RC impedance properties; Foster and Cauer synthesis

Properties of RC impedance function

An RC driving-point impedance $Z_{RC}(s)$ has these properties:

1. All poles and zeros are simple and lie on the negative real axis of the $s$-plane (including the origin).
2. Poles and zeros alternate along the negative real axis.
3. The critical frequency nearest the origin (lowest) is a pole; it may be at $s=0$.
4. The critical frequency farthest from the origin is a zero; it may be at $s=\infty$.
5. $Z_{RC}(0) > Z_{RC}(\infty)$, and $Z_{RC}(\infty)\ge 0$ is a constant (no pole at infinity).
6. Residues of $Z_{RC}(s)$ at its poles are real and positive; on the real axis $dZ(\sigma)/d\sigma<0$.
7. The same form describes an RL admittance $Y_{RL}(s)$.

Checking the given function

$Z(s) = \dfrac{3(s+2)(s+4)}{s(s+3)} = \dfrac{3s^2+18s+24}{s^2+3s}$: poles at $0, -3$; zeros at $-2, -4$. Order p 0, z 2, p 3, z 4 — alternating, pole nearest origin, zero farthest; $Z(0)=\infty > Z(\infty)=3$. Valid RC impedance.

Foster (Foster-I) form

Expand $Z(s)$ in partial fractions; every residue must be positive:

$$\begin{aligned} k_\infty &= Z(\infty) = 3 \\ k_0 &= [s\,Z(s)]_{s=0} = 8 \\ k_{3} &= [(s+3)Z(s)]_{s=-3} = 1 \end{aligned}$$ $$Z(s) = 3 + \frac{8}{s} + \frac{1}{s + 3}$$

o--[R1]--[C2]--[R3||C3]--o

Cauer (Cauer-I) form

Divide with descending powers (continued fraction about $s=\infty$). For an RC impedance the quotients alternate between a constant (series $R$) and $ks$ (shunt $C$):

1. $Z_1(s) = \frac{3 s^{2} + 18 s + 24}{s^{2} + 3 s}$ → remove $3$, giving series $R_1$ = 3 Ω; remainder $\frac{3 \left(3 s + 8\right)}{s \left(s + 3\right)}$.
2. $Y_2(s) = \frac{s^{2} + 3 s}{9 s + 24}$ → remove $\frac{s}{9}$, giving shunt $C_2$ = 1/9 F (≈ 0.1111 F); remainder $\frac{s}{9 \left(3 s + 8\right)}$.
3. $Z_3(s) = \frac{27 s + 72}{s}$ → remove $27$, giving series $R_3$ = 27 Ω; remainder $\frac{72}{s}$.
4. $Y_4(s) = \frac{s}{72}$ → remove $\frac{s}{72}$, giving shunt $C_4$ = 1/72 F (≈ 0.01389 F); remainder $0$.

o--[R1]----+---[R3]----+
           |           |
         [C2]        [C4]
           |           |
o----------+-----------+