School of Electrical and Computer Engineering ECE2040 Dr. George F. Riley Summer 2007, GT Lorraine Analysis of LRC with Sinusoidal Sources
|
|
- Kelly Mitchell
- 5 years ago
- Views:
Transcription
1 School of Electrical and Computer Engineering ECE2040 Dr. George F. Riley Summer 2007, GT Lorraine Analysis of LRC with Sinusoidal Sources In chapter 8 in the textbook, we determined that if the forcing function (the voltage V s (t) or the current I s (t)) is of the form V s (t) = cosωt then forced response v o (t) of an RC circuit or RL circuit is of the form: v f = A cosωt B sin ωt If we multiply the right side of the above equation by A 2 B 2 A we get: 2 B 2 v f = ( ) A A 2 B 2 A2 B cosωt B 2 A2 B sin ωt 2 Now, if we define θ = tan 1 (B/A), we can say that: So sin θ = B A2 B 2 cosθ = A A2 B 2 By the trigonometric identity: we get: v f = cosθ cosωt sin θ sin ωt cosα β = cosαcosβ sin α sin β v f = A 2 B 2 cos(ωt θ) (1) The above result will be used in the next set of equation. The previous analysis is found in section 10.3 in Dorf and Svoboda. 1
2 Now consider the circuit below where v s (t) = cosωt and we want to find the forced response i f (t). This analysis is found in section 10.4 of Dorf and Svoboda. R vs(t) L i(t) Using KVL: L di dt Ri = cosωt From chapter 8, we know that the forced response i f (t) will be of the form: and: thus: di dt i f (t) = A cosωt B sin ωt = ωa sin ωt ωb cosωt L(ωA sin ωt ωb cosωt) R(A cosωt B sin ωt) = cosωt We can find A and B by chosing a value for ωt that makes sin ωt = 0 and cosωt = 1. ωlb RA = V b and a value for ωt that makes sin ωt = 1 and cos ωt = 0. Solving for A and B we get: ωla RB = 0; A = B = R R 2 ω 2 L 2 ωl R 2 ω 2 L 2 2
3 From equation 1, we know that: So we need to find A 2 B 2 : So finally: A 2 B 2 = v f = A cosωt B sin ωt = A 2 B 2 (cosωt β) R 2 V 2 m (R 2 ω 2 L 2 )(R 2 ω 2 L 2 ) ω 2 L 2 V 2 m (R 2 ω 2 L 2 )(R 2 ω 2 L 2 ) A 2 B 2 = V 2 m (R2 ω 2 L 2 ) (R 2 ω 2 L 2 )(R 2 ω 2 L 2 ) A2 B 2 = R2 ω 2 L 2 where: i(t) = A 2 B 2 cos(ωt β) = Z cos(ωt β) (2) Z = R 2 ω 2 L 2 β = tan 1 (B/A) = tan 1 ωl R Equation 2 above is the solution for the forced response to an LR circuit with sinusoidal forcing function, and should be memorized. 3
4 We now turn out attention to the RC circuit shown below, where v s (t) = cosωt. This analysis is not in the textbook, but is inportant and should be studied and understood. R vs(t) C vo(t) We know that the current is the same across all elements of this circuit, and that the current through a capacitor is: KVL around the loop gives: C dv dt V s (t) = cosωt = RC dv dt v o(t) As before, assume the response v o (t) is of the form: so substituting above for v o (t) and dv dt v o (t) = A cosωt B sin ωt we get: cosωt = RCBω cosωt RCAω sin ωt A cosωt B sin ωt Again choosing appropriate values for cosωt and sin ωt, we get: B = RCAω Solving for A and B we get: Again finding A 2 B 2 : A = RCBω A = 1 R 2 C 2 ω 2 B = RCω 1 R 2 C 2 ω 2 A 2 B 2 = V 2 m R2 C 2 ω 2 V 2 m (1 R 2 C 2 ω 2 )(1 R 2 C 2 ω 2 ) = V 2 m (1 R2 C 2 ω 2 ) (1 R 2 C 2 ω 2 )(1 R 2 C 2 ω 2 ) = V 2 m 1 R 2 C 2 ω 2 4
5 Finally, the forced response v o (t) is: A2 B 2 = 1 R2 C 2 ω 2 where v o (t) = P cos(ωt β) (3) P = 1 R 2 C 2 ω 2, β = tan 1 (B/A) = tan 1 (RCω) Equation 3 above is the response to an RC circuit with a sinusoidal forcing function and should be memorized. 5
6 Next, we will analyze an RL circuit with a complex exponential forcing function. The circuit below is the same as the LR circuit used previously, and the forcing function v s (t) is again cos ωt. However, in this case we will see that the analysis is a bit simpler. R vs(t) L i(t) We start by observing that: v s (t) = cos ωt = Re { e jωt} Where the Re notation indicates the real part of the complex variable. For the remainder of this analysis, we will drop the Re notation, and simply take the real part of the computed forced response when we are done. Using KVL: v s (t) = L di dt Ri We know that the response to an exponential forcing function is of the form: So: Solving for A: where: i o (t) = Ae jωt v s (t) = e jωt = AjωLe jωt RAe jωt = Ae jωt (jωl R) A = R jωl = Z ejβ Z = R 2 ω 2 L 2, β = tan 1 ωl R So the response is: { } Vm i o (t) = Re Z ejβ e jωt = Z ejωtβ = cos(ωt β) Z which matches the result earlier in equation 2. We could do a similar analysis for an RC circuit using the same technique, which would give the same results as in equation 3. 6
