# 3.8: Exercises

- Page ID
- 41274

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- A coaxial line is short-circuited at one end and is filled with a dielectric with a relative permittivity of \(64\). [Parallels Example 3.1.1]
- What is the free-space wavelength at \(18\text{ GHz}\)?
- What is the wavelength in the dielectricfilled coaxial line at \(18\text{ GHz}\)?
- The first resonance of the coaxial resonator is at \(18\text{ GHz}\). What is the physical length of the resonator?

- A transmission line has the following \(RLGC\) parameters: \(R = 100\:\Omega\text{/m},\: L = 85\text{ nH/m},\: G = 1\text{ S/m,}\) and \(C = 150\text{ pF/m}\). Consider a traveling wave on the transmission line with a frequency of \(1\text{ GHz}\). [Parallels Example 3.2.2]
- What is the attenuation constant?
- What is the phase constant?
- What is the phase velocity?
- What is the characteristic impedance of the line?
- What is the group velocity?

- A transmission line has the per-unit length parameters \(L = 85\text{ nH/m},\: G = 1\text{ S/m},\) and \(C = 150\text{ pF/m}\). Use a frequency of \(1\text{ GHz}\). [Parallels Example 3.2.2]
- What is the phase velocity if \(R = 0\:\Omega\text{/m}\)?
- What is the group velocity if \(R = 0\:\Omega\text{/m}\)?
- If \(R = 10\text{ k}\Omega\text{/m}\) what is the phase velocity?
- If \(R = 10\text{ k}\Omega\text{/m}\) what is the group velocity?

- A line is \(10\text{ cm}\) long and at the operating frequency the phase constant \(\beta\) is \(40\text{ rad/m}\). What is the electrical length of the line? [Parallels Example 3.2.1]
- A coaxial transmission line is filled with lossy dielectric material with a relative permittivity of \(5 −\jmath 0.2\). If the line is air-filled it would have a characteristic impedance of \(100\:\Omega\). What is the input impedance of the line if it is \(1\text{ km}\) long? Use reasonable approximations. [Hint: Does the termination matter?]
- A transmission line has the per unit length parameters \(R = 2\:\Omega\text{/cm},\: L=100\text{ nH/m},\: G = 1\text{ mS/M},\: C=200\text{ pF/m}\).
- What is the propagation constant of the line at \(5\text{ GHz}\)?
- What is the characteristic impedance of the line at \(5\text{ GHz}\)?
- Plot the magnitude of the characteristic impedance versus frequency from \(100\text{ MHz}\) to \(10\text{ GHz}\).

- A line is \(20\text{ cm}\) long and at \(1\text{ GHz}\) the phase constant \(\beta\) is \(20\text{ rad/m}\). What is the electrical length of the line in degrees?
- What is the electrical length of a line that is a quarter of a wavelength long,
- in degrees?
- in radians?

- A lossless transmission line has an inductance of \(8\text{ nH/cm}\) and a capacitance of \(40\text{ pF/cm}\).
- What is the characteristic impedance of the line?
- What is the phase velocity on the line at \(1\text{ GHz}\)?

- A transmission line has an attenuation of \(2\text{ dB/m}\) and a phase constant of \(25\text{ radians/m}\) at \(2\text{ GHz}\). [Parallels Example 3.2.3]
- What is the complex propagation constant of the transmission line?
- If the capacitance of the line is \(50\text{ pF}\cdot\text{m}^{−1}\) and \(G = 0\), what is the characteristic impedance of the line?

- A very low-loss microstrip transmission line has the following per unit length parameters: \(R = 2\:\Omega\text{/m},\: L = 80\text{ nH/m},\: C = 200\text{ pF/m},\) and \(G = 1\text{ µS/m}\).
- What is the characteristic impedance of the line if loss is ignored?
- What is the attenuation constant due to conductor loss?
- What is the attenuation constant due to dielectric loss?

- A lossless transmission line carrying a \(1\text{ GHz}\) signal has the following per unit length parameters: \(L = 80\text{ nH/m},\: C = 200\text{ pF/m}\).
- What is the attenuation constant?
- What is the phase constant?
- What is the phase velocity?
- What is the characteristic impedance of the line?

- A transmission line has a characteristic impedance \(Z_{0}\) and is terminated in a load with a reflection coefficient of \(0.8\angle 45^{\circ}\). A forward-traveling voltage wave on the line has a power of \(1\text{ dBm}\).
- How much power is reflected by the load?
- What is the power delivered to the load?

- A transmission line has an attenuation of \(0.2\text{ dB/cm}\) and a phase constant of \(50\text{ radians/m}\) at \(1\text{ GHz}\).
- What is the complex propagation constant of the transmission line?
- If the capacitance of the line is \(100\text{ pF/m}\) and \(G = 0\), what is the complex characteristic impedance of the line?
- If the line is driven by a source modeled as an ideal voltage and a series impedance, what is the impedance of the source for maximum transfer of power to the transmission line?
- If \(1\text{ W}\) is delivered (i.e. in the forwardtraveling wave) to the transmission line by the generator, what is the power in the forward-traveling wave on the line at \(2\text{ m}\) from the generator?

