LC impedance check; Cauer II realization

Testing the functions

Tests for a valid LC impedance: (1) even/odd or odd/even ratio, (2) simple poles and zeros only on the $j\omega$ axis, (3) poles and zeros alternate, (4) a pole or a zero at both $s=0$ and $s=\infty$ (degrees differ by exactly 1), (5) positive residues.

Cauer-II realization of the valid function

$Z(s) = \dfrac{s(s^2+4)}{2(s^2+1)(s^2+9)} = \dfrac{s^3+4s}{2s^4+20s^2+18}$.

$Z(s)$ 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. $Y_1(s) = \frac{18 + 20 s^{2} + 2 s^{4}}{4 s + s^{3}}$ → quotient $\frac{9}{2 s}$ gives shunt $L_1$ = 2/9 H (≈ 0.2222 H); remainder $\frac{s \left(4 s^{2} + 31\right)}{2 \left(s^{2} + 4\right)}$.
2. $Z_2(s) = \frac{8 + 2 s^{2}}{31 s + 4 s^{3}}$ → quotient $\frac{8}{31 s}$ gives series $C_2$ = 31/8 F (≈ 3.875 F); remainder $\frac{30 s}{31 \left(4 s^{2} + 31\right)}$.
3. $Y_3(s) = \frac{961 + 124 s^{2}}{30 s}$ → quotient $\frac{961}{30 s}$ gives shunt $L_3$ = 30/961 H (≈ 0.03122 H); remainder $\frac{62 s}{15}$.
4. $Z_4(s) = \frac{15}{62 s}$ → quotient $\frac{15}{62 s}$ gives series $C_4$ = 62/15 F (≈ 4.133 F); remainder $0$.

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