RC impedance properties; Foster I and Cauer I

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 $Y_{RL}(s)$.

Testing the functions

Realize $Z(s) = \dfrac{(s+3)(s+7)}{(s+1)(s+5)} = \dfrac{s^2+10s+21}{s^2+6s+5}$.

Foster-I realization

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

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

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

Cauer-I realization

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. $Z_1(s) = \frac{s^{2} + 10 s + 21}{s^{2} + 6 s + 5}$ → remove $1$, giving series $R_1$ = 1 Ω; remainder $\frac{4 \left(s + 4\right)}{\left(s + 1\right) \left(s + 5\right)}$.
2. $Y_2(s) = \frac{s^{2} + 6 s + 5}{4 s + 16}$ → remove $\frac{s}{4}$, giving shunt $C_2$ = 1/4 F (≈ 0.25 F); remainder $\frac{2 s + 5}{4 \left(s + 4\right)}$.
3. $Z_3(s) = \frac{4 s + 16}{2 s + 5}$ → remove $2$, giving series $R_3$ = 2 Ω; remainder $\frac{6}{2 s + 5}$.
4. $Y_4(s) = \frac{2 s + 5}{6}$ → remove $\frac{s}{3}$, giving shunt $C_4$ = 1/3 F (≈ 0.3333 F); remainder $\frac{5}{6}$.
5. $Z_5(s) = \frac{6}{5}$ → remove $\frac{6}{5}$, giving series $R_5$ = 6/5 Ω (≈ 1.2 Ω); remainder $0$.

o--[R1]----+---[R3]----+---[R5]-+
           |           |        |
         [C2]        [C4]       |
           |           |        |
o----------+-----------+--------+