# Fields and Waves I Lecture - Rensselaer Polytechnic Institute

Fields and Waves I Lecture 5 Lossy Transmission Lines K. A. Connor Electrical, Computer, and Systems Engineering Department Rensselaer Polytechnic Institute, Troy, NY http://www.tvhistory.tv/ February 12, 2020 Fields and Waves I 2 Ulaby February 12, 2020 Fields and Waves I 3 Overview Henry Farny Song of the Talking Wire Taft Museum of Art

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 February 12, 2020 Fields and Waves I 4 Why do we use phasors? j t Example Ohms Law for Resistors February 12, 2020 Fields and Waves I

5 Why do we use phasors? February 12, 2020 Fields and Waves I j t 6 Lossless/Lossy Models of TL Lossless Model of TL has no R or G: R L G Lossy Model of TL: R C Loss effects due to Resistances:

L G C R - resistance of conductors G - conductivity of insulators - both are ideally small February 12, 2020 Fields and Waves I 7 Workspace February 12, 2020 Fields and Waves I 8 Effects on Zc - Characteristic Impedance l Zo c For Lossless system,

Zo represents V I R L G Replace jl with r + jl Characteristic Impedance February 12, 2020 C Replace jc with g + jc r j l Z0 g jc Fields and Waves I 9 Review of Lossless Transmission Lines Parameters Z o

u L General Solution February 12, 2020 Fields and Waves I 10 Attenuation Factor For lossless systems: lc For lossy systems: j (r jl )( g jc) The phasors have the factor: e

z e z e j z Attenuation/loss factor due to resistance February 12, 2020 Fields and Waves I 11 Lossless vs. Lossy Lines r isj l For a lossy line, the series impedance l a lossless whilej for line it is g For j c ja clossy line, the parallel admittance is

while for a lossless line it is Z L becomes Z o t a n h d The input impedance Z Z in February 12, 2020 o Z o Z Fields and Waves I L t a n h d 12 Lossy Transmission Lines v(z) V

e i(z) I e z z V e I e v(z) V V i(z) Z February 12, 2020 v(z) V e V i(z) Z

z o e e z z z z Le Le Fields and Waves I e o

z z z V e V Z z e z o 13 Attenuation Factor

Finding the attenuation factor Ulaby February 12, 2020 Fields and Waves I 14 Low Loss Lines Using the Binomial Theorem1 x 1 x<<1. Z o r j l g j c r j l

j c ( r j l)( g j c ) r 1 j l ( r j l )( j c ) j j February 12, 2020 j l j c x 2 for l r 1 j c 2l ( j l )( j c ) 1

r lc 1 j 2l Fields and Waves I 15 r j l Low Loss Lines The propagation and attenuation constants become j j lc r lc 2l

r 2 r 2Z l c o Most practical lines are low loss February 12, 2020 Fields and Waves I 16 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

February 12, 2020 Fields and Waves I 17 Low Loss Lines February 12, 2020 Exact and Approximate Expressions are Plotted Fields and Waves I 18 Low Loss Lines For the previous case r 0 .1 j l j 2 x 1 0 6 0 . 2 5 x 1 0 6

j 2 Consider another case r .5 j l j 2 0 . 2 x 1 0 February 12, 2020 6 Fields and Waves I . 2 5 x 1 0 6 j 10 19 Low Loss Lines Wavelength is about right but the attenuation is too large Low Loss Approximation

February 12, 2020 Fields and Waves I 20 Determining Loss Loss in the conductors February 12, 2020 Fields and Waves I R ? 21 Estimation of R l R A , if constant cross-section On a per meter basis, Outer 1 1

r A inner A outer because inner and outer conductors are in series Inner February 12, 2020 At high frequencies, not all the copper is used for conducting Current only flows in outer portion due to skin depth effects Fields and Waves I 22 Estimation of G (we will do this after electrostatics) The 1/G component represents radial current flow, due to small of insulator I 1 the cross-sectional area is not constant G G

1 A R l j ds 1 I Estimation of G: G R V Vab From Electrostatics, D ds Q C Vab Vab j E D February 12, 2020 Also, g

G c l Fields and Waves I 23 Transmission Line Parameters Types of transmission lines Ulaby February 12, 2020 Fields and Waves I 24 Transmission Line Parameters Resistance per unit length: r Ohms/m Rs 1 1 2 a b

Rs a where f c c February 12, 2020 R s 2R w c , Fields and Waves I c s are for the conductors, not the insulators

25 Transmission Line Parameters For high frequency, the area for resistance for a circular wire is A 2 a February 12, 2020 1 f c A 2 b c Fields and Waves I Ulaby 26 Transmission Line Parameters Inductance per unit length: l H/m

b ln 2 a , , February 12, 2020 d ln 2a 2 d 1 2a d ln a d w for d >> 2a

are for the insulating material between the conductors Fields and Waves I 27 Transmission Line Parameters Capacitance per unit length: c F/m 2 b ln a Also, February 12, 2020 w d 2 d d ln

1 2a 2 a d for d >> 2a ln a Fields and Waves I g G c l 28 Paper and Pencil Analysis Calculate the skin depth of copper at 1kHz and 15MHz 1 f c c

0 .0 6 6 f 5 .8 x 1 0 S m 7 For an RG58 cable with polyethylene dielectric, a 0 .4 m m find r and g. r 2 .3 February 12, 2020 b 1 . 4 m m 1 .5 3 m m 10 13 Fields and Waves I S m 29

