The Transmission Line Wave Equation

Size: px
Start display at page:

Download "The Transmission Line Wave Equation"

Transcription

1 1//9 The Transmission Line Wave Equation.doc 1/8 The Transmission Line Wave Equation Let s assume that v ( t, ) and (, ) i t each have the timeharmonic form: j t { V e ω j t = } and i (, t) = Re { I( ) e ω } v ( t, ) Re ( ) The time-derivative of these functions are: vt (, ) t i(, t) t e = Re V( ) = Re j V( ) e t { ω } e = Re I ( ) = Re j Ie ( ) t { ω } Inserting these results into the telegrapher s equations, we find: V ( ) Re e j t Re { ( R j L ) I ( ) e ω = ω } I ( ) Re e = Re ( G j C ) V ( ) e { ω } Simplifying, we have the complex form of telegrapher s equations:

2 1//9 The Transmission Line Wave Equation.doc /8 V( ) = ( R jω L) I( ) I ( ) = ( G jω C ) V( ) Note that these complex differential equations are not a function of time t! * The functions I( ) and V ( ) are complex, where the magnitude and phase of the complex functions describe the j t magnitude and phase of the sinusoidal time function e ω. * Thus, I( ) and V ( ) describe the current and voltage along the transmission line, as a function as position. * Remember, not just any function I( ) and V ( ) can exist on a transmission line, but rather only those functions that satisfy the telegraphers equations. Our task, therefore, is to solve the telegrapher equations and V! find all solutions I( ) and ( )

3 1//9 The Transmission Line Wave Equation.doc 3/8 Q: So, what functions I( ) and V ( ) do satisfy both telegrapher s equations?? A: To make this easier, we will combine the telegrapher V and equations to form one differential equation for ( ) another for I( ). First, take the derivative with respect to of the first telegrapher equation: V( ) = ( R jω L) I( ) V( ) I( ) = = ( R jω L) Note that the second telegrapher equation expresses the V : derivative of I( ) in terms of ( ) I ( ) = ( G jωc ) V( ) Combining these two equations, we get an equation involving V only: ( ) V( ) = ( R jωl)( G jωc ) V( ) We can simplify this equation by defining the complex value : = ( R jωl)( G jωc )

4 1//9 The Transmission Line Wave Equation.doc 4/8 So that: V( ) = V( ) In a similar manner (i.e., begin by taking the derivative of the second telegrapher equation), we can derive the differential equation: I ( ) = I ( ) We have decoupled the telegrapher s equations, such that we now have two equations involving one function only: V( ) = V( ) I ( ) = I ( ) These are known as the transmission line wave equations. Note that value is complex, and is determined by taking the square-root of a complex value. Likewise, is a complex value. Do you know how to square a complex number? Can you determine the square root of a complex number?

5 1//9 The Transmission Line Wave Equation.doc 5/8 Note only special functions satisfy these wave equations; if we take the double derivative of the function, the result is the original function (to within a constant )! Q: Yeah right! Every function that I know is changed after a double differentiation. What kind of magical function could possibly satisfy this differential equation? A: Such functions do exist! For example, the functions V ( ) = e and V ( ) e = each satisfy this transmission line wave equation (insert these into the differential equation and see for yourself!). Likewise, since the transmission line wave equation is a linear differential equation, a weighted superposition of the two solutions is also a solution (again, insert this solution to and see for yourself!): V = V e V e ( ) In fact, it turns out that any and all possible solutions to the differential equations can be expressed in this simple form! Therefore, the general solution to these complex wave equations (and thus the telegrapher equations) are:

6 1//9 The Transmission Line Wave Equation.doc 6/8 V ( ) = V e V e I( ) = I e I e where V,V,I, and I are complex constants. It is unfathomably important that you understand what this result means! It means that the functions V ( ) and I( ), describing the current and voltage at all points along a transmission line, can always be completely specified with just four complex constants ( V,V,I,I )!! We can alternatively write these solutions as: V ( ) = V ( ) V ( ) where: I( ) = I ( ) I ( ) V ( ) V e V ( ) V e I ( ) I e I ( ) I e