- A lossless transmission line is driven by a \(1\text{ GHz}\) generator having a Thevenin equivalent impedance of \(50\:\Omega\). The transmission line is lossless, has a characteristic impedance of \(75\:\Omega\), and is infinitely long. The maximum power that can be delivered to a load attached to the generator is \(2\text{ W}\).
- What is the total (phasor) voltage at the input to the transmission line?
- What is the magnitude of the forwardtraveling voltage wave at the generator side of the line?
- What is the magnitude of the forwardtraveling current wave at the generator side of the line?

- A transmission line is terminated in a short circuit. What is the ratio of the forward- and backward-traveling voltage waves at the termination? [Parallels Example 3.3.1]
- A \(50\:\Omega\) transmission line is terminated in a \(40\:\Omega\) load. What is the ratio of the forward- to the backward-traveling voltage waves at the termination? [Parallels Example 3.3.1]
- A \(50\:\Omega\) transmission line is terminated in an open circuit. What is the ratio of the forwardto the backward-traveling voltage waves at the termination? [Parallels Example 3.3.1]
- A line has a characteristic impedance \(Z_{0}\) and is terminated in a load with a reflection coefficient of \(0.8\). A forward-traveling voltage wave on the line has a power of \(1\text{ W}\).
- How much power is reflected by the load?
- What is the power delivered to the load?

- A load consists of a shunt connection of a capacitor of \(10\text{ pF}\) and a resistor of \(25\:\Omega\). The load terminates a lossless \(50\:\Omega\) transmission line. The operating frequency is \(1\text{ GHz}\). [Parallels Example 3.3.2]
- What is the impedance of the load?
- What is the normalized impedance of the load (normalized to the characteristic impedance of the line)?
- What is the reflection coefficient of the load?
- What is the current reflection coefficient of the load?
- What is the standing wave ratio (SWR)?
- What is the current standing wave ratio (ISWR)?

- An amplifier is connected to a load by a transmission line matched to the amplifier. If the SWR on the line is \(1.5\), what percentage of the available amplifier power is absorbed by the load?
- A load has a reflection coefficient of \(0.5\) when referred to \(50\:\Omega\). The load is at the end of a line with a \(50\:\Omega\) characteristic impedance.
- If the line has an electrical length of \(45^{\circ}\), what is the reflection coefficient calculated at the input of the line?
- What is the VSWR on the \(50\:\Omega\) line?

- A \(100\:\Omega\) resistor in parallel with a \(5\text{ pF}\) capacitor terminates a \(100\:\Omega\) transmission line. Calculate the SWR on the line at \(2\text{ GHz}\).
- A lossless \(50\:\Omega\) transmission line has a \(50\:\Omega\) generator at one end and is terminated in \(100\:\Omega\). What is the VSWR on the line?
- A lossless \(75\:\Omega\) line is driven by a \(75\:\Omega\) generator. The line is terminated in a load that with a reflection coefficient (referred to \(50\:\Omega\)) of \(0.5 +\jmath 0.5\). What is the VSWR on the line?
- A load with a \(20\text{ pF}\) capacitor in parallel with a \(50\:\Omega\) resistor terminates a \(25\:\Omega\) line. The operating frequency is \(5\text{ GHz}\). [Parallels Example 3.3.3]
- What is the VSWR?
- What is ISWR?

- A load \(Z_{L} = 55−\jmath 55\:\Omega\) and the system reference impedance, \(Z_{0}\), is \(50\:\Omega\). [Parallels Example 3.3.4]
- What is the load reflection coefficient \(\Gamma_{L}\)?
- What is the current reflection coefficient?
- What is the VSWR on the line?
- What is the ISWR on the line?
- Now consider a source connected directly to the load. The source has a Thevenin equivalent impedance \(Z_{G} = 60\:\Omega\) and an available power of \(1\text{ W}\). Use \(\Gamma_{L}\) to find the power delivered to \(Z_{L}\).
- What is the total power absorbed by \(Z_{G}\)?

- A load of \(100\:\Omega\) is to be matched to a transmission line with a characteristic impedance of \(50\:\Omega\). Use a quarter-wave transformer. What is the characteristic impedance of the quarterwave transformer?
- Determine the characteristic impedance of a quarter-wave transformer used to match a load of \(50\:\Omega\) to a generator with a Thevenin equivalent impedance of \(75\:\Omega\).
- A transmission line is to be inserted between a \(5\:\Omega\) line and a \(50\:\Omega\) load so that there is maximum power transfer to the \(50\:\Omega\) load at \(20\text{ GHz}\).
- How long is the inserted line in terms of wavelengths at \(20\text{ GHz}\)?
- What is the characteristic impedance of the line at \(20\text{ GHz}\)?

## 3.8.1 Exercises By Section

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

\(§3.1 1\)

\(§3.2 2†, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12†, 13\)

\(§3.3 14†, 15, 16, 17, 18, 19†, 20†, 21†, 22, 23, 24, 25, 26, 27\)

\(§3.4 28, 29, 30\)

## 3.8.2 Answers to Selected Exercises

- \(0.23+\jmath 25\text{ m}^{-1}\)

- \(61.2\:\Omega\)