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Chapter 10: Problem 19

A lossless transmission line is \(50 \mathrm{~cm}\) in length and operates at a frequency of \(100 \mathrm{MHz}\). The line parameters are \(L=0.2 \mu \mathrm{H} / \mathrm{m}\) and \(C=80 \mathrm{pF} / \mathrm{m}\). The line is terminated in a short circuit at \(z=0\), and there is a load \(Z_{L}=50+j 20 \Omega\) across the line at location \(z=-20 \mathrm{~cm} .\) What average power is delivered to \(Z_{L}\) if the input voltage is \(100 \angle 0^{\circ} \mathrm{V} ?\)

Short Answer

Expert verified
Answer: The average power delivered to the load is \(50W\).

Step by step solution

01

Calculate the characteristic impedance and propagation constant

First, we need to find the characteristic impedance \(Z_0\) and the propagation constant \(\gamma\) of the transmission line: $$Z_0 = \sqrt{\frac{L}{C}}$$ $$\gamma=\sqrt{LC}\omega$$ Here, \(\omega = 2\pi f\), where \(f\) is the frequency. Using the given values, we have: $$Z_0 = \sqrt{\frac{0.2\times10^{-6}}{80\times10^{-12}}}$$ $$Z_0 \approx 50\Omega$$ $$\gamma = \sqrt{(0.2\times10^{-6})(80\times10^{-12})} (2\pi\times100\times10^{6})$$ $$\gamma \approx j20\, \text{rad/m}$$
02

Calculate the reflection coefficient at the load

We can now calculate the reflection coefficient \(\Gamma\) at the load using: $$\Gamma = \frac{Z_L - Z_0}{Z_L + Z_0}$$ With \(Z_L = 50 + j20 \Omega\) and \(Z_0 = 50\Omega\), we obtain: $$\Gamma = \frac{(50+j20)-50}{(50+j20)+50}$$ $$\Gamma \approx \frac{j20}{100+j20}$$
03

Calculate input impedance

Next, we need to find the input impedance \(Z_{in}\) of the transmission line: $$Z_{in} = Z_0 \frac{Z_L + j Z_0 \tan(\gamma z)}{Z_0 + j Z_L\tan(\gamma z)}$$ Substituting the values for \(z = -20cm = -0.2m\), \(Z_L\), \(Z_0\), and \(\gamma\), we get: $$Z_{in} = 50 \frac{(50+j20) + j50\tan(-0.2j20)}{50 + j(50+j20)\tan(-0.2j20)}$$ $$Z_{in} \approx 50 + j50\Omega$$
04

Calculate the current

Using Ohm's Law, we can calculate the current delivered to the input impedance as follows: $$I = \frac{V}{Z_{in}}$$ Using the given input voltage \(V = 100 \angle 0^{\circ}\) and calculated input impedance \(Z_{in}\approx 50+j50\Omega\), we have: $$I = \frac{100 \angle 0^{\circ}}{50 + j50}$$ $$I \approx 1 - j1\, \text{A}$$
05

Calculate average power delivered to the load

Finally, we can calculate the average power \(P_{avg}\) delivered to the load using the current and load impedance: $$P_{avg} = \frac{1}{2} \cdot |I|^2 \cdot Re(Z_L)$$ With \(I = 1-j1\, \text{A}\) and \(Re(Z_L) = 50\Omega\), we get: $$P_{avg} = \frac{1}{2} \cdot (1^2 + (-1)^2) \cdot 50$$ $$P_{avg} = 50\, \text{W}$$ The average power delivered to the load \(Z_L\) is \(50W\).

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Most popular questions from this chapter

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?

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

In a circuit in which a sinusoidal voltage source drives its internal impedance in series with a load impedance, it is known that maximum power transfer to the load occurs when the source and load impedances form a complex conjugate pair. Suppose the source (with its internal impedance) now drives a complex load of impedance \(Z_{L}=R_{L}+j X_{L}\) that has been moved to the end of a lossless transmission line of length \(\ell\) having characteristic impedance \(Z_{0}\). If the source impedance is \(Z_{g}=R_{g}+j X_{g}\), write an equation that can be solved for the required line length, \(\ell\), such that the displaced load will receive the maximum power.

In Figure \(10.39, R_{L}=Z_{0}\) and \(R_{g}=Z_{0} / 3\). The switch is closed at \(t=0\). Determine and plot as functions of time \((a)\) the voltage across \(R_{L} ;(b)\) the voltage across \(R_{g} ;(c)\) the current through the battery.

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.
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