International Journal of Microwave Science and Technology

Volume 2015, Article ID 630131, 18 pages

http://dx.doi.org/10.1155/2015/630131

## The Spiral Coaxial Cable

Department of Physics, University of Milan, Via Celoria 16, 20133 Milan, Italy

Received 18 September 2014; Revised 10 January 2015; Accepted 12 January 2015

Academic Editor: Kamya Yekeh Yazdandoost

Copyright © 2015 I. M. Fabbri. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A new concept of metal spiral coaxial cable is introduced. The solution to
*Maxwell’s* equations for the fundamental propagating *TEM* eigenmode, using a
generalization of the *Schwarz-Christoffel* conformal mapping of the spiral transverse
section, is provided together with the analysis of the impedances and the
Poynting vector of the line. The new cable may find application as a medium for
telecommunication and networking or in the sector of the Microwave Photonics.
A spiral plasmonic coaxial cable could be used to propagate subwavelength surface
plasmon polaritons at optical frequencies. Furthermore, according to the present model, the myelinated nerves can be considered natural examples of spiral coaxial cables. This study suggests that a malformation of the Peters angle, which determines the power of the neural signal in the *TEM* mode, causes higher/lower power to be transmitted in the neural networks
with respect to the natural level. The formulas of the myelin sheaths thickness, the
diameter of the axon, and the spiral factor of the lipid bilayers, which are mathematically
related to the impedances of the spiral coaxial line, can make it easier to analyze the neural line impedance mismatches and the signal disconnections typical of the neurodegenerative diseases.

#### 1. Introduction

The coaxial cable invented by Heaviside [1] is a transmission line composed of an inner conductor surrounded by an insulating layer and an outer conducting shield. The inner and the outer conductors share the same geometrical axis.

Nowadays, there exist several types of transmission lines, the coaxial cables, the hollow waveguides (rectangular, circular, elliptical, and parallel plates [2, 3]), the two wires [3, 4], and the channel waveguides (buried, strip-loaded, ridge, rib, diffused, and graded-dielectric index [5]). The two-wire transmission line used in conventional circuits is inefficient for transferring electromagnetic energy at microwave frequencies because the fields are not confined in all directions, the energy escapes by radiation, and it has large copper losses due to its relatively small surface area.

On the other hand the larger surface area of the boundary makes in general the waveguides more efficient with respect to the coaxials on reducing the copper losses.

Dielectric losses in two wires and coaxial lines, caused by the heating of the insulation between the conductors, are also lower in waveguides. The insulation behaves as the dielectric of a capacitor and the breakdown of the insulation between the conductors of a transmission line is more frequently a problem than is the dielectric loss in practical applications [3].

Waveguides are also subject to the dielectric breakdown caused by the stationary voltage spikes at the nodes of the standing waves.

In spite of the dielectric in waveguides is air, it causes arcing which decreases the efficiency of energy transfer and can severely damage the waveguide.

The spiral coaxial cable (SCC) discussed in this paper has many advantages with respect to the other types of transmission lines; first of all the elm energy can be distributed efficiently over a larger area, reducing all the undesired aforementioned effects.

The power handling versus frequency is consequently higher for the metal spiral coaxial line (MSCC) with respect to the other lines.

The Microwave Photonics (MWP) [6], a new discipline that joins together the radio-frequency engineering and optoelectronics, represents the future in many civil and defense applications like speed of the digital signal processors, cable television, and optical signal processing.

The MSCC could become one of the key objects in the field of MWP for its characteristics in terms of both power handling and energy transfer efficiency.

Recently, it has also been demonstrated experimentally that metal coaxial waveguide nanostructures perform at optical frequencies [7], opening up new research pursuits unexpected in the area of nanoscale waveguiding, field enhancement, imaging with coaxial cavities, and negative-index metamaterials [8].

Several different plasmonic waveguiding structures have been proposed such as metallic nanowires [9, 10], metal-dielectric-metal (MDM) structures [11], and metallic nanoparticle arrays [12] for achieving compact integrated photonic devices.

