Chapter 5, Synthesis of Lossless Two-ports Video Solutions, Synthesis of electrical networks

Problem 1

Verify that the following is a Foster impedance matrix and find a realization
$$
[Z(p)]=\left[\begin{array}{cc}
\frac{1+p^2}{2 p} & \frac{4+p^2}{8 p} \\
\frac{4+p^2}{8 p} & \frac{16+p^2}{32 p}
\end{array}\right]
$$

Prachita Kush

Numerade Educator

Problem 2

Realize the Foster matrix,
$$
[Z(p)]=\left[\begin{array}{cc}
\frac{p^4+3 p^2+1}{p^3+p} & \frac{2 p^4+2 p^2-1}{p^3+p^4} \\
\frac{2 p^4+2 p^2+1}{p^3+p} & \frac{4 p^4+6 p^2+1}{p^3+p}
\end{array}\right]
$$
and show that the resulting lossless two-port is consact.

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03:13

Problem 3

Evaluate the scattering matrix, referred to $1 \Omega$ terminations, for each of the lossless two-ports of Problems 5.1 and 5.2.

Prachita Kush

Numerade Educator

Problem 4

Consider a passive lossless $n$-port as shown in "ig. P5.4. Show that it possesses a scattering matrix [S], referred to real terminations, which satisfies the same properties of Theorem 5.2 except that [S] is now an $n \times n$ matrix. In particular, show that the entries of $[S] \varepsilon$ defined by
(a) $S_{i t}=$ reflection coefficient of the one-port formed by closing all the other ports on their reference resistors.
(b) $S_{i j}(i \neq j)=$ transmission coefficient from port $j$ to port $i$ under reference terminating conditions.

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Problem 5

Consider the non-reciprocal three-port shown in Fig. P5.5(a). This is formed by adding a pair of terminals to an ideal gyrator. Find its scattering matrix referred to $1 \Omega$ resistors and show that the main diagonal entries are zero. Also show that when a signal is incident at port $i(i=1 \rightarrow 3)$ with all ports matched, none is reflected, none is transmitted to port $(i-1)$ but all the signal is transmitted to port $(i+1)$ without loss. Thus, the three-port has a cyclic power transmission property and is called a three-port circulator. The symbol for a circulator is hown in Fig. P5.5(b).

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