchhoff second rule Close a battery on a resstor: smplest crcut! = When the current flows n a resstor there s a voltage drop = How much current flows n the crcut? Ohm s law: Krchhoff s second law: Around
More informationMAE140 - Linear Circuits - Winter 16 Final, March 16, 2016
ME140 - Lnear rcuts - Wnter 16 Fnal, March 16, 2016 Instructons () The exam s open book. You may use your class notes and textbook. You may use a hand calculator wth no communcaton capabltes. () You have
More informationUNIVERSITY OF UTAH ELECTRICAL & COMPUTER ENGINEERING DEPARTMENT. 10k. 3mH. 10k. Only one current in the branch:
UNIERSITY OF UTH ELECTRICL & COMPUTER ENGINEERING DEPRTMENT ECE 70 HOMEWORK #6 Soluton Summer 009. fter beng closed a long tme, the swtch opens at t = 0. Fnd (t) for t > 0. t = 0 0kΩ 0kΩ 3mH Step : (Redraw
More informationChapter 6 Electrical Systems and Electromechanical Systems
ME 43 Systems Dynamcs & Control Chapter 6: Electrcal Systems and Electromechancal Systems Chapter 6 Electrcal Systems and Electromechancal Systems 6. INTODUCTION A. Bazoune The majorty of engneerng systems
More informationLecture 13 APPROXIMATION OF SECOMD ORDER DERIVATIVES
COMPUTATIONAL FLUID DYNAMICS: FDM: Appromaton of Second Order Dervatves Lecture APPROXIMATION OF SECOMD ORDER DERIVATIVES. APPROXIMATION OF SECOND ORDER DERIVATIVES Second order dervatves appear n dffusve
More informationPhysics 4B. Question and 3 tie (clockwise), then 2 and 5 tie (zero), then 4 and 6 tie (counterclockwise) B i. ( T / s) = 1.74 V.
Physcs 4 Solutons to Chapter 3 HW Chapter 3: Questons:, 4, 1 Problems:, 15, 19, 7, 33, 41, 45, 54, 65 Queston 3-1 and 3 te (clockwse), then and 5 te (zero), then 4 and 6 te (counterclockwse) Queston 3-4
More informationPHY2049 Exam 2 solutions Fall 2016 Solution:
PHY2049 Exam 2 solutons Fall 2016 General strategy: Fnd two resstors, one par at a tme, that are connected ether n SERIES or n PARALLEL; replace these two resstors wth one of an equvalent resstance. Now
More informationDC Circuits. Crossing the emf in this direction +ΔV
DC Crcuts Delverng a steady flow of electrc charge to a crcut requres an emf devce such as a battery, solar cell or electrc generator for example. mf stands for electromotve force, but an emf devce transforms
More informationPhysics 114 Exam 2 Fall 2014 Solutions. Name:
Physcs 114 Exam Fall 014 Name: For gradng purposes (do not wrte here): Queston 1. 1... 3. 3. Problem Answer each of the followng questons. Ponts for each queston are ndcated n red. Unless otherwse ndcated,
More informationELECTRONICS. EE 42/100 Lecture 4: Resistive Networks and Nodal Analysis. Rev B 1/25/2012 (9:49PM) Prof. Ali M. Niknejad
A. M. Nknejad Unversty of Calforna, Berkeley EE 100 / 42 Lecture 4 p. 1/14 EE 42/100 Lecture 4: Resstve Networks and Nodal Analyss ELECTRONICS Rev B 1/25/2012 (9:49PM) Prof. Al M. Nknejad Unversty of Calforna,
More informationMAE140 - Linear Circuits - Fall 13 Midterm, October 31
Instructons ME140 - Lnear Crcuts - Fall 13 Mdterm, October 31 () Ths exam s open book. You may use whatever wrtten materals you choose, ncludng your class notes and textbook. You may use a hand calculator
More informationDEMO #8 - GAUSSIAN ELIMINATION USING MATHEMATICA. 1. Matrices in Mathematica
demo8.nb 1 DEMO #8 - GAUSSIAN ELIMINATION USING MATHEMATICA Obectves: - defne matrces n Mathematca - format the output of matrces - appl lnear algebra to solve a real problem - Use Mathematca to perform
More informationMAE140 - Linear Circuits - Winter 16 Midterm, February 5
Instructons ME140 - Lnear Crcuts - Wnter 16 Mdterm, February 5 () Ths exam s open book. You may use whatever wrtten materals you choose, ncludng your class notes and textbook. You may use a hand calculator
More informationIntroduction to Vapor/Liquid Equilibrium, part 2. Raoult s Law:
CE304, Sprng 2004 Lecture 4 Introducton to Vapor/Lqud Equlbrum, part 2 Raoult s Law: The smplest model that allows us do VLE calculatons s obtaned when we assume that the vapor phase s an deal gas, and
