RC impedance properties; Foster form

Properties of RC driving-point impedance

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 functions

Foster form of (a)

Foster-I (series) form: Expand $Z(s)$ in partial fractions; every residue must be positive:

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

o--[R1]--[R2||C2]--[R3||C3]--o

Foster-II (parallel) form: Take $Y(s)=1/Z(s) = \frac{s^{2} + 6 s + 5}{s^{2} + 9 s + 18}$. Expand $Y(s)/s$ in partial fractions and multiply back by $s$:

$$\begin{aligned} k_0 &= Y(0) = \frac{5}{18} \\ k_{2} &= \left[\frac{(s+3)Y(s)}{s}\right]_{s=-3} = \frac{4}{9} \\ k_{3} &= \left[\frac{(s+6)Y(s)}{s}\right]_{s=-6} = \frac{5}{18} \end{aligned}$$ $$Y(s) = \frac{5}{18} + \frac{4 s}{9 \left(s + 3\right)} + \frac{5 s}{18 \left(s + 6\right)}$$

o--+-------+-------+
   |       |       |
  [R1]    [R2]    [R3]
   |      [C2]    [C3]
   |       |       |
o--+-------+-------+