7 Now we look at using the concept of Complex Phasors to solve LR and RC circuit responses with sinusoidal forcing functions. In the previous analysis, we found that the term e jωt occurred often, and was unchanged from the forcing function to the respose function. For ease of notation, we use the notation: V = Re { e jβ e jωt} = β The boldface V indicates the value is a Phasor, which has a magnitude and a phase angle. Phasors also have an e jωt term which is not written but assumed present. To see how this is useful, consider the circuit below with v s (t) = cos(ωt φ). R vs(t) L i(t) Assume the response i(t) is of the form: v s (t) = cos(ωt φ) = Re { e j(ωtφ)} By KVL: i(t) = I m cos(ωt β) = Re { I m e jωtβ} v s (t) = L di dt Ri e j(ωtφ) = (jωli m RI m )e j(ωtβ) To convert to phasor notation, first remove the e jωt from every term: e jφ = (jωli m RI m )e jβ Thus in phasor notation, we have: V s = e jφ, I = I m e jβ (jωl R)I = V s I = V s jωl R 7
8 If we let φ = 10, ω = 100rad/s, R = 200Ω, L = 2H, we get: V s I = jωl R = j = = Converting back to time domain representation, we get: V s i(t) = 283 cos(100t 35 ) As another example, use Phasor notation to solve for v o (T) in the figure below, where i(t) = 10 cosωt. i(t) R C vo(t) By KCL: i(t) = 10 cosωt = I = 10 0 i(t) = v o(t) R C dv dt i(t) = 10Re { e jωt} v o (t) = Re { e jωtβ} R ej(ωtβ) jωc e j(ωtβ) = 10e jωt Suppress the e jωt term to convert to phasor notation: ( 1 R jωc ( 1 R jωc Let R = 1Ω, C = 10mF, ω = 100, we get: ) e jβ = 10e j0 ) V = I (1 j1)v = I, V = I 1 j1, V = Converting back to time domain representation, we get: = v o (t) = 10 2 cos(100t 45 ) 8
Sinusoids and Phasors
CHAPTER 9 Sinusoids and Phasors We now begins the analysis of circuits in which the voltage or current sources are time-varying. In this chapter, we are particularly interested in sinusoidally time-varying
More informationElectric 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 informationCircuit Analysis-III. Circuit Analysis-II Lecture # 3 Friday 06 th April, 18
Circuit Analysis-III Sinusoids Example #1 ü Find the amplitude, phase, period and frequency of the sinusoid: v (t ) =12cos(50t +10 ) Signal Conversion ü From sine to cosine and vice versa. ü sin (A ± B)
More informationFall 2011 ME 2305 Network Analysis. Sinusoidal Steady State Analysis of RLC Circuits
Fall 2011 ME 2305 Network Analysis Chapter 4 Sinusoidal Steady State Analysis of RLC Circuits Engr. Humera Rafique Assistant Professor humera.rafique@szabist.edu.pk Faculty of Engineering (Mechatronics)
More informationReview of 1 st Order Circuit Analysis
ECEN 60 Circuits/Electronics Spring 007-7-07 P. Mathys Review of st Order Circuit Analysis First Order Differential Equation Consider the following circuit with input voltage v S (t) and output voltage
More informationTo find the step response of an RC circuit
To find the step response of an RC circuit v( t) v( ) [ v( t) v( )] e tt The time constant = RC The final capacitor voltage v() The initial capacitor voltage v(t ) To find the step response of an RL circuit
More informationPhasors: Impedance and Circuit Anlysis. Phasors
Phasors: Impedance and Circuit Anlysis Lecture 6, 0/07/05 OUTLINE Phasor ReCap Capacitor/Inductor Example Arithmetic with Complex Numbers Complex Impedance Circuit Analysis with Complex Impedance Phasor
More informationEE292: 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 informationPhasor Diagram. Figure 1: Phasor Diagram. A φ. Leading Direction. θ B. Lagging Direction. Imag. Axis Complex Plane. Real Axis
1 16.202: PHASORS Consider sinusoidal source i(t) = Acos(ωt + φ) Using Eulers Notation: Acos(ωt + φ) = Re[Ae j(ωt+φ) ] Phasor Representation of i(t): = Ae jφ = A φ f v(t) = Bsin(ωt + ψ) First convert the
More informationSinusoidal Steady-State Analysis
Chapter 4 Sinusoidal Steady-State Analysis In this unit, we consider circuits in which the sources are sinusoidal in nature. The review section of this unit covers most of section 9.1 9.9 of the text.
