Input impedance of lossless transmission line

The appropriate impedance Z0 of a manual band is the arrangement of the amplitude of a alone voltage beachcomber to its accepted wave. Since a lot of manual ambit aswell accept a reflected wave, the appropriate impedance is about not the impedance that is abstinent on the line.

For a lossless manual line, it can be apparent that the impedance abstinent at a accustomed position l from the amount impedance ZL is

Z_\mathrm{in} (l)=Z_0 \frac{Z_L + jZ_0\tan(\beta l)}{Z_0 + jZ_L\tan(\beta l)}

where \beta=\frac{2\pi}{\lambda} is the wavenumber.

In artful β, the amicableness is about altered central the manual band to what it would be in free-space and the acceleration affiliated of the actual the manual band is fabricated of needs to be taken into annual if accomplishing such a calculation.

edit Appropriate cases

edit Bisected beachcomber length

For the appropriate case area βl = nπ area n is an accumulation (meaning that the breadth of the band is a assorted of bisected a wavelength), the announcement reduces to the amount impedance so that

Z_\mathrm{in}=Z_L \,

for all n. This includes the case if n = 0, acceptation that the breadth of the manual band is negligibly baby compared to the wavelength. The concrete acceptation of this is that the manual band can be abandoned (i.e. advised as a wire) in either case.

edit Division beachcomber length

Main article: division beachcomber impedance transformer

For the case area the breadth of the band is one division amicableness long, or an odd assorted of a division amicableness long, the ascribe impedance becomes

Z_\mathrm{in}=\frac{{Z_0}^2}{Z_L}. \,

edit Akin load

Another appropriate case is if the amount impedance is according to the appropriate impedance of the band (i.e. the band is matched), in which case the impedance reduces to the appropriate impedance of the band so that

Z_\mathrm{in}=Z_L=Z_0 \,

for all l and all λ.

edit Short

Main article: stub

For the case of a shorted amount (i.e. ZL = 0), the ascribe impedance is absolutely abstract and a alternate action of position and amicableness (frequency)

Z_\mathrm{in} (l)=j Z_0 \tan(\beta l). \,

edit Open

Main article: stub

For the case of an accessible amount (i.e. Z_L=\infty), the ascribe impedance is already afresh abstract and periodic

Z_\mathrm{in} (l)=-j Z_0 \cot(\beta l). \,

edit Stepped manual line

A simple archetype of stepped manual band consisting of three segments.

A stepped manual line2 is acclimated for ample ambit impedance matching. It can be advised as assorted manual band segments affiliated in series, with the appropriate impedance of anniversary alone aspect to be Z0,i. The ascribe impedance can be acquired from the alternating appliance of the alternation relation

Z_\mathrm{i+1}=Z_\mathrm{0,i} \frac{Z_i + jZ_\mathrm{0,i}\tan(\beta_i l_i)}{Z_\mathrm{0,i} + jZ_i\tan(\beta_i l_i)}

where βi is the beachcomber amount of the ith manual band articulation and li is the breadth of this segment, and Zi is the front-end impedance that endless the ith segment.

The impedance transformation amphitheater forth a manual band whose appropriate impedance Z0,i is abate than that of the ascribe cable Z0. And as a result, the impedance ambit is off-centered appear the -x axis. Conversely, if Z0,i > Z0, the impedance ambit should be off-centered appear the +x axis.

Because the appropriate impedance of anniversary manual band articulation Z0,i is generally altered from that of the ascribe cable Z0, the impedance transformation amphitheater is off centered forth the x arbor of the Smith Chart whose impedance representation is usually normalized adjoin Z0.