Chapter 6, Synthesis of RLC Driving-point Impedances: the Link with Filter Synthesis Video Solutions, Synthesis of electrical networks

Problem 1

Consider the p.r. impedance
$$
Z(p)=\frac{p^4+2 p^3+6 p^2+2 p+4}{p^3+p^2+4 p}
$$
(a) Perform the Brune preamble on $Z(p)$, i.e. extract all $j \omega$-axis poles of $Z$ and $Y$, including those at $p=0, p=\infty$.
(b) Evaluate the even-part of the remainder impedatce.
(c) Use cascade synthesis to realize the remainder, hence the entire impedance.

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01:11

Problem 2

Realize the following minimum reactance and minimum susceptance p.r.f. using the technique of cascade synthesis.
$$
Z(p)=\frac{p^4+4 p^3+3 p^2+4 p+1}{p^4+p^3+3 p^2+p+1}
$$

Narayan Hari

Numerade Educator

15:53

Problem 3

Given the polynomial matrix
$$
[t(p)]=\left[\begin{array}{cc}
9 p^2+1 & 2 p^3+3 p \\
18 p^3+3 p & 4 p^4+8 p^2+
\end{array}\right]
$$
verify that $[t(p)]$ is realizable as the polynomial tr nsmission matrix of a lossless reciprocal two-port. Find a realization by caicade synthesis.

Millie Lopez

Numerade Educator

Problem 4

A lossless two-port has the transmission coefficient
$$
S_{21}(p)=\frac{3}{p^2+3 p+3}
$$
referred to $1 \Omega$ terminations. Calculate the input pedance of the $1 \Omega$ terminated lossless two-port, and hence find a realition.

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Problem 5

Realize the following Foster matrices by casca'e synthesis. $\mathrm{r}^{+\cdots}$
$$
\begin{aligned}
& {[Z(p)]=\left[\begin{array}{cc}
\frac{2+p^2}{p} & \frac{2-2 p^2}{p} \\
\frac{2-2 p^2}{p} & \frac{2+4 p^2}{p}
\end{array}\right]} \\
& {[Z(p)]=\left[\begin{array}{ll}
\frac{p^4+3 p^2+1}{p^3+p} & \frac{2 p^4+2 p^2+1}{p^3+p} \\
\frac{2 p^4+2 p^2+1}{p^3+p} & \frac{4 p^4+6 p^2+1}{p^3+p}
\end{array}\right]}
\end{aligned}
$$
(c)
(d)
$$
\begin{aligned}
& {[Y(p)]=\left[\begin{array}{cc}
\frac{p^2+1}{p} & -\frac{p^2+2}{p} \\
-\frac{p^2+2}{p} & \frac{p^2+4}{p}
\end{array}\right]} \\
& {[Y(p)]=\left[\begin{array}{cc}
\frac{p^4+4 p^2+1}{p^3+p} & -\frac{p^4+p^2+2}{p^3+p} \\
-\frac{p^4+p^2+2}{p^3+p} & \frac{p^4+7 p^2+4}{p^3+p}
\end{array}\right]}
\end{aligned}
$$
117

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04:45

Problem 6

Consider the all-pass transmission coefficient, referred to $1 \Omega$ terminations,
$$
S_{21}(p)=\frac{4-6 p+4 p^2-p^3}{4+6 p+4 p^2+p^3}
$$

Find a reciprocal realization by cascade synthesis.

Prachita Kush

Numerade Educator

Problem 7

Consider the singly terminated two-port $N$ shown in Fig. P6.7. Define the transfer impedance as
$$
Z_{21}(p)=\frac{V_2(p)}{I_1(p)}
$$

Derive the necessary and sufficient conditions under which $Z_{21}$ is realizable as the transfer impedance of a lossless reciprocal two-port, hence develop a synthesis technique (Hint: express $Z_{21}$ in terms of the transmission parameters of $N$ and compare with the conditions of Theorem 5.4 and Corollary 5.3.)
Fig. P6.7 Problem 6.7 .

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03:04

Problem 8

Using the results of Problem 6.7, realize the transfer impedance
$$
Z_{21}(p)=\frac{2 p^2+1}{4 p^2+p+1}
$$
as that of a $1 \Omega$-terminated lossless reciprocal two-port.

Prachita Kush

Numerade Educator