7 1//9 The Transmission Line Wave Equation.doc 7/8 The two terms in each solution describe two waves propagating V I ) propagating in the transmission line, one wave ( ( ) or ( ) in one direction ( ) and the other wave ( V ( ) or I ( ) propagating in the opposite direction ( ). ) V ( ) V e V ( ) V e Q: So just what are the complex values V, V, I, I? A: Consider the wave solutions at one specific point on the transmission line the point =. For example, we find that: ( ) V = = V e ( = ) ( ) = V e = V ( 1) = V In other words, V is simply the complex value of the wave V at the point = on the transmission line! function ( ) Likewise, we find:

8 1//9 The Transmission Line Wave Equation.doc 8/8 ( ) V = V = ( ) I = I = ( ) I = I = Again, the four complex values V, I, V, I are all that is needed to determine the voltage and current at any and all points on the transmission line. More specifically, each of these four complex constants completely specifies one of the four transmission line wave V I V I. functions ( ), ( ), ( ), ( ) Q: But what determines these wave functions? How do we find the values of constants V, I, V, I? A: As you might expect, the voltage and current on a transmission line is determined by the devices attached to it on either end (e.g., active sources and/or passive loads)! The precise values of V, I, V, I are therefore determined by satisfying the boundary conditions applied at each end of the transmission line much more on this later!

The Transmission Line Wave Equation

The Transmission Line Wave Equation 1//5 Th Transmission Lin Wav Equation.doc 1/6 Th Transmission Lin Wav Equation Q: So, what functions I (z) and V (z) do satisfy both tlgraphr s quations?? A: To mak this asir, w will combin th tlgraphr

More information

II Transmitter and Receiver Design

II Transmitter and Receiver Design 8/3/6 transmission lines 1/7 II Transmitter and Receiver Design We design radio systems using RF/microwave components. Q: Why don t we use the usual circuit components (e.g., resistors, capacitors, op-amps,

More information

Two-Port Networks Admittance Parameters CHAPTER16 THE LEARNING GOALS FOR THIS CHAPTER ARE THAT STUDENTS SHOULD BE ABLE TO:

Two-Port Networks Admittance Parameters CHAPTER16 THE LEARNING GOALS FOR THIS CHAPTER ARE THAT STUDENTS SHOULD BE ABLE TO: CHAPTER16 Two-Port Networks THE LEARNING GOALS FOR THIS CHAPTER ARE THAT STUDENTS SHOULD BE ABLE TO: Calculate the admittance, impedance, hybrid, and transmission parameter for two-port networks. Convert

More information

EECS 117 Lecture 3: Transmission Line Junctions / Time Harmonic Excitation

EECS 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 information

Superposition (take 3)

Superposition (take 3) Superposition (take 3) Previously, we looked at how to analyze an AC to DC converter and a Buck converter. To solve these circuits, we changed the problem so that the input contained A DC signal, and An

More information

Transmission line equations in phasor form

Transmission line equations in phasor form Transmission line equations in phasor form Kenneth H. Carpenter Department of Electrical and Computer Engineering Kansas State University November 19, 2004 The text for this class presents transmission

More information

1.3 Sinusoidal Steady State

1.3 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 information

DC and AC Impedance of Reactive Elements

DC and AC Impedance of Reactive Elements 3/6/20 D and A Impedance of Reactive Elements /6 D and A Impedance of Reactive Elements Now, recall from EES 2 the complex impedances of our basic circuit elements: ZR = R Z = jω ZL = jωl For a D signal

More information

Designing Information Devices and Systems II Fall 2018 Elad Alon and Miki Lustig Discussion 5A

Designing Information Devices and Systems II Fall 2018 Elad Alon and Miki Lustig Discussion 5A EECS 6B Designing Information Devices and Systems II Fall 208 Elad Alon and Miki Lustig Discussion 5A Transfer Function When we write the transfer function of an arbitrary circuit, it always takes the

More information

Experiment 06 - Extraction of Transmission Line Parameters

Experiment 06 - Extraction of Transmission Line Parameters ECE 451 Automated Microwave Measurements Laboratory Experiment 06 - Extraction of Transmission Line Parameters 1 Introduction With the increase in both speed and complexity of mordern circuits, modeling