More informationSinusoidal Steady-State Analysis
Sinusoidal Steady-State Analysis Mauro Forti October 27, 2018 Constitutive Relations in the Frequency Domain Consider a network with independent voltage and current sources at the same angular frequency
More informationd n 1 f dt n 1 + K+ a 0f = C cos(ωt + φ)
Tutorial TUTOR: THE PHASOR TRANSFORM All voltages currents in linear circuits with sinusoidal sources are described by constant-coefficient linear differential equations of the form (1) a n d n f dt n
More informationEE292: Fundamentals of ECE
EE292: Fundamentals of ECE Fall 2012 TTh 10:00-11:15 SEB 1242 Lecture 20 121101 http://www.ee.unlv.edu/~b1morris/ee292/ 2 Outline Chapters 1-3 Circuit Analysis Techniques Chapter 10 Diodes Ideal Model
More informationP A R T 2 AC CIRCUITS. Chapter 9 Sinusoids and Phasors. Chapter 10 Sinusoidal Steady-State Analysis. Chapter 11 AC Power Analysis
P A R T 2 AC CIRCUITS Chapter 9 Sinusoids and Phasors Chapter 10 Sinusoidal Steady-State Analysis Chapter 11 AC Power Analysis Chapter 12 Three-Phase Circuits Chapter 13 Magnetically Coupled Circuits Chapter
More informationSinusoidal 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 information09/29/2009 Reading: Hambley Chapter 5 and Appendix A
EE40 Lec 10 Complex Numbers and Phasors Prof. Nathan Cheung 09/29/2009 Reading: Hambley Chapter 5 and Appendix A Slide 1 OUTLINE Phasors as notation for Sinusoids Arithmetic with Complex Numbers Complex
More informationAC analysis - many examples
AC analysis - many examples The basic method for AC analysis:. epresent the AC sources as complex numbers: ( ). Convert resistors, capacitors, and inductors into their respective impedances: resistor Z
More informationChapter 5 Steady-State Sinusoidal Analysis
Chapter 5 Steady-State Sinusoidal Analysis Chapter 5 Steady-State Sinusoidal Analysis 1. Identify the frequency, angular frequency, peak value, rms value, and phase of a sinusoidal signal. 2. Solve steady-state
More informationSinusoidal steady-state analysis
Sinusoidal steady-state analysis From our previous efforts with AC circuits, some patterns in the analysis started to appear. 1. In each case, the steady-state voltages or currents created in response
More informationSCHOOL OF MATHEMATICS MATHEMATICS FOR PART I ENGINEERING. Self-paced Course
SCHOOL OF MATHEMATICS MATHEMATICS FOR PART I ENGINEERING Self-paced Course MODULE 26 APPLICATIONS TO ELECTRICAL CIRCUITS Module Topics 1. Complex numbers and alternating currents 2. Complex impedance 3.
