Coplanar waveguide designs are easy to route in an RF PCB layout, and they are normally used for controlled impedance routing. This is quite important for RF applications, where a signal often needs to be routed to an amplifier, mixer, antenna, or other RF component. In this brief guide, I’ve compiled the important coplanar waveguide equations you’ll need when designing for controlled impedance in your PCB layout.

Getting Started with Coplanar Waveguide Design

All coplanar waveguides come in two varieties: standard coplanar waveguide and grounded coplanar waveguide. The standard coplanar waveguide can be placed above a ground plane, or it can be placed without ground (i.e., on top of a 2-layer of a PCB with no internal ground). Both are designed for routing on the surface layer, although you can create variations that operate in an internal layer.

There are some advantages to using a coplanar waveguide design over a standard microstrip or stripline design:

• High isolation. This is probably the most common reason for using a coplanar waveguide design. Although the waveguide is routed on the surface layer, it will have some lateral isolation thanks to the surrounding ground planes. This is the same reason ground pour is used to fill in the surface layer routed with low speed traces: it provides additional EMI shielding and terminates field lines emanating from the trace.


• Lower loss than striplines. A stripline and some other waveguides, such as substrate integrated waveguides, confine the field entirely to the substrate, so the wave sees entirely a lossy PCB substrate. In a coplanar waveguide, much of the field lines pass through the region above the dielectric, so they will not experience as much loss.


• Tunable bandwidth. The bandwidth of the structure can be tuned by adjusting the geometry. Coplanar waveguides used at ~5 GHz can be quite large, but the bandwidth can be confined to just the frequency range needed for single-mode propagation.


• Single mode propagation. The fact that the bandwidth is tunable means that we can limit the bandwidth to just the value needed for a single mode (normally TE-mode in fenced waveguides).


• Compatible with existing PCB manufacturing processes. These structures do not need special manufacturing processes for production. Current manufacturing processes are useful for fabricating coplanar waveguide designs with carrier frequencies up to ~100 GHz and higher.

 

The three standard coplanar waveguide geometries.

 

Coplanar Waveguide Design Equations

As part of coplanar waveguide design, there are two primary quantities that need to be calculated: impedance and effective dielectric constant (this has basically the same meaning as that for microstrip traces). The primary mathematical tool used for these calculations is the elliptical integral of the first kind, which arises due to the use of conformal mapping for deriving wave impedance in these structures. Let’s look a bit more closely at each type of structure to see how these calculations are performed.

There is a long set of equations needed here involving elliptical integrals. Rather than repeat this list of equations here, interested readers should look at Brian C. Wadell’s Transmission Line Design Handbook. This excellent resource provides closed-form design equations for a huge range of RF structures and should be required reading for any RF designer. Pages 73-75 provide equations for the standard coplanar trace shown above; in total there are 18 equations involving a host of parameters and elliptical integrals for calculating the impedance, propagation constant, and effective dielectric constant.

Coplanar Waveguide with Ground

This structure is something of a transition from a microstrip to a true coplanar waveguide. In the limit of substrate height h approximately equal to b, we now have approximately microstrip behavior. Otherwise, we can use the coplanar waveguide equations from Wadell with reasonable accuracy.

Here, the field lines are confined between the central conductor and the nearby ground on the surface layer, although field lines can now terminate at the lower ground. The effective dielectric constant is approximately independent of the waveguide’s geometry and is approximately the average of the dielectric constants of air and the substrate. For the impedance, we need to use the elliptical integral K(k) shown below.

 

 

In this set of equation, a is the width of the central trace and b is the distance between the edges of the grounded copper pour.

There are many calculators online and some open source tools that will handle the above calculation for you. See also Wadell’s textbook, beginning on page 79, for more information.

Grounded Coplanar Waveguide Equations

Here, we have a simple structure that adds vias for mode selection and greater suppression of radiation (i.e., higher isolation). This structure is used on a PCB as an alternative to a microstrip line. The gaps between the strip and ground must be less than the signal’s carrier frequency, but greater than the thickness of the substrate, so the field is contained between the strip and the lower ground plane. Finally, TEM propagation is ensured when the via gap is less than half the signal’s carrier wavelength.

To date, I have not seen closed form equations for grounded coplanar waveguide design impedance, either as approximations of elliptical integrals or using the elliptical integrals directly. The calculators you’ll find online use curve fitting results that are only valid within certain parameter ranges. Outside the given range (some do not state the valid range), these curve fitting approximations fail. A field solver that uses method of moments or boundary element method will provide a highly accurate solution for the impedance, propagation constant, and effective dielectric constant. Although there may not be closed form equations in Wadell's textbook, there are some design principles that can be used to design these structures:

1. The 1st higher-order (non-TEM) mode will arise when the lateral span of the waveguide is approximately 1/2 the signal wavelength.
2. The via fence pitch must be less than about 1/8th the signal wavelength.
3. The TEM propagation bandwidth can be increased by making the overall size of the waveguide smaller, or by using a substrate material with lower Dk value.