More information

EE 40: Introduction to Microelectronic Circuits Spring 2008: Midterm 2

EE 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 information

Time-Harmonic Solutions for Transmission Lines

Time-Harmonic Solutions for Transmission Lines 1/20/2012 Time Harmonic Solutions for Transmission Lines present 1/10 Time-Harmonic Solutions for Transmission Lines There are an unaccountably infinite number of solutions v ( zt, ) and (, ) the telegrapher

More information

Transmission Line Theory

Transmission Line Theory S. R. Zinka zinka@vit.ac.in School of Electronics Engineering Vellore Institute of Technology April 26, 2013 Outline 1 Free Space as a TX Line 2 TX Line Connected to a Load 3 Some Special Cases 4 Smith

More information

ECE 6340 Intermediate EM Waves. Fall Prof. David R. Jackson Dept. of ECE. Notes 7

ECE 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 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

Harman Outline 1A CENG 5131

Harman 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 information

Lecture 17 Date:

Lecture 17 Date: Lecture 17 Date: 09.03.017 The Quadrature Hybrid We began our discussion of dividers and couplers by considering important general properties of three- and four-port networks. This was followed by an analysis

More information

Design of Narrow Band Filters Part 1

Design of Narrow Band Filters Part 1 E.U.I.T. Telecomunicación 2010, Madrid, Spain, 27.09 30.09.2010 Design of Narrow Band Filters Part 1 Thomas Buch Institute of Communications Engineering University of Rostock Th. Buch, Institute of Communications

More information

Fall 2011 ME 2305 Network Analysis. Sinusoidal Steady State Analysis of RLC Circuits

Fall 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 information

Lecture 19 Date:

Lecture 19 Date: Lecture 19 Date: 8.10.014 The Quadrature Hybrid We began our discussion of dividers and couplers by considering important general properties of three- and fourport networks. This was followed by an analysis

More information

3.1 Superposition theorem

3.1 Superposition theorem Many electric circuits are complex, but it is an engineer s goal to reduce their complexity to analyze them easily. In the previous chapters, we have mastered the ability to solve networks containing independent

More information

Mutual Inductance: This is the magnetic flux coupling of 2 coils where the current in one coil causes a voltage to be induced in the other coil.

Mutual Inductance: This is the magnetic flux coupling of 2 coils where the current in one coil causes a voltage to be induced in the other coil. agnetically Coupled Circuits utual Inductance: This is the magnetic flux coupling of coils where the current in one coil causes a voltage to be induced in the other coil. st I d like to emphasize that

More information

Transmission Lines in the Frequency Domain

Transmission 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 information

Electric Circuits I FINAL EXAMINATION

Electric 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 information

ECE 604, Lecture 13. October 16, 2018

ECE 604, Lecture 13. October 16, 2018 ECE 604, Lecture 13 October 16, 2018 1 Introduction In this lecture, we will cover the following topics: Terminated Transmission Line Smith Chart Voltage Standing Wave Ratio (VSWR) Additional Reading:

More information

Sinusoidal Steady-State Analysis

Sinusoidal 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 information

Chapter 2 Circuit Elements

Chapter 2 Circuit Elements Chapter Circuit Elements Chapter Circuit Elements.... Introduction.... Circuit Element Construction....3 Resistor....4 Inductor...4.5 Capacitor...6.6 Element Basics...8.6. Element Reciprocals...8.6. Reactance...8.6.3

More information

Phasors: Impedance and Circuit Anlysis. Phasors

Phasors: 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 information

2. The following diagram illustrates that voltage represents what physical dimension?

2. The following diagram illustrates that voltage represents what physical dimension? BioE 1310 - Exam 1 2/20/2018 Answer Sheet - Correct answer is A for all questions 1. A particular voltage divider with 10 V across it consists of two resistors in series. One resistor is 7 KΩ and the other

More information

Lecture 2 - Transmission Line Theory

Lecture 2 - Transmission Line Theory Lecture 2 - Transmission Line Theory Microwave Active Circuit Analysis and Design Clive Poole and Izzat Darwazeh Academic Press Inc. Poole-Darwazeh 2015 Lecture 2 - Transmission Line Theory Slide1 of 54