More informationK.K. Gan L3: R-L-C AC Circuits. amplitude. Volts. period. -Vo
Lecture 3: R-L-C AC Circuits AC (Alternative Current): Most of the time, we are interested in the voltage at a point in the circuit will concentrate on voltages here rather than currents. We encounter
More informationCIRCUIT ANALYSIS II. (AC Circuits)
Will Moore MT & MT CIRCUIT ANALYSIS II (AC Circuits) Syllabus Complex impedance, power factor, frequency response of AC networks including Bode diagrams, second-order and resonant circuits, damping and
More informationChapter 10: Sinusoidal Steady-State Analysis
Chapter 10: Sinusoidal Steady-State Analysis 1 Objectives : sinusoidal functions Impedance use phasors to determine the forced response of a circuit subjected to sinusoidal excitation Apply techniques
More informationEE221 Circuits II. Chapter 14 Frequency Response
EE22 Circuits II Chapter 4 Frequency Response Frequency Response Chapter 4 4. Introduction 4.2 Transfer Function 4.3 Bode Plots 4.4 Series Resonance 4.5 Parallel Resonance 4.6 Passive Filters 4.7 Active
More informationSinusoidal Steady State Analysis
Sinusoidal Steady State Analysis 9 Assessment Problems AP 9. [a] V = 70/ 40 V [b] 0 sin(000t +20 ) = 0 cos(000t 70 ).. I = 0/ 70 A [c] I =5/36.87 + 0/ 53.3 =4+j3+6 j8 =0 j5 =.8/ 26.57 A [d] sin(20,000πt
More informationEE221 Circuits II. Chapter 14 Frequency Response
EE22 Circuits II Chapter 4 Frequency Response Frequency Response Chapter 4 4. Introduction 4.2 Transfer Function 4.3 Bode Plots 4.4 Series Resonance 4.5 Parallel Resonance 4.6 Passive Filters 4.7 Active
More informationComplex Numbers Review
Complex Numbers view ference: Mary L. Boas, Mathematical Methods in the Physical Sciences Chapter 2 & 4 George Arfken, Mathematical Methods for Physicists Chapter 6 The real numbers (denoted R) are incomplete
More informationSINUSOIDAL STEADY STATE CIRCUIT ANALYSIS
SINUSOIDAL STEADY STATE CIRCUIT ANALYSIS 1. Introduction A sinusoidal current has the following form: where I m is the amplitude value; ω=2 πf is the angular frequency; φ is the phase shift. i (t )=I m.sin
More informationName: Lab: M8 M2 W8 Th8 Th11 Th2 F8. cos( θ) = cos(θ) sin( θ) = sin(θ) sin(θ) = cos. θ (radians) θ (degrees) cos θ sin θ π/6 30
Name: Lab: M8 M2 W8 Th8 Th11 Th2 F8 Trigonometric Identities cos(θ) = cos(θ) sin(θ) = sin(θ) sin(θ) = cos Cosines and Sines of common angles Euler s Formula θ (radians) θ (degrees) cos θ sin θ 0 0 1 0
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 informationLecture 4: R-L-C Circuits and Resonant Circuits
Lecture 4: R-L-C Circuits and Resonant Circuits RLC series circuit: What's V R? Simplest way to solve for V is to use voltage divider equation in complex notation: V X L X C V R = in R R + X C + X L L
More informationSinusoidal Steady-state Analysis
Siusoidal Steady-state Aalysis Complex umber reviews Phasors ad ordiary differetial equatios Complete respose ad siusoidal steady-state respose Cocepts of impedace ad admittace Siusoidal steady-state aalysis
More informationChapter 9 Objectives
Chapter 9 Engr8 Circuit Analysis Dr Curtis Nelson Chapter 9 Objectives Understand the concept of a phasor; Be able to transform a circuit with a sinusoidal source into the frequency domain using phasor
More informationEE 230. Lecture 4. Background Materials
EE 230 Lecture 4 Background Materials Transfer Functions Test Equipment in the Laboratory Quiz 3 If the input to a system is a sinusoid at KHz and if the output is given by the following expression, what
More informationFrequency Response. Re ve jφ e jωt ( ) where v is the amplitude and φ is the phase of the sinusoidal signal v(t). ve jφ
27 Frequency Response Before starting, review phasor analysis, Bode plots... Key concept: small-signal models for amplifiers are linear and therefore, cosines and sines are solutions of the linear differential
More informationR-L-C Circuits and Resonant Circuits
P517/617 Lec4, P1 R-L-C Circuits and Resonant Circuits Consider the following RLC series circuit What's R? Simplest way to solve for is to use voltage divider equation in complex notation. X L X C in 0
More informationREACTANCE. By: Enzo Paterno Date: 03/2013
REACTANCE REACTANCE By: Enzo Paterno Date: 03/2013 5/2007 Enzo Paterno 1 RESISTANCE - R i R (t R A resistor for all practical purposes is unaffected by the frequency of the applied sinusoidal voltage or
More information15-884/484 Electric Power Systems 1: DC and AC Circuits
15-884/484 Electric Power Systems 1: DC and AC Circuits J. Zico Kolter October 8, 2013 1 Hydro Estimated U.S. Energy Use in 2010: ~98.0 Quads Lawrence Livermore National Laboratory Solar 0.11 0.01 8.44
More informationAn op amp consisting of a complex arrangement of resistors, transistors, capacitors, and diodes. Here, we ignore the details.
