7.10: Exercises

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A load is modeled as a \(50\:\Omega\) resistance in series with a reactance of \(50\:\Omega\). This load is to be matched to a source with a Thevenin equivalent resistance of \(50\:\Omega\). Use the Fano-Bode criteria to determine the upper limit on the matching network bandwidth when the average in-band reflection coefficient is \(−10\text{ dB}\).
A load is modeled as a \(50\:\Omega\) resistance in series with a reactance of \(50\:\Omega\). This load is to be matched to a source with a Thevenin equivalent resistance of \(50\:\Omega\). Use the Fano-Bode criteria to determine the upper limit on the matching network bandwidth when the average in-band reflection coefficient is \(−20\text{ dB}\).
The output of a transistor amplifier operating at \(1\text{ GHz}\) is modeled as a \(100\:\Omega\) resistor in parallel with a \(10\text{ pF}\) capacitor. The amplifier must drive the input of a \(\lambda/2\) dipole antenna with an input resistance of \(73\:\Omega\). To do this efficiently a matching network is required. Consider that the input resistance of the antenna is independent of frequency, and assume that the matching network is lossless. This is the same as assuming that its bandwidth is much greater than the bandwidth required. If the required fractional bandwidth of the matching network is \(5\%\), and using the FanoBode criteria, determine the following:
1. The lower limit on the average in-band reflection coefficient of the matching network.
2. The upper limit on the average transmission coefficient of the matching network.
Design a broadband matching network at \(1\text{ GHz}\) to match a source \(Z_{S} = 80+\jmath 50\:\Omega\) to a load with an impedance \(Z_{L} = 60.0+\jmath 20.0\:\Omega\). Maintain the maximum bandwidth possible with this source and load. [Parallels Example 7.3.1]
Design a broadband matching network at \(1\text{ GHz}\) to match a source \(Z_{S} = 45+\jmath 10\:\Omega\) to a load with an impedance \(Z_{L} = 50.0+\jmath 80.0\:\Omega\). Maintain the maximum bandwidth possible with this source and load. [Parallels Example 7.3.1]
Consider the problem of matching a source with a Thevenin equivalent impedance of \(25\:\Omega\) to a load of admittance \(0.035 + \jmath 0.035\).
1. What is the minimum \(Q\) that can be achieved for the network, and what is the topology of the matching network that will yield the match with the widest bandwidth?
2. Design the matching network with the widest bandwidth possible if the matching network can have at most four elements.
Develop the electrical design of a three-section quarter-wave transformer to match a \(50\:\Omega\) cable to an antenna with a \(10\:\Omega\) input impedance. [Parallels Example 7.3.2]
Design of a two-section quarter-wave transformer to match a \(50\:\Omega\) cable to a \(75\:\Omega\) cable. [Parallels Example 7.3.2]
Develop the electrical design of a two-section quarter-wave transformer to match a \(50\:\Omega\) cable to a \(75\:\Omega\) cable. [Parallels Example 7.4.1]
Develop the electrical design of a three-section maximally flat stepped-impedance transformer to match a source \(Z_{S} = 20\:\Omega\) to a load \(Z_{L} = 50\:\Omega\) load. [Parallels Example 7.4.2]
Design a stepped impedance transmission line transformer with two transmission line sections to match a \(50\:\Omega\) source to a load with an impedance of \(25\:\Omega\). Design for a maximally flat response. [Parallels Example 7.4.2]
Design a maximally flat four-section stepped impedance transmission line transformer matching a basestation amplifier with a \(2\:\Omega\) output impedance to a \(50\:\Omega\) cable. [Parallels Example 7.4.2]
Develop the electrical design of a \(100\%\) bandwidth three-section Chebyshev stepped-impedance transformer in microstrip to connect a power amplifier with an output impedance of \(2\:\Omega\) to a \(50\:\Omega\) cable. [Parallels Example 7.4.3]
Design a microstrip Klopfenstein taper to match a \(Z_{S} = 15\:\Omega\) source to a \(Z_{L} = 75\:\Omega\) load. The maximum transmission ripple is to be \(0.5\text{ dB}\) and the minimum passband frequency is \(60\text{ GHz}\). Only an electrical design is required but draw the microstrip layout. [Parallels Example 7.5.1]

7.10.1 Exercises by Section

\(†\)challenging, \(‡\)very challenging

\(§7.2\: 1, 2, 3 \)

\(§7.3\: 4, 5, 6\)

\(§7.4\: 7, 8, 9, 10, 11, 12, 13\)

\(§7.5\: 14 \)

7.10.2 Answers to Selected Exercises

\(41.36\text{ meV}\)
\(662.6\text{ fJ}\)
\(3.25\text{ cm}\)