# 7.8: Exercises

- Page ID
- 41305

- A load has a reflection coefficient of \(0.5 −\jmath 0.1\) in a \(75\:\Omega\) reference system. What is the reflection coefficient in a \(50\:\Omega\) reference system?
- The \(50\:\Omega\) S parameters of a two-port are \(S_{11} = 0.5+\jmath 0.5,\: S_{12} = 0.95+\jmath 0.25,\: S_{21} = 0.15−\jmath 0.05,\) and \(S_{22} = 0.5 − \jmath 0.5\). Port 1 is connected to a \(50\:\Omega\) source with an available power of \(1\text{ W}\) and Port 2 is terminated in \(50\:\Omega\). What is the power reflected from Port 1?
- The scattering parameters of a certain two-port are \(S_{11} = 0.5 + \jmath 0.5,\: S_{12} = 0.95 + \jmath 0.25,\: S_{21} = 0.15 −\jmath 0.05,\) and \(S_{22} = 0.5 − \jmath 0.5\). The system reference impedance is \(50\:\Omega\).
- Is the two-port reciprocal? Explain.
- Consider that Port 1 is connected to a \(50\:\Omega\) source with an available power of \(1\text{ W}\). What is the power delivered to a \(50\:\Omega\) load placed at Port 2?
- What is the reflection coefficient of the load required for maximum power transfer at Port 2?

- In characterizing a two-port, power could only be applied at Port 1. The signal reflected was measured and the signal at a \(50\:\Omega\) load at Port 2 was also measured. This yielded two \(S\) parameters referenced to \(50\:\Omega\): \(S_{11} = 0.3 − \jmath 0.4\) and \(S_{21} = 0.5\).
- If the network is reciprocal, what is \(S_{12}\)?
- Is the two-port lossless?
- What is the power delivered into the \(50\:\Omega\) load at Port 2 when the available power at Port 1 is \(0\text{ dBm}\)?

- A matched lossless transmission line has a length of one-quarter wavelength. What are the scattering parameters of the two-port?
- A connector has the scattering parameters \(S_{11} = 0.05,\: S_{21} = 0.9,\: S_{12} = 0.9,\) and \(S_{22} = 0.04\) and the reference impedance is \(50\:\Omega\). What is the return loss in \(\text{dB}\) of the connector at Port 1 in a \(50\:\Omega\) system?
- The scattering parameters of an amplifier are \(S_{11} = 0.5,\: S_{21} = 2.,\: S_{12} = 0.1,\) and \(S_{22} = −0.2\) and the reference impedance is \(50\:\Omega\). If the amplifier is terminated at Port 2 in a resistance of \(25\:\Omega\), what is the return loss in \(\text{dB}\) at Port 1?
- A two-port network has the scattering parameters \(S_{11} = −0.5,\: S_{21} = 0.9,\: S_{12} = 0.8,\) and \(S_{22} = 0.04\) and the reference impedance is \(50\:\Omega\).
- What is the return loss in \(\text{dB}\) of the connector at Port 1 in a \(50\:\Omega\) system?
- Is the two-port reciprocal and why?

- A two-port network has the scattering parameters \(S_{11} = −0.2,\: S_{21} = 0.8,\: S_{12} = 0.7,\) and \(S_{22} = 0.5\) and the reference impedance is \(75\:\Omega\).
- What is the return loss in \(\text{dB}\) of the connector at Port 1 in a \(75\:\Omega\) system?
- Is the two-port reciprocal and why?

- A cable has the scattering parameters \(S_{11} = 0.1,\: S_{21} = 0.7,\: S_{12} = 0.7,\) and \(S_{22} = 0.1\). At Port 2 is a \(55\:\Omega\) load and the \(S\) parameters and reflection coefficients are referred to \(50\:\Omega\).
- What is the load’s reflection coefficient?
- What is the input reflection coefficient of the terminated cable?
- What is the return loss, at Port 1 and in \(\text{dB}\), of the cable terminated in the load?