More information

EE40 Lecture 11 Josh Hug 7/19/2010

EE40 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 information

FINAL EXAM IN FYS-3007

FINAL EXAM IN FYS-3007 Page 1 of 4 pages + chart FINAL EXAM IN FYS-007 Exam in : Fys-007 Microwave Techniques Date : Tuesday, May 1, 2011 Time : 09.00 1.00 Place : Åsgårdveien 9 Approved remedies : All non-living and non-communicating

More information

Statistical Analysis of Voltage and Current Fluctuations in a Transmission Line with Distributed Parameters Varying Randomly with Space and Time

Statistical Analysis of Voltage and Current Fluctuations in a Transmission Line with Distributed Parameters Varying Randomly with Space and Time International Journal of Electronic and Electrical Engineering. ISSN 0974-2174 Volume 7, Number 8 (2014), pp. 851-855 International Research Publication House http://www.irphouse.com Statistical Analysis

More information

ECE 45 Average Power Review

ECE 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 information

R-L-C Circuits and Resonant Circuits

R-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 information

Sinusoids and Phasors

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 information

Chapter 5 Steady-State Sinusoidal Analysis

Chapter 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 information

Lecture 13 Date:

Lecture 13 Date: ecture 3 Date: 6.09.204 The Signal Flow Graph (Contd.) Impedance Matching and Tuning Tpe Matching Network Example Signal Flow Graph (contd.) Splitting Rule Now consider the three equations SFG a a b 2

More information

Complex Numbers, Phasors and Circuits

Complex Numbers, Phasors and Circuits Complex Numbers, Phasors and Circuits Transmission Lines Complex numbers are defined by points or vectors in the complex plane, and can be represented in Cartesian coordinates or in polar (exponential)

More information

Electric Circuits II Sinusoidal Steady State Analysis. Dr. Firas Obeidat

Electric 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 information

Network Graphs and Tellegen s Theorem

Network 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 information

Chapter 9. Electromagnetic Waves

Chapter 9. Electromagnetic Waves Chapter 9. Electromagnetic Waves 9.1 Waves in One Dimension 9.1.1 The Wave Equation What is a "wave?" Let's start with the simple case: fixed shape, constant speed: How would you represent such a string

More information

Frequency Bands. ω the numeric value of G ( ω ) depends on the frequency ω of the basis

Frequency 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 information

Circuit Theorems Overview Linearity Superposition Source Transformation Thévenin and Norton Equivalents Maximum Power Transfer

Circuit Theorems Overview Linearity Superposition Source Transformation Thévenin and Norton Equivalents Maximum Power Transfer Circuit Theorems Overview Linearity Superposition Source Transformation Thévenin and Norton Equivalents Maximum Power Transfer J. McNames Portland State University ECE 221 Circuit Theorems Ver. 1.36 1

More information

Incident, Reflected, and Absorbed Power

Incident, Reflected, and Absorbed Power /7/9 Incident Reflected and Absorbed ower.doc /8 Incident, Reflected, and Absorbed ower We have discovered that two waves propagate along a V z and V z transmission line, one in each direction ( ( ) (

More information

EXPERIMENT 12 OHM S LAW

EXPERIMENT 12 OHM S LAW EXPERIMENT 12 OHM S LAW INTRODUCTION: We will study electricity as a flow of electric charge, sometimes making analogies to the flow of water through a pipe. In order for electric charge to flow a complete

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

Frequency Response Prof. Ali M. Niknejad Prof. Rikky Muller

Frequency Response Prof. Ali M. Niknejad Prof. Rikky Muller EECS 105 Spring 2017, Module 4 Frequency Response Prof. Ali M. Niknejad Department of EECS Announcements l HW9 due on Friday 2 Review: CD with Current Mirror 3 Review: CD with Current Mirror 4 Review:

More information

Review of 1 st Order Circuit Analysis

Review 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 information

The Impedance Matrix

The Impedance Matrix 0/0/09 The mpedance Matrix.doc /7 The mpedance Matrix Consider the -port microwave device shown below: z ( z ) z z port z z port 0 -port 0 microwave 0 device P z z P z port z P z ( z ) z port 0 ( z ) z

