1. Fields and Waves I Lecture 5 K. A. Connor Lossy Transmission Lines
Electrical, Computer, and Systems Engineering Department
Rensselaer Polytechnic Institute, Troy, NY
Welcome to Fields and Waves I
Before I start, can those of you with pagers and cell phones please turn them off? Thanks.

2.
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3. Ulaby
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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 Wire Taft Museum of Art
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5. Example – Ohms Law for Resistors
Why do we use phasors?
Example – Ohms Law for Resistors
Show that we can use formulas from resistive circuits to analyze reactive circuits.
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6. Why do we use phasors? 7 November 2018 Fields and Waves I
Show that we can use formulas from resistive circuits to analyze reactive circuits.
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7. Lossless/Lossy Models of TL
Lossless Model of TL has no R or G:
Lossy Model of TL:
Loss effects due to Resistances:
Show this as two elements, one in series and one in parallel
R - resistance of conductors
G - conductivity of insulators
- both are ideally small
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8. Workspace
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9. Effects on Zc - Characteristic Impedance
For Lossless system,
Zo represents
Replace jwl with r + jwl
Replace jwc with g + jwc
Characteristic Impedance
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10. Review of Lossless Transmission Lines
Parameters
General Solution
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11. Attenuation Factor For lossless systems: For lossy systems:
The phasors have the factor:
Attenuation/loss factor due to resistance
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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
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13. Lossy Transmission Lines
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14. Finding the attenuation factor
If the amplitude is 1 at z= and .9 at z=100, what is alpha? ans 0.001
Ulaby
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15. Low Loss Lines Using the Binomial Theorem for x<<1.
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16. The propagation and attenuation constants become
Low Loss Lines
The propagation and attenuation constants become
Most practical lines are low loss
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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
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18. Low Loss Lines Exact and Approximate Expressions are Plotted
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19. Low Loss Lines For the previous case Consider another case
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20. Low Loss Lines Low Loss Approximation
Wavelength is about right but the attenuation is too large
Low Loss Approximation
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21. Determining Loss Loss in the conductors 7 November 2018
Qualitatively introduce resistive loss in cylindrical conductors and maybe striplines.
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22. Estimation of R On a per meter basis,
, if constant cross-section
On a per meter basis,
Inner
Outer
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
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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 I
Estimation of G:
From Electrostatics,
Also,
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24. Transmission Line Parameters
Types of transmission lines
Ulaby
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25. Transmission Line Parameters
Resistance per unit length: r Ohms/m
are for the conductors,
not the insulators
where
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26. Transmission Line Parameters
For high frequency, the area for resistance for a circular wire is
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Ulaby

27. Transmission Line Parameters
Inductance per unit length: l H/m
for d >> 2a
are for the insulating material between the conductors
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28. Transmission Line Parameters
Capacitance per unit length: c F/m
for d >> 2a
Also,
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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.
Skin depth 2.1mm and 17 micrometers
At 1kHz r=0.05 ohms per meter, g = 5e-13 S/m
At 15MHz r=0.5 ohms/m g=the same
Zo=50-j.55
alpha = beta = lossless value
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30. Workspace
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31. Note that the propagation constant varies with frequency
Distortionless Lines
Note that the propagation constant varies with frequency
Zo is also frequency dependent and not purely resistive
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32. Distortionless Lines
Example:
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33. Distortionless Lines Square and Gaussian pulses are distorted
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34. Distortionless Lines Distorted at the input and due to propagation
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35. Distortionless Lines
Add a capacitor to the input to partially compensate for the input distortion
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36. Distortionless Lines There remains distortion due to propagation
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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.
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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
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39. For this combination of parameters
Distortionless Lines
For this combination of parameters
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40. The characteristic impedance also simplifies
Distortionless Lines
The characteristic impedance also simplifies
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41. Distortionless Lines Result: no distortion but smaller pulses
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42. Distortionless Lines
Expanded view
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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.
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44. Oliver Heaviside
He reduced Maxwell’s 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
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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.
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46. Distortionless Lines References
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47. Project 1: RF Notch Filter
AKA – Channel Blocker
Basic Configuration
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48. If the extra cable had a short circuit load
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.
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49. You can choose from 3 types of analysis
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)
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50. Project 1
For Matlab – see old project information and link to Design with Matlab
For PSpice – see link to Design with PSpice
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51. Project 1 Campus Cable (might be slightly out of date) Channels 2-6
Campus Cable (might be slightly out of date)
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52. Project 1 Using Old Spectrum Analyzer
Note channels are reasonably distinct
Using Old Spectrum Analyzer
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53. Project 1 Using Old Spectrum Analyzer
More than one channel is affected
Using Old Spectrum Analyzer
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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
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55. Alan Dumont RPI graduate First practical TV Wikipedia Info
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