7. Generalization to lossy transmission lines with frequency-dependent parameters

So far, we only considered lossy transmission lines with frequency-independent parameters. However, many models of transmission line with frequency-dependent parameters can be found in the literature. In this section, we give the main ingredients to generalize Theorem 1 to the most common models encountered in the literature.

Such models may contain poles in $Y_{l}\left(s\right)$, associated with the time constant of a dielectric mechanism in the PCB material (see [28], [36])

where $G$ is the per-unit-length DC conductance, $C$ is the per-unit-length static capacitance and $\overline{\epsilon}(s)$ is the normalized complex permittivity for the dielectric material. The poles of $\overline{\epsilon}(s)$ correspond to time-constants associated with relaxation of the dipoles in the dielectric and lie in the left half-plane. These models also take the skin effect into account. The latter is usually modeled by introducing terms in $\sqrt{s}$, thereby introducing a branch-point of order 1 in $s=0$ in the expression for $Z_{l}\left(s\right)$:

where $R$, $L$, $M$ are positive constants, and $f(u)$ is analytic with $\Re(f(u))\geq 0$ on a neighborhood of $\{u: u=r\, e^{j\theta}, r\geq 0, -\pi/4\leq \theta \leq \pi/4\}$, and asymptotic behavior at $u=\infty$ of the form (see [28])

The principal branch of $\sqrt{s}$ is used so that $Z_{l}\left(s\right)$ is strictly positive real on $\overline{\Pi}^+$. The branch-point at $s=0$ in $Z_{l}\left(s\right)$ induces a branch-point at $0$ in the Y and Z-matrices (4.4) of the transmission line. The poles as well as the zeros of $Y_{l}\left(s\right)$ and $Z_{l}\left(s\right)$ also give rise to branch-points for $Z(s)$ and $Y(s)$, but these lie in the open left half-plane. Thus, the Y and Z-matrices are meromorphic on a two-sheeted Riemann surface ${\mathcal{S}}_\epsilon$ above $\{s; \Re(s)\geq -\epsilon \}$, for some $\epsilon>0$, with a branching of order $1$ at $0$.

Adapting the proofs of Theorem A.1, Lemma 1 and Theorem 1 to the present case, where (4.2) gets replaced by (7.1) and (7.2), one can show the following results.

Firstly, properties (i) and (ii) of Definition 1 hold on the principal sheet. Secondly, if $G(s)$ is a partial frequency response of a realistic linearized circuit, including transmission line models with frequency dependent parameters as above, then it is meromorphic on a Riemann surface $S_{\tilde{\epsilon}}$ as above, for some $\tilde\epsilon>0$ depending on the circuit. On the principal sheet, it has a Puiseux expansion at $0$ of the form

Let $\tilde{\Pi}_{0}^{+}$ (resp. $\tilde \Pi^+$) denote the subset of this sheet defined by $\Re(s) \geq 0$ (resp. $\Re(s) > 0$). Then, on the subset of $\tilde{\Pi}_{0}^{+}$ corresponding to $|s|>K$ for some $K>0$, $G(s)$ is bounded. Consequently, on the imaginary axis viewed as a subset of $\tilde{\Pi}_{0}^{+}$, we have a decomposition

where $h(s)\in \mathcal{H}^{\infty}$ and $\mathfrak{r}(s)$ is a strictly proper rational function having poles in the closed right half-plane only. Specifically, to get $h(s)$ we substract from $G(s)$ the singular part of the Puiseux expansion, and the singular part at the possible unstable poles which are finite in number, by the above mentioned generalization of Theorem 4. The resulting function $h(s)$ is analytic in $\tilde{\Pi}^+$ and bounded on compact subsets of $\tilde{\Pi}_{0}^{+}$. Finally, it is bounded on $\tilde{\Pi}_{0}^{+}$ because $G(s)$ is bounded for $|s|\geq K$ and so is the singular part there.

Equation (7.3) indicates that for circuits involving transmission lines with frequency varying parameters, some unstable singularity may occur at the zero frequency, which is of square root rather than polar type. Detecting this instability may be done systematically since it occurs at the known place $\omega=0$, and if it is not present, then the stability analysis procedure described in Section 2 applies.