Bandwidth in Grounded Coplanar Waveguides

One important aspect of coplanar waveguide design is engineering the bandwidth of the structure to allow your signal to propagate in the desired mode. It is generally desired to have TEM propagation through a coplanar waveguide, meaning that all higher order modes are suppressed when the structure is excited. When excited with a harmonic signal, ideally only the TEM mode would be excited and the electromagnetic field distribution along the axis of the coplanar waveguide would be of constant amplitude. When excited with a broadband signal, the waveguide's bandwidth in the TEM mode should be large enough to contain the entire bandwidth of the injected signal, which would prevent signal and transmission in the required frequency range. This means that, if a grounded coplanar waveguide is made small enough, it can certainly allow wideband digital signals to propagate.

To a first order approximation at moderate frequencies, a coplanar waveguide is similar to a rectangular waveguide. The eigenmodes and their resonant frequencies depend on the geometry of the structure and the dielectric constant of the PCB substrate material. Due to the typical PCB substrate thickness and the typical dimensions of a coplanar waveguide, the lateral resonances will be excited first because they have lower frequencies than the vertical resonances.

To see what happens when higher order modes are excited in a coplanar waveguide, take a look at the data in the following images. These images show S-parameter data for coplanar waveguides of various sizes. The structure's geometric parameters are also shown. When a higher order mode is excited such that destructive interference occurs the structure will appear to be very lossy. The result is significant power loss and a large dip in the insertion loss spectrum (S21). There will be a corresponding peak in the return loss spectrum (S11).

 

 

Just as we would expect for a typical rectangular waveguide, the eigenmodes in a coplanar waveguide will depend on the geometry of the structure, and a smaller structure will generally have eigenmodes with higher frequencies. The bandwidth can be extended to 100 GHz or higher, but this begins to push the structure's size to the limits of conventional PCB manufacturing capabilities.

Although other designers who are not experts in RF PCB design will tell you that only TEM propagation should be allowed, there are more advanced RF structures where excitation of a higher order mode in a coplanar waveguide may be preferred; we generally call a waveguide operating in a higher order mode a mode-selective transmission line. One example is when exciting a resonant cavity, antenna, or another waveguide. Regarding the substrate integrated waveguide, we have built these structures that include a slot antenna operating at a frequency matching a higher order coplanar waveguide mode. These structures can be built rather large to reduce loss and ensure manufacturability with standard processing, yet they can transmit very strongly at at high frequencies reaching into the mmWave range.

More Advanced RF Design with Waveguides

There are other waveguide geometries that can be used for high frequency routing. These waveguides can provide TE-mode propagation and can provide isolation thanks to the nearby ground shielding. These other waveguide routing styles can be found in another article on our blog. These other routing styles can also be fabricated with standard processes, making them easy to include in your next RF PCB.

Just like a coplanar waveguide design, these other routing styles do not provide TEM mode propagation. They also provide the same advantages of high isolation, tunable bandwidth, and single-mode propagation. In addition, by selecting the appropriate termination, a standing mode can form in the structure, which is useful for coupling to an antenna or other passive RF component on the PCB. If you have access to a program like Mathematica, you can calculate the elliptical integrals listed above with high precision and determine the electromagnetic behavior of your coplanar waveguide design. More complex structures may need a field solver to get an accurate description of wave propagation in the device.

Currently, the most advanced high speed and high frequency routing standards involve broadband design, where the causal network parameters and dispersion in the system are considered throughout the signal bandwidth. To learn more about this important area of RF design, take a look at one of my recent technical presentations from the 2020 IEEE EPEPS conference. By looking at the wave impedance in the waveguide, the structure can be designed with desired impedance throughout the signal bandwidth. Upcoming high speed digital standards (e.g., USB 4.0) are taking this approach to interconnect design, and RF designers can do the same to eliminate signal distortion in their waveguide designs.

 

The experienced PCB design and layout team at NWES can help you create your next advanced RF PCB layout with any of the coplanar waveguide design types shown in this article. We help private clients, aerospace companies, and the US military stay at the cutting edge with advanced PCB design and layout services. We've also partnered directly with EDA companies and advanced PCB manufacturers, and we'll make sure your next design is fully manufacturable at scale. Contact NWES for a consultation.

 

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