CHAPTER 5 Operational Amplifiers In this chapter, we learn how to use a new circuit element called op amp to build circuits that can perform various kinds of mathematical operations. Op amp is a building
More informationAC analysis. EE 201 AC analysis 1
AC analysis Now we turn to circuits with sinusoidal sources. Earlier, we had a brief look at sinusoids, but now we will add in capacitors and inductors, making the story much more interesting. What are
More informationEE 40: Introduction to Microelectronic Circuits Spring 2008: Midterm 2
EE 4: Introduction to Microelectronic Circuits Spring 8: Midterm Venkat Anantharam 3/9/8 Total Time Allotted : min Total Points:. This is a closed book exam. However, you are allowed to bring two pages
More informationChapter 10: Sinusoids and Phasors
Chapter 10: Sinusoids and Phasors 1. Motivation 2. Sinusoid Features 3. Phasors 4. Phasor Relationships for Circuit Elements 5. Impedance and Admittance 6. Kirchhoff s Laws in the Frequency Domain 7. Impedance
More informationSource-Free RC Circuit
First Order Circuits Source-Free RC Circuit Initial charge on capacitor q = Cv(0) so that voltage at time 0 is v(0). What is v(t)? Prof Carruthers (ECE @ BU) EK307 Notes Summer 2018 150 / 264 First Order
More informationChapter 10: Sinusoidal Steady-State Analysis
Chapter 0: Sinusoidal Steady-State Analysis Sinusoidal Sources If a circuit is driven by a sinusoidal source, after 5 tie constants, the circuit reaches a steady-state (reeber the RC lab with t τ). Consequently,
More informationLecture 9 Time Domain vs. Frequency Domain
. Topics covered Lecture 9 Time Domain vs. Frequency Domain (a) AC power in the time domain (b) AC power in the frequency domain (c) Reactive power (d) Maximum power transfer in AC circuits (e) Frequency
More informationENGR 2405 Chapter 8. Second Order Circuits
ENGR 2405 Chapter 8 Second Order Circuits Overview The previous chapter introduced the concept of first order circuits. This chapter will expand on that with second order circuits: those that need a second
More information1 Phasors and Alternating Currents
Physics 4 Chapter : Alternating Current 0/5 Phasors and Alternating Currents alternating current: current that varies sinusoidally with time ac source: any device that supplies a sinusoidally varying potential
More informationCircuits and Systems I
Circuits and Systems I LECTURE #2 Phasor Addition lions@epfl Prof. Dr. Volkan Cevher LIONS/Laboratory for Information and Inference Systems License Info for SPFirst Slides This work released under a Creative
More informationPhasor mathematics. Resources and methods for learning about these subjects (list a few here, in preparation for your research):
Phasor mathematics This worksheet and all related files are licensed under the Creative Commons Attribution License, version 1.0. To view a copy of this license, visit http://creativecommons.org/licenses/by/1.0/,
More informationAnnouncements: Today: more AC circuits
Announcements: Today: more AC circuits I 0 I rms Current through a light bulb I 0 I rms I t = I 0 cos ωt I 0 Current through a LED I t = I 0 cos ωt Θ(cos ωt ) Theta function (is zero for a negative argument)
More informationEE100Su08 Lecture #11 (July 21 st 2008)
EE100Su08 Lecture #11 (July 21 st 2008) Bureaucratic Stuff Lecture videos should be up by tonight HW #2: Pick up from office hours today, will leave them in lab. REGRADE DEADLINE: Monday, July 28 th 2008,
More information1.3 Sinusoidal Steady State
1.3 Sinusoidal Steady State Electromagnetics applications can be divided into two broad classes: Time-domain: Excitation is not sinusoidal (pulsed, broadband, etc.) Ultrawideband communications Pulsed
More informationCircuits. Fawwaz T. Ulaby, Michel M. Maharbiz, Cynthia M. Furse. Solutions to the Exercises
Circuits by Fawwaz T. Ulaby, Michel M. Maharbiz, Cynthia M. Furse Solutions to the Exercises Chapter 1: Circuit Terminology Chapter 2: Resisitive Circuits Chapter 3: Analysis Techniques Chapter 4: Operational
More informationHarman Outline 1A CENG 5131
