Fundamentals of Electromagnetics for Teaching and Learning: A Two-Week Intensive Course for Faculty in Electrical-, Electronics-, Communication-, and Computer- Related Engineering Departments in Engineering Colleges in India by Nannapaneni Narayana Rao Edward C. Jordan Professor Emeritus of Electrical and Computer Engineering University of Illinois at Urbana-Champaign, USA Distinguished Amrita Professor of Engineering Amrita Vishwa Vidyapeetham, India

7-1 Program for Hyderabad Area and Andhra Pradesh Faculty Sponsored by IEEE Hyderabad Section, IETE Hyderabad Center, and Vasavi College of Engineering IETE Conference Hall, Osmania University Campus Hyderabad, Andhra Pradesh June 3 – June 11, 2009 Workshop for Master Trainer Faculty Sponsored by IUCEE (Indo-US Coalition for Engineering Education) Infosys Campus, Mysore, Karnataka June 22 – July 3, 2009

7-2 Module 7 Transmission Line Analysis in Time Domain 7.1 Line terminated by a resistive load 7.2 Transmission-line discontinuity 7.3 Lines with reactive terminations and discontinuities 7.4 Lines with initial conditions 7.5 Lines with nonlinear elements

7-3 Instructional Objectives 49. Find the voltage and current variations at a location on a lossless transmission line a functions of time and at an instant of time as functions of distance, and the steady state values of the line voltage and current, for a line terminated by a resistive load and excited by turning on a constant voltage source, by using the bounce-diagram technique 50. Design a lossless transmission line system by determining its parameters from information specified concerning the voltage and/or current variations on the line 51. Design a system of three lines in cascade for achieving a specified unit impulse response

7-4 Instructional Objectives (Continued) 52. Compute the reflected power for a wave incident on a junction of multiple lossless transmission lines from one of the lines and the values of power transmitted into each of the other lines, where the junction may consist of lines connected in series, parallel, or series-parallel, and include resistive elements 53. Find the solutions for voltage and current along a transmission-line system excited by a constant voltage source and having reactive elements as terminations/discontinuities 54. Find the voltage and current variations at a location on a lossless transmission-line system as functions of time and at an instant of time as functions of distance, for specified nonzero initial voltage and/or current distributions along the system

7-5 Instructional Objectives (Continued) 55. Analyze a transmission line terminated by a nonlinear element by using the load line technique 56. Understand the effect of time delay in interconnections between logic gates

Line Terminated by Resistive Load (EEE, Sec. 6.2; FEME, Sec. 7.4)

7-7 Notation

7-8

7-9 Assuming Z 0 = 50 Ω,

7-10 Assuming Z 0 = 50 Ω,

7-11 Excitation by Constant Voltage Source Semi-infinite Line, No Source Resistance

7-12

7-13

7-14 E7.1 t,  s

7-15 t = 1  s

7-16 Effect of Source Resistance B.C. (+) Wave

7-17

7-18 Line Terminated by Resistance

7-19 B.C:

7-20 Define Voltage Reflection Coefficient, Then, Current Reflection Coefficient

7-21

7-22

7-23

7-24

7-25

7-26 For constant voltage source, Actual Situation in the Steady State One (+) Wave and One (–) Wave

7-27 Four equations for the four unknowns

7-28 E7.2

7-29 Solving, we obtain

7-30 Bounce Diagram Technique: Constant Voltage Source E7.3

7-31 Voltage

7-32 Current

7-33 VoltageCurrent

7-34 t,  s

7-35 t,  s

7-36 t,  s

7-37

7-38 Rectangular Pulse Source Use superposition. E7.4 t,  s

7-39 t,  s

7-40 Review Questions 7.1. Discuss the general solutions for the line voltage and line current and the notation associated with their interpretation in concise form What is the fundamental distinction between the occurrence of the response in one branch of a lumped circuit to the application of an excitation in a different branch of the circuit and the occurrence of the response at one location on a transmission line to the application of an excitation at a different location on the line? 7.3. Describe the phenomenon of the bouncing back and forth of transient waves on a transmission line excited by a constant voltage source in series with internal resistance and terminated by a resistance.

7-41 Review Questions (Continued) 7.4. What is the nature of the formula for the voltage reflection coefficient? Discuss its values for some special cases What is the steady state equivalent of a line excited by a constant voltage source? What is the actual situation in the steady state? 7.6. Discuss the bounce diagram technique of keeping track of the bouncing back and forth of transient waves on a transmission line for a constant voltage source Discuss the bounce diagram technique of keeping track of the bouncing back and forth of transient waves on a transmission line for a pulse voltage source.