Most of these structures support a highly confined mode near the surface plasmon frequency [11]. This study could become the reference point to introduce the MSCC as a valid alternative to guide subwavelength surface plasmon polaritons (SPPs).

The extraordinary transmission of light through an array of subwavelength apertures, enhancement which arises from the coupling of the incident light with the SPPs through the surface grating in metal film [13], could result particularly efficiently on the spiral metal dielectric interface with periodic holes.

High sensitivity spiral biosensors and spiral photonic integrated circuits based on nonlinear surface plasmon polariton optics [14, 15] may be implemented.

The aim of this pioneering work on the SCC is to represent an initial landmark in the continuously growing sector of the microwave research.

The popularity of Video on Demand (VoD) and Over the Top Technology (OTT) services to access high definition videos over home interconnected devices of the hybrid fibre-coax (HFC) networks is driving the research toward more efficient and cost-effective cables. Particularly, the development of Converged Cable Access Platforms (CCAP) that combine video and data transmission supporting simultaneous network access of multiple users over a single coaxial cable is flourishing and sustaining the demand for new high speed transmission media.

In a metallic guide, the reflection mechanism responsible for confining the energy is due to the reflection from the conductors at the boundary [16], whose geometry is strictly related to the propagating modes.

Coaxial cables were designed to propagate high frequency radio signals. The principal constraints on performance of a coaxial are attenuation, thermal noise, and passive intermodulation noise (PIM).

In RF applications, the wave propagates essentially in the fundamental transverse electric magnetic () mode; that is, the electric and magnetic fields are both perpendicular to the direction of propagation.

In the ideal case, the conductors can be considered to have infinite conductivity and the TEM eigenmode is the basic propagating wave (see [17] page 110) along the transmission line.

Practical lines have finite conductivity, and this results in a perturbation or change of the TEM mode (see [18] page 119).

Above the cutoff frequency, transverse electric () or transverse magnetic () modes [19] can also propagate with different velocities within a practical cylindrical coax, interfering with each other producing distortion of the signal.

The frequency of operation for a specific outer conductor size is then limited by the highest usable cutoff frequency before undesirable modes of propagation occur.

In order to prevent higher order modes from being launched, the radiuses of the coaxial conductors must be reduced, diminishing the amount of power that can be transmitted.

On the other hand at high frequencies it is impossible to make the cylindrical coaxial line in the small size necessary to propagate the mode alone.

The research described in this paper demonstrates the propagation on the fundamental wave along the ideal MSCC.

Since the mode of transmission on an ideal line is the TEM wave, the relations for input impedance, reflection coefficient, return loss (RL), standing wave ratio (SWR), and so forth, given afterward in the next sections, are applicable in general to the spiral transmission lines (see [18] Chapter 3).

The metal double spiral coaxial cable or MDSCC, resulting from the superposition of two spiral conductors that share the same geometrical axis, can be made multi-turn. The amount of heat generated by the losses for heating can be distributed over a larger area and this would lower the temperature and raise the reliability of the line.

In fact, operation at higher temperatures results in a reduction in the life expectancy and reliability of the transmission line relative to the lower temperature performance.

Applications like nanoscale optical components for integration on semiconductor chips could benefit from these characteristics of the MSCC.

Where signal integrity is important, coaxial cables are needed to be shielded against radio frequency noise (RF noise). The multiturn MDSCC is naturally shielded because the highest part of the elm energy can be distributed on the inner part of the cable, which protects small signals from interference due to external electric fields.

A new class of spiral passive components, computer-aided engineering (CAE) tools as well as electromagnetic (EM) simulators, is required before new high-frequency spiral RF/microwave circuits will be implemented.

The spiral geometry occurs widely in nature; examples like the* spiral galaxies* are found at the universe level while the* myelin bundles* are common in the microcosm of the neuron cells.

Recently, a new spiral optical fibre has been proposed both in the fundamental mode [20] and in the higher order modes [21] operation.