Harman Outline 1A CENG 5131 Numbers Real and Imaginary PDF In Chapter 2, concentrate on 2.2 (MATLAB Numbers), 2.3 (Complex Numbers). A. On R, the distance of any real number from the origin is the magnitude,
More informationEE292: Fundamentals of ECE
EE292: Fundamentals of ECE Fall 2012 TTh 10:00-11:15 SEB 1242 Lecture 14 121011 http://www.ee.unlv.edu/~b1morris/ee292/ 2 Outline Review Steady-State Analysis RC Circuits RL Circuits 3 DC Steady-State
More informationECE 205: Intro Elec & Electr Circuits
ECE 205: Intro Elec & Electr Circuits Final Exam Study Guide Version 1.00 Created by Charles Feng http://www.fenguin.net ECE 205: Intro Elec & Electr Circuits Final Exam Study Guide 1 Contents 1 Introductory
More informationChapter 10 AC Analysis Using Phasors
Chapter 10 AC Analysis Using Phasors 10.1 Introduction We would like to use our linear circuit theorems (Nodal analysis, Mesh analysis, Thevenin and Norton equivalent circuits, Superposition, etc.) to
More informationECE 6340 Intermediate EM Waves. Fall Prof. David R. Jackson Dept. of ECE. Notes 7
ECE 634 Intermediate EM Waves Fall 16 Prof. David R. Jackson Dept. of ECE Notes 7 1 TEM Transmission Line conductors 4 parameters C capacitance/length [F/m] L inductance/length [H/m] R resistance/length
More informationECE 201 Fall 2009 Final Exam
ECE 01 Fall 009 Final Exam December 16, 009 Division 0101: Tan (11:30am) Division 001: Clark (7:30 am) Division 0301: Elliott (1:30 pm) Instructions 1. DO NOT START UNTIL TOLD TO DO SO.. Write your Name,
More informationC R. Consider from point of view of energy! Consider the RC and LC series circuits shown:
ircuits onsider the R and series circuits shown: ++++ ---- R ++++ ---- Suppose that the circuits are formed at t with the capacitor charged to value. There is a qualitative difference in the time development
More informationElectric Circuits II Sinusoidal Steady State Analysis. Dr. Firas Obeidat
Electric Circuits II Sinusoidal Steady State Analysis Dr. Firas Obeidat 1 Table of Contents 1 2 3 4 5 Nodal Analysis Mesh Analysis Superposition Theorem Source Transformation Thevenin and Norton Equivalent
More informationRefinements to Incremental Transistor Model
Refinements to Incremental Transistor Model This section presents modifications to the incremental models that account for non-ideal transistor behavior Incremental output port resistance Incremental changes
More informationECE 421/521 Electric Energy Systems Power Systems Analysis I 2 Basic Principles. Instructor: Kai Sun Fall 2013
ECE 41/51 Electric Energy Systems Power Systems Analysis I Basic Principles Instructor: Kai Sun Fall 013 1 Outline Power in a 1-phase AC circuit Complex power Balanced 3-phase circuit Single Phase AC System
More information15 n=0. zz = re jθ re jθ = r 2. (b) For division and multiplication, it is handy to use the polar representation: z = rejθ. = z 1z 2.
Professor Fearing EECS0/Problem Set v.0 Fall 06 Due at 4 pm, Fri. Sep. in HW box under stairs (st floor Cory) Reading: EE6AB notes. This problem set should be review of material from EE6AB. (Please note,
More informationPhasor Young Won Lim 05/19/2015
Phasor Copyright (c) 2009-2015 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version
More informationLecture 1a. Complex numbers, phasors and vectors. Introduction. Complex numbers. 1a.1
1a.1 Lecture 1a Comple numbers, phasors and vectors Introduction This course will require ou to appl several concepts ou learned in our undergraduate math courses. In some cases, such as comple numbers
More informationECE 45 Discussion 2 Notes
UC San Diego J. Connelly Frequency Response ECE 45 Discussion Notes The inputs and outputs of RLC circuits are generally either voltages or currents. The output of the circuit depends on the frequency
More informationSinusoidal Steady State Analysis (AC Analysis) Part II
Sinusoidal Steady State Analysis (AC Analysis) Part II Amin Electronics and Electrical Communications Engineering Department (EECE) Cairo University elc.n102.eng@gmail.com http://scholar.cu.edu.eg/refky/
