EC-2.4 Transmission Lines
You can compute the characteristic impedance, reflection coefficient and standing wave ratio of a line, transform an impedance along it, design a simple matching arrangement, and read a Smith chart.
A transmission line is where circuit theory and field theory meet: it is described by the same voltages and currents as a circuit, but only because the wave picture has been folded into the model as distributed inductance and capacitance. The single idea that organises the whole topic is the reflection that occurs when the line's own impedance and the load's impedance disagree, and everything after it is either measuring that reflection or removing it. The Smith chart is not a relic, because it makes the impedance transformation a rotation you can see rather than a formula you can only evaluate.
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Why a line is not a wire: distributed elements and the transmission line equations
Once a connection is a noticeable fraction of a wavelength, voltage differs along its length and it must be modelled as inductance and capacitance spread out rather than lumped. The two coupled equations that follow are the wave equation again in different clothes.
NPTEL: Transmission Lines and EM Waves · CourseCharacteristic impedance and propagation constant, and what makes a line lossless
The characteristic impedance is the ratio a wave sees while travelling with nothing to reflect from, and it is a property of the geometry rather than of the length. A lossless line has a purely real one, which is why cable is specified by a single number.
NPTEL: Transmission Lines and EM Waves · CourseThe reflection coefficient, and standing waves as its visible consequence
A load that does not equal the line impedance sends part of the wave back, and the forward and returning waves add to a pattern fixed in space. The ratio of its maximum to its minimum is what an instrument measures.
Impedance transformation along a line, and the half and quarter wavelength cases
What a source sees depends on how far it is from the load in wavelengths, repeating every half wavelength and inverting every quarter. Those two special cases solve a surprising share of practical problems on their own.
MIT OpenCourseWare 6.013: Electromagnetics and Applications · CourseThe Smith chart: impedance as a point, and length as a rotation
The chart maps every impedance onto a disc so that moving along a line becomes a rotation about its centre. Learning to read one is quicker than it looks and makes matching a drawing problem rather than an algebra problem.
NPTEL: Transmission Lines and EM Waves · CourseMatching: the quarter-wave transformer and the single stub
A quarter-wave section of the right impedance matches two real impedances, and a stub of the right length placed at the right distance cancels a reactance. Both are the same goal reached differently, which is to stop the reflection.
Scattering parameters, and why they are the ones measured at high frequency
Opening and shorting a port is not practical at high frequency, so networks are described by how much of an incident wave is reflected and how much passes through. These are the numbers a network analyser actually reports.
MIT OpenCourseWare 6.013: Electromagnetics and Applications · Course
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