Spirals are also of extreme interest to the field of the new metamaterials and invisible cloaking [22].

Myelinated nerve fibers are micro-spiral coaxial cables () whose electric behaviour is still today described by neurophysiologists using* W. Thomson’s* (later known as* Lord Kelvin*)* cable formula* [23] of the 1860s, which determines the velocity of the signal propagating in saltatory conduction [23, 24].

Cable theory in neurobiology has a long history, having first been applied to neurons in 1863 by* C. Matteucci* [25] who discovered that if a constant current flows through a portion of a platinum wire covered with a sheath saturated with fluid, extra-polar current can be led off which corresponds to the electrotonic current of nerves.

Since the 1950s–60s myelinated nerves have been recognized to have a spiral structure and to behave like a high loss coaxial cable [26, 27] with negligible inductance.

The mathematical model presented in this paper can be used to refine the elm theory of the myelinated nerves by taking into account their spiral geometry.

In a coaxial guide, the determination of the electromagnetic fields within any region of the guide is dependent upon one’s ability to explicitly solve the* Maxwell field equations* in an appropriate coordinate system [28].

Let us consider* Maxwell’s equations*
where the time variation of the fields is assumed to be .

In view of the nature of the boundary surface, it is convenient to separate these field equations into components parallel and transverse to the waveguide -axis.

This is achieved by scalar and vector multiplication of (1) with , a unit vector in the direction, thus obtaining
Since the transmission line description of the electromagnetic field within uniform guides is independent of the particular form of the coordinate system employed to describe the cross section, no reference to cross-sectional coordinates is made on deriving the* telegrapher’s equation* [28, 29].

Substituting (2) into (3) we obtain Vector notation is employed with the following meanings for the symbols:the rms electric field intensity transverse to the -axis. the rms magnetic field intensity transverse to the -axis. intrinsic impedance of the medium . = propagation constant in medium or the wavenumber (see [28] page 3). gradient operator transverse to -axis . unit dyadic defined such that .

Equations (4) and (2), which are fully equivalent to the* Maxwell equations*, make evident the separate dependence of the field on the cross-sectional coordinates and on the longitudinal coordinate . The cross-sectional dependence may be integrated out of (4) by means of a suitable set of vector orthogonal functions provided they satisfy appropriate conditions on the boundary curve or curves of the cross section.

Such vector functions are known to be of two types: the* E-mode* functions defined by
where
and the* H-mode* functions defined by
where
where denotes a double index and is the outward normal to in the cross-section plane.

The constants and are defined as the cutoff wave numbers or eigenvalues associated with the guide cross section.

The functions possess the vector orthogonality properties with the integration extended over the entire guide cross section with surface .

The total average power flow along the guide in the direction is where all quantities are rms and the asterisk denotes the complex conjugate.

In modes, both and vanish, and the fields are fully transverse. Their cutoff condition or (where is the speed of the light) is equivalent to the following relation [28]: between the electric and magnetic transverse fields, where is the medium impedance so that and .

The electric field is determined from the rest of* Maxwell’s* equations which read
These are recognized as the field equations of an equivalent two-dimensional electrostatic problem.

Once the electrostatic solution is found, the magnetic field is constructed from (11).

Because of the relationship between and , the Poynting vector will be

#### 2. The Spiral Differential Geometry

For the MSCC structures it is difficult to construct solutions for* Laplace’s equation* with polar or cartesian coordinates.

The conformal mapping technique is a powerful method for solving two-dimensional potential problems and mapping the boundaries into a simpler configuration for which solutions to* Laplace’s equation* are easily found [17, 18].

For the specific purposes of the MSCC, the following spiral coordinates based on a generalization of the* Schwarz-Christoffel* mapping (see appendix) are introduced:
where , represent the spiral coordinates and is a constant which characterizes the transformation (see appendix).

As it can be seen in Figure 1, the equation . represented by a vertical line in the plane corresponds to a logarithmic spiral into the plane and a constant coordinate line of the spiral mapping.