4. Realistic Transmission lines

We will now look into realistic models for transmission lines. They are passive components, so a realistic model should satisfy Definition 3.1. We start by constructing the $Y$-matrix of a transmission line and will then show that a lossy transmission line is indeed realistic.

Using the Telegrapher’s equations, we describe the behavior of an infinitesimal piece of transmission line as a function of its per-unit-length parameters:

where $Z_{l}\left(s\right)$ is the per-unit-length longitudinal impedance and $Y_{l}\left(s\right)$ is the per-unit-length transverse admittance [28]. For a transmission line with frequency-independent parameters, we have the following expressions for $Z_{l}\left(s\right)$ and $Y_{l}\left(s\right)$:

with $L,G,C,R\in\R^{+}$. In a lossless, frequency-independent line, $R\!=\!G\!=\!0$. In general, $Z_{l}\left(s\right)$ and $Y_{l}\left(s\right)$ are more complicated strictly positive real functions which satisfy the Kronig-Kramers relations [29][28]. We will discuss transmission lines with frequency-dependent parameters later.

From the per-unit-length properties of the line, the propagation constant $\gamma\left(s\right)$ and characteristic impedance $z_{0}\left(s\right)$ of the transmission line are defined as:

and the impedance matrix of a transmission line of length $\ell$ is then given by

To show that a transmission line model is realistic, we first have to show that it is meromorphic on $\mathbb{C}$. For a transmission line with frequency-independent parameters, the elements of the Z-matrix of the transmission line are meromorphic on the whole complex plane. Indeed, since $\gamma\left(s\right)=\sqrt{Z_{l}\left(s\right)Y_{l}\left(s\right)}$ is a function with a branchpoint of order 1 at each of its zeros, while $\cosh$ is entire and even, we see that $\cosh\left(\gamma\left(s\right)\ell\right)$ is an entire function. To show that $z_{0}^{-1}\sinh\left(\gamma\left(s\right)\ell\right)$ is meromorphic, we re-write it

The $\sinh$ function is entire and odd. Therefore, $\gamma\left(s\right)\sinh\left(\gamma\left(s\right)\ell\right)$ is an entire function. Because $\cosh\left(\gamma\left(s\right)\ell\right)$, $\gamma\left(s\right)\sinh\left(\gamma\left(s\right)\ell\right)$ and $Z_{l}\left(s\right)$ are all meromorphic on the whole complex plane, we have that $z_{0}\left(s\right)\coth\left(\gamma\left(s\right)\ell\right)$ is meromorphic on the whole complex plane, which shows that the Z-matrix of a transmission line with frequency-independent parameters is meromorphic on the whole complex plane. In the appendix, we prove that a transmission line for which $R,L,G,C$ are strictly positive is realistic.