Fields and Waves I

1
Fields and Waves I

• Lecture 5
• Lossy Transmission Lines
• K. A. Connor
• Electrical, Computer, and Systems Engineering
  Department
• Rensselaer Polytechnic Institute, Troy, NY

2
http//www.tvhistory.tv/
3
Ulaby
4
Overview

• Incorporating lossy circuit elements in the line
  model
• Estimating resistance and conductance per unit
  length
• Per unit length parameters for transmission lines
• Distortionless lines
• Project 1

Henry Farny Song of the Talking WireTaft Museum
of Art
5
Why do we use phasors?

• Example Ohms Law for Resistors

6
Why do we use phasors?
7
Lossless/Lossy Models of TL
Lossless Model of TL has no R or G
Lossy Model of TL
Loss effects due to Resistances
R - resistance of conductors
G - conductivity of insulators
- both are ideally small
8
Workspace
9
Effects on Zc - Characteristic Impedance
Replace jwl with r jwl
Replace jwc with g jwc
Characteristic Impedance
10
Review of Lossless Transmission Lines

• Parameters
• General Solution

11
Attenuation Factor
For lossless systems
For lossy systems
The phasors have the factor
Attenuation/loss factor due to resistance
12
Lossless vs. Lossy Lines

• For a lossy line, the series impedance is
  while for a lossless
  line it is
• For a lossy line, the parallel admittance is
• while for a lossless line it is
• The input impedance becomes

13
Lossy Transmission Lines
14
Attenuation Factor

• Finding the attenuation factor

Ulaby
15
Low Loss Lines

• Using the Binomial Theorem
  for xltlt1.

16
Low Loss Lines

• The propagation and attenuation constants become
• Most practical lines are low loss

17
Low Loss Lines

• Example -- Assume the following f 1MHz
  standard RG58 cable parameters r per unit
  length of 0.1 Ohm per meter, the wave is seen to
  attenuate markedly in 2000 meters.
• Plot the voltage wave both exactly and using the
  low loss approximation

18
Low Loss Lines
Exact and Approximate Expressions are Plotted
19
Low Loss Lines

• For the previous case
• Consider another case

20
Low Loss Lines
Wavelength is about right but the attenuation is
too large
Low Loss Approximation
21
Determining Loss
Loss in the conductors
22
Estimation of R
On a per meter basis,
because inner and outer conductors are in series

• At high frequencies, not all the copper is used
  for conducting
• Current only flows in outer portion due to skin
  depth effects

23
Estimation of G (we will do this after
electrostatics)
The 1/G component represents radial current flow,
due to small s of insulator

• the cross-sectional area is not constant

Estimation of G
24
Transmission Line Parameters

• Types of transmission lines

Ulaby
25
Transmission Line Parameters

• Resistance per unit length r Ohms/m

are for the conductors, not the insulators
where
26
Transmission Line Parameters

• For high frequency, the area for resistance for a
  circular wire is

Ulaby
27
Transmission Line Parameters

• Inductance per unit length l H/m

for d gtgt 2a
are for the insulating material between the
conductors
28
Transmission Line Parameters

• Capacitance per unit length c F/m

for d gtgt 2a
29
Paper and Pencil Analysis

• Calculate the skin depth of copper at 1kHz and
  15MHz
• For an RG58 cable with polyethylene dielectric,
  find r and g.

30
Workspace
31
Distortionless Lines

• Note that the propagation constant varies with
  frequency
• Zo is also frequency dependent and not purely
  resistive

32
Distortionless Lines

• Example

33
Distortionless Lines

• Square and Gaussian pulses are distorted

34
Distortionless Lines

• Distorted at the input and due to propagation

35
Distortionless Lines

• Add a capacitor to the input to partially
  compensate for the input distortion

36
Distortionless Lines

• There remains distortion due to propagation

37
Distortionless Lines

• Distortion in a transmission line limits its
  useful length. Attenuation can be addressed by
  adding amplification. However, distorted signals
  cannot generally be undistorted, so a method
  needed to be found to eliminate it.
• Remarkably, lines can be made distortionless by
  adding loss. That is, we can trade additional
  attenuation for clarity of signal.

38
Distortionless Lines

• Recall that, for practical lines, the conductance
  per unit length g is negligible. Thus, we will
  add loss between the conductors so that
• For 2-wire lines, this can be done by adding
  lumped resistors periodically

39
Distortionless Lines

• For this combination of parameters

40
Distortionless Lines

• The characteristic impedance also simplifies

41
Distortionless Lines

• Result no distortion but smaller pulses

42
Distortionless Lines

• Expanded view

43
Distortionless Lines

• In the early days of telephony, Heaviside
  proposed making lines distortionless. This was
  done by adding inductance rather than conductance
  since the losses were not increased
  significantly.

http//www.du.edu/jcalvert/tech/cable.htm
44
Oliver Heaviside

• He reduced Maxwells equations from 20 with 20
  unknowns to 2 with 2 unknowns.
• From Cats -- Journey to the Heaviside Layer Up
  up up past the Russell hotel,Up up up to the
  Heaviside layer

http//www-gap.dcs.st-and.ac.uk/history/BigPictur
es/
45
Distortionless Lines

• Adding these components made it possible for
  phone calls to go from NY to Chicago.
• This is maybe the very best example of why a
  solid, math-based education can produce some
  non-intuitive results in engineering. To add
  resistance and make the signal better is hard to
  accept without some serious theoretical basis.

46
Distortionless Lines

• References
• http//www.hep.princeton.edu/mcdonald/examples/di
  stortionless.pdf
• http//www.du.edu/jcalvert/tech/cable.htm

47
Project 1 RF Notch Filter
AKA Channel Blocker

• Basic Configuration

48
Project 1

• If the extra cable had a short circuit load
• At particular frequencies, the input impedance
  would be very small and short out the signal. At
  other frequencies, the input impedance would be
  very large and have no effect.

49
Project 1

• For the analysis, you need to find the parameters
  of standard 75 Ohm CATV cables (RG59 or RG6 are
  used) Tessco has good information
• You can choose from 3 types of analysis
• Matlab
• PSpice
• Smith Charts (next lecture)

50
Project 1

• For Matlab see old project information and link
  to Design with Matlab http//hibp.ecse.rpi.edu/7
  Econnor/education/Fields/matlab_analysis.pdf
• For PSpice see link to Design with PSpice
  http//hibp.ecse.rpi.edu/7Econnor/education/Field
  s/pspice_analysis.pdf

51
Project 1
Channels 2-6
http//hibp.ecse.rpi.edu/7Econnor/education/Field
s/cable-channels.xls

• Campus Cable (might be slightly out of date)

52
Project 1
Note channels are reasonably distinct

• Using Old Spectrum Analyzer

53
Project 1
More than one channel is affected

• Using Old Spectrum Analyzer

54
Project 1

• Using your choice for analysis, select two
  blocker designs and analyze them
• Analyze CATV channel rejection
• Analyze 0-15MHz noise rejection
• Lossless analysis is due on 7 February
• Lossy analysis and physical testing due on 14
  February.
• There are two choices for testing
• Test CATV channel blocker
• Test 0-15MHz noise rejection using studio
  equipment

55
Alan Dumont

• RPI graduate
• First practical TV
• Wikipedia Info

http//www.tvhistory.tv/