Editing Transmission line (section)

==Input impedance of transmission line==
[[File:SmithChartLineLength.svg|thumb|350px|Looking towards a load through a length <math>\ell</math> of lossless transmission line, the impedance changes as <math>\ell</math> increases, following the blue circle on this [[Smith chart|impedance Smith chart]]. (This impedance is characterized by its [[reflection coefficient]], which is the reflected voltage divided by the incident voltage.) The blue circle, centred within the chart, is sometimes called an ''SWR circle'' (short for ''constant [[standing wave ratio]]'').]]

The [[characteristic impedance]] <math>Z_0</math> of a transmission line is the ratio of the amplitude of a ''single'' voltage wave to its current wave. Since most transmission lines also have a reflected wave, the characteristic impedance is generally not the impedance that is measured on the line.

The impedance measured at a given distance <math>\ell</math> from the load impedance <math>Z_\mathrm{L}</math> may be expressed as

:<math>Z_\mathrm{in}\left(\ell\right)=\frac{V(\ell)}{I(\ell)} = Z_0 \frac{1 + \mathit{\Gamma}_\mathrm{L} e^{-2 \gamma \ell}}{1 - \mathit{\Gamma}_\mathrm{L} e^{-2 \gamma \ell}}</math>,

where <math>\gamma</math> is the propagation constant and <math>\mathit{\Gamma}_\mathrm{L} = \frac{\,Z_\mathrm{L} - Z_0\,}{Z_\mathrm{L} + Z_0}</math> is the voltage [[reflection coefficient]] measured at the load end of the transmission line. Alternatively, the above formula can be rearranged to express the input impedance in terms of the load impedance rather than the load voltage reflection coefficient:

:<math>Z_\mathrm{in}(\ell) = Z_0\,\frac{Z_\mathrm{L} + Z_0 \tanh\left(\gamma \ell\right)}{Z_0 + Z_\mathrm{L}\,\tanh\left(\gamma \ell \right)}</math>.

===Input impedance of lossless transmission line===
For a lossless transmission line, the propagation constant is purely imaginary, <math>\gamma = j\,\beta</math>, so the above formulas can be rewritten as
:<math>
Z_\mathrm{in}(\ell) = Z_0 \frac{Z_\mathrm{L} + j\,Z_0\,\tan(\beta \ell)}{Z_0 + j\,Z_\mathrm{L}\tan(\beta \ell)}
</math>

where <math>\beta = \frac{\,2 \pi\,}{\lambda}</math> is the [[wavenumber]].

In calculating <math>\beta,</math> the wavelength is generally different ''inside'' the transmission line to what it would be in free-space. Consequently, the velocity factor of the material the transmission line is made of needs to be taken into account when doing such a calculation.

===Special cases of lossless transmission lines===
====Half wave length====
For the special case where <math>\beta\,\ell= n\,\pi</math> where n is an integer (meaning that the length of the line is a multiple of half a wavelength), the expression reduces to the load impedance so that
:<math>Z_\mathrm{in} = Z_\mathrm{L} \,</math>

for all <math>n\,.</math> This includes the case when <math>n=0</math>, meaning that the length of the transmission line is negligibly small compared to the wavelength. The physical significance of this is that the transmission line can be ignored (i.e. treated as a wire) in either case.

====Quarter wave length====
{{Main|quarter-wave impedance transformer}}
For the case where the length of the line is one quarter wavelength long, or an odd multiple of a quarter wavelength long, the input impedance becomes
:<math>
Z_\mathrm{in}=\frac{Z_0^2}{Z_\mathrm{L}} ~\,.
</math>

====Matched load====
Another special case is when the load impedance is equal to the characteristic impedance of the line (i.e. the line is ''matched''), in which case the impedance reduces to the characteristic impedance of the line so that
:<math>Z_\mathrm{in}=Z_\mathrm{L}=Z_0 \,</math>
for all <math>\ell</math> and all <math>\lambda</math>.

====Short====
[[File:Transmission line animation open short2.gif|thumb|right|300px|[[Standing wave]]s on a transmission line with an open-circuit load (top), and a short-circuit load (bottom). Black dots represent electrons, and the arrows show the electric field.]]
{{main|stub (electronics)#Short circuited stub|l1=stub}}
For the case of a shorted load (i.e. <math>Z_\mathrm{L} = 0</math>), the input impedance is purely imaginary and a periodic function of position and wavelength (frequency)

:<math>Z_\mathrm{in}(\ell) = j\,Z_0\,\tan(\beta \ell). \,</math>

====Open====
{{main|stub (electronics)#Open_circuited_stub|l1=stub}}
For the case of an open load (i.e. <math>Z_\mathrm{L} = \infty</math>), the input impedance is once again imaginary and periodic

:<math>Z_\mathrm{in}(\ell) = -j\,Z_0 \cot(\beta \ell). \,</math>