# 7.8: Exercises

1. 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?
2. 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?
3. 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 $$50\:\Omega$$.
1. Is the two-port reciprocal? Explain.
2. 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?
3. What is the reflection coefficient of the load required for maximum power transfer at Port 2?
4. 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 $$S_{21} = 0.5$$.
1. If the network is reciprocal, what is $$S_{12}$$?
2. Is the two-port lossless?
3. What is the power delivered into the $$50\:\Omega$$ load at Port 2 when the available power at Port 1 is $$0\text{ dBm}$$?
5. A matched lossless transmission line has a length of one-quarter wavelength. What are the scattering parameters of the two-port?
6. 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?
7. 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?
8. 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 $$50\:\Omega$$.
1. What is the return loss in $$\text{dB}$$ of the connector at Port 1 in a $$50\:\Omega$$ system?
2. Is the two-port reciprocal and why?
9. 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 $$75\:\Omega$$.
1. What is the return loss in $$\text{dB}$$ of the connector at Port 1 in a $$75\:\Omega$$ system?
2. Is the two-port reciprocal and why?
10. 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 $$50\:\Omega$$.
1. What is the load’s reflection coefficient?
2. What is the input reflection coefficient of the terminated cable?
3. What is the return loss, at Port 1 and in $$\text{dB}$$, of the cable terminated in the load?
11. 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.
12. 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.
1. What is the return loss of the cable in the $$55\:\Omega$$ system? (Hint see Section sec:input:terminated:two:port and consider finding $$Z_{i}n$$.]
2. What is the insertion loss of the cable in the $$55\:\Omega$$ system? Follow the procedure in Example 7.2.1.
3. What is the return loss of the cable in a $$50\:\Omega$$ system?
4. What is the insertion loss of the cable in a $$50\:\Omega$$ system?
13. 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.
1. What is the return loss of the cable?
2. What is the insertion loss of the cable?
14. A lossy directional coupler has the following $$50\:\Omega\: S$$ parameters:
$S=\left[\begin{array}{cccc}{0}&{-0.95\jmath}&{0.005}&{0.1}\\{-0.95\jmath}&{0}&{0.1}&{0.005}\\{0.005}&{0.1}&{0}&{-0.95\jmath}\\{0.1}&{0.005}&{-0.95\jmath}&{0}\end{array}\right]\nonumber$
1. What are the through (transmission) paths (identify two paths)? That is, identify the pairs of ports at the ends of the through paths.
2. What is the coupling in decibels?
3. What is the isolation in decibels?
4. What is the directivity in decibels?
15. 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 $$S_{11} =0= S_{22} = S_{33} = S_{44}$$.
1. What is the isolation factor in decibels?
2. Determine the power dissipated in the directional coupler if the input power to Port 1 is $$1\text{ W}$$.
16. A lossy directional coupler has the following $$50\:\Omega\: S$$ parameters:
$S=\left[\begin{array}{cccc}{0}&{0.25}&{-0.9\jmath}&{0.01}\\{0.25}&{0}&{0.01}&{-0.9\jmath}\\{-0.9\jmath}&{0.01}&{0}&{0.25}\\{0.01}&{-0.9\jmath}&{0.25}&{0}\end{array}\right]\nonumber$
1. Which port is the input port (there could be more than one answer)?
2. What is the coupling in decibels?
3. What is the isolation in decibels?
4. What is the directivity factor in decibels?
17. 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

1. $$0.638-\jmath 0.079$$
1. (d) $$50+\jmath 100\:\Omega$$
1. $$26\text{ dB}$$
1. (b) $$2.99\text{ dB}$$
1. (b) $$187\text{ mW}$$
2. (d) $$28\text{ dB}$$