RC impedance properties; LC Cauer II realization

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 $Y_{RL}(s)$.

Example: $Z(s) = \dfrac{(s+2)(s+4)}{(s+1)(s+3)}$ has poles at $-1,-3$ and zeros at $-2,-4$ (p 1, z 2, p 3, z 4): the lowest is a pole, the highest a zero, they alternate, and $Z(0) = 8/3 > Z(\infty) = 1$. Its expansion $Z = 1 + \dfrac{1.5}{s+1} + \dfrac{0.5}{s+3}$ has positive residues, giving $1\ \Omega$ in series with $1.5\ \Omega \parallel \tfrac{2}{3}$ F and $\tfrac{1}{6}\ \Omega \parallel 2$ F.

Cauer-II realization of the LC function

$Z(s) = \dfrac{s(s^2+3)}{(s^2+1)(s^2+4)} = \dfrac{s^3+3s}{s^4+5s^2+4}$ (z 0, p 1, z 1.732, p 2, z ∞ — valid LC).

$Z(s)$ has a zero at $s=0$, so $Y(s)=1/Z(s)$ has a pole at the origin and the first element is a shunt inductor. Arrange numerator and denominator in ascending powers of $s$ and divide repeatedly (continued fraction about $s=0$); each quotient is $k/s$:

1. $Y_1(s) = \frac{4 + 5 s^{2} + s^{4}}{3 s + s^{3}}$ → quotient $\frac{4}{3 s}$ gives shunt $L_1$ = 3/4 H (≈ 0.75 H); remainder $\frac{s \left(3 s^{2} + 11\right)}{3 \left(s^{2} + 3\right)}$.
2. $Z_2(s) = \frac{9 + 3 s^{2}}{11 s + 3 s^{3}}$ → quotient $\frac{9}{11 s}$ gives series $C_2$ = 11/9 F (≈ 1.222 F); remainder $\frac{6 s}{11 \left(3 s^{2} + 11\right)}$.
3. $Y_3(s) = \frac{121 + 33 s^{2}}{6 s}$ → quotient $\frac{121}{6 s}$ gives shunt $L_3$ = 6/121 H (≈ 0.04959 H); remainder $\frac{11 s}{2}$.
4. $Z_4(s) = \frac{2}{11 s}$ → quotient $\frac{2}{11 s}$ gives series $C_4$ = 11/2 F (≈ 5.5 F); remainder $0$.

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