More information

Lecture 11 Date:

Lecture 11 Date: Lecture 11 Date: 11.09.014 Scattering Parameters and Circuit Symmetry Even-mode and Odd-mode Analysis Generalized S-Parameters Example T-Parameters Q: OK, but how can we determine the scattering matrix

More information

analyse and design a range of sine-wave oscillators understand the design of multivibrators.

analyse and design a range of sine-wave oscillators understand the design of multivibrators. INTODUTION In this lesson, we investigate some forms of wave-form generation using op amps. Of course, we could use basic transistor circuits, but it makes sense to simplify the analysis by considering

More information

Chapter 9 Objectives

Chapter 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 information

Complex Numbers Review

Complex 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 information

Phasor Diagram. Figure 1: Phasor Diagram. A φ. Leading Direction. θ B. Lagging Direction. Imag. Axis Complex Plane. Real Axis

Phasor 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 information

Name. Section. Short Answer Questions. 1. (20 Pts) 2. (10 Pts) 3. (5 Pts) 4. (10 Pts) 5. (10 Pts) Regular Questions. 6. (25 Pts) 7.

Name. Section. Short Answer Questions. 1. (20 Pts) 2. (10 Pts) 3. (5 Pts) 4. (10 Pts) 5. (10 Pts) Regular Questions. 6. (25 Pts) 7. Name Section Short Answer Questions 1. (20 Pts) 2. (10 Pts) 3. (5 Pts). (10 Pts) 5. (10 Pts) Regular Questions 6. (25 Pts) 7. (20 Pts) Notes: 1. Please read over all questions before you begin your work.

More information

Schedule. ECEN 301 Discussion #20 Exam 2 Review 1. Lab Due date. Title Chapters HW Due date. Date Day Class No. 10 Nov Mon 20 Exam Review.

Schedule. ECEN 301 Discussion #20 Exam 2 Review 1. Lab Due date. Title Chapters HW Due date. Date Day Class No. 10 Nov Mon 20 Exam Review. Schedule Date Day lass No. 0 Nov Mon 0 Exam Review Nov Tue Title hapters HW Due date Nov Wed Boolean Algebra 3. 3.3 ab Due date AB 7 Exam EXAM 3 Nov Thu 4 Nov Fri Recitation 5 Nov Sat 6 Nov Sun 7 Nov Mon

More information

Sophomore Physics Laboratory (PH005/105)

Sophomore Physics Laboratory (PH005/105) CALIFORNIA INSTITUTE OF TECHNOLOGY PHYSICS MATHEMATICS AND ASTRONOMY DIVISION Sophomore Physics Laboratory (PH5/15) Analog Electronics Active Filters Copyright c Virgínio de Oliveira Sannibale, 23 (Revision

More information

AC analysis - many examples

AC 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 information

09/29/2009 Reading: Hambley Chapter 5 and Appendix A

09/29/2009 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 information

Thevenin equivalent circuits

Thevenin equivalent circuits Thevenin equivalent circuits We have seen the idea of equivalency used in several instances already. 1 2 1 2 same as 1 2 same as 1 2 R 3 same as = 0 V same as 0 A same as same as = EE 201 Thevenin 1 The

More information

Solutions to Problems in Chapter 5

Solutions to Problems in Chapter 5 Appendix E Solutions to Problems in Chapter 5 E. Problem 5. Properties of the two-port network The network is mismatched at both ports (s 0 and s 0). The network is reciprocal (s s ). The network is symmetrical

More information

ECE 107: Electromagnetism

ECE 107: Electromagnetism ECE 107: Electromagnetism Set 2: Transmission lines Instructor: Prof. Vitaliy Lomakin Department of Electrical and Computer Engineering University of California, San Diego, CA 92093 1 Outline Transmission

More information

TRANSMISSION LINES AND MATCHING

TRANSMISSION LINES AND MATCHING TRANSMISSION LINES AND MATCHING for High-Frequency Circuit Design Elective by Michael Tse September 2003 Contents Basic models The Telegrapher s equations and solutions Transmission line equations The

