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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The radial extent of the electric field is dependent only on line impedance and not on conductor diameter or wavelength. Because displacement current is constant, conductor diameter affects the electric-field magnitude at the surface of the conductor but not the contour it follows in the surrounding dielectric medium.

Contrary to previous belief in regard to surface waves on G-Line, the launcher need not be large. Because most of the electric field is quite close to the conductor, both in the TEM region and in the TM region, the majority of the terminating field lines and real current also occur quite close to the center axis. The field solutions show that the magnitude of the radial component follows a 1/r curve and that the majority of the propagated energy is within a few conductor diameters of the center axis.

Longitudinally, the electric field is dependent on wavelength since each field line must have a termination point, which can be up to one-half wavelength away. Therefore, the conductor must be at least one-half wavelength to support the TM mode.

The line impedance in the vicinity of the launcher is lower than that of the line in free space. This is because field lines producing real current in the coaxial (TEM) region are present along with the lines terminating on the conductor in the TM mode.

Figure 4 shows a vector-networkanalyzer (VNA) time-domain measurement of a simple system constructed with a pair of circular brass planar launchers 68 mm in diameter and spaced 680 mm. The conductor is circular, 0.5-mm-diameter bare copper conductor (burnished No. 24 copper wire) connected between the center pins of SMA connectors each mounted at the center of one of the launchers. The left Y axis has labels for the equivalent line impedance, as it relates to the real part of the reflection coefficient plotted over a range from 0 to 1 with a system reference impedance of 50 Ohms.

Within about the first centimeter from the excitation port, approximately 20 wire diameters, the impedance rises very rapidly from the initial 50-Ohm value at the SMA connector. Beyond that, it rises much more slowly and asymptotically approaches the free-space value of 377 Ohms. The value of the reflection coefficient at the marker corresponds to a line impedance of about 366 Ohms. The discontinuity at 4.5 ns is at the location of the second SMA connector.

A practical launcher should provide the transition from TEM00 to TM00 waves as effectively as possible. Generally, this transition is between different impedances as well as between different modes. Any launcher represents a discontinuity to the propagating waves. This discontinuity can produce radiation away from the region. In the TM portion of a system, as shown in Figure 3 and Figure 4, there is complete symmetry of the electric field; every field line is one of a pair of lines of equal magnitude but opposite sign. This symmetry is present both axially and longitudinally. Therefore, at distances of more than a few wavelengths, these fields add to zero and no net field and no radiation results. However, for the region near a launcher, there is no longer longitudinal symmetry and incomplete cancellation of fields results at large distances. This produces radiation away from the launcher with the radiated wave linearly polarized parallel to the conductor.

Figure 5 shows a measurement of the same system with planar launchers that was measured in Fig. 4. The lower trace is of S21, which displays the ripple or "beat" between the discontinuities produced by the launchers at each end of the line. In addition to the ripple there is a large amount of mismatch loss between the 50- Ohm impedance of the VNA and the impedance presented by the TM system at each port. The upper trace is a calculation of GAmax, which effectively removes the extra attenuation due to port mismatch and allows just the Ohmic and radiation losses to be evaluated. In addition to attenuation due to Ohmic losses in the copper conductor, approximately 2-dB loss is apparent near 1 GHz. This is almost entirely radiation loss due to the discontinuities at the launchers. Because of the large standing waves present on the line due to mismatch, the radiation loss is greater than it would be for the situation of a perfectly impedance-matched launcher. While a planar launcher of the type shown in these examples is useful for analysis, even with impedance matching added at the ports, it is not generally the best design to achieve minimum system attenuation. The modal discontinuities of this type of launcher generally produce both unwanted radiation and reflection.

Figure 6 shows measurements of a system with somewhat better launchers. These are also 68 mm in diameter but of the "forward conical horn" rather than the planar type. These were fabricated from a section of a circular brass disk folded and soldered so as to create a 90-deg. cone. An SMA bulkhead connector was soldered to the narrow end of the cone and the same type and length of bare copper conductor used for Figure 5 and Figure 6 was soldered to the center pin of the connector. Two measurements of GAmax are shown: these are with and without a small polyethylene dielectric compensator added to help reduce the discontinuity and consequent reflection and radiation. Each compensator was fabricated from an approximately 30-mm long section of polyethylene dielectric removed from conventional RG/8 coaxial cable and placed a few mm away from the SMA connector. Material was removed to taper the diameter of the compensator linearly from the wire diameter at each end to a maximum diameter of about 8 mm at its middle. As can be seen by the measurement, this small amount of compensation is only sufficient to make significant improvement above about 6 GHz.

Figure 7 shows a measurement of GAmax for the same type of compensated launcher but the line length has been increased to 3.4 m. Additionally a measurement of an equivalent length of 0.085-in.-diameter semirigid coaxial cable with Teflon dielectric has been included. The coaxial center conductor is of about the same diameter as the conductor of the TM line but is silver plated. In spite of the better conductivity of the coaxial line and the radiation due to the launchers, the lower attenuation of the TM wave system is obvious. A better launcher design can provide even more contrast between the attenuations of the TEM and TM modes. Even with only crude techniques, it is possible to reduce total loss for a single launcher to less than 0.25 dB. These and other launcher possibilities and designs have been described elsewhere.8 Because the displacement current in TM line is much less than the real current in conventional coax, the Ohmic losses in TM line are less than for coax.

Even with simple launchers it is possible to achieve three or more decades of low attenuation performance. The lower frequency limit is primarily limited by the diameter of the launcher and by the ability to effectively match to the impedance the line presents. The launcher acts as a sort of "capacitor to space" and provides a return path for displacement current. As the launcher gets very small, the total load presented by the line plus launcher becomes more reactive and makes broadband impedance matching more difficult. However, a 60-cmdiameter planar launcher has proven quite usable to below 20 MHz.

The upper frequency limit is affected mainly by the detail of the transition from the coaxial connector to the line itself. The same 60-cm-diameter planar launcher described above easily provided good performance from 20 MHz to 20 GHz, which was the upper limit of the measurement equipment. It seems very probable that performance was excellent well beyond this limit. As the line diameter becomes large compared to a wavelength more care needs to be taken to assure that unwanted discontinuities and resultant radiation do not occur. However it is possible to support the TM mode on lines that are large compared to a wavelength. Work between 30 and 500 GHz indicates that the mode is useful at least that high.9

Next month, the final installment of this three-part article will describe further measurements of the TM mode on a single conductor. In particular, the use of the mode with overhead power line distribution and transmission systems will be shown.

7. S-Parameter Design, Application Note AN-154, Agilent Technologies, Santa Rosa, CA,
8. Glenn Elmore, United States Patent Application, No. 20080211727. 9. Kanglin Wang and Daniel M. Mittleman, "Dispersion of Surface Plasmon Polaritons on Metal Wires in the Terahertz Frequency Range," Physical Review Letters, PRL 96, 157401, April 21, 2006.