Workspace February 12, 2020 Fields and Waves I 30 Distortionless Lines Note that the propagation constant varies with frequency j ( r j l)( g j c ) Zo is also frequency dependent and not purely resistive Z February 12, 2020 o r j l g j c Fields and Waves I 31

Distortionless Lines H Example: r 0 . 0 2 5 l 0 . 1 9 5 m m d 1 0 ,0 0 0 m g 0 S m c 78 pF m R 1 T1 LO S S Y 50 V1 = 0 T D = 3 .5 u s V2 = 1 T R = .5 u s

PW = 2us T F = .5 u s PER = 500us R 2 V1 50 R 3 C 1 1M eg 13pF 0 February 12, 2020 Fields and Waves I 32 Distortionless Lines

Square and Gaussian pulses are distorted 600mV 500mV 400mV 300mV 200mV 100mV 0V 0s V(R22:1) 5us V(T1:B+) February 12, 2020 10us V(R1:2) 15us V(R21:2)

20us 25us 30us 35us 40us 45us 50us Time Fields and Waves I 33 Distortionless Lines Distorted at the input and due to propagation 600mV Output Pulse 400mV 200mV

SEL>> 0V 600mV V(R2:1)*11 Input Pulse 400mV 200mV 0V 0s V(R1:2) 10us February 12, 2020 20us 30us 40us 50us 60us

70us 80us 90us 100us Time Fields and Waves I 34 Distortionless Lines Add a capacitor to the input to partially compensate for the input distortion R 41 C 42 50 350n T4 LO S S Y

V1 = 0 T D = 3 .5 u s V2 = 1 T R = .5 u s PW = 2us T F = .5 u s PER = 500us R 42 V4 50 R 43 C 41 1M eg 13pF 0 February 12, 2020

Fields and Waves I 35 Distortionless Lines There remains distortion due to propagation 550mV Output Pulse 400mV 200mV 0V 550mV V(T4:B+)*11 Input Pulse 400mV 200mV SEL>> -50mV 0s

V(C42:2) 10us February 12, 2020 20us 30us 40us 50us 60us 70us 80us 90us 100us Time Fields and Waves I 36

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. February 12, 2020 Fields and Waves I 37 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 r g l

c For 2-wire lines, this can be done by adding lumped resistors periodically February 12, 2020 Fields and Waves I 38 Distortionless Lines For this combination of parameters rc ( r j l)( j c ) l j r February 12, 2020 c l c r j l

l Fields and Waves I lc 39 Distortionless Lines The characteristic impedance also simplifies Z o r j l g j c February 12, 2020 r j l rc

j c l Fields and Waves I l c r j l r j l l c 40 Distortionless Lines Result: no distortion but smaller pulses 600mV 500mV 400mV

300mV 200mV 100mV 0V 0s V(T1:B+) 5us V(R22:1) February 12, 2020 10us V(R1:2) 15us V(R21:2) 20us 25us 30us 35us

40us 45us 50us Time Fields and Waves I 41 Distortionless Lines Expanded view 10mV 8mV 6mV 4mV 2mV 0V 0s V(T1:B+)

5us V(R22:1) February 12, 2020 10us V(R1:2) 15us V(R21:2) 20us 25us 30us 35us 40us 45us 50us Time Fields and Waves I 42

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 February 12, 2020 Fields and Waves I 43 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 February 12, 2020 http://www-gap.dcs.st-and.ac.uk/~history/BigPicture s/ Fields and Waves I 44 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. February 12, 2020 Fields and Waves I 45 Distortionless Lines References http://www.hep.princeton.edu/~mcdona ld/examples/distortionless.pdf

http://www.du.edu/~jcalvert/tech/ cable.htm February 12, 2020 Fields and Waves I 46 Project 1: RF Notch Filter AKA Channel Blocker Basic Configuration Source VOFF = 0 VAMPL = 1 FREQ = 1e7 R1 V1 75 Tee T1 T2

V V R2 75 0 T3 0 Load 0 R3 100Meg 0 February 12, 2020 Open Circuit Line Fields and Waves I 47 Project 1

If the extra cable had a short circuit load Z in Z o Z Z L o jZ o ta n d jZ o ta n d jZ L ta n d 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. February 12, 2020

Fields and Waves I 48 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) February 12, 2020 Fields and Waves I 49 Project 1 For Matlab see old project information and link to Design with Matlab http://hibp.ecse.rpi.edu/%7Econnor/edu

cation/Fields/matlab_analysis.pdf For PSpice see link to Design with PSpice http://hibp.ecse.rpi.edu/%7Econnor/edu cation/Fields/pspice_analysis.pdf February 12, 2020 Fields and Waves I 50 Project 1 Channels 2-6 http://hibp.ecse.rpi.edu/%7Econnor/education/ Fields/cable-channels.xls Campus Cable (might be slightly out of date) Campus Cable Channel Frequencies 15 10 5 0 Channe l Num be r 15 10 5

0 54.00 78.00 102.00 126.00 150.00 174.00 198.00 222.00 246.00 270.00 294.00 318.00 342.00 366.00 Channe l Fre que ncie s

February 12, 2020 Fields and Waves I 51 Project 1 Note channels are reasonably distinct Using Old Spectrum Analyzer February 12, 2020 Fields and Waves I 52 Project 1 More than one channel is affected Using Old Spectrum Analyzer February 12, 2020 Fields and Waves I

53 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 February 12, 2020 Fields and Waves I 54 Alan Dumont

RPI graduate First practical TV Wikipedia Info http://www.tvhistory.tv/ February 12, 2020 Fields and Waves I 55

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