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 \(Z_{\text {in }}=R_{\text {in }}+j 0 ?\)

Short Answer

Expert verified
Question: Calculate the inductance (L) and capacitance (C) of a lossless transmission line given its characteristic impedance (Z0) and propagation constant (γ) at a specific frequency (f). Then, determine the shortest distance from the load to a point where the input impedance (Zin) becomes purely resistive (Rin). Answer: The inductance (L) of the transmission line is approximately \(77.35 \times 10^{-9} H\) and the capacitance (C) is approximately \(46.41 \times 10^{-12} F\). The shortest distance from the load to a point where the input impedance is purely resistive is approximately 2.31 meters.

Step by step solution

01

Relation between L, C, Z0, and γ

For a lossless transmission line, we can establish the following relations: - Characteristic impedance (Z0) = \(Z_{0}=\sqrt{\frac{L}{C}}\) - Propagation constant (γ) = \(0 + j\beta\), where \(β = 2πf\sqrt{LC}\)
02

Calculate L and C

We're given \(Z_{0} = 50\Omega\) and \(\gamma = 0 + j0.2\pi m^{-1}\). Now let's find L and C using the provided information and equations: From the propagation constant, we can write: \(β = 0.2\pi m^{-1}\) As mentioned before, \(β = 2\pi f\sqrt{L C}\) Plugging in the given values (f = 60 MHz) into the equation, we can find LC: \((0.2\pi) = 2\pi(60\times10^6)\sqrt{L C}\) \(LC = \frac{1}{(60\times10^6)^2 4\pi^2}\) Now, using the relation between Z0 and L, C: \(50 = \sqrt{\frac{L}{C}}\) Squaring both sides: \(50^2 = \frac{L}{C}\) Now we have a system of two equations with two unknowns (L and C). Solving this system, we get: \(L = 77.35 \times 10^{-9} H\) \(C = 46.41 \times 10^{-12} F\) Part (b):
03

Input impedance equation

The input impedance for a transmission line can be written as: \(Z_{in} = Z_{0} \frac{Z_L + jZ_{0}\tan(\beta l)}{Z_{0}+jZ_L\tan(\beta l)}\) In our case, \(Z_{0} = 50\Omega\) and \(Z_{L} = 60 + j80\Omega\)
04

Find the shortest distance for purely resistive Zin

We want to find the shortest distance (l) at which \(Z_{in} = R_{in} + j0\). In other words, the imaginary part of Zin should be zero. For this condition, the imaginary part of the denominator in Zin should be equal to the imaginary part of the numerator. So, \(Z_0\tan(\beta l) = 80\) Previously we found \(β = 0.2\pi m^{-1}\). Plugging this into the equation and solving for l, we get: \(\tan(0.2\pi l) = \frac{80}{50}\) \(l = \frac{1}{0.2\pi} \tan^{-1}\left(\frac{80}{50}\right)\) \(l \approx 2.31m\) The shortest distance from the load to a point where the input impedance is purely resistive is approximately 2.31 meters.

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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?

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 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.

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.

A transmitter and receiver are connected using a cascaded pair of transmission lines. At the operating frequency, line 1 has a measured loss of \(0.1 \mathrm{~dB} / \mathrm{m}\), and line 2 is rated at \(0.2 \mathrm{~dB} / \mathrm{m}\). The link is composed of \(40 \mathrm{~m}\) of line 1 joined to \(25 \mathrm{~m}\) of line 2 . At the joint, a splice loss of \(2 \mathrm{~dB}\) is measured. If the transmitted power is \(100 \mathrm{~mW}\), what is the received power?

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