More informationElectrical Circuits Lab Series RC Circuit Phasor Diagram
Electrical Circuits Lab. 0903219 Series RC Circuit Phasor Diagram - Simple steps to draw phasor diagram of a series RC circuit without memorizing: * Start with the quantity (voltage or current) that is
More informationElectric Circuits I FINAL EXAMINATION
EECS:300, Electric Circuits I s6fs_elci7.fm - Electric Circuits I FINAL EXAMINATION Problems Points.. 3. 0 Total 34 Was the exam fair? yes no 5//6 EECS:300, Electric Circuits I s6fs_elci7.fm - Problem
More informationProf. Shayla Sawyer CP08 solution
What does the time constant represent in an exponential function? How do you define a sinusoid? What is impedance? How is a capacitor affected by an input signal that changes over time? How is an inductor
More informationRevision: January 9, E Main Suite D Pullman, WA (509) Voice and Fax
.7.: Sinusoidal steady-state system response evision: January 9, 0 E Main Suite D ullman, WA 9963 (09 334 6306 oice and Fax Overview In this module, the concepts presented in chapters.7.0 and.7. are used
More informationNetwork Graphs and Tellegen s Theorem
Networ Graphs and Tellegen s Theorem The concepts of a graph Cut sets and Kirchhoff s current laws Loops and Kirchhoff s voltage laws Tellegen s Theorem The concepts of a graph The analysis of a complex
More informationECE 5260 Microwave Engineering University of Virginia. Some Background: Circuit and Field Quantities and their Relations
ECE 5260 Microwave Engineering University of Virginia Lecture 2 Review of Fundamental Circuit Concepts and Introduction to Transmission Lines Although electromagnetic field theory and Maxwell s equations
More informationElectric Circuits I Final Examination
EECS:300 Electric Circuits I ffs_elci.fm - Electric Circuits I Final Examination Problems Points. 4. 3. Total 38 Was the exam fair? yes no //3 EECS:300 Electric Circuits I ffs_elci.fm - Problem 4 points
More informationSinusoidal Steady State Power Calculations
10 Sinusoidal Steady State Power Calculations Assessment Problems AP 10.1 [a] V = 100/ 45 V, Therefore I = 20/15 A P = 1 (100)(20)cos[ 45 (15)] = 500W, 2 A B Q = 1000sin 60 = 866.03 VAR, B A [b] V = 100/
More informationResponse of Second-Order Systems
Unit 3 Response of SecondOrder Systems In this unit, we consider the natural and step responses of simple series and parallel circuits containing inductors, capacitors and resistors. The equations which
More informationEECS 117 Lecture 3: Transmission Line Junctions / Time Harmonic Excitation
EECS 117 Lecture 3: Transmission Line Junctions / Time Harmonic Excitation Prof. Niknejad University of California, Berkeley University of California, Berkeley EECS 117 Lecture 3 p. 1/23 Transmission Line
More informationReview of DC Electric Circuit. DC Electric Circuits Examples (source:
Review of DC Electric Circuit DC Electric Circuits Examples (source: http://hyperphysics.phyastr.gsu.edu/hbase/electric/dcex.html) 1 Review - DC Electric Circuit Multisim Circuit Simulation DC Circuit
More informationModule 4. Single-phase AC Circuits
Module 4 Single-phase AC Circuits Lesson 14 Solution of Current in R-L-C Series Circuits In the last lesson, two points were described: 1. How to represent a sinusoidal (ac) quantity, i.e. voltage/current
More informationPhysics 116A Notes Fall 2004
Physics 116A Notes Fall 2004 David E. Pellett Draft v.0.9 Notes Copyright 2004 David E. Pellett unless stated otherwise. References: Text for course: Fundamentals of Electrical Engineering, second edition,
More informationProf. Anyes Taffard. Physics 120/220. Voltage Divider Capacitor RC circuits
Prof. Anyes Taffard Physics 120/220 Voltage Divider Capacitor RC circuits Voltage Divider The figure is called a voltage divider. It s one of the most useful and important circuit elements we will encounter.
More informationConsider a simple RC circuit. We might like to know how much power is being supplied by the source. We probably need to find the current.
