1_4TL equation in phasor form

1_4TL equation in phasor form - Department of Electrical...

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Department of Electrical and Computer Engineering ECSE 352 Electromagnetic Waves and Optics 4 Transmission line equations in phasor form 1.4 Transmission line equations in phasor form References: Hayt and Buck 11.6, 11.7 Whats on the web? Andrew Kirk 2006 1.4-1 Wn w ? http://webct.mcgill.ca The telegraph: The first transmission line application ©AGK 2006 ECSE 352 1.4-2 Overview In the last class we introduced the concept of single frequency or harmonic waves, and showed how they could be described either as sinusoidal signals or as complex phasors. In this class we will apply this concept to the transmission line equations. Based on thi sa n alysi sw ewill obtai ne q u a tions for the propagation of signals on lossless and low loss lines. We will also consider the impedance of lossy lines. p y ©AGK 2006 ECSE 352 1.4-3 Transmission line classes A. Transmission line basics Introduction Lossless transmission lines Harmonic waves on transmission lines Time independent (phasor) notation ower transmission and loss Power transmission and loss B. Analog signal transmission 6. Resistive loads and reflections 7. Voltage standing wave ratio 8. Finite lines and input impedance 9. Smith chart calculation techniques 0 ingle stub matching 10. Single stub matching C. Digital signal transmission 11. Transient analysis ©AGK 2006 ECSE 352 1.4-4 12. Pulse propagation and initially charged lines
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Contents • Transmission line equations in phasor form • Complex impedance • Lossless and low-loss lines i t ti l li Distortionless lines ©AGK 2006 ECSE 352 1.4-5 Learning outcomes After taking this class you should be able to: • Define the propagation constant and explain its significance alculate the propagation constant of lossless and Calculate the propagation constant of lossless and low loss lines • Define the attenuation coefficient for a transmission line • Recognize the significance of a distortionless line Recognize that impedance can be complex ©AGK 2006 ECSE 352 1.4-6 Review: numerical solution to wave equation + R g V I + + () 22 VV V LC LG RC RGV t ∂∂ =+ ++ V 0 - V + =V 0 - t zt l V ( 0 , t ) ( , ) L (H/m), C (F/m), R ( Ω /m), G (S/m) z + Δ Δ z ( , ) ( + Δ , ) I ( , ) ( z + Δ , ) N C Δ G Δ R Δ L Δ ©AGK 2006 ECSE 352 1.4-7 Lossy (R,G >0) Δ Review: Harmonic waves on transmission lines • We assume that waves have a single frequency ω • They are steady-state • There are two representations: ( ) [ ] 0 ,c o s V t z ω βφ + V Instantanenous (sinusoidal): ( ) 0 j z s Vz V e β ± = Complex (phasor): 00 j e φ = where t V 0 And time dependence is: jt e ©AGK 2006 ECSE 352 1.4-8 -V 0 t 0
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Propagation constant • In order to deal with lossy lines, we introduce a new parameter: the propagation constant γ • Use this to represent the spatial wave evolution •R e w r i t e : ( ) 00 j z j z s Vz Ve Ve β +− =+ •A s : • We will define γ more precisely in a moment ©AGK 2006 ECSE 352 1.4-9 Transmission line equations () 22 VV V LC LG RC RGV t ∂∂ ++ From 1.1-20: t zt R g + I + V 0 R L - - V R,C,L,G • Using phasor notation, we find: ©AGK 2006 ECSE 352 1.4-10 • So that: Propagation constant • Since we have: ( ) R j LG j C γω ω +
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This note was uploaded on 01/27/2011 for the course ECSE 352 taught by Professor Mi during the Spring '10 term at McGill.

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1_4TL equation in phasor form - Department of Electrical...

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