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Chapter 10: Problem 27

The characteristic admittance \(\left(Y_{0}=1 / Z_{0}\right)\) of a lossless transmission line is \(20 \mathrm{mS}\). The line is terminated in a load \(Y_{L}=40-j 20 \mathrm{mS}\). Use the Smith chart to find \((a) s ;(b) Y_{\text {in }}\) if \(l=0.15 \lambda ;(c)\) the distance in wavelengths from \(Y_{I}\) to the nearest voltage maximum.

Short Answer

Expert verified
Question: Calculate the reflection coefficient (s), input admittance (Y_in), and the distance in wavelengths from Y_in to the nearest voltage maximum using the Smith chart if the characteristic admittance is 20 mS, load admittance is 40-j20 mS, and the transmission line is 0.15 wavelengths long. Answer: a) The reflection coefficient (s) is approximately 0.71 + j0.28, b) the input admittance (Y_in) is approximately 28 - j10 mS, and c) the distance in wavelengths from Y_in to the nearest voltage maximum is approximately 0.045λ.

Step by step solution

01

Normalize the load admittance

To normalize the load admittance, divide the given load admittance (Y_L) by the characteristic admittance (Y_0). \(Y_L' = \frac{Y_L}{Y_0} = \frac{40-j20}{20} mS\) \(Y_L' = 2 - j1\)
02

Find the reflection coefficient (s) using the Smith chart

Locate Y_L' on the Smith chart and read the corresponding reflection coefficient (s) value: \(s \approx 0.71 + j0.28\)
03

Determine the input admittance (Y_in) using the Smith chart

Starting with the reflection coefficient found in Step 2, move clockwise along the constant VSWR circle by an electrical distance of \(0.15\lambda\) (as the transmission line length, \(l\), is 0.15 wavelengths). The new point indicates the normalized input admittance (\(Y_{in'}\)) on the chart. \(Y_{in'} \approx 1.4 - j0.5\)
04

Calculate the distance to the nearest voltage maximum

Locate Y_in' on the Smith chart to find the electrical distance to the nearest voltage maximum (either clockwise or counterclockwise). Read the value: \(\Delta l \approx 0.045\lambda\)
05

Convert the input admittance back to the actual value

Multiply the normalized input admittance (Y_in') with the characteristic admittance (Y_0) to obtain the actual input admittance (Y_in). \(Y_{\text{in}} = Y_{0} \cdot Y_{in'} = 20(1.4 - j0.5) mS\) \(Y_{\text{in}} = 28 - j10 mS\) Therefore, we have found the following values: a) \(s \approx 0.71 + j0.28\) b) \(Y_{\text{in}} \approx 28 - j10 mS\) c) \(\Delta l \approx 0.045\lambda\)

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Most popular questions from this chapter

A lossless line having an air dielectric has a characteristic impedance of \(400 \Omega\). The line is operating at \(200 \mathrm{MHz}\) and \(Z_{\text {in }}=200-j 200 \Omega\). Use analytic methods or the Smith chart (or both) to find \((a) s ;(b) Z_{L}\), if the line is \(1 \mathrm{~m}\) long; \((c)\) the distance from the load to the nearest voltage maximum.

An absolute measure of power is the \(\mathrm{dBm}\) scale, in which power is specified in decibels relative to one milliwatt. Specifically, \(P(\mathrm{dBm})=10 \log _{10}[P(\mathrm{~mW}) / 1 \mathrm{~mW}]\). Suppose that a receiver is rated as having a sensitivity of \(-20 \mathrm{dBm}\), indicating the mimimum power that it must receive in order to adequately interpret the transmitted electronic data. Suppose this receiver is at the load end of a \(50-\Omega\) transmission line having \(100-\mathrm{m}\) length and loss rating of \(0.09 \mathrm{~dB} / \mathrm{m}\). The receiver impedance is \(75 \Omega\), and so is not matched to the line. What is the minimum required input power to the line in \((a) \mathrm{dBm},(b) \mathrm{mW} ?\)

The normalized load on a lossless transmission line is \(2+j 1\). Let \(\lambda=20 \mathrm{~m}\) and make use of the Smith chart to find \((a)\) the shortest distance from the load to a point at which \(z_{\text {in }}=r_{\text {in }}+j 0\), where \(r_{\text {in }}>0 ;\) (b) \(z_{\text {in }}\) at this point. (c) The line is cut at this point and the portion containing \(z_{L}\) is thrown away. A resistor \(r=r_{\text {in }}\) of part \((a)\) is connected across the line. What is \(s\) on the remainder of the line? \((d)\) What is the shortest distance from this resistor to a point at which \(z_{\text {in }}=2+j 1 ?\)

The characteristic impedance of a certain lossless transmission line is \(72 \Omega\). If \(L=0.5 \mu \mathrm{H} / \mathrm{m}\), find \((a) C ;(b) v_{p} ;(c) \beta\) if \(f=80 \mathrm{MHz} .(d)\) The line is terminated with a load of \(60 \Omega\). Find \(\Gamma\) and \(s\).

A \(50-\Omega\) lossless line is of length \(1.1 \lambda\). It is terminated by an unknown load impedance. The input end of the \(50-\Omega\) line is attached to the load end of a lossless \(75-\Omega\) line. A VSWR of 4 is measured on the \(75-\Omega\) line, on which the first voltage maximum occurs at a distance of \(0.2 \lambda\) in front of the junction between the two lines. Use the Smith chart to find the unknown load impedance.
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