More informationGeorgia Tech PHYS 6124 Mathematical Methods of Physics I
Georga Tech PHYS 624 Mathematcal Methods of Physcs I Instructor: Predrag Cvtanovć Fall semester 202 Homework Set #7 due October 30 202 == show all your work for maxmum credt == put labels ttle legends
More information8.022 (E&M) Lecture 8
8.0 (E&M) Lecture 8 Topcs: Electromotve force Crcuts and Krchhoff s rules 1 Average: 59, MS: 16 Quz 1: thoughts Last year average: 64 test slghtly harder than average Problem 1 had some subtletes math
More informationCHAPTER 14 GENERAL PERTURBATION THEORY
CHAPTER 4 GENERAL PERTURBATION THEORY 4 Introducton A partcle n orbt around a pont mass or a sphercally symmetrc mass dstrbuton s movng n a gravtatonal potental of the form GM / r In ths potental t moves
More information8.6 The Complex Number System
8.6 The Complex Number System Earler n the chapter, we mentoned that we cannot have a negatve under a square root, snce the square of any postve or negatve number s always postve. In ths secton we want
More informationMoments of Inertia. and reminds us of the analogous equation for linear momentum p= mv, which is of the form. The kinetic energy of the body is.
Moments of Inerta Suppose a body s movng on a crcular path wth constant speed Let s consder two quanttes: the body s angular momentum L about the center of the crcle, and ts knetc energy T How are these
More informationFrom Biot-Savart Law to Divergence of B (1)
From Bot-Savart Law to Dvergence of B (1) Let s prove that Bot-Savart gves us B (r ) = 0 for an arbtrary current densty. Frst take the dvergence of both sdes of Bot-Savart. The dervatve s wth respect to
More informationElectrical Circuits II (ECE233b)
Electrcal Crcuts (ECE33b SteadyState Power Analyss Anests Dounas The Unersty of Western Ontaro Faculty of Engneerng Scence SteadyState Power Analyss (t AC crcut: The steady state oltage and current can
More informationLecture #4 Capacitors and Inductors Energy Stored in C and L Equivalent Circuits Thevenin Norton
EES ntro. electroncs for S Sprng 003 Lecture : 0/03/03 A.R. Neureuther Verson Date 0/0/03 EES ntroducton to Electroncs for omputer Scence Andrew R. Neureuther Lecture # apactors and nductors Energy Stored
More informationI. INTRODUCTION. 1.1 Circuit Theory Fundamentals
I. INTRODUCTION 1.1 Crcut Theory Fundamentals Crcut theory s an approxmaton to Maxwell s electromagnetc equatons n order to smplfy analyss of complcated crcuts. A crcut s made of seeral elements (boxes
More informationWeek3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity
Week3, Chapter 4 Moton n Two Dmensons Lecture Quz A partcle confned to moton along the x axs moves wth constant acceleraton from x =.0 m to x = 8.0 m durng a 1-s tme nterval. The velocty of the partcle
More informationPhysics 2102 Spring 2007 Lecture 10 Current and Resistance
esstance Is Futle! Physcs 0 Sprng 007 Jonathan Dowlng Physcs 0 Sprng 007 Lecture 0 Current and esstance Georg Smon Ohm (789-854) What are we gong to learn? A road map lectrc charge lectrc force on other
More informationElectricity and Magnetism Lecture 13 - Physics 121 Electromagnetic Oscillations in LC & LCR Circuits,
lectrcty and Magnetsm ecture 3 - Physcs lectromagnetc Oscllatons n & rcuts, Y&F hapter 3, Sec. 5-6 Alternatng urrent rcuts, Y&F hapter 3, Sec. - Summary: and crcuts Mechancal Harmonc Oscllator rcut Oscllatons
More informationLectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix
Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could
More informationComplex Numbers. x = B B 2 4AC 2A. or x = x = 2 ± 4 4 (1) (5) 2 (1)
Complex Numbers If you have not yet encountered complex numbers, you wll soon do so n the process of solvng quadratc equatons. The general quadratc equaton Ax + Bx + C 0 has solutons x B + B 4AC A For
More informationBernoulli Numbers and Polynomials
Bernoull Numbers and Polynomals T. Muthukumar tmk@tk.ac.n 17 Jun 2014 The sum of frst n natural numbers 1, 2, 3,..., n s n n(n + 1 S 1 (n := m = = n2 2 2 + n 2. Ths formula can be derved by notng that
