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2.12: Exercises

  • Page ID
    41088
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    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 \(50\:\Omega\text{ S}\) 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 \(\mathsf{1}\) is connected to a \(50\:\Omega\) source with an available power of \(1\text{ W}\) and Port \(\mathsf{2}\) is terminated in \(50\:\Omega\). What is the power reflected from Port \(\mathsf{1}\)?
    3. Derive the two-port \(50\:\Omega\text{ S}\) parameters for the resistive circuit below.

    clipboard_e6752dc8e282a6c430169a3b255a951e4.png

    Figure \(\PageIndex{1}\)

    1. Derive the two-port \(50\:\Omega\text{ S}\) parameters at \(1\text{ GHz}\) for the circuit below.

    clipboard_e516d538b65f6b66072363ab30f6e848d.png

    Figure \(\PageIndex{2}\)

    1. Derive the two-port \(50\:\Omega\text{ S}\) parameters at \(1\text{ GHz}\) for the circuit below.

    clipboard_e56e9086bd40863bb7a33d5052d5a3828.png

    Figure \(\PageIndex{3}\)

    1. Derive the two-port \(S\) parameters of a shunt \(25\:\Omega\) resistor in a \(50\:\Omega\) reference system using the method presented in Section 2.3.5.
    2. Derive the two-port \(S\) parameters of a series \(25\:\Omega\) resistor in a \(100\:\Omega\) reference system using the method presented in Section 2.3.5.
    3. A two-port consists of a \(π\) resistor network as shown. Derive the scattering parameters referenced to \(50\:\Omega\) using the method in Section 2.3.5.

    clipboard_e0902f2f8b34fe7631cf1b65d732a2259.png

    Figure \(\PageIndex{4}\)

    1. A two-port consists of a \(\text{T}\) resistor network as shown. Derive the scattering parameters referenced to \(1\:\Omega\) using the method in 2.3.5.

    clipboard_e31556d965819fb9862875a6fc58bb211.png

    Figure \(\PageIndex{5}\)

    1. Derive the \(50\:\Omega\text{ S}\) parameters of the following.

    clipboard_e849d143b0e607ed228e79d4cec31e0d5.png

    Figure \(\PageIndex{6}\)

    1. Derive the \(50\:\Omega\text{ S}\) parameters of the following.

    clipboard_eab6ab9d87edbd1884c5268e8ec1229c7.png

    Figure \(\PageIndex{7}\)

    1. Derive the \(100\:\Omega\text{ S}\) parameters of the following.

    clipboard_e312914c94bb8b3632b67dc083c5dd5ea.png

    Figure \(\PageIndex{8}\)

    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 \(50\:\Omega\).
      1. Is the two-port reciprocal? Explain.
      2. Consider that Port \(\mathsf{1}\) 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 \(\mathsf{2}\)?
      3. What is the reflection coefficient of the load required for maximum power transfer at Port \(\mathsf{2}\)?
    2. In characterizing a two-port, power could only be applied at Port \(\mathsf{1}\). The signal reflected was measured and the signal at a \(50\:\Omega\) load at Port \(\mathsf{2}\) 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 \(\mathsf{2}\) when the available power at Port \(\mathsf{1}\) is \(0\text{ dBm}\)?
    3. The \(S\) parameters of a two-port are \(S_{11} = 0.25,\: S_{12} = 0,\: S_{21} = 1.2,\) and \(S_{22} = 0.5\). The system reference impedance is \(50\:\Omega\) and \(Z_{G} = 50\:\Omega\). The power available from the source is \(1\text{ mW}\). \(Z_{L} = 25\:\Omega\).

    clipboard_ee63849bd59b9ffbaeb2eb9539950dfd0.png

    Figure \(\PageIndex{9}\)

