Realizability conditions and lossless two-port properties

Properties for a function to be realizable

A driving-point immittance $F(s)$ is realizable with passive $R$, $L$, $C$ (and ideal transformers) if and only if it is a positive real (PR) function: $F(s)$ is real for real $s$, and $\text{Re}\,F(s) \ge 0$ whenever $\text{Re}\,s \ge 0$. In practice this is tested through these properties:

1. $F(s) = N(s)/D(s)$ is a ratio of polynomials with real, positive coefficients.
2. Poles and zeros lie in the left half plane or on the $j\omega$ axis; $N(s)$ and $D(s)$ are Hurwitz.
3. Poles and zeros on the $j\omega$ axis are simple, with real and positive residues.
4. The degrees of $N$ and $D$ differ by at most 1, and so do their lowest powers (no multiple pole or zero at $s=\infty$ or $s=0$).
5. $\text{Re}\,F(j\omega) \ge 0$ for all $\omega$.

For a transfer function to be realizable, it must be stable (poles strictly in the left half plane, denominator strictly Hurwitz), and the numerator degree must not exceed the denominator degree.

Properties of a lossless two-port

For a lossless (LC) reciprocal two-port described by $z_{11}, z_{12}=z_{21}, z_{22}$ (or the $y$-parameters):

1. $z_{11}$ and $z_{22}$ are LC driving-point functions (odd rational functions with alternating simple poles and zeros on the $j\omega$ axis, positive residues).
2. $z_{21}$ is also an odd rational function; its poles lie on the $j\omega$ axis and are simple.
3. Every pole of $z_{21}$ is also a pole of $z_{11}$ and $z_{22}$. $z_{11}$ or $z_{22}$ may have extra private poles not present in $z_{21}$.
4. Residue condition at every pole: $k_{11} \ge 0$, $k_{22} \ge 0$, $k_{21}$ real (may be negative), and

$$k_{11}k_{22} - k_{21}^2 \ge 0$$

When equality holds at every pole the network is called compact. 5. The zeros of transmission (zeros of $z_{21}$) are the roots of an even or odd polynomial, so they are symmetric about both axes. In ladder realizations they lie on the $j\omega$ axis, including $0$ and $\infty$. They need not alternate with the poles. 6. Since no power is lost, $|S_{11}(j\omega)|^2 + |S_{21}(j\omega)|^2 = 1$ when the two-port is resistively terminated.