More informationI. INTRODUCTION. There are two other circuit elements that we will use and are special cases of the above elements. They are:
I. INTRODUCTION 1.1 Crcut Theory Fundamentals In ths course we study crcuts wth non-lnear elements or deces (dodes and transstors). We wll use crcut theory tools to analyze these crcuts. Snce some of tools
More informationS-Domain Analysis. s-domain Circuit Analysis. EE695K VLSI Interconnect. Time domain (t domain) Complex frequency domain (s domain) Laplace Transform L
EE695K S nterconnect S-Doman naly -Doman rcut naly Tme doman t doman near rcut aplace Tranform omplex frequency doman doman Tranformed rcut Dfferental equaton lacal technque epone waveform aplace Tranform
More informationOne-sided finite-difference approximations suitable for use with Richardson extrapolation
Journal of Computatonal Physcs 219 (2006) 13 20 Short note One-sded fnte-dfference approxmatons sutable for use wth Rchardson extrapolaton Kumar Rahul, S.N. Bhattacharyya * Department of Mechancal Engneerng,
More informationMAE140 - Linear Circuits - Fall 10 Midterm, October 28
M140 - Lnear rcuts - Fall 10 Mdterm, October 28 nstructons () Ths exam s open book. You may use whatever wrtten materals you choose, ncludng your class notes and textbook. You may use a hand calculator
More informationON MECHANICS WITH VARIABLE NONCOMMUTATIVITY
ON MECHANICS WITH VARIABLE NONCOMMUTATIVITY CIPRIAN ACATRINEI Natonal Insttute of Nuclear Physcs and Engneerng P.O. Box MG-6, 07725-Bucharest, Romana E-mal: acatrne@theory.npne.ro. Receved March 6, 2008
More information9. Complex Numbers. 1. Numbers revisited. 2. Imaginary number i: General form of complex numbers. 3. Manipulation of complex numbers
9. Comple Numbers. Numbers revsted. Imagnar number : General form of comple numbers 3. Manpulaton of comple numbers 4. The Argand dagram 5. The polar form for comple numbers 9.. Numbers revsted We saw
More informationFoundations of Arithmetic
Foundatons of Arthmetc Notaton We shall denote the sum and product of numbers n the usual notaton as a 2 + a 2 + a 3 + + a = a, a 1 a 2 a 3 a = a The notaton a b means a dvdes b,.e. ac = b where c s an
More informationPHYSICS - CLUTCH CH 28: INDUCTION AND INDUCTANCE.
!! www.clutchprep.com CONCEPT: ELECTROMAGNETIC INDUCTION A col of wre wth a VOLTAGE across each end wll have a current n t - Wre doesn t HAVE to have voltage source, voltage can be INDUCED V Common ways
More informationSelected Student Solutions for Chapter 2
/3/003 Assessment Prolems Selected Student Solutons for Chapter. Frst note that we know the current through all elements n the crcut except the 6 kw resstor (the current n the three elements to the left
More informationCircuits II EE221. Instructor: Kevin D. Donohue. Instantaneous, Average, RMS, and Apparent Power, and, Maximum Power Transfer, and Power Factors
Crcuts II EE1 Unt 3 Instructor: Ken D. Donohue Instantaneous, Aerage, RMS, and Apparent Power, and, Maxmum Power pp ransfer, and Power Factors Power Defntons/Unts: Work s n unts of newton-meters or joules
More informationChapter 6. Operational Amplifier. inputs can be defined as the average of the sum of the two signals.
6 Operatonal mpler Chapter 6 Operatonal mpler CC Symbol: nput nput Output EE () Non-nvertng termnal, () nvertng termnal nput mpedance : Few mega (ery hgh), Output mpedance : Less than (ery low) Derental
More informationHow Differential Equations Arise. Newton s Second Law of Motion
page 1 CHAPTER 1 Frst-Order Dfferental Equatons Among all of the mathematcal dscplnes the theory of dfferental equatons s the most mportant. It furnshes the explanaton of all those elementary manfestatons
More informationDifference Equations
Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1
More informationElectricity and Magnetism Lecture 07 - Physics 121 Current, Resistance, DC Circuits: Y&F Chapter 25 Sect. 1-5 Kirchhoff s Laws: Y&F Chapter 26 Sect.