- A cable has the \(50\:\Omega\) scattering parameters \(S_{11} = 0.05,\: S_{21} = 0.5,\: S_{12} = 0.5,\) and \(S_{22} = 0.05\). What is the insertion loss of the cable if the source at Port 1 has a \(50\:\Omega\) Thevenin impedance and the termination at Port 2 is \(50\:\Omega\)? Express your answer in decibels.
- A \(1\text{ m}\) long cable has the \(50\:\Omega\) scattering parameters \(S_{11} = 0.1,\: S_{21} = 0.7,\: S_{12} = 0.7,\) and \(S_{22} = 0.1\). The cable is used in a \(55\:\Omega\) system. Express your answers in decibels.
- What is the return loss of the cable in the \(55\:\Omega\) system? (Hint see Section sec:input:terminated:two:port and consider finding \(Z_{i}n\).]
- What is the insertion loss of the cable in the \(55\:\Omega\) system? Follow the procedure in Example 7.2.1.
- What is the return loss of the cable in a \(50\:\Omega\) system?
- What is the insertion loss of the cable in a \(50\:\Omega\) system?

- A \(1\text{ m}\) long cable has the \(50\:\Omega\) scattering parameters \(S_{11} = 0.05,\: S_{21} = 0.5,\: S_{12} = 0.5,\) and \(S_{22} = 0.05\). The Thevenin equivalent impedance of the source and terminating load impedances of the cable are \(50\:\Omega\). Express your answers in decibels.
- What is the return loss of the cable?
- What is the insertion loss of the cable?

- A lossy directional coupler has the following \(50\:\Omega\: S\) parameters:

\[S=\left[\begin{array}{cccc}{0}&{-0.95\jmath}&{0.005}&{0.1}\\{-0.95\jmath}&{0}&{0.1}&{0.005}\\{0.005}&{0.1}&{0}&{-0.95\jmath}\\{0.1}&{0.005}&{-0.95\jmath}&{0}\end{array}\right]\nonumber\]- What are the through (transmission) paths (identify two paths)? That is, identify the pairs of ports at the ends of the through paths.
- What is the coupling in decibels?
- What is the isolation in decibels?
- What is the directivity in decibels?

- A directional coupler has the following characteristics: coupling factor \(C = 20\), transmission factor \(0.9\), and directivity factor \(25\text{ dB}\). Also, the coupler is matched so that \(S_{11} =0= S_{22} = S_{33} = S_{44}\).
- What is the isolation factor in decibels?
- Determine the power dissipated in the directional coupler if the input power to Port 1 is \(1\text{ W}\).

- A lossy directional coupler has the following \(50\:\Omega\: S\) parameters:

\[S=\left[\begin{array}{cccc}{0}&{0.25}&{-0.9\jmath}&{0.01}\\{0.25}&{0}&{0.01}&{-0.9\jmath}\\{-0.9\jmath}&{0.01}&{0}&{0.25}\\{0.01}&{-0.9\jmath}&{0.25}&{0}\end{array}\right]\nonumber\]- Which port is the input port (there could be more than one answer)?
- What is the coupling in decibels?
- What is the isolation in decibels?
- What is the directivity factor in decibels?

- A directional coupler using coupled lines has a coupling factor of \(3.38\), a transmission factor of \(−\jmath 0.955\), and infinite directivity and isolation. The input port is Port 2 and the through port is Port 2. Write down the \(4\times 4 S\) parameter matrix of the coupler.

## 7.8.1 Exercises by Selection

\(†\)challenging

\(§7.3 1, 2, 3†, 4†, 5†\)

\(§7.4 6, 7, 8, 9, 10, 11, 12†, 13\)

\(§7.5 14†, 15†, 16†, 17†\)

## 7.8.2 Answers to Selected Exercises

- \(0.638-\jmath 0.079\)

- (d) \(50+\jmath 100\:\Omega\)

- \(26\text{ dB}\)

- (b) \(2.99\text{ dB}\)

- (b) \(187\text{ mW}\)
- (d) \(28\text{ dB}\)