module 1b final - 1/20/2009 The traveling wave equations...

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1/20/2009 1 Lecture 3 (module 1b) The traveling wave equations Taking the position derivative of both sides of the V s(z) expression leads to 2 2 2 () ( ) 0, s s d V z Vz dz  where propagation constant  ' ' ' ' . R j L G j C j Solving the differential equation, we arrive at a general solution . zz oo s V z V e V e       The instantaneous form is ( , ) cos cos . v z t V e t z V e t z      In a like manner, we find ( , ) cos cos . i z t I e t z I e t z     traveling wave equations the mathematical expression for an attenuated wave!!!! What did we just do? From the assumption of harmonic time dependence, and application of elementary circuit theory for a structure with distributed R,L,G, and C We have shown that the general solution for time varying currents and voltages on transmission lines are attenuated sinusoidal waves along the length of this structure! The time harmonic condition is very general and corresponds simply to harmonic forcing at one end Characteristic impedance Z o Given the traveling wave equations: ( , ) cos cos v z t V e t z V e t z     ( , ) cos cos i z t I e t z I e t z     Characteristic impedance Z o is the ratio of voltage to current wave amplitudes in one direction, or o VV Z II    Note the sign change for the backward wave. This actually comes out of the calculation. It shows that for the backward wave the current is flowing in the “wrong” direction. ...
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1/20/2009 2 Calculation of line impedance Insert V s =V + 0 e - z + V - 0 e + z and I s =I + 0 e - z + I - 0 e + z into Must be equal everywhere (equalize terms with exp(- z)) yields : V + 0 /I + 0 = ( R’+j L ’)/ = Z 0 N.B. Equalizing terms with exp(+ z) gives V - 0 /I - 0 = - Z 0 Characteristic impedance Z o : summary Given the traveling wave equations: ( , ) cos cos zz oo v z t V e t z V e t z      ( , ) cos cos i z t I e t z I e t z     Characteristic impedance Z o is the ratio of voltage to current wave amplitudes in one direction, or o VV Z II    Inserting the wave equations into the phasor-form Telegraphist’s equations we find '' o R j L Z G j C For low loss line, this becomes ' ' o L Z C In general this is a complex number i.e. A real number and V 0 is in phase with I 0 Low-loss and lossless line Low loss lines feature high conductivity conductors and low loss dielectrics such that R’ << L’ and G’ << C’. For this case, j L C j and ' . ' o L Z C phase constant LC  propagation velocity 1 p u  For low-loss coaxial cable , we find p r c u and 60 ln o r b Z a    Working with transmission lines Low loss and lossless lines Power flow in transmission lines Distortionless lines Impedance matching
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1/20/2009 3 From u
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This note was uploaded on 09/27/2011 for the course ENGINEERIN 3600 taught by Professor Victor during the Spring '11 term at Carleton CA.

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module 1b final - 1/20/2009 The traveling wave equations...

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