Editing Transmission line (section)

====Stepped transmission line====
{{see also|Waveguide filter#Impedance matching}}
[[Image:Segments.jpg|thumb|right|350px|A simple example of stepped transmission line consisting of three segments]]
A stepped transmission line is used for broad range impedance matching. It can be considered as multiple transmission line segments connected in series, with the characteristic impedance of each individual element to be <math>Z_\mathrm{0,i}</math>.<ref>{{cite journal |last1=Qian |first1=Chunqi |last2=Brey |first2=William W. |year=2009 |title=Impedance matching with an adjustable segmented transmission line |journal=Journal of Magnetic Resonance |volume=199 |issue=1 |pages=104–110 |bibcode=2009JMagR.199..104Q |doi=10.1016/j.jmr.2009.04.005 |pmid=19406676}}</ref> The input impedance can be obtained from the successive application of the chain relation

:<math>Z_\mathrm{i+1} = Z_\mathrm{0,i}\,\frac{\,Z_\mathrm{i} + j\,Z_\mathrm{0,i}\,\tan(\beta_\mathrm{i} \ell_\mathrm{i})\,}{Z_\mathrm{0,i} + j\,Z_\mathrm{i}\,\tan(\beta_\mathrm{i} \ell_\mathrm{i})}\,</math>

where <math>\beta_\mathrm{i}</math> is the wave number of the <math>\mathrm{i}</math>-th transmission line segment and <math>\ell_\mathrm{i}</math> is the length of this segment, and <math>Z_\mathrm{i}</math> is the front-end impedance that loads the <math>\mathrm{i}</math>-th segment.

[[Image:PolarSmith.jpg|thumb|The impedance transformation circle along a transmission line whose characteristic impedance <math>Z_\mathrm{0,i}</math> is smaller than that of the input cable <math>Z_0</math>. And as a result, the impedance curve is off-centred towards the <math>-x</math> axis. Conversely, if <math>Z_\mathrm{0,i} > Z_0</math>, the impedance curve should be off-centred towards the <math>+x</math> axis.]]
Because the characteristic impedance of each transmission line segment <math>Z_\mathrm{0,i}</math> is often different from the impedance <math>Z_0</math> of the fourth, input cable (only shown as an arrow marked <math>Z_0</math> on the left side of the diagram above), the impedance transformation circle is off-centred along the <math>x</math> axis of the [[Smith Chart]] whose impedance representation is usually normalized against <math>Z_0</math>.