Electrcty and Magnetsm Lecture 07 - Physcs Current, esstance, DC Crcuts: Y&F Chapter 5 Sect. -5 Krchhoff s Laws: Y&F Chapter 6 Sect. Crcuts and Currents Electrc Current Current Densty J Drft Speed esstance,
More informationSolution Thermodynamics
Soluton hermodynamcs usng Wagner Notaton by Stanley. Howard Department of aterals and etallurgcal Engneerng South Dakota School of nes and echnology Rapd Cty, SD 57701 January 7, 001 Soluton hermodynamcs
More informationECE 320 Energy Conversion and Power Electronics Dr. Tim Hogan. Chapter 1: Introduction and Three Phase Power
ECE 3 Energy Conerson and Power Electroncs Dr. Tm Hogan Chapter : ntroducton and Three Phase Power. eew of Basc Crcut Analyss Defntons: Node - Electrcal juncton between two or more deces. Loop - Closed
More informationUnit 1. Current and Voltage U 1 VOLTAGE AND CURRENT. Circuit Basics KVL, KCL, Ohm's Law LED Outputs Buttons/Switch Inputs. Current / Voltage Analogy
..2 nt Crcut Bascs KVL, KCL, Ohm's Law LED Outputs Buttons/Swtch Inputs VOLTAGE AND CRRENT..4 Current and Voltage Current / Voltage Analogy Charge s measured n unts of Coulombs Current Amount of charge
More informationBoise State University Department of Electrical and Computer Engineering ECE 212L Circuit Analysis and Design Lab
Bose State Unersty Department of Electrcal and omputer Engneerng EE 1L rcut Analyss and Desgn Lab Experment #8: The Integratng and Dfferentatng Op-Amp rcuts 1 Objectes The objectes of ths laboratory experment
More informationEE215 FUNDAMENTALS OF ELECTRICAL ENGINEERING
EE215 FUNDAMENTALS OF ELECTRICAL ENGINEERING TaChang Chen Unersty of Washngton, Bothell Sprng 2010 EE215 1 WEEK 8 FIRST ORDER CIRCUIT RESPONSE May 21 st, 2010 EE215 2 1 QUESTIONS TO ANSWER Frst order crcuts
More informationWeek 11: Differential Amplifiers
ELE 0A Electronc rcuts Week : Dfferental Amplfers Lecture - Large sgnal analyss Topcs to coer A analyss Half-crcut analyss eadng Assgnment: hap 5.-5.8 of Jaeger and Blalock or hap 7. - 7.3, of Sedra and
More information1 GSW Iterative Techniques for y = Ax
1 for y = A I m gong to cheat here. here are a lot of teratve technques that can be used to solve the general case of a set of smultaneous equatons (wrtten n the matr form as y = A), but ths chapter sn
More information1. Estimation, Approximation and Errors Percentages Polynomials and Formulas Identities and Factorization 52
ontents ommonly Used Formulas. Estmaton, pproxmaton and Errors. Percentages. Polynomals and Formulas 8. Identtes and Factorzaton. Equatons and Inequaltes 66 6. Rate and Rato 8 7. Laws of Integral Indces
More informationPHYSICS - CLUTCH 1E CH 28: INDUCTION AND INDUCTANCE.
!! www.clutchprep.com CONCEPT: ELECTROMAGNETIC INDUCTION A col of wre wth a VOLTAGE across each end wll have a current n t - Wre doesn t HAVE to have voltage source, voltage can be INDUCED V Common ways
More informationChapter - 2. Distribution System Power Flow Analysis
Chapter - 2 Dstrbuton System Power Flow Analyss CHAPTER - 2 Radal Dstrbuton System Load Flow 2.1 Introducton Load flow s an mportant tool [66] for analyzng electrcal power system network performance. Load
More informationDigital Signal Processing
Dgtal Sgnal Processng Dscrete-tme System Analyss Manar Mohasen Offce: F8 Emal: manar.subh@ut.ac.r School of IT Engneerng Revew of Precedent Class Contnuous Sgnal The value of the sgnal s avalable over
More informationIntegrals and Invariants of Euler-Lagrange Equations
Lecture 16 Integrals and Invarants of Euler-Lagrange Equatons ME 256 at the Indan Insttute of Scence, Bengaluru Varatonal Methods and Structural Optmzaton G. K. Ananthasuresh Professor, Mechancal Engneerng,
More informationwhere the sums are over the partcle labels. In general H = p2 2m + V s(r ) V j = V nt (jr, r j j) (5) where V s s the sngle-partcle potental and V nt
Physcs 543 Quantum Mechancs II Fall 998 Hartree-Fock and the Self-consstent Feld Varatonal Methods In the dscusson of statonary perturbaton theory, I mentoned brey the dea of varatonal approxmaton schemes.
