LC lossless check; Foster and Cauer synthesis

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.

Synthesize $Z_2(s) = \dfrac{s^4+12s^2+20}{s^3+5s}$.

Foster (Foster-I) form

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

$$\begin{aligned} k_\infty &= \left[\frac{Z(s)}{s}\right]_{s\to\infty} = 1 \\ k_0 &= \left[s\,Z(s)\right]_{s=0} = 4 \\ 2k_{3} &= \left[\frac{(s^2+5)\,Z(s)}{s}\right]_{s^2=-5} = 3 \end{aligned}$$ $$Z(s) = s + \frac{4}{s} + \frac{3 s}{s^{2} + 5}$$

o--[L1]--[C2]--[L3||C3]--o

Cauer (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. $Z_1(s) = \frac{s^{4} + 12 s^{2} + 20}{s^{3} + 5 s}$ → quotient $s$ gives series $L_1$ = 1 H; remainder $\frac{7 s^{2} + 20}{s \left(s^{2} + 5\right)}$.
2. $Y_2(s) = \frac{s^{3} + 5 s}{7 s^{2} + 20}$ → quotient $\frac{s}{7}$ gives shunt $C_2$ = 1/7 F (≈ 0.1429 F); remainder $\frac{15 s}{7 \left(7 s^{2} + 20\right)}$.
3. $Z_3(s) = \frac{49 s^{2} + 140}{15 s}$ → quotient $\frac{49 s}{15}$ gives series $L_3$ = 49/15 H (≈ 3.267 H); remainder $\frac{28}{3 s}$.
4. $Y_4(s) = \frac{3 s}{28}$ → quotient $\frac{3 s}{28}$ gives shunt $C_4$ = 3/28 F (≈ 0.1071 F); remainder $0$.

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