# 2.12: Exercises

- Page ID
- 41088

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 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 \(50\:\Omega\text{ S}\) 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 \(\mathsf{1}\)?
- Derive the two-port \(50\:\Omega\text{ S}\) parameters for the resistive circuit below.

Figure \(\PageIndex{1}\)

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

Figure \(\PageIndex{2}\)

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

Figure \(\PageIndex{3}\)

- 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.
- 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.
- 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.

Figure \(\PageIndex{4}\)

- 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.

Figure \(\PageIndex{5}\)

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

Figure \(\PageIndex{6}\)

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

Figure \(\PageIndex{7}\)

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

Figure \(\PageIndex{8}\)

- 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 \(\mathsf{1}\) 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 \(\mathsf{2}\)?
- What is the reflection coefficient of the load required for maximum power transfer at Port \(\mathsf{2}\)?

- 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 \(S_{21} = 0.5\).
- 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 \(\mathsf{2}\) when the available power at Port \(\mathsf{1}\) is \(0\text{ dBm}\)?

- 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 \(1\text{ mW}\). \(Z_{L} = 25\:\Omega\).

Figure \(\PageIndex{9}\)

- Is the two-port reciprocal and why?
- What is the voltage of the source?
- What is the power reflected from Port \(\mathsf{1}\)?
- Determine the \(z\) parameters of the two-port.
- Using \(z\) parameters, what is the power dissipated by the load at Port \(\mathsf{2}\)?

- 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?
- A matched lossless transmission line has a length of one-quarter wavelength. What are the scattering parameters of the two-port?
- Consider a two-port comprising a \(100\:\Omega\) resistor connected in series between the ports.
- Write down the \(S\) parameters of the two-port using a \(50\:\Omega\) reference impedance.
- From the scattering parameters derive the \(ABCD\) parameters of the two-port.

- Consider a two-port comprising a \(25\:\Omega\) resistor connected in shunt.
- Write down the scattering parameters of the two-port using a \(50\:\Omega\) reference impedance.
- From the scattering parameters derive the \(ABCD\) parameters of the two-port.

- What are the scattering transfer \(T\) parameters of a two port with the scattering parameters \(S_{11} = 0 = S_{22}\) and \(S_{12} = −\jmath = S_{21}\)?
- 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 \(10\:\Omega\)?
- 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.)
- What is the transmission coefficient in the \(75\:\Omega\) system?
- What is the attenuation (i.e., the insertion loss) in decibels in the \(75\:\Omega\) system?
- What is the input reflection coefficient at Port \(\mathsf{1}\) including the \(75\:\Omega\) at Port \(\mathsf{2}\)?

- 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.)
- What is the power delivered to a \(50\:\Omega\) load at port \(\mathsf{2}\)?
- What are the generalized scattering parameters with a \(50\:\Omega\) reference at port \(\mathsf{1}\) and a \(75\:\Omega\) reference at port \(\mathsf{2}\)?
- Using the generalized scattering parameters, calculate the power delivered to the \(50\:\Omega\) load at port \(\mathsf{2}\).

- 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 \(S_{11} =0= S_{22}\)?
- 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.
- 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 \(b_{2}\)?
- 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)?
- 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?
- 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 \(\mathsf{1}\)?
- 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 \(\mathsf{1}\) 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 \(\mathsf{1}\) 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 \(\mathsf{2}\) 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 \(\mathsf{1}\) 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 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 \(50\:\Omega\)?
- 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.
- 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 \(Z_{\text{in}}\).)
- What is the insertion loss in \(\text{dB}\) of the cable in the \(55\:\Omega\) system? Follow the procedure in Example 2.8.1.
- What is the return loss in \(\text{dB}\) of the cable in a \(50\:\Omega\) system?
- What is the insertion loss in \(\text{dB}\) of the cable in a \(50\:\Omega\) system?

- 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.
- What is the return loss in \(\text{dB}\) of the cable in the \(55\:\Omega\) system?
- What is the insertion loss in \(\text{dB}\) of the cable in the \(55\:\Omega\) system?

- 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.
- What is the return loss of the cable in the \(75\:\Omega\) system?
- What is the insertion loss of the cable in the \(75\:\Omega\) 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 \(50\:\Omega\).
- What is the return loss in \(\text{dB}\) of the cable?
- What is the insertion lossin dB of the cable?

- 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.
- What is the load’s \(\Gamma\)?
- What is \(a_{1}\) (see Equation (2.4.9)?
- What is \(b_{2}\)?
- What is \(a_{2}\)?
- What is the power, in \(\text{dBm}\), delivered to the load?
- What is the power, in \(\text{dBm}\), delivered to the load if the cable is removed and replaced by a direct connection?
- Hence what is the insertion loss, in \(\text{dB}\), of the cable?

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

\[S=\left[\begin{array}{cccc}{0}&{-0.95\jmath}&{0.005}&{0.1}\\{-0.95\jmath}&{0}&{0.1}&{0.005}\\{0.005}&{0.1}&{0}&{-0.95\jmath}\\{0.1}&{0.005}&{-0.95\jmath}&{0}\end{array}\right]\nonumber \]- 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 \(\mathsf{1}\) is \(1\text{ W}\).

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

\[S=\left[\begin{array}{cccc}{0}&{0.25}&{-0.9\jmath}&{0.01}\\{0.25}&{0}&{0.01}&{-0.9\jmath}\\{-0.9\jmath}&{0.01}&{0}&{0.25}\\{0.01}&{-0.9\jmath}&{0.25}&{0}\end{array}\right]\nonumber \]- 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?
- Draw the signal flow graph of the directional coupler.

- 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

- \(0.638 − \jmath 0.079\)

- \(\left[\begin{array}{cc}{0.96\angle -115^{\circ}}&{.995\angle -12.6^{\circ}}\\{.995\angle -12.6^{\circ}}&{0.97\angle -91^{\circ}}\end{array}\right]\)

- \(S_{21}=0.2222\)

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

- (c) \(-0.1807\)

- \(b_{2}=0.1155\)

- \(26\text{ dB}\)

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

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