Last month, the first part of this three-part article series introduced a transversemagnetic (TM) propagation mode that is present with the better-known transverseelectromagnetic (TEM) mode in conventional coaxial transmission line. Put simply, waves move along a conductor having no outer shielding and no insulation or special surface conditioning. Part 2 will describe the TM mode around an unconditioned conductor, backed by models and measurements of the fields and transmission characteristics of practical systems.

To understand the usefulness of the TM mode, it is necessary to provide some insight into practical noncoaxial application of the TM mode in realworld situations. Since it is necessary to couple to and from the TM mode, developing a visualization of the associated electric field may be useful.

The solution to the wave equation for the propagating TM mode produces a nonzero longitudinal component of the electric field. This is in contrast to the solution for the TEM mode in coax that produces only a transverse electric field. Whereas the TEM mode is excited by real current, the TM wave is excited by the displacement current. The potential on the central conductor increases as line impedance increases. As these increase, the magnitude of the electric field increases as well. However, a nearby conductor other than the line itself may provide a termination point and thereby reduce energy coupled into the TM wave. This is the case with the shield of conventional coaxial cable of common geometry. The nearness of a shield reduces the TEM impedance, provides a return path for electric-field lines, increases real current, reduces displacement current and correspondingly reduces the power coupled into the TM wave. The result is that as the geometry is reduced, propagation in conventional coaxial cable rapidly becomes dominated by the TEM mode to the exclusion of the TM mode. When the geometry has reached 50 Ohms in an ideal coaxial environment, b/a  2.3, the TM mode has been almost entirely suppressed. To examine this mode, it is necessary to consider a central conductor apart from nearby shielding or other conductors.

Figure 2 shows a plot of the electric field generated from a numeric solution of Maxwell’s equations by a three-dimensional (3D) electromagnetic (EM) field solver software (HFSS from Ansoft). The model is of a finite, perfectly conducting circular disk on the left, having a central hole through which passes a perfectly conducting wire that extends continuously from left to right. The short region inside this hole is equivalent to a section of ideal coax and excitation of this port is configured to be coaxial at this location. The rest of the region in the illustration is vacuum wherein the short lines indicate the direction of the electric field that results when the port is driven by a sinusoidal signal through a port impedance equivalent to that of the TEM mode at the coaxial input at the plane of the disk.

It is important to recognize that because the TM mode has not previously been deemed important or even existent, computer analysis tools may make assumptions about the conditions at the port of a model. Although in the analysis itself, a full numerical solution of Maxwell’s equations may be performed, the port excitation for the model may not include this.

For the model and plot shown above, analysis was performed with the assumption that conditions to the left of the launcher port, that region “inside” the modeler, is a TEM extension of the port. No longitudinal electric field component is present there and as such it only models excitation from a TEM source. Because a TM wave does exist this causes some error. However, in this example the port geometry has been chosen to provide a relatively low impedance, in the vicinity of 50 Ohms, and the TM contribution to the propagating wave is small and the error is negligible.

This same problem exists with conventional scalar and vector network measurement and analysis of coaxial systems in general. All commercial systems of which the author is aware presently make the implicit assumption that only a TEM wave is present in coax.

For 50-Ohm systems this assumption has been, and continues to be, almost entirely adequate with the possible exception of characterization of precision coaxial calibration standards for vector network analysis. Recent characterization of precision coaxial line standards in slightly lossy line for use in vector network analysis at 50 Ohms have found the effect of the TM₀₀ to be small. In calculating its effect, the H-field and wave admittance associated only with the radial component of the electric field were included. Apparently, this is due to the a priori assumption that no propagating TM mode, or at least no significant mode, exists in coaxial cable and any longitudinal component of the E-field is only evanescent or so small that it could be neglected.

In coax, the TM mode is so well suppressed that for almost all practical measurements and applications the errors due to this assumption are small. The conductive planar surface with the coaxial port (the left-hand side of Fig. 2) is called a “launcher” and serves to couple energy from the coaxial stimulus into the TM wave propagating along the central conductor.

From Fig. 2, it can be seen that close to the excitation port, the electric fields extend from the central conductor to the launcher and are normal to the surfaces of each conductor immediately adjacent to the conductor. Perfect conductivity forces tangential components of the electric field to be zero and only a field component at right angle to the conductor surface is possible. In this region near the port, real current flows in the plane and returns by way of the outer conductor of the input coax port. Further to the right, away from the launcher, close examination of Fig. 2 will reveal that the electric-field lines terminate along the conductor. Here also they leave the conductor normal to its surface but curve in the enclosing (vacuum) dielectric and return at a different location along the same conductor, up to one half wavelength away. In this region the resulting wave is TM. In essence, the launcher serves as a transition between the predominantly TEM mode in the coax and the predominantly TM mode on the conductor in the region far from the launcher.

The field solution to the wave equation for coaxial line shows that the peak magnitude for the longitudinal electric field is displaced from the peak magnitude for the radial field by one quarter wavelength. The peak longitudinal fields occur at the location of voltage minima on the central conductor. The phase of the excitation in Fig. 2 has placed the voltage maximum at or near the input port. Careful examination of the field lines will show that the first clearly discernible maximum of the longitudinal electric fields occurs slightly to the left of the center of the central conductor and approximately three quarter wavelengths away from the maximum occurring near the excitation port. The first longitudinal maximum occurs one quarter wavelength from the port but is difficult to discern because of the other field lines returning to the launcher.

Although Fig. 2 gives insight into electric-field direction, it gives almost no information about electric-field amplitude or even relative magnitude. To help provide this, contours of constant electric-field magnitude for a different modeled two-port system are shown in Fig. 3. These lines are contours of constant magnitude so Figure 2 and Figure 3 must be taken together in order to visualize the complete electric-field vectors, which contain both amplitude and direction information. The launchers here are 100 mm square and the central conductor is 400 mm long, also square but tapered from 4 mm at each end to 0.04 mm at the center. The stimulus is 1875 MHz where the structure is 2.5 wavelengths.

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