Validate LC impedance; Foster I and Cauer II

Checking validity

A given $Z(s)$ is checked as follows:

1. Factor numerator and denominator. Each must be a product of factors $s$ and $(s^2+\omega_i^2)$ only, so that one is even and the other odd.
2. Write the critical frequencies in increasing order of $\omega$ and check that poles and zeros alternate.
3. Check that there is a pole or zero at $s=0$ and at $s=\infty$ (degrees differ by exactly 1).
4. Check that all residues of the partial-fraction expansion are positive (this follows when steps 1–3 hold).

For the given function:

$$Z(s) = \frac{8s^3+10s}{s^4+6s^2+5} = \frac{8s(s^2+1.25)}{(s^2+1)(s^2+5)}$$

Odd/even; zeros at $\omega = 0,\ 1.118,\ \infty$ (degree 3 over 4), poles at $\omega = 1,\ 2.236$; order z 0, p 1, z 1.118, p 2.236, z ∞ — they alternate, with a zero at both $s=0$ and $s=\infty$. Valid LC impedance. The partial fractions below also give positive residues, which confirms it.

Foster-I form

Expand $Z(s)$ in partial fractions (each term is an impedance; the elements are in series):

$$\begin{aligned} 2k_{1} &= \left[\frac{(s^2+1)\,Z(s)}{s}\right]_{s^2=-1} = \frac{1}{2} \\ 2k_{2} &= \left[\frac{(s^2+5)\,Z(s)}{s}\right]_{s^2=-5} = \frac{15}{2} \end{aligned}$$ $$Z(s) = \frac{s}{2 \left(s^{2} + 1\right)} + \frac{15 s}{2 \left(s^{2} + 5\right)}$$

o--[L1||C1]--[L2||C2]--o

Cauer-II form

$Z(s)$ has a zero at $s=0$, so $Y(s)=1/Z(s)$ has a pole at the origin and the first element is a shunt inductor. Arrange numerator and denominator in ascending powers of $s$ and divide repeatedly (continued fraction about $s=0$); each quotient is $k/s$:

1. $Y_1(s) = \frac{5 + 6 s^{2} + s^{4}}{10 s + 8 s^{3}}$ → quotient $\frac{1}{2 s}$ gives shunt $L_1$ = 2 H; remainder $\frac{s \left(s^{2} + 2\right)}{2 \left(4 s^{2} + 5\right)}$.
2. $Z_2(s) = \frac{10 + 8 s^{2}}{2 s + s^{3}}$ → quotient $\frac{5}{s}$ gives series $C_2$ = 1/5 F (≈ 0.2 F); remainder $\frac{3 s}{s^{2} + 2}$.
3. $Y_3(s) = \frac{2 + s^{2}}{3 s}$ → quotient $\frac{2}{3 s}$ gives shunt $L_3$ = 3/2 H (≈ 1.5 H); remainder $\frac{s}{3}$.
4. $Z_4(s) = \frac{3}{s}$ → quotient $\frac{3}{s}$ gives series $C_4$ = 1/3 F (≈ 0.3333 F); remainder $0$.

o----+---[C2]----+---[C4]-+
     |           |        |
   [L1]        [L3]       |
     |           |        |
o----+-----------+--------+