Fields and Waves I

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Fields and Waves I

• Lecture 4
• Pulses on Transmission Lines
• K. A. Connor
• Electrical, Computer, and Systems Engineering
  Department
• Rensselaer Polytechnic Institute, Troy, NY

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These Slides Were Prepared by Prof. Kenneth A.
Connor Using Original Materials Written Mostly by
the Following

• Kenneth A. Connor ECSE Department, Rensselaer
  Polytechnic Institute, Troy, NY
• J. Darryl Michael GE Global Research Center,
  Niskayuna, NY
• Thomas P. Crowley National Institute of
  Standards and Technology, Boulder, CO
• Sheppard J. Salon ECSE Department, Rensselaer
  Polytechnic Institute, Troy, NY
• Lale Ergene ITU Informatics Institute,
  Istanbul, Turkey
• Jeffrey Braunstein Chung-Ang University, Seoul,
  Korea

Materials from other sources are referenced where
they are used. Those listed as Ulaby are figures
from Ulabys textbook.
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http//www.ohiodiary1872.com/maps.htm
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Review So Far Example

• Handout Page 1
• Fill out all information
• Method of Solution
• Example
• Find Parameters
• Follow Method of Solution

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http//www.telegraph-history.org/transcontinental-
telegraph/index.html
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Example Telegraph Line

• Parameters (Some are realistic and some are not)
• Inductance per meter 1.6x10-6
• Capacitance per meter 6.8x10-12
• Characteristic Impedance about 500 Ohms
• Velocity ?
• Assume lossless (bad assumption)
• Source 60V (small internal impedance)
• Frequency 1000Hz
• Large load impedance

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Example
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Pulses on Transmission Lines
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Experiment from the first class
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Pulses Measured with the Reels of RG58/U Cable
50 Ohm source 50 Ohm line long reel of cable
terminated in 50 Ohms
Look at details of individual pulses
Improperly terminated cable connecting input to
scope
Properly terminated cable connecting input to
scope
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Overview

• Review the derivation of the wave equation
• PSpice simulation
• General form of voltages and currents
• Initial conditions
• Reflection at the load and the source
• Bounce diagrams

Henry Farny Song of the Talking WireTaft Museum
of Art
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Transmission Line Representation
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Transmission Line Representation
Similarly,
Obtain the following PDE
These are functions that move with velocity u
Solutions are
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Wave Equation Solutions Can Have Any Shape

• Pulses will look the same at the input and output
  except for a delay

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Pulse Input and Output Voltages
Source
Load
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General Form of Voltage and Current on the Line

• The representation is very general

or

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The Shape of the Pulse Does Not Matter
For Lossless Lines
Output
Input
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General Form of Solution

• In general, both positive and negative traveling
  pulses will exist on a line.

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Workspace
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Simplifying the Solution

• As with the time harmonic case, we can use the
  voltage solution to obtain the current solution.
  Applying
• We have

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General Solution Again

• Using the voltage information

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Compare with Steady State

• So far, the solution looks like the solution for
  steady state.
• Pos traveling current looks like pos traveling
  voltage divided by the characteristic impedance
• Neg traveling current looks like neg traveling
  voltage divided by minus the characteristic
  impedance
• Why should this be the case?

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Steady State Transients

• Time varying signals can be broken down into
  individual frequencies (principle behind Fourier
  and Laplace analysis).
• We can analyze a pulse by first finding these
  frequencies, analyzing what happens to each one
  and then combining the results.
• A simple example would be a signal with two
  frequencies, one of which is filtered out by some
  circuit, with the result that only one frequency
  will remain.

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Spectrum Examples from EE 352 Univ of
Saskatchewan
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Spectrum Examples
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Steady State Transients

• We can just use what we learned for steady state
  or develop transient analysis independently.
• It is best to do this independently so that you
  can be further convinced that our transmission
  line solutions make sense.