More informationLecture 12: Discrete Laplacian
Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly
More informationWeek 9 Chapter 10 Section 1-5
Week 9 Chapter 10 Secton 1-5 Rotaton Rgd Object A rgd object s one that s nondeformable The relatve locatons of all partcles makng up the object reman constant All real objects are deformable to some extent,
More informationProf. Paolo Colantonio a.a
Pro. Paolo olantono a.a. 3 4 Let s consder a two ports network o Two ports Network o L For passve network (.e. wthout nternal sources or actve devces), a general representaton can be made by a sutable
More informationModule B3 3.1 Sinusoidal steady-state analysis (single-phase), a review 3.2 Three-phase analysis. Kirtley
Module B3 3.1 Snusodal steady-state analyss (sngle-phase), a reew 3. hree-phase analyss Krtley Chapter : AC oltage, Current and Power.1 Sources and Power. Resstors, Inductors, and Capactors Chapter 4:
More informationCOMPLEX NUMBERS AND QUADRATIC EQUATIONS
COMPLEX NUMBERS AND QUADRATIC EQUATIONS INTRODUCTION We know that x 0 for all x R e the square of a real number (whether postve, negatve or ero) s non-negatve Hence the equatons x, x, x + 7 0 etc are not
More information(b) i(t) for t 0. (c) υ 1 (t) and υ 2 (t) for t 0. Solution: υ 2 (0 ) = I 0 R 1 = = 10 V. υ 1 (0 ) = 0. (Given).
Problem 5.37 Pror to t =, capactor C 1 n the crcut of Fg. P5.37 was uncharged. For I = 5 ma, R 1 = 2 kω, = 5 kω, C 1 = 3 µf, and C 2 = 6 µf, determne: (a) The equvalent crcut nvolvng the capactors for
More informationChapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems
Numercal Analyss by Dr. Anta Pal Assstant Professor Department of Mathematcs Natonal Insttute of Technology Durgapur Durgapur-713209 emal: anta.bue@gmal.com 1 . Chapter 5 Soluton of System of Lnear Equatons
More informationMechanics Physics 151
Mechancs Physcs 151 Lecture 3 Lagrange s Equatons (Goldsten Chapter 1) Hamlton s Prncple (Chapter 2) What We Dd Last Tme! Dscussed mult-partcle systems! Internal and external forces! Laws of acton and
More informationChapter 3 Differentiation and Integration
MEE07 Computer Modelng Technques n Engneerng Chapter Derentaton and Integraton Reerence: An Introducton to Numercal Computatons, nd edton, S. yakowtz and F. zdarovsky, Mawell/Macmllan, 990. Derentaton
More informationRevision: December 13, E Main Suite D Pullman, WA (509) Voice and Fax
.9.1: AC power analyss Reson: Deceber 13, 010 15 E Man Sute D Pullan, WA 99163 (509 334 6306 Voce and Fax Oerew n chapter.9.0, we ntroduced soe basc quanttes relate to delery of power usng snusodal sgnals.
