LC impedance properties; identify valid function

Properties of LC driving-point impedance

An LC (lossless) driving-point immittance $Z(s)$ or $Y(s)$ has these properties:

1. It is the ratio of an even polynomial to an odd polynomial, or odd to even, with real, positive coefficients.
2. All poles and zeros are simple and lie on the $j\omega$ axis (they occur in conjugate pairs $\pm j\omega_i$).
3. Poles and zeros alternate (interlace) along the $j\omega$ axis.
4. The highest powers of numerator and denominator differ by exactly 1; so do the lowest powers.
5. There is always either a pole or a zero at $s=0$, and either a pole or a zero at $s=\infty$.
6. The residues at all poles are real and positive.
7. On the $j\omega$ axis $Z(j\omega)=jX(\omega)$ is purely reactive and $dX/d\omega>0$ (Foster's reactance theorem).

Identifying the LC function

(1)

$$Z(s) = \frac{8s^3+10s}{s^4+6s^2+5} = \frac{8s(s^2+1.25)}{(s^2+1)(s^2+5)}$$

Odd/even; zeros at $\omega = 0,\ 1.118,\ \infty$ (degree 3 over 4), poles at $\omega = 1,\ 2.236$; order z 0, p 1, z 1.118, p 2.236, z ∞ — they alternate, with a zero at both $s=0$ and $s=\infty$. Valid LC impedance.

(2)

$$Z(s) = \frac{s^4+5s^2+4}{s^3+9s} = \frac{(s^2+1)(s^2+4)}{s(s^2+9)}$$

Even/odd and degrees differ by 1, but the critical frequencies are p 0, z 1, z 2, p 3, p ∞: two zeros (1 and 2) are adjacent and two poles (3 and ∞) are adjacent, so they do not alternate. The residue at $s^2=-9$ is $\left[\frac{(s^2+1)(s^2+4)}{s^2}\right]_{s^2=-9} = \frac{(-8)(-5)}{-9} = -\frac{40}{9} < 0$. Not LC.

Realization of the valid function (Foster-I)

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

$$\begin{aligned} 2k_{1} &= \left[\frac{(s^2+1)\,Z(s)}{s}\right]_{s^2=-1} = \frac{1}{2} \\ 2k_{2} &= \left[\frac{(s^2+5)\,Z(s)}{s}\right]_{s^2=-5} = \frac{15}{2} \end{aligned}$$ $$Z(s) = \frac{s}{2 \left(s^{2} + 1\right)} + \frac{15 s}{2 \left(s^{2} + 5\right)}$$

o--[L1||C1]--[L2||C2]--o