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Module 1a - Module 1 Transmission lines Part a Module 1...

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1/13/2009 1 Module 1: Transmission lines Part a What is a field (scalar, vectorial, examples)? Review of waves in one dimension, use of phasors When does a piece of wire become a transmission line? Applications: coax cable, parallel wire line, microstrip lines Voltages and currents on transmission lines Circuit view of transmission line: distributed R, G, L and C . Calculation of distributed parameters for Coax cable (only, for other lines, results just presented) Telegraphist’s equations. Wave equation for V and I waves on transmission lines Lossless and lossy lines Wave impedance (V/I), phase velocity, group velocity Frequency-wavelength relationships: length and cross-section scales Power flux on transmission lines How to deal with wave impedance in combination with lumped sources and loads Reflection and transmission Voltage standing wave ratio Termination of lines: impedance matching Power transfer into loads Transients Module 1: Transmission lines Part b Scalar and vector fields Example of scalar field 3 examples of vector fields Gravitational (WAVES, MEDIUM?) Acoustic (waves, medium) EM (incl. Optical), (waves=yes, medium=?) A mathematical expression for wave behavior of a quantity The general equation for a sinusoidal wave traveling in the + z direction is where is the amplitude is the amplitude at = 0 is the attenuation constant in Nepers/meter is the phase is the angular frequency ( 2 ) in radians/second i , cos z o o z o A e A z t z f A z t A e t z s the phase constant in radians/meter is the phase shift If there is one. This depends on the choice of time and space origin Spatial dependence of wave at t=0 z (meters) A(z,0) A o -A o l 2 l is the propagation constant or phase constant Elements of Electromagnetics Fourth Edition Sadiku 6 Figure 10.1 Plot of E ( z , t ) = A sin( t z) (a) with constant t , (b) with constant z .
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1/13/2009 2 Sinusoidal plot at t = 0 with attenuation z (meters) A(z,0) A -A o o e z  Elements of Electromagnetics Fourth Edition Sadiku 8 Figure 10.2 Plot of E ( z , t ) = A sin( t z) at time (a) t 0, (b) t T /4, (c) t = T /2; P moves in the + z - direction with velocity u . Define a “phase” velocity:u p The crests of the wave move towards positive z at a certain speed. One of the crests correspond to ( t- z+   2 ..or equivalently z = ( t+ -2 / is the position of the crest at time t At a later time t’ = t+ D t , the new value of z will be z’ = z+ D z = ( t’+ -2 / = ( t+ -2 / + D t /  z+ D t / Wave propagation z (meters) A(z,t) A o -A o propagation velocity p u f l This is an important result. Usually the phase velocity depends on the kind of wave, the medium in which the wave propagates, and the frequency. This will be covered later.
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