More information

Electric Circuit Theory

Electric Circuit Theory Electric Circuit Theory Nam Ki Min nkmin@korea.ac.kr 010-9419-2320 Chapter 18 Two-Port Circuits Nam Ki Min nkmin@korea.ac.kr 010-9419-2320 Contents and Objectives 3 Chapter Contents 18.1 The Terminal Equations

More information

ELECTROMAGNETISM SUMMARY. Maxwell s equations Transmission lines Transmission line transformers Skin depth

ELECTROMAGNETISM SUMMARY. Maxwell s equations Transmission lines Transmission line transformers Skin depth ELECTROMAGNETISM SUMMARY Maxwell s equations Transmission lines Transmission line transformers Skin depth 1 ENGN4545/ENGN6545: Radiofrequency Engineering L#4 Magnetostatics: The static magnetic field Gauss

More information

Alternating Current (AC) Circuits

Alternating Current (AC) Circuits Alternating Current (AC) Circuits We have been talking about DC circuits Constant currents and voltages Resistors Linear equations Now we introduce AC circuits Time-varying currents and voltages Resistors,

More information

Frequency Response. Re ve jφ e jωt ( ) where v is the amplitude and φ is the phase of the sinusoidal signal v(t). ve jφ

Frequency 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 information

Four Ports. 1 ENGN4545/ENGN6545: Radiofrequency Engineering L#16

Four Ports. 1 ENGN4545/ENGN6545: Radiofrequency Engineering L#16 Four Ports Theorems of four port networks 180 o and 90 o Hybrids Directional couplers Transmission line transformers, Four port striplines Rat-races, Wilkinsons, Magic-T s Circulators, Diplexers and Filter

More information

DEPARTMENT OF COMPUTER ENGINEERING UNIVERSITY OF LAHORE

DEPARTMENT OF COMPUTER ENGINEERING UNIVERSITY OF LAHORE DEPARTMENT OF COMPUTER ENGINEERING UNIVERSITY OF LAHORE NAME. Section 1 2 3 UNIVERSITY OF LAHORE Department of Computer engineering Linear Circuit Analysis Laboratory Manual 2 Compiled by Engr. Ahmad Bilal

More information

Topic 5: Transmission Lines

Topic 5: Transmission Lines Topic 5: Transmission Lines Profs. Javier Ramos & Eduardo Morgado Academic year.13-.14 Concepts in this Chapter Mathematical Propagation Model for a guided transmission line Primary Parameters Secondary

More information

Notes on Mutual Inductance and Transformers J. McCalley

Notes on Mutual Inductance and Transformers J. McCalley ωωωωωωωω ωωωωωωωω otes on Mutual nductance and Transformers J. McCalley.0 Use of Transformers Transformers are one of the most common electrical devices. Perhaps the most familiar application today is

More information

ECE 308 SIGNALS AND SYSTEMS SPRING 2013 Examination #2 14 March 2013

ECE 308 SIGNALS AND SYSTEMS SPRING 2013 Examination #2 14 March 2013 ECE 308 SIGNALS AND SYSTEMS SPRING 2013 Examination #2 14 March 2013 Name: Instructions: The examination lasts for 75 minutes and is closed book, closed notes. No electronic devices are permitted, including

More information

Solution: K m = R 1 = 10. From the original circuit, Z L1 = jωl 1 = j10 Ω. For the scaled circuit, L 1 = jk m ωl 1 = j10 10 = j100 Ω, Z L

Solution: K m = R 1 = 10. From the original circuit, Z L1 = jωl 1 = j10 Ω. For the scaled circuit, L 1 = jk m ωl 1 = j10 10 = j100 Ω, Z L Problem 9.9 Circuit (b) in Fig. P9.9 is a scaled version of circuit (a). The scaling process may have involved magnitude or frequency scaling, or both simultaneously. If R = kω gets scaled to R = kω, supply

More information

6 Lectures 3 Main Sections ~2 lectures per subject

6 Lectures 3 Main Sections ~2 lectures per subject P5-Electromagnetic ields and Waves Prof. Andrea C. errari 1 1 6 ectures 3 Main Sections ~ lectures per subject Transmission ines. The wave equation.1 Telegrapher s Equations. Characteristic mpedance.3

