In the potential-current representation, transmission-line parameters in the Transmission-Line Super Theory (TLST) do not have a direct physical meaning – they are gauge dependent, i.e.: they are different in the Lorenz and Coulomb gauge. However, they retain traces of their classical origin: They are constituted of capacitances and inductances for forward and backward running waves along the lines. Therefore their corresponding matrices are not symmetrical as in the case of classical transmission-line theory. In the charge-current representation the parameter matrices have a physical meaning: their elements consist of damping functions due to the non-uniformities of the lines and of the propagation functions along the lines, incorporating conductor and radiation losses. The transmission line parameters also contribute to the total radiated power of the lines. The attempt to quantize radiation locally, fails because radiation describes a long-range (integral) interaction, and therefore affects all conductor parts of all lines. However, it can be stated that at stronger inhomogeneities the local contributions to radiation increase, and are particularly recognizable along the risers.

Haase and Nitsch

On the other hand, it was shown

For the notation in this paper the reader is referred to the publications

The solution of Eq. (

Now, in the low-frequency limit (

For the general solution – without any restriction to frequencies – Eq. (4) is inserted for the block column vector

Equation (11) together with the continuity equation can now be written in the
desired form Eq. (1) where one obtains for the parameter block matrix

At this point it seems appropriate to give a brief interpretation of the previous results. The starting point of the above considerations was the MPIE Eq. (5). From this it could be demonstrated that – for low frequencies – it obeyed a wave Eq. (8) for the current. Assuming now that for any frequencies the current fulfills a wave Eq. (2) (forward – and backward running waves for the homogeneous solution), then one obtains (again with the aid of the continuity equation) the generalized line equations in the charge-current representation Eq. (1). However, the line parameters and the sources are still unknown in this equation. These unknowns are eventually derived by insertion of the solution ansatz Eq. (4) in Eq. (1) and are represented by Eq. (13a, b).

The iterative solution procedures of Eq. (13a, b) are described in detail in

Flow-diagram of the iterative solution procedure for the TLST parameter matrix.

The parameters are derived from the homogeneous problem of Eq. (1), i.e., without
exterior distributed sources. Therefore, they do not depend on those sources
or on boundary conditions. After knowing the parameters in an appropriate
approximation, the renormalized sources are calculated. The enforced (by the
exterior distributed sources) solution of Eq. (2) (besides the two fundamental
solutions) is hidden in the renormalized sources. The renormalization of the
exterior source

In many cases it is convenient to deal with voltages or potentials instead of
a per-unit length charge. For instance, if the transmission line is embedded
into a circuit or if other elements are connected to the line, the voltages
at the terminals must be known to perform the calculations. The connection
between the potential vector

Taking the partial derivative of Eq. (

Equations (

The next step refers to the derivation of the line parameters in the

The fundamental solutions of the homogeneous MPIE in classical TLT for the
current are forward- and backward running waves (after the first iteration):

However, the parameters in the Coulomb gauge differ from those in the Lorenz
gauge. The denominators of the Coulomb parameters are all real whereas the
parameters which contain inductance terms are complex-valued. Due to the
instantaneous relation of potential and charge in the Coulomb gauge, the
parameter

With the knowledge of the parameter matrix (in the Lorenz gauge)

It is well known that radiation is an integral problem. If one, however,
assumes for a moment that the radiation of individual line segments would
have a meaning, then it would make sense to calculate the radiated power of
these segments. Especially at strong inhomogenities of the conductor one
would expect a stronger radiation than in largely homogeneous zones
(experimentally shown in

The first line configuration starts with a parallel conductor above perfectly
electrical conducting (PEC) ground, with two risers terminated with 50 Ohms
at both ends and fed by a 1 V source (see Fig.

Parallel line above PEC ground with risers at each end.

Total radiated power in the frequency range of 100 MHz to 1 GHz.

Figure

Figures

Different radiation contributions along the line from Fig.

Capacitive and inductive radiation contributions along the line from Fig.

One can see that most radiation

Figures

Damping function

Magnitude of the damping function

Comparison of the graphs for the damping function

The magnitude of the propagation function

Obviously, damping almost vanishes in the asymptotic regime of the line. It
is strongest at the ends of the line, due to the strong change of the
capacitance (see Eq. 8). And, as can be seen in Fig.

It might be interesting to compare the shape of the damping function with
that of the radiated power stemming from the diagonal parameter elements. In
both cases the local change of the line configuration is important. In Fig.

The second measurable quantity is the propagation function

The next numerical example deals with a parallel line with two risers and a
central local scatterer above PEC ground, terminated at both ends with 50 Ohms (see Fig.

Parallel line over conducting ground with two risers and a central local scatterer (90

Total radiated power for a parallel line with and without a central local scatterer (90

The bend in the line increases the radiated power (red curve in Fig.

Figure

Different radiation contributions along the line from Fig.

Damping function

Similar arguments hold for the damping function

Magnitude of the damping function

Magnitude of the damping function

In Fig.

Magnitude of the propagation function

As in the previous line configuration, the current waves propagate with the
speed of light

This paper focused on the question of the physical meaning of the
line parameters in the TLST. The meaning of the parameters depends on the
representation of the TLST equations. In the charge-current representation
the two line parameters were directly correlated to measurable quantities:
namely, the propagation function

The authors would like to thank Günter Wollenberg for his permanent interest in our work and for fruitful discussions. Edited by: F. Gronwald Reviewed by: three anonymous referees