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Chapter 10: Problem 26

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.

Short Answer

Expert verified
The unknown load impedance, \(Z_{load}\), can be found by following these steps: 1. Calculate the reflection coefficient at the junction using the given VSWR: \(\Gamma = 0.6\). 2. Find the angle associated with the reflection coefficient: \(72^\circ\). 3. Plot the reflection coefficient and angle on the Smith chart to obtain the normalized impedance at the junction. 4. Move \(1.1\lambda\) toward the load on the Smith chart to find the impedance at the input of the \(50-\Omega\) line. 5. Multiply the normalized impedance obtained in step 4 by the characteristic impedance of the \(50-\Omega\) line (\(Z_0 = 50\Omega\)) to find the unknown load impedance: \(Z_{load} = Z_{normalized} \times 50\Omega\). Use the normalized impedance value obtained from the Smith chart in step 4 and multiply it by \(50\Omega\) to find the unknown load impedance.

Step by step solution

01

Find the reflection coefficient at the junction

Using the given VSWR of 4, we can calculate the reflection coefficient at the junction between the two transmission lines: \(\Gamma = \frac{VSWR - 1}{VSWR + 1} = \frac{4 - 1}{4 + 1} = \frac{3}{5} = 0.6\)
02

Find the angle associated with the reflection coefficient

The first voltage maximum occurs at \(0.2\lambda\) in front of the junction, which corresponds to an angle of \(72^\circ\). As we move toward the load, the angle increases. So, the angle associated with the reflection coefficient is \(72^\circ\).
03

Plot the reflection coefficient on the Smith chart

Now, we can plot the reflection coefficient we calculated in step 1 with the angle from step 2 on the Smith chart. This will give us the normalized impedance at the junction between the two transmission lines.
04

Find the impedance at the input of the \(50-\Omega\) line

To find the impedance at the input of the \(50-\Omega\) line, we need to move \(1.1\lambda\) toward the load on the Smith chart. Since \(1.1\lambda\) is equivalent to \(1\lambda + 0.1\lambda\), we first need to move \(1\lambda\). Moving \(1\lambda\) in the direction of the load on the Smith chart will result in the same point where we initially plotted our reflection coefficient. Now, move an additional \(0.1\lambda\) toward the load to find the impedance at the input of the \(50-\Omega\) line. On the Smith chart, we can use the wavelength scale to move this additional \(0.1\lambda\).
05

Find the unknown load impedance

Now that we have obtained the normalized impedance at the input of the \(50-\Omega\) line, we can find the unknown load impedance by simply multiplying the normalized impedance by the characteristic impedance of the \(50-\Omega\) line (denoted as \(Z_0\)). \(Z_{load} = Z_{normalized} \times Z_0 = Z_{normalized} \times 50\Omega\) Use the normalized impedance value obtained from the Smith chart in step 4 and multiply it by \(50\Omega\) to find the unknown load impedance.

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Most popular questions from this chapter

A sinusoidal voltage source drives the series combination of an impedance, \(Z_{g}=50-j 50 \Omega\), and a lossless transmission line of length \(L\), shorted at the load end. The line characteristic impedance is \(50 \Omega\), and wavelength \(\lambda\) is measured on the line. \((a)\) Determine, in terms of wavelength, the shortest line length that will result in the voltage source driving a total impedance of \(50 \Omega .(b)\) Will other line lengths meet the requirements of part \((a)\) ? If so, what are they?

Two characteristics of a certain lossless transmission line are \(Z_{0}=50 \Omega\) and \(\gamma=0+j 0.2 \pi \mathrm{m}^{-1}\) at \(f=60 \mathrm{MHz}(a)\) find \(L\) and \(C\) for the line. \((b) \mathrm{A}\) load \(Z_{L}=60+j 80 \Omega\) is located at \(z=0 .\) What is the shortest distance from the load to a point at which \(Z_{\text {in }}=R_{\text {in }}+j 0 ?\)

A \(300-\Omega\) transmission line is short-circuited at \(z=0\). A voltage maximum, \(|V|_{\max }=10 \mathrm{~V}\), is found at \(z=-25 \mathrm{~cm}\), and the minimum voltage, \(|V|_{\min }=\) 0 , is at \(z=-50 \mathrm{~cm}\). Use the Smith chart to find \(Z_{L}\) (with the short circuit replaced by the load) if the voltage readings are \((a)|V|_{\max }=12 \mathrm{~V}\) at \(z=\) \(-5 \mathrm{~cm}\), and \(|V|_{\min }=5 \mathrm{~V} ;(b)|V|_{\max }=17 \mathrm{~V}\) at \(z=-20 \mathrm{~cm}\), and \(|V|_{\min }=0 .\)

A transmission line having primary constants \(L, C, R\), and \(G\) has length \(\ell\) and is terminated by a load having complex impedance \(R_{L}+j X_{L}\). At the input end of the line, a dc voltage source, \(V_{0}\), is connected. Assuming all parameters are known at zero frequency, find the steady-state power dissipated by the load if \((a) R=G=0 ;(b) R \neq 0, G=0 ;(c) R=0\), \(G \neq 0 ;(d) R \neq 0, G \neq 0 .\)

A standing wave ratio of \(2.5\) exists on a lossless \(60 \Omega\) line. Probe measurements locate a voltage minimum on the line whose location is marked by a small scratch on the line. When the load is replaced by a short circuit, the minima are \(25 \mathrm{~cm}\) apart, and one minimum is located at a point \(7 \mathrm{~cm}\) toward the source from the scratch. Find \(Z_{L}\).
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