AC power Consider a simple RC circuit We might like to know how much power is being supplied by the source We probably need to find the current R 10! R 10! is VS Vmcosωt Vm 10 V f 60 Hz V m 10 V C 150
More informationECE 45 Average Power Review
UC San Diego J. Connelly Complex Power ECE 45 Average Power Review When dealing with time-dependent voltage and currents, we have to consider a more general definition of power. We can calculate the instantaneous
More informationReview of Linear Time-Invariant Network Analysis
D1 APPENDIX D Review of Linear Time-Invariant Network Analysis Consider a network with input x(t) and output y(t) as shown in Figure D-1. If an input x 1 (t) produces an output y 1 (t), and an input x
More informationInductive & Capacitive Circuits. Subhasish Chandra Assistant Professor Department of Physics Institute of Forensic Science, Nagpur
Inductive & Capacitive Circuits Subhasish Chandra Assistant Professor Department of Physics Institute of Forensic Science, Nagpur LR Circuit LR Circuit (Charging) Let us consider a circuit having an inductance
More informationNote 11: Alternating Current (AC) Circuits
Note 11: Alternating Current (AC) Circuits V R No phase difference between the voltage difference and the current and max For alternating voltage Vmax sin t, the resistor current is ir sin t. the instantaneous
More informationEIT Quick-Review Electrical Prof. Frank Merat
CIRCUITS 4 The power supplied by the 0 volt source is (a) 2 watts (b) 0 watts (c) 2 watts (d) 6 watts (e) 6 watts 4Ω 2Ω 0V i i 2 2Ω 20V Call the clockwise loop currents i and i 2 as shown in the drawing
More informationChapter 33. Alternating Current Circuits
Chapter 33 Alternating Current Circuits 1 Capacitor Resistor + Q = C V = I R R I + + Inductance d I Vab = L dt AC power source The AC power source provides an alternative voltage, Notation - Lower case
More informationEE40 Lecture 11 Josh Hug 7/19/2010
EE40 Lecture Josh 7/9/200 Logistical Things Lab 4 tomorrow Lab 5 (active filter lab) on Wednesday Prototype for future lab for EE40 Prelab is very short, sorry. Please give us our feedback Google docs
More informationTransmission Lines in the Frequency Domain
Berkeley Transmission Lines in the Frequency Domain Prof. Ali M. Niknejad U.C. Berkeley Copyright c 2016 by Ali M. Niknejad August 30, 2017 1 / 38 Why Sinusoidal Steady-State? 2 / 38 Time Harmonic Steady-State
More informationMath Assignment 6
Math 2280 - Assignment 6 Dylan Zwick Fall 2013 Section 3.7-1, 5, 10, 17, 19 Section 3.8-1, 3, 5, 8, 13 Section 4.1-1, 2, 13, 15, 22 Section 4.2-1, 10, 19, 28 1 Section 3.7 - Electrical Circuits 3.7.1 This
More informationComplex Numbers Review
Complex Numbers view ference: Mary L. Boas, Mathematical Methods in the Physical Sciences Chapter 2 & 14 George Arfken, Mathematical Methods for Physicists Chapter 6 The real numbers (denoted R) are incomplete
More informationBFF1303: ELECTRICAL / ELECTRONICS ENGINEERING. Alternating Current Circuits : Basic Law
BFF1303: ELECTRICAL / ELECTRONICS ENGINEERING Alternating Current Circuits : Basic Law Ismail Mohd Khairuddin, Zulkifil Md Yusof Faculty of Manufacturing Engineering Universiti Malaysia Pahang Alternating
More informationFrequency Bands. ω the numeric value of G ( ω ) depends on the frequency ω of the basis
1/28/2011 Frequency Bands lecture 1/9 Frequency Bands The Eigen value G ( ω ) of a linear operator is of course dependent on frequency ω the numeric value of G ( ω ) depends on the frequency ω of the basis
More informationLectures 16 & 17 Sinusoidal Signals, Complex Numbers, Phasors, Impedance & AC Circuits. Nov. 7 & 9, 2011
Lectures 16 & 17 Sinusoidal Signals, Complex Numbers, Phasors, Impedance & AC Circuits Nov. 7 & 9, 2011 Material from Textbook by Alexander & Sadiku and Electrical Engineering: Principles & Applications,
More informationChapter 2: Complex numbers
Chapter 2: Complex numbers Complex numbers are commonplace in physics and engineering. In particular, complex numbers enable us to simplify equations and/or more easily find solutions to equations. We
More informationECE 421/521 Electric Energy Systems Power Systems Analysis I 2 Basic Principles. Instructor: Kai Sun Fall 2014
ECE 41/51 Electric Energy Systems Power Systems Analysis I Basic Princiles Instructor: Kai Sun Fall 014 1 Outline Power in a 1-hase AC circuit Comlex ower Balanced 3-hase circuit Single Phase AC System
More informationElectric Circuits I Final Examination
The University of Toledo s8fs_elci7.fm - EECS:300 Electric Circuits I Electric Circuits I Final Examination Problems Points.. 3. Total 34 Was the exam fair? yes no The University of Toledo s8fs_elci7.fm
More informationExperiment 3: Resonance in LRC Circuits Driven by Alternating Current
Experiment 3: Resonance in LRC Circuits Driven by Alternating Current Introduction In last week s laboratory you examined the LRC circuit when constant voltage was applied to it. During this laboratory
More information