Editing Transmission line (section)

===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>