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Chapter 10: Problem 9

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?

Short Answer

Expert verified
Answer: The shortest length of the transmission line that will result in a total impedance of 50Ω is L = λ/8. The other line lengths that meet the requirements are given by L = (λ/8) + n(λ/2), where n is an integer.

Step by step solution

01

Understand the given information

We are given the generator impedance \(Z_{g}=50-j50\Omega\), and a lossless transmission line with characteristic impedance \(Z_{0}=50\Omega\) and length \(L\). The transmission line is shorted at the load end.
02

Finding the input impedance of the transmission line

The input impedance of a transmission line can be found using the equation: \[Z_{in}=Z_{0}\frac{Z_{L}+jZ_{0}\tan(\beta L)}{Z_{0}+jZ_{L}\tan(\beta L)}\] However, since the transmission line is shorted at the load end, the load impedance \(Z_{L}=0\). Therefore, the equation simplifies to: \[Z_{in}=jZ_{0}\tan(\beta L)\]
03

Calculate the total impedance driven by the voltage source

The total impedance driven by the voltage source is given by: \[Z_{total}=Z_{g}+Z_{in}\] We are given that \(Z_{g}=50-j50\Omega\), and \(Z_{in}=j50\tan(\beta L)\). So, the equation will be: \[Z_{total}=50-j50+j50\tan(\beta L)\]
04

Find the line length for the total impedance to be \(50\Omega\)

We need to find the shortest line length \(L\) such that \(Z_{total}=50\Omega\). To do this, set the imaginary part of the total impedance to zero and solve for \(L\), i.e. \[-50+50\tan(\beta L)=0\] \[tan(\beta L)=1\] Now, \(\beta=\frac{2\pi}{\lambda}\). So, we can write: \[tan\left(\frac{2\pi}{\lambda} L\right)=1\] The shortest line length which satisfies this condition is when the argument inside the tangent function is \(\frac{\pi}{4}\), i.e. \[\frac{2\pi}{\lambda} L=\frac{\pi}{4}\] Now, we can solve for the shortest length \(L\) in terms of the wavelength, which is: \[L=\frac{\lambda}{8}\]
05

Determine if other line lengths will meet the requirements

Now, we must check if other values of line length \(L\) will satisfy the conditions. From the tangent function's properties, we know that \(tan(x+\pi)=tan(x)\). So, we need to check if the following length values also meet the requirements: \[L=\frac{\lambda}{8} + n\frac{\lambda}{2}\] where \(n\) is an integer. Yes, other line lengths will meet the requirements, and they are given as: \[L=\frac{\lambda}{8} + n\frac{\lambda}{2}\] where \(n\) is an integer.

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Most popular questions from this chapter

A sinusoidal voltage wave of amplitude \(V_{0}\), frequency \(\omega\), and phase constant \(\beta\) propagates in the forward \(z\) direction toward the open load end in a lossless transmission line of characteristic impedance \(Z_{0}\). At the end, the wave totally reflects with zero phase shift, and the reflected wave now interferes with the incident wave to yield a standing wave pattern over the line length (as per Example 10.1). Determine the standing wave pattern for the current in the line. Express the result in real instantaneous form and simplify.

A lossless transmission line having characteristic impedance \(Z_{0}=50 \Omega\) is driven by a source at the input end that consists of the series combination of a 10 -V sinusoidal generator and a \(50-\Omega\) resistor. The line is one- quarter wavelength long. At the other end of the line, a load impedance, \(Z_{L}=50-j 50 \Omega\) is attached. (a) Evaluate the input impedance to the line seen by the voltage source-resistor combination; \((b)\) evaluate the power that is dissipated by the load; \((c)\) evaluate the voltage amplitude that appears across the load.

A \(100-\Omega\) lossless transmission line is connected to a second line of \(40-\Omega\) impedance, whose length is \(\lambda / 4\). The other end of the short line is terminated by a \(25-\Omega\) resistor. A sinusoidal wave (of frequency \(f\) ) having \(50 \mathrm{~W}\) average power is incident from the \(100-\Omega\) line. \((a)\) Evaluate the input impedance to the quarter-wave line. (b) Determine the steady-state power that is dissipated by the resistor. \((c)\) Now suppose that the operating frequency is lowered to one-half its original value. Determine the new input impedance, \(Z_{i n}^{\prime}\), for this case. \((d)\) For the new frequency, calculate the power in watts that returns to the input end of the line after reflection.

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 \(z_{\text {in }}=2+j 1 ?\)

Two lossless transmission lines having different characteristic impedances are to be joined end to end. The impedances are \(Z_{01}=100 \Omega\) and \(Z_{03}=25 \Omega\). The operating frequency is \(1 \mathrm{GHz}\). \((a)\) Find the required characteristic impedance, \(Z_{02}\), of a quarter-wave section to be inserted between the two, which will impedance-match the joint, thus allowing total power transmission through the three lines. \((b)\) The capacitance per unit length of the intermediate line is found to be \(100 \mathrm{pF} / \mathrm{m}\). Find the shortest length in meters of this line that is needed to satisfy the impedance-matching condition. ( \(c\) ) With the three-segment setup as found in parts \((a)\) and \((b)\), the frequency is now doubled to \(2 \mathrm{GHz}\). Find the input impedance at the line-1-to-line- 2 junction, seen by waves incident from line \(1 .(d)\) Under the conditions of part \((c)\), and with power incident from line 1 , evaluate the standing wave ratio that will be measured in line 1 , and the fraction of the incident power from line 1 that is reflected and propagates back to the line 1 input.
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