More information

Kimmo Silvonen, Transmission lines, ver

Kimmo Silvonen, Transmission lines, ver Kimmo Silvonen, Transmission lines, ver. 13.10.2008 1 1 Basic Theory The increasing operating and clock frequencies require transmission line theory to be considered more and more often! 1.1 Some practical

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

Sinusoidal steady-state analysis

Sinusoidal 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 information

Proceedings of Meetings on Acoustics

Proceedings of Meetings on Acoustics Proceedings of Meetings on Acoustics Volume 9, 203 http://acousticalsociety.org/ ICA 203 Montreal Montreal, Canada 2-7 June 203 Engineering Acoustics Session peab: Transduction, Transducers, and Energy

More information

Lecture 4: R-L-C Circuits and Resonant Circuits

Lecture 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 information

Scattering Parameters

Scattering Parameters Berkeley Scattering Parameters Prof. Ali M. Niknejad U.C. Berkeley Copyright c 2016 by Ali M. Niknejad September 7, 2017 1 / 57 Scattering Parameters 2 / 57 Scattering Matrix Voltages and currents are

More information

EE105 Fall 2015 Microelectronic Devices and Circuits Frequency Response. Prof. Ming C. Wu 511 Sutardja Dai Hall (SDH)

EE105 Fall 2015 Microelectronic Devices and Circuits Frequency Response. Prof. Ming C. Wu 511 Sutardja Dai Hall (SDH) EE05 Fall 205 Microelectronic Devices and Circuits Frequency Response Prof. Ming C. Wu wu@eecs.berkeley.edu 5 Sutardja Dai Hall (SDH) Amplifier Frequency Response: Lower and Upper Cutoff Frequency Midband

More information

EE 205 Dr. A. Zidouri. Electric Circuits II. Frequency Selective Circuits (Filters) Low Pass Filter. Lecture #36

EE 205 Dr. A. Zidouri. Electric Circuits II. Frequency Selective Circuits (Filters) Low Pass Filter. Lecture #36 EE 05 Dr. A. Zidouri Electric ircuits II Frequency Selective ircuits (Filters) ow Pass Filter ecture #36 - - EE 05 Dr. A. Zidouri The material to be covered in this lecture is as follows: o Introduction

More information

CHAPTER 45 COMPLEX NUMBERS

CHAPTER 45 COMPLEX NUMBERS CHAPTER 45 COMPLEX NUMBERS EXERCISE 87 Page 50. Solve the quadratic equation: x + 5 0 Since x + 5 0 then x 5 x 5 ( )(5) 5 j 5 from which, x ± j5. Solve the quadratic equation: x x + 0 Since x x + 0 then

More information

Transmission Lines. Author: Michael Leddige

Transmission Lines. Author: Michael Leddige Transmission Lines Author: Michael Leddige 1 Contents PCB Transmission line structures Equivalent Circuits and Key Parameters Lossless Transmission Line Analysis Driving Reflections Systems Reactive Elements

More information

INTRODUCTION. Description

INTRODUCTION. Description INTRODUCTION "Acoustic metamaterial" is a label that encompasses for acoustic structures that exhibit acoustic properties not readily available in nature. These properties can be a negative mass density,

More information

A Dash of Maxwell s. In Part 4, we derived our third form of Maxwell s. A Maxwell s Equations Primer. Part 5 Radiation from a Small Wire Element

A Dash of Maxwell s. In Part 4, we derived our third form of Maxwell s. A Maxwell s Equations Primer. Part 5 Radiation from a Small Wire Element A Dash of Maxwell s by Glen Dash Ampyx LLC A Maxwell s Equations Primer Part 5 Radiation from a Small Wire Element In Part 4, we derived our third form of Maxwell s Equations, which we called the computational

More information

Network Synthesis. References :

Network Synthesis. References : References : Network ynthesis Gabor C. Temes & Jack W. Lapatra, Introduction to Circuit ynthesis and Design, McGraw-Hill Book Company. M.E. Van Valkenburg, Introduction to Modern Network ynthesis, John

More information

Homework 6 Solutions and Rubric

Homework 6 Solutions and Rubric Homework 6 Solutions and Rubric EE 140/40A 1. K-W Tube Amplifier b) Load Resistor e) Common-cathode a) Input Diff Pair f) Cathode-Follower h) Positive Feedback c) Tail Resistor g) Cc d) Av,cm = 1/ Figure

