LC impedance check; Foster parallel and Cauer I

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.

• $Z(s) = \dfrac{s^4+10s^2+9}{s^3+4s} = \dfrac{(s^2+1)(s^2+9)}{s(s^2+4)}$: p 0, z 1, p 2, z 3, p ∞ — alternate. Valid LC.
• $Z(s) = \dfrac{s^3+4s}{s^4+5s^2+6} = \dfrac{s(s^2+4)}{(s^2+2)(s^2+3)}$: z 0, p 1.414, p 1.732, z 2 — two poles adjacent; the residue at $s^2=-3$ is $\left[\frac{s^2+4}{s^2+2}\right]_{s^2=-3} = \frac{1}{-1} = -1 < 0$. Not LC.

Foster parallel (Foster-II) form

Take $Y(s) = 1/Z(s) = \frac{s^{3} + 4 s}{s^{4} + 10 s^{2} + 9}$ and expand $Y(s)$ in partial fractions (each term is an admittance; the branches are in parallel):

$$\begin{aligned} 2k_{1} &= \left[\frac{(s^2+1)\,Y(s)}{s}\right]_{s^2=-1} = \frac{3}{8} \\ 2k_{2} &= \left[\frac{(s^2+9)\,Y(s)}{s}\right]_{s^2=-9} = \frac{5}{8} \end{aligned}$$ $$Y(s) = \frac{3 s}{8 \left(s^{2} + 1\right)} + \frac{5 s}{8 \left(s^{2} + 9\right)}$$

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

Cauer-I form

$Z(s)$ has a pole at $s=\infty$ (numerator degree is higher), so the first element is a series inductor. Divide repeatedly with polynomials in descending powers (continued fraction about $s=\infty$), inverting the remainder each time:

1. $Z_1(s) = \frac{s^{4} + 10 s^{2} + 9}{s^{3} + 4 s}$ → quotient $s$ gives series $L_1$ = 1 H; remainder $\frac{3 \left(2 s^{2} + 3\right)}{s \left(s^{2} + 4\right)}$.
2. $Y_2(s) = \frac{s^{3} + 4 s}{6 s^{2} + 9}$ → quotient $\frac{s}{6}$ gives shunt $C_2$ = 1/6 F (≈ 0.1667 F); remainder $\frac{5 s}{6 \left(2 s^{2} + 3\right)}$.
3. $Z_3(s) = \frac{12 s^{2} + 18}{5 s}$ → quotient $\frac{12 s}{5}$ gives series $L_3$ = 12/5 H (≈ 2.4 H); remainder $\frac{18}{5 s}$.
4. $Y_4(s) = \frac{5 s}{18}$ → quotient $\frac{5 s}{18}$ gives shunt $C_4$ = 5/18 F (≈ 0.2778 F); remainder $0$.

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