More informationFE REVIEW OPERATIONAL AMPLIFIERS (OP-AMPS)
FE EIEW OPEATIONAL AMPLIFIES (OPAMPS) 1 The Opamp An opamp has two nputs and one output. Note the opamp below. The termnal labeled wth the () sgn s the nvertng nput and the nput labeled wth the () sgn
More informationPoisson brackets and canonical transformations
rof O B Wrght Mechancs Notes osson brackets and canoncal transformatons osson Brackets Consder an arbtrary functon f f ( qp t) df f f f q p q p t But q p p where ( qp ) pq q df f f f p q q p t In order
More informationKernel Methods and SVMs Extension
Kernel Methods and SVMs Extenson The purpose of ths document s to revew materal covered n Machne Learnng 1 Supervsed Learnng regardng support vector machnes (SVMs). Ths document also provdes a general
More informationSection 3.6 Complex Zeros
04 Chapter Secton 6 Comple Zeros When fndng the zeros of polynomals, at some pont you're faced wth the problem Whle there are clearly no real numbers that are solutons to ths equaton, leavng thngs there
More informationVoltage and Current Laws
CHAPTER 3 Voltage and Current Laws KEY CONCEPTS INTRODUCTION In Chap. 2 we were ntroduced to ndependent voltage and current sources, dependent sources, and resstors. We dscovered that dependent sources
More informationAPPENDIX A Some Linear Algebra
APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,
More informationDynamics of a Superconducting Qubit Coupled to an LC Resonator
Dynamcs of a Superconductng Qubt Coupled to an LC Resonator Y Yang Abstract: We nvestgate the dynamcs of a current-based Josephson juncton quantum bt or qubt coupled to an LC resonator. The Hamltonan of
More informationTHE SUMMATION NOTATION Ʃ
Sngle Subscrpt otaton THE SUMMATIO OTATIO Ʃ Most of the calculatons we perform n statstcs are repettve operatons on lsts of numbers. For example, we compute the sum of a set of numbers, or the sum of the
More informationPhysics Courseware Electronics
Physcs ourseware Electroncs ommon emtter amplfer Problem 1.- In the followg ommon Emtter mplfer calculate: a) The Q pot, whch s the D base current (I ), the D collector current (I ) and the voltage collector
More informationTransfer Functions. Convenient representation of a linear, dynamic model. A transfer function (TF) relates one input and one output: ( ) system
Transfer Functons Convenent representaton of a lnear, dynamc model. A transfer functon (TF) relates one nput and one output: x t X s y t system Y s The followng termnology s used: x y nput output forcng
More informationWaveguides and resonant cavities
Wavegudes and resonant cavtes February 8, 014 Essentally, a wavegude s a conductng tube of unform cross-secton and a cavty s a wavegude wth end caps. The dmensons of the gude or cavty are chosen to transmt,
More informationProblem Solving in Math (Math 43900) Fall 2013
Problem Solvng n Math (Math 43900) Fall 2013 Week four (September 17) solutons Instructor: Davd Galvn 1. Let a and b be two nteger for whch a b s dvsble by 3. Prove that a 3 b 3 s dvsble by 9. Soluton:
More informationPhysics 5153 Classical Mechanics. D Alembert s Principle and The Lagrangian-1
P. Guterrez Physcs 5153 Classcal Mechancs D Alembert s Prncple and The Lagrangan 1 Introducton The prncple of vrtual work provdes a method of solvng problems of statc equlbrum wthout havng to consder the
More information4.1 The Ideal Diode. Reading Assignment: pp Before we get started with ideal diodes, let s first recall linear device behavior!
1/25/2012 secton3_1the_ideal_ode 1/2 4.1 The Ideal ode Readng Assgnment: pp.165-172 Before we get started wth deal dodes, let s frst recall lnear dece behaor! HO: LINEAR EVICE BEHAVIOR Now, the deal dode
More informationBoise State University Department of Electrical and Computer Engineering ECE 212L Circuit Analysis and Design Lab
Bose State Unersty Department of Electrcal and omputer Engneerng EE 1L rcut Analyss and Desgn Lab Experment #8: The Integratng and Dfferentatng Op-Amp rcuts 1 Objectes The objectes of ths laboratory experment
More informationFundamental loop-current method using virtual voltage sources technique for special cases
Fundamental loop-current method usng vrtual voltage sources technque for specal cases George E. Chatzaraks, 1 Marna D. Tortorel 1 and Anastasos D. Tzolas 1 Electrcal and Electroncs Engneerng Departments,
More informationEE 2006 Electric Circuit Analysis Fall September 04, 2014 Lecture 02
EE 2006 Electrc Crcut Analyss Fall 2014 September 04, 2014 Lecture 02 1 For Your Informaton Course Webpage http://www.d.umn.edu/~jngba/electrc_crcut_analyss_(ee_2006).html You can fnd on the webpage: Lecture:
More informationMath1110 (Spring 2009) Prelim 3 - Solutions
Math 1110 (Sprng 2009) Solutons to Prelm 3 (04/21/2009) 1 Queston 1. (16 ponts) Short answer. Math1110 (Sprng 2009) Prelm 3 - Solutons x a 1 (a) (4 ponts) Please evaluate lm, where a and b are postve numbers.
More information