7-42 Problem S7.1. Plotting line voltage and line current on a transmission-line system involving two lines

7-43 Problem S7.2. Finding several quantities associated with a transmission-line system from given observations

7-44 Problem S7.3. Time-domain analysis of a transmission- line system using the bounce diagram technique

7-45 Problem S7.4. Time-domain analysis of a transmission- line system for a sinusoidal excitation

Transmission-Line Discontinuity (EEE, Sec. 6.3)

7-47 Transmission-Line Discontinuity

7-48

7-49

7-50 Current Transmission Coefficient, Define Voltage Transmission Coefficient,

7-51 Note that

7-52 Three Lines in Cascade E7.5 (t)(t)

7-53 t,  s

7-54 (t)(t)

7-55

7-56

7-57

7-58 

7-59 Junction of Three Lines E7.6

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7-61

7-62

7-63 Review Questions 7.8. Discuss the phenomenon of reflection and transmission for a wave incident on a junction between two transmission lines How are the voltage and current transmission coefficients at a junction between two lines related to voltage reflection coefficient? Explain how it is possible for the transmitted voltage or current at a junction between two transmission lines to exceed the incident voltage or current, respectively Discuss the determination of the unit impulse response of a system of three lines in cascade Outline the procedure for the determination of the frequency response of a system of three lines in cascade from its unit impulse response.

7-64 Review Questions (Continued) Discuss the determination of the reflection and transmission coefficients at a junction connecting a transmission line to two lines in parallel How would you determine the reflection and transmission coefficients at a junction connecting a transmission line to two lines in series?

7-65 Problem S7.5. Finding three parameters for a system of three media from unit impulse response

7-66 Problem S7.5. Finding three parameters for a system of three media from unit impulse response (Continued)

7-67 Problem S7.6. Analysis of a system of three transmission lines excited by a pulse voltage source

7-68 Problem S7.6. Analysis of a system of three transmission lines excited by a pulse voltage source (Continued)

7-69 Problem S7.7. A system of three transmission lines with a resistive network at the junction

Lines with Reactive Terminations and Discontinuities (EEE, Sec. 6.4)

7-71 Line Terminated by an Inductor E7.7 t = T+

7-72 Using I.C.,

7-73

7-74 Voltage  = 0

7-75 –  = 0

7-76

7-77

7-78 Line Terminated by a Capacitor E7.8 t = T+

7-79

7-80 Using I.C., t > T

7-81 Review Questions Discuss the transient analysis of a line driven by a constant voltage source in series with a resistance equal to Z 0 of the line and terminated by an inductor Why is the concept of reflection coefficient not useful for studying the transient behavior of lines with reactive terminations and discontinuities? Outline the transient analysis of a line driven by a constant voltage source in series with a resistance equal to Z 0 of the line and terminated by a capacitor.

7-82 Problem S7.8. Transient analysis of a transmission line terminated by a capacitive network

7-83 Problem S7.9. Finding the nature of a discontinuity in a transmission line system

Lines with Initial Conditions (EEE, Sec. 6.5; FEME, Sec. 7.5)

7-85 Line with Initial Conditions

7-86

7-87 E7.9

7-88

7-89

7-90

7-91

7-92

7-93 Uniform Distribution E7.10

7-94

7-95

7-96

7-97 Bounce Diagram Technique for Uniform Distribution

7-98

7-99

7-100 Energy Storage in Transmission Lines w e, Electric stored energy density = W e, Electric stored energy =

7-101 w m, Magnetic stored energy density = W m, Magnetic stored energy =

7-102 Check of Energy Balance Initial stored energy

7-103 Energy dissipated in R L

7-104 E7.11 System in steady state at t = 0–.

7-105 t = 0–: steady state

7-106 t = 0+:

7-107 Solving, we obtain

7-108 Voltage  = 0  V = 1

7-109 Current  = 0  = 0,  C = 1  C eff = 0.5

7-110 New steady state

7-111

7-112 Review Questions Discuss the determination of the voltage and current distributions on an initially charged line for any given time from the knowledge of the initial voltage and current distributions Discuss with the aid of an example the discharging of an initially charged line into a resistor Discuss the bounce-diagram technique of transient analysis of a line with uniform initial voltage and current distributions How do you check the energy balance for the case of a line with initial voltage and/or current distribution(s) and discharged into a resistor?

7-113 Problem S7.10. For the analysis of an initially-charged transmission line discharging into a resistor

7-114 Problem S7.10. For the analysis of an initially-charged transmission line discharging into a resistor (Continued)

7-115 Problem S7.11. Analysis of a system of an initially-charged line connected to another initially-charged line

7-116 Problem S7.12. For the analysis of an initially charged transmission line connected to a capacitor

Lines with Nonlinear Elements (EEE, Sec. 6.6; FEME, Sec. 7.6)

7-118 Nonlinear Termination: Load line Technique E7.12

7-119

7-120

7-121

7-122

7-123

7-124 Interconnection Between Logic Gates

7-125

7-126

7-127

7-128

7-129

7-130 Review Questions Discuss the load-line technique of obtaining the time variations of the voltages and currents at the source and load ends of a line from the knowledge of the terminal V-I characteristics Discuss the analysis of interconnection between logic gates, using transmission line.

7-131 Problem S7.13. Application of load-line technique for an initially charged line discharging into a nonlinear resistor

The End