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.

http://www.tvhistory.tv/ 7 November 2018 Fields and Waves I

Ulaby 7 November 2018 Fields and Waves I

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 7 November 2018 Fields and Waves I

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. 7 November 2018 Fields and Waves I

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. 7 November 2018 Fields and Waves I

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 7 November 2018 Fields and Waves I

Workspace 7 November 2018 Fields and Waves I

Effects on Zc - Characteristic Impedance For Lossless system, Zo represents Replace jwl with r + jwl Replace jwc with g + jwc Characteristic Impedance 7 November 2018 Fields and Waves I

Review of Lossless Transmission Lines Parameters General Solution 7 November 2018 Fields and Waves I

Attenuation Factor For lossless systems: For lossy systems: The phasors have the factor: Attenuation/loss factor due to resistance 7 November 2018 Fields and Waves I

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 7 November 2018 Fields and Waves I

Lossy Transmission Lines 7 November 2018 Fields and Waves I

Finding the attenuation factor If the amplitude is 1 at z= and .9 at z=100, what is alpha? ans 0.001 Ulaby 7 November 2018 Fields and Waves I

Low Loss Lines Using the Binomial Theorem for x<<1. 7 November 2018 Fields and Waves I

The propagation and attenuation constants become Low Loss Lines The propagation and attenuation constants become Most practical lines are low loss 7 November 2018 Fields and Waves I

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 7 November 2018 Fields and Waves I

Low Loss Lines Exact and Approximate Expressions are Plotted 7 November 2018 Fields and Waves I

Low Loss Lines For the previous case Consider another case 7 November 2018 Fields and Waves I

Low Loss Lines Low Loss Approximation Wavelength is about right but the attenuation is too large Low Loss Approximation 7 November 2018 Fields and Waves I

Determining Loss Loss in the conductors 7 November 2018 Qualitatively introduce resistive loss in cylindrical conductors and maybe striplines. 7 November 2018 Fields and Waves I

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 7 November 2018 Fields and Waves I

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, 7 November 2018 Fields and Waves I

Transmission Line Parameters Types of transmission lines Ulaby 7 November 2018 Fields and Waves I

Transmission Line Parameters Resistance per unit length: r Ohms/m are for the conductors, not the insulators where 7 November 2018 Fields and Waves I

Transmission Line Parameters For high frequency, the area for resistance for a circular wire is 7 November 2018 Fields and Waves I Ulaby

Transmission Line Parameters Inductance per unit length: l H/m for d >> 2a are for the insulating material between the conductors 7 November 2018 Fields and Waves I

Transmission Line Parameters Capacitance per unit length: c F/m for d >> 2a Also, 7 November 2018 Fields and Waves I

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 = .005 beta = lossless value 7 November 2018 Fields and Waves I

Workspace 7 November 2018 Fields and Waves I

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 7 November 2018 Fields and Waves I

Distortionless Lines Example: 7 November 2018 Fields and Waves I

Distortionless Lines Square and Gaussian pulses are distorted 7 November 2018 Fields and Waves I

Distortionless Lines Distorted at the input and due to propagation 7 November 2018 Fields and Waves I

Distortionless Lines Add a capacitor to the input to partially compensate for the input distortion 7 November 2018 Fields and Waves I

Distortionless Lines There remains distortion due to propagation 7 November 2018 Fields and Waves I

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. 7 November 2018 Fields and Waves I

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 7 November 2018 Fields and Waves I

For this combination of parameters Distortionless Lines For this combination of parameters 7 November 2018 Fields and Waves I

The characteristic impedance also simplifies Distortionless Lines The characteristic impedance also simplifies 7 November 2018 Fields and Waves I

Distortionless Lines Result: no distortion but smaller pulses 7 November 2018 Fields and Waves I

Distortionless Lines Expanded view 7 November 2018 Fields and Waves I

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 7 November 2018 Fields and Waves I

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 http://www-gap.dcs.st-and.ac.uk/~history/BigPictures/ 7 November 2018 Fields and Waves I

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. 7 November 2018 Fields and Waves I

Distortionless Lines References http://www.hep.princeton.edu/~mcdonald/examples/distortionless.pdf http://www.du.edu/~jcalvert/tech/cable.htm 7 November 2018 Fields and Waves I

Project 1: RF Notch Filter AKA – Channel Blocker Basic Configuration 7 November 2018 Fields and Waves I

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. 7 November 2018 Fields and Waves I

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) 7 November 2018 Fields and Waves I

Project 1 For Matlab – see old project information and link to Design with Matlab http://hibp.ecse.rpi.edu/%7Econnor/education/Fields/matlab_analysis.pdf For PSpice – see link to Design with PSpice http://hibp.ecse.rpi.edu/%7Econnor/education/Fields/pspice_analysis.pdf 7 November 2018 Fields and Waves I

Project 1 Campus Cable (might be slightly out of date) Channels 2-6 http://hibp.ecse.rpi.edu/%7Econnor/education/Fields/cable-channels.xls Campus Cable (might be slightly out of date) 7 November 2018 Fields and Waves I

Project 1 Using Old Spectrum Analyzer Note channels are reasonably distinct Using Old Spectrum Analyzer 7 November 2018 Fields and Waves I

Project 1 Using Old Spectrum Analyzer More than one channel is affected Using Old Spectrum Analyzer 7 November 2018 Fields and Waves I

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 7 November 2018 Fields and Waves I

Alan Dumont RPI graduate First practical TV Wikipedia Info http://www.tvhistory.tv/ 7 November 2018 Fields and Waves I