More information

Keysight Technologies Heidi Barnes

Keysight Technologies Heidi Barnes Keysight Technologies 2018.03.29 Heidi Barnes 1 S I G N A L I N T E G R I T Y A N D P O W E R I N T E G R I T Y Hewlett-Packard Agilent Technologies Keysight Technologies Bill and Dave s Company and the

More information

Numerical Solution of BLT Equation for Inhomogeneous Transmission Line Networks

Numerical Solution of BLT Equation for Inhomogeneous Transmission Line Networks 656 PIERS Proceedings, Kuala Lumpur, MALAYSIA, March 27 30, 2012 Numerical Solution of BLT Equation for Inhomogeneous Transmission Line Networks M. Oumri, Q. Zhang, and M. Sorine INRIA Paris-Rocquencourt,

More information

EIT Review. Electrical Circuits DC Circuits. Lecturer: Russ Tatro. Presented by Tau Beta Pi The Engineering Honor Society 10/3/2006 1

EIT Review. Electrical Circuits DC Circuits. Lecturer: Russ Tatro. Presented by Tau Beta Pi The Engineering Honor Society 10/3/2006 1 EIT Review Electrical Circuits DC Circuits Lecturer: Russ Tatro Presented by Tau Beta Pi The Engineering Honor Society 10/3/2006 1 Session Outline Basic Concepts Basic Laws Methods of Analysis Circuit

More information

Circuit Analysis Using Fourier and Laplace Transforms

Circuit Analysis Using Fourier and Laplace Transforms EE2015: Electrical Circuits and Networks Nagendra Krishnapura https://wwweeiitmacin/ nagendra/ Department of Electrical Engineering Indian Institute of Technology, Madras Chennai, 600036, India July-November

More information

Sinusoidal Steady-State Analysis

Sinusoidal 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 information

Transmission Lines, Stacked Insulated Washers Lines, Tesla Coils and the Telegrapher s Equation

Transmission Lines, Stacked Insulated Washers Lines, Tesla Coils and the Telegrapher s Equation Transmission Lines, Stacked Insulated Washers Lines, Tesla oils and the Telegrapher s Equation Kurt Nalty August, 2008 Abstract Tesla coils have more in common with transmission lines than transformers.

More information

Transmission lines. Shouri Chatterjee. October 22, 2014

Transmission lines. Shouri Chatterjee. October 22, 2014 Transmission lines Shouri Chatterjee October 22, 2014 The transmission line is a very commonly used distributed circuit: a pair of wires. Unfortunately, a pair of wires used to apply a time-varying voltage,

More information

CLOSED FORM SOLUTIONS FOR NONUNIFORM TRANSMISSION LINES

CLOSED FORM SOLUTIONS FOR NONUNIFORM TRANSMISSION LINES Progress In Electromagnetics Research B, Vol. 2, 243 258, 2008 CLOSED FORM SOLUTIONS FOR NONUNIFORM TRANSMISSION LINES M. Khalaj-Amirhosseini College of Electrical Engineering Iran University of Science

More information

Stepped-Impedance Low-Pass Filters

Stepped-Impedance Low-Pass Filters 4/23/27 Stepped Impedance Low Pass Filters 1/14 Stepped-Impedance Low-Pass Filters Say we know te impedance matrix of a symmetric two-port device: 11 21 = 21 11 Regardless of te construction of tis two

More information

EE 212 PASSIVE AC CIRCUITS

EE 212 PASSIVE AC CIRCUITS EE 212 PASSIVE AC CIRCUITS Condensed Text Prepared by: Rajesh Karki, Ph.D., P.Eng. Dept. of Electrical Engineering University of Saskatchewan About the EE 212 Condensed Text The major topics in the course

More information

Chapter 5 Impedance Matching and Tuning

Chapter 5 Impedance Matching and Tuning 3/25/29 section 5_1 Match with umped Elements 1/3 Chapter 5 Impedance Match and Tun One of the most important and fundamental two-port networks that microwave eneers des is a lossless match network (otherwise

More information