Lossless one-port properties; Foster I and Foster II

Properties of lossless one-port network

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).

Given $F(s) = \dfrac{s(s^2+2)(s^2+4)}{(s^2+1)(s^2+3)} = \dfrac{s^5+6s^3+8s}{s^4+4s^2+3}$: z 0, p 1, z 1.414, p 1.732, z 2, p ∞ — a valid LC impedance.

Foster-I realization

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

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

o--[L1]--[L2||C2]--[L3||C3]--o

Foster-II realization

Take $Y(s) = 1/Z(s) = \frac{s^{4} + 4 s^{2} + 3}{s^{5} + 6 s^{3} + 8 s}$ and expand $Y(s)$ in partial fractions (each term is an admittance; the branches are in parallel):

$$\begin{aligned} k_0 &= \left[s\,Y(s)\right]_{s=0} = \frac{3}{8} \\ 2k_{2} &= \left[\frac{(s^2+2)\,Y(s)}{s}\right]_{s^2=-2} = \frac{1}{4} \\ 2k_{3} &= \left[\frac{(s^2+4)\,Y(s)}{s}\right]_{s^2=-4} = \frac{3}{8} \end{aligned}$$ $$Y(s) = \frac{3}{8 s} + \frac{s}{4 \left(s^{2} + 2\right)} + \frac{3 s}{8 \left(s^{2} + 4\right)}$$

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  [L1]    [L2]    [L3]
   |      [C2]    [C3]
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