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Reflection Coefficient at the Load
When the incident pulse reaches the load, it will
reflect if the load is not matched to the line
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Workspace
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Reflection Coefficient at the Load

• The impedance at the load should equal the ratio
  of the voltage to the current at the end of the
  line

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Launching the Pulse

• At the source end the line is driven by something
  like a function generator

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Launching the Pulse

• One very large difference between the transient
  case and steady state is that, when the pulse is
  first launched on the line, there can be no
  negative traveling pulse since the line is
  assumed to have no voltages before the first
  pulse is launched.
• In steady state, positive and negative waves
  always exist simulataneously

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Launching the Pulse

• Like steady state, it is necessary to determine
  the input impedance of the line to see how a
  source interacts with it.

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Finding Zin

• The input impedance will be the ratio of the
  voltage to the current at the input end for only
  the positive traveling signal

Why?
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Input Voltage to the Line

• The input voltage to the line is, thus,
  determined from the voltage divider relation

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Pulse Analysis

• Use the voltage divider relationship to find the
  initial voltage on the line. The pulse is
  launched and propagates to the load.

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Pulse Analysis

• The pulse then is either totally absorbed by the
  load or is partially reflected. If the latter, it
  then propagates back toward the source.

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Pulse Analysis

• The pulse then is either totally absorbed by the
  source impedance or partially reflected and
  propagates back to the load.

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Reflection Coefficient at the Source

• The pulse sees the source impedance as the same
  as the load impedance. Thus,

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Bounce Diagram

• There is a systematic method for applying this
  information using what is called a bounce diagram
  or lattice diagram
• Each step of the process is included
• Space and time information are included

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Bounce Diagram
Td/u is the time to transit from one end of the
line to the other
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Bounce Diagram
See Unit XII for examples
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Bounce Diagram what happens when everything is
matched?
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Several Kinds of Transients

• Short Pulses (like in HW2)
• Switching on DC sources (example to follow)
• Long Pulses whose duration exceeds the line
  transit time (basically a combination of the
  other two)
• Combinations of all types of pulses

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Example 1

• Switching on a 10 Volt DC source
• Question What voltage will eventually appear
  across the output? That is, what will the voltage
  become for large time? Hint How would you have
  answered this question before taking Fields
  Waves I?

Extra switches to maintain ground for PSpice
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Example 1

• Voltage divider gives 8.333 V launched on line
• Reflection coefficient at the load is 1
• Reflection coefficient at the source is -2/3
• Transit time is 800ns (arbitrary for this
  example, but for completeness, assume a velocity
  of 2x108 m/s)

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Example 1
-5.555V
5.56V
-5.555V
11.11V
8.333V
16.67V
8.333V
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Example 1
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Example 1
Longer time scale (up to 0.1 sec)
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Example 2

• If the pulse width is much less than the transit
  time T, then only a single incident and reflected
  pulse will occur at the load or source end while
  reflection occurs.
• This is much simpler to consider and is the case
  for HW 2.

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Example 2

• Let us first begin with the PSpice analysis
• Use precisely the same line, load, etc. except
  for the source, which is now a 50ns pulse. A time
  delay has been added to match the turn-on time of
  the switch.

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Example 2
Comparison of two cases
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Example 2

• Expanding the pulse width to 100ns show how
  reflection occurs. A small Matlab program was
  written to show the pulses.

Incident
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Example 2
Both Incident and Reflected Pulses Must Exist
Simultaneously
Reflected
Incident
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Example 2
Reflected
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Example 2
Both Incident and Reflected Pulses Must Exist
Simultaneously
Reflected
Incident
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Example 2
Incident
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Example 2
Again, expanding the pulse for clarity, we see
that incident and reflected pulses exist
simultaneously
Leading Edge
Trailing Edge
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Example 3

• Non-Resistive Load
• If the load includes either a capacitor or
  inductor of significant size, one observes the
  charging and discharging time of these elements.
• This analysis can be done analytically, but we
  will only use PSpice

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Example 3

• Source and Line Matched (RsZo), Capacitive
  Load (C0.1 microfarad). Length 2564meters and
  velocity 2.564x108 meters per second

Source
Load
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Java Applet

• From Georgia Tech http//users.ece.gatech.edu/7Ew
  rscott/applet_bounce/Reflect1.html
• Waves and Bounce Diagram