    1. Is the two-port reciprocal and why?
    2. What is the voltage of the source?
    3. What is the power reflected from Port \(\mathsf{1}\)?
    4. Determine the \(z\) parameters of the two-port.
    5. Using \(z\) parameters, what is the power dissipated by the load at Port \(\mathsf{2}\)?
    1. The scattering parameters of a two-port network are \(S_{11} = 0.25,\: S_{21} = 2.0,\: S_{21} = 0.1,\) and \(S_{22} = 0.5\) and the reference impedance is \(50\:\Omega\). What are the scattering transfer \((T)\) parameters of the two-port?
    2. A matched lossless transmission line has a length of one-quarter wavelength. What are the scattering parameters of the two-port?
    3. Consider a two-port comprising a \(100\:\Omega\) resistor connected in series between the ports.
      1. Write down the \(S\) parameters of the two-port using a \(50\:\Omega\) reference impedance.
      2. From the scattering parameters derive the \(ABCD\) parameters of the two-port.
    4. Consider a two-port comprising a \(25\:\Omega\) resistor connected in shunt.
      1. Write down the scattering parameters of the two-port using a \(50\:\Omega\) reference impedance.
      2. From the scattering parameters derive the \(ABCD\) parameters of the two-port.
    5. What are the scattering transfer \(T\) parameters of a two port with the scattering parameters \(S_{11} = 0 = S_{22}\) and \(S_{12} = −\jmath = S_{21}\)?
    6. The scattering parameters of a two-port amplifier referred to \(50\:\Omega\) are \(S_{11} = 0.5,\: S_{21} = 2,\: S_{12} = 0.1,\) and \(S_{22} = −0.01\). What are the generalized scattering parameters of the two-port network if the reference impedance at Port \(\mathsf{1}\) is \(100\:\Omega\) and at Port \(\mathsf{2}\) is \(10\:\Omega\)?
    7. A \(50\:\Omega\), \(10\text{ dB}\) attenuator is inserted in a \(75\:\Omega\) system. (That is, the attenuator is a two-port network and using a \(50\:\Omega\) reference the \(S\)-parameters of the attenuator are \(S_{11} =0= S_{22}\) and the insertion loss of the two-port in the \(50\:\Omega\) system is \(10\text{ dB}\). Now consider the same two-port in a \(75\:\Omega\) system.)
      1. What is the transmission coefficient in the \(75\:\Omega\) system?
      2. What is the attenuation (i.e., the insertion loss) in decibels in the \(75\:\Omega\) system?
      3. What is the input reflection coefficient at Port \(\mathsf{1}\) including the \(75\:\Omega\) at Port \(\mathsf{2}\)?
    8. A two-port has the \(50-\Omega\) scattering parameters \(S_{11} = 0.1;\: S_{12} = 0.9 = S_{21};\: S_{22} = 0.2\). The \(50-\Omega\) source at port \(\mathsf{1}\) has an available power of \(1\text{ W}\). (This is the power that would be delivered to a \(50\:\Omega\) termination at the source.)
      1. What is the power delivered to a \(50\:\Omega\) load at port \(\mathsf{2}\)?
      2. What are the generalized scattering parameters with a \(50\:\Omega\) reference at port \(\mathsf{1}\) and a \(75\:\Omega\) reference at port \(\mathsf{2}\)?
      3. Using the generalized scattering parameters, calculate the power delivered to the \(50\:\Omega\) load at port \(\mathsf{2}\).
    9. A two-port is matched to a source with a Thevenin impedance of \(50\:\Omega\) connected at Port \(\mathsf{1}\) and a load of \(25\:\Omega\) at Port \(\mathsf{2}\). If the two-port is represented by generalized scattering parameters, what should the normalization impedances at the ports be for \(S_{11} =0= S_{22}\)?
    10. A two-port is terminated at Port \(\mathsf{2}\) in \(50\:\Omega\). At Port \(\mathsf{1}\) is a source with a Thevenin equivalent impedance of \(75\:\Omega\). The generalized scattering parameters of the two-port are \(S_{11} = 0,\: S_{21} = 2.0,\: S_{12} = 0.1,\) and \(S_{22} = 0.5\). The reference impedances are \(75\:\Omega\) at Port \(\mathsf{1}\) and \(50\:\Omega\) at Port \(\mathsf{2}\). If the Thevenin equivalent source has a peak voltage of \(1\text{ V}\), write down the root power waves at each port.
    11. A two-port is terminated at Port \(\mathsf{2}\) in \(10\:\Omega\). At Port \(\mathsf{1}\) is a source with a Thevenin equivalent impedance of \(100\:\Omega\). The generalized scattering parameters of the two-port are \(S_{11} = 0.2,\: S_{21} = 0.5,\: S_{12} = 0.1,\) and \(S_{22} = 0.3\). The reference impedances are \(100\:\Omega\) at Port \(\mathsf{1}\) and \(10\:\Omega\) at Port \(\mathsf{2}\). If the Thevenin equivalent source has a peak voltage of \(50\text{ V}\), what are \(a_{1},\: b_{1},\: a_{2},\) and \(b_{2}\)?
    12. The scattering parameters of a two-port network are \(S_{11} = 0.6,\: S_{21} = 0.8,\: S_{12} = 0.5,\) and \(S_{22} = 0.3\) and the reference impedance is \(50\:\Omega\). At Port \(\mathsf{1}\) a \(50\:\Omega\) transmission line with an electrical length of \(90^{\circ}\) is connected and at Port \(\mathsf{2}\) a \(50\:\Omega\) transmission line with an electrical length of \(180^{\circ}\) is connected. What are the scattering parameters of the cascaded system (transmission line–original two-port–transmission line)?
    13. 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 \(\mathsf{1}\) in a \(50\:\Omega\) system?
    14. The scattering parameters of an amplifier are \(S_{11} = 0.5,\: S_{21} = 2.0,\: S_{12} = 0.1,\) and \(S_{22} = −0.2\) and the reference impedance is \(50\:\Omega\). If the amplifier is terminated at Port \(\mathsf{2}\) in a resistance of \(25\:\Omega\), what is the return loss in \(\text{dB}\) at Port \(\mathsf{1}\)?
    15. 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 \(\mathsf{1}\) in a \(50\:\Omega\) system?
      2. Is the two-port reciprocal and why?
    16. 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 \(\mathsf{1}\) in a \(75\:\Omega\) system?
      2. Is the two-port reciprocal and why?
    17. 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 \(\mathsf{2}\) 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 \(\mathsf{1}\) and in \(\text{dB}\), of the cable terminated in the load?
    18. 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 in \(\text{dB}\) of the cable if the source at Port \(\mathsf{1}\) has a \(50\:\Omega\) Thevenin impedance and the termination at Port \(\mathsf{2}\) is \(50\:\Omega\)?
    19. 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.
      1. What is the return loss in \(\text{dB}\) of the cable in the \(55\:\Omega\) system? (Hint see Section 2.3.4 and consider finding \(Z_{\text{in}}\).)
      2. What is the insertion loss in \(\text{dB}\) of the cable in the \(55\:\Omega\) system? Follow the procedure in Example 2.8.1.
      3. What is the return loss in \(\text{dB}\) of the cable in a \(50\:\Omega\) system?
      4. What is the insertion loss in \(\text{dB}\) of the cable in a \(50\:\Omega\) system?
    20. 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. Follow the procedure in Example 2.8.2.
      1. What is the return loss in \(\text{dB}\) of the cable in the \(55\:\Omega\) system?
      2. What is the insertion loss in \(\text{dB}\) of the cable in the \(55\:\Omega\) system?
    21. A \(10\text{ m}\) long cable has the \(50\:\Omega\) scattering parameters \(S_{11} = −0.1,\: S_{21} = 0.5,\: S_{12} = 0.5,\) and \(S_{22} = −0.1\). The cable is used in a \(75\:\Omega\) system. Use the change of \(S\) parameter reference impedance method [Parallels Example 2.8.2]. Express your answers in decibels.
      1. What is the return loss of the cable in the \(75\:\Omega\) system?
      2. What is the insertion loss of the cable in the \(75\:\Omega\) system?
    22. 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 \(50\:\Omega\).
      1. What is the return loss in \(\text{dB}\) of the cable?
      2. What is the insertion lossin dB of the cable?
    23. A cable in a \(50\:\Omega\) system has the \(S\) parameters \(S_{11} = 0.1,\: S_{21} = 0.7,\: S_{12} = 0.7,\) and \(S_{22} = 0.1\). The available power at Port \(\mathsf{1}\) is \(0\text{ dBm}\) and at Port \(\mathsf{2}\) is a \(55\:\Omega\) load. \(\Gamma\) is reflection coefficient and \(a_{n}\) and \(b_{n}\) are the root power waves at the \(n\)th port.
      1. What is the load’s \(\Gamma\)?
      2. What is \(a_{1}\) (see Equation (2.4.9)?
      3. What is \(b_{2}\)?
      4. What is \(a_{2}\)?
      5. What is the power, in \(\text{dBm}\), delivered to the load?
      6. What is the power, in \(\text{dBm}\), delivered to the load if the cable is removed and replaced by a direct connection?
      7. Hence what is the insertion loss, in \(\text{dB}\), of the cable?
    24. A lossy directional coupler has the following \(50\:\Omega\text{ S}\) 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?
    25. 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 \(\mathsf{1}\) is \(1\text{ W}\).
    26. A lossy directional coupler has the following \(50\:\Omega\text{ S}\) 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?
      5. Draw the signal flow graph of the directional coupler.
    27. 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 \(\mathsf{2}\) and the through port is Port \(\mathsf{2}\). Write down the \(4\times 4\:S\) parameter matrix of the coupler.

