Synthesis of electrical networks

H. Baher

Chapter 2 The Key Concepts: Fundamental Properties of Passive One-ports - all with Video Answers

Educators

Chapter Questions

Problem 1

Test the following polynomials for Hurwitz character,
(a) $p^3+6 p^2+10 p+8$
(b) $p^4+2 p^3+3 p^2+4 p+3$
(c) $p^5+6 p^4+12 p^3+12 p^2+11 p+5$
(d) $p^5+7 p^4+16 p^3+18 p^2+10 p+2$
(c) $p^6+7 p^4+14 p^2+8$
(f) $p^8+1$

Ankit Gupta

Numerade Educator

12:29

Problem 2

Test the following polynomials for non-negativity for all real $\omega$. The solution may require the use of a computer.)
(a) $\omega^{10}+\omega^4+3 \omega^6-3 \omega^4+4 \omega^2+10$
(b) $\omega^{10}-3 \omega^5+4 \omega^6-4 \omega^2+4$
(c) $\omega^{11}+5 \omega^{\mathrm{B}}+15 \omega^6-25 \omega^4+24 \omega^2+10$
(d) $\omega^{30}+\omega^5+3 \omega^6-3 \omega^4+4 \omega^2+10$

Uma Kumari

Numerade Educator

Problem 3

Show that if $Z(p)$ is a p.r.f., then $Z(k / p)$ is also p.r., where $k$ is a positive constant.

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Problem 4

Show that if $Z(p)$ and $W(p)$ are both p.r, then $Z(W)$ is also p.r.

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06:04

Problem 5

Test the following functions for p.r. character
(a) $\frac{1}{p^2+3 p+2}$
(b) $\frac{p^2+7 p+12}{p^2+3 p+2}$
(c) $\frac{p^2+p+2}{2 p^2+2 p+1}$
(d) $\frac{2 p^2+2 p+1}{2 p^3+2 p^2+2 p+1}$
(e) $\frac{2 p^3+3 p^2+6 p+1}{p^2+p+1}$
(f) $\frac{p^4+4 p^3+3 p^2-4 p+1}{p^4+p^3+3 p^2} \frac{4}{p+1}$
(g) $\frac{p(p+3)(p+5)}{(p+1)(p+4)}$
(h) $\frac{p^3+5 p^2+4 p}{p^4+8 p^2+15}$

Ma. Theresa Alin

Numerade Educator

Problem 6

Let $Z(p)$ be a rational p.r.f, and let $k$ be a posit :e constant.
(a) Show that the function
$$
\begin{aligned}
Z_1(p) & =\frac{k Z(p)-p Z(k)}{k Z(p)-p Z(p)} \\
& =\frac{N_1(p)}{D_1(p)}
\end{aligned}
$$
is also p.r.
(b) Show that $N_1$ and $D_1$ bave a common factor $(p-k)$.
(c) Show that $N_1$ and $D_1$ will have a common factor $\left(p^2-k^2\right)$ if $p=k$ is a zero of the even-part of $Z(p)$, i.e. if
$$
\left\{Z(p)+Z_s(p)\right\}_{p=k}=0
$$

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