Students of electromagnetics are familiar with wave propagation modes, namely Transverse Electric (TE) and Transverse Magnetic (TM). In standard transmission line geometries used in circuit boards, the wave propagation mode is in the quasi-TEM mode, where the quasi-static approximation tells us the wave propagation mode is very similar to TEM within the bulk of a signal's spatial extent.

Next, if you happen to be familiar with how waveguides operate, you'll know that TM mode will eventually end, and a non-TEM mode will begin propagating. The recognition of this eventually led to the naming of a new type of transmission line, called the Mode Selective Transmission Line (MSTL). In reality, this is just a grounded coplanar waveguide with the geometry designed such that the TEM cut-off is known by the designer.

This is quite important in some of the applications we work in, particularly in RF design. It is also important in digital design for the fastest interfaces spanning up to 56 GHz, specifically 224G PAM4 lanes. In both cases, the geometry of the waveguide will determine the TEM bandwidth limits and thus the extent of broadband operation for digital signals. I'll examine this topic in this article and present some basic results one can expect as a function of grounded coplanar waveguide geometry.

## Resonances in Grounded Coplanar Waveguides

Grounded coplanar waveguides are partially closed rectangular structures that have an infinite set of orthogonal eigenmodes. The eigenmodes for a grounded coplanar waveguide can be approximated as rectangular modes as long as two geometric constraints are satisfied in the structure shown in the image below. The important geometric constraints are:

- The via pitch in the waveguide is much smaller than the wavelength, with 1/8 wavelength typically taken as the upper limit.
- The spacing between the trace edge and the coplanar ground is much smaller than the signal wavelength.

In the above list of assumptions, we've assumed a harmonic signal, but if necessary, we could make these limits dependent on the distance traveled by a signal during its rise time. In this case, we would be dealing with a digital signal, which is a broadband signal and is also subject to the TEM bandwidth limit in waveguides.

In fact, if you look at test boards for the fastest digital interfaces, which require broadband signal integrity, you will see that they use grounded coplanar differential waveguides, where a differential pair is routed between coplanar ground. An alternative but related configuration is skip layer routing, which is normally done in coaxial configuration as a grounded coplanar waveguide.

*Top view and cross-section view of skip-layer routing, which is equivalent to a microstrip or stripline differential grounded coplanar waveguide.*

In these digital cases, the bandwidth available for broadband propagation ends when TEM propagation ends. So in either case, we need to determine a TEM bandwidth limit based on the geometry.

## TEM Bandwidth Estimation

Luckily, TEM bandwidth can be estimated using the same equation we would use to estimate the frequencies for the eigenmodes in a rectangular waveguide. Under the above assumptions, the eigenmode frequencies are indexed as follows:

*Wavenumbers for propagating modes in a rectangular waveguide or approximately rectangular waveguide.*

To illustrate some values of TEM cut-off in typical geometries, I've prepared the tables shown below. In these tables, I've calculated the cut-off for the quasi-TM mode on excitation of resonance in either the x or y directions along the cross-section. It should be clear that, on thin PCB substrates with narrow via pitch, the TEM bandwidth cut-offs can be quite high.

*Mode cutoff/excitation frequencies.*

On standard thickness PCBs with large via spacing and 50 ohm grounded coplanar waveguides, the TEM cut-off values could be much lower. I'll leave determination of some values as an exercise for the reader.

From a design standpoint, in practical cases there are two levers a designer can pull to change the TEM cut-off frequency in a grounded coplanar waveguide:

- Dielectric constant of the substrate material
- Either thickness of dielectric or span between the via fence

In the second point, a designer will usually adjust only one of these parameters. This applies regardless of dealing with RF or digital. This also applies if our digital interconnect is differential.

## 3D Simulation Data from the Literature

There are simulation results in the literature specifically showing mode cut-offs for single-ended grounded coplanar waveguides. The images below show a simulation model and a set of results showing bandwidth limits based on S-parameter data.

The S-parameter data below clearly shows when we would expect the lowest order propagating mode to be cut off. In this parameter plot, this manifests itself as strong losses at a specific frequency, effectively illustrating when the quasi-TEM mode stops propagating and a higher order mode is excited. We can see this in an insertion loss plot, or this can be converted to a total power loss plot as shown below.

The takeaway from this should be quite clear: for certain values of geometric parameters, it is possible to extend the TEM bandwidth in grounded coplanar waveguides to 100 GHz or higher. This is supported by the estimation from rectangular waveguide resonant frequencies. In the simulation data above, the grounded coplanar waveguide with the highest bandwidth reaching above 100 GHz has via size, via pitch, and trace to ground spacing that is near the limits of conventional PCB fabrication capabilities. However, minor adjustments to this would still provide a mass-manufacturable structure on standard material sets with negligible change in performance.

It should now be clear why differential grounded coplanar waveguides are used in microstrip and stripline configurations to support the highest bandwidth digital signaling standards, namely 224G PAM4 lanes. The bandwidth of these structures is very high and can still support the next data rate doubling, which will take channel bandwidth requirements beyond 56 GHz. It remains to be seen what modulation format a 448G interface will use, but I am skeptical that PAM4 will be that modulation format. Instead, I think it will be an alternative multi-level signaling scheme, such as PAM6 or PAM8. This could allow doubling of data rates without doubling the channel bandwidth requirements and would thus allow continued use of the structures I've discussed in this article.

**Whether you're designing high-speed PCBs for mil-aero embedded systems or a complex RF product, you should work with a design and development firm that can ensure your product will be reliable and manufacturable at scale. NWES helps aerospace OEMs, defense primes, and private companies in multiple industries design modern PCBs and create cutting-edge embedded technology, including power systems for high reliability applications and precision control systems. We've also partnered directly with EDA companies and advanced ITAR-compliant PCB manufacturers, and we'll make sure your design is fully manufacturable at scale. Contact NWES for a consultation.**