    2.12.1 Exercises by Section

    \(†\)challenging

    \(§2.3\: 1, 2, 3†, 4†, 5†, 6†, 7†, 8†, 9†, 10, 11, 12, 13†, 14†, 15†, 16, 17†, 18†, 19†, 20, 21, 22†\)

    \(§2.4\: 23†, 24, 25†, 26†\)

    \(§2.6\: 27\)

    \(§2.8\: 28, 29, 30, 31, 32, 33, 34†, 35, 36, 37, 38\)

    \(§2.9\: 39†, 40†, 41†, 42†\)

    2.12.2 Answers to Selected Exercises

    1. \(0.638 − \jmath 0.079\)
    1. \(\left[\begin{array}{cc}{0.96\angle -115^{\circ}}&{.995\angle -12.6^{\circ}}\\{.995\angle -12.6^{\circ}}&{0.97\angle -91^{\circ}}\end{array}\right]\)
    1. \(S_{21}=0.2222\)
    1. (d) \(50+\jmath 100\:\Omega\)
    1. (c) \(-0.1807\)
    1. \(b_{2}=0.1155\)
    1. \(26\text{ dB}\)
    1. (b) \(2.99\text{ dB}\)
    1. (b) \(187\text{mW}\)
    2. (d) \(28\text{ dB}\)

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