Question

The input impedance of a loss-less transmission line is 25 ohms when terminated in a short-circuit, and 49 ohms when terminated in an open circuit. The characteristic impedance of the line is __________ *​

Answer 1

Explanation:

First, let us consider the question at hand: What is the input impedance when the transmission line is open- or short-circuited? For a short circuit, ZL=0" role="presentation" style="display: inline-table; font-style: normal; font-weight: normal; line-height: normal; font-size: 17.6px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">ZL=0ZL=0, Γ=−1" role="presentation" style="display: inline-table; font-style: normal; font-weight: normal; line-height: normal; font-size: 17.6px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">Γ=−1Γ=−1, so we find

Zin(l)=Z01+Γe−j2βl1−Γe−j2βl=Z01−e−j2βl1+e−j2βl" role="presentation" style="display: inline-table; font-style: normal; font-weight: normal; line-height: normal; font-size: 17.6px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">Zin(l)=Z01+Γe−j2βl1−Γe−j2βl=Z01−e−j2βl1+e−j2βlZin(l)=Z01+Γe−j2βl1−Γe−j2βl=Z01−e−j2βl1+e−j2βl

Multiplying numerator and denominator by e+jβl" role="presentation" style="display: inline-table; font-style: normal; font-weight: normal; line-height: normal; font-size: 17.6px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">e+jβle+jβl we obtain

(3.16.1)Zin(l)=Z0e+jβl−e−jβle+jβl+e−jβl" role="presentation" style="display: inline-table; font-style: normal; font-weight: normal; line-height: normal; font-size: 17.6px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; width: 10000em !important; position: relative;">Zin(l)=Z0e+jβl−e−jβle+jβl+e−jβl(3.16.1)(3.16.1)Zin(l)=Z0e+jβl−e−jβle+jβl+e−jβl

Now we invoke the following trigonometric identities:

(3.16.2)cos⁡θ=12[e+jθ+e−jθ]sin⁡θ=1j2[e+jθ−e−jθ]" role="presentation" style="display: inline-table; font-style: normal; font-weight: normal; line-height: normal; font-size: 17.6px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; width: 10000em !important; position: relative;">cosθ=12[e+jθ+e−jθ]sinθ=1j2