One Dielectric Constant Is Not Enough
I first started designing microwave filters back in the late 1970s. In those days, I always used a single value for the dielectric constant of the substrate. There was no EM analysis software. We used Matthaei, Young, and Jones’ Microwave Filters, Impedance-Matching Networks, and Coupling Structures… and a slide rule. We would build the first filter and then tweak the dimensions, and after a couple more prototypes, we might have a filter that meets our requirements.
Today, with EM analysis, our hope is to get the filter right on, first time. Sometimes that even happens. Often, we are still a bit off. Another prototype is needed. Experienced users come up with empirical rules like, “Design the filter for a bit wider bandwidth and use a specially tuned dielectric constant, then when we build it, it should be pretty close.”
Electromagnetic analysis should converge to the exact answer as the mesh is made finer. But so many times it does not quite make it. Why? One reason is that using a single value for the dielectric constant — that of an “isotropic” dielectric — is wrong. Two numbers work better.
Most substrates are “anisotropic.” The dielectric constant depends on direction. A full description of anisotropy (when there are no magnetic complications) requires nine numbers, which fill a 3 × 3 matrix. Good topic for a Ph. D. dissertation.
For many practical substrates, using two numbers can give really good answers. We use one dielectric constant for vertical electric field (imagine the substrate flat on the test bench). The second dielectric constant is for the horizontal electric field. This kind of anisotropy is called “uniaxial.”
What happens when we are restricted to just one dielectric constant? Most often, we adjust that dielectric constant in our analysis so that the EM result shows the same center frequency as we measure. The filter center frequency is determined to a large degree by the vertical dielectric constant. But if our EM analysis restricts us to isotropic, then the EM analysis uses the same value for the horizontal dielectric constant. The horizontal dielectric constant strongly influences the coupling between resonators (it is the horizontal electric field that couples resonators). This, in turn, determines the filter bandwidth. If we are forced to use the wrong horizontal dielectric constant, we have the wrong bandwidth. So we need to design the filter for a wider (or narrower) bandwidth than we will get when we build the filter. When we don’t want to guess what bandwidth we need to design, including anisotropy is important.
So what substrates are anisotropic? Turns out, just about everything. Composite substrates, for example, have a fiber weave, often glass, embedded in an epoxy. Given that it is too difficult to include the actual fiber weave in the EM analysis, we can approximate its effect using anisotropy. The glass has a high dielectric constant, and the epoxy has a low dielectric constant. The horizontal electric field, which is parallel to about half of the glass fibers, preferentially experiences the higher glass dielectric constant. The vertical electric field, which is crossways to the glass fiber, sees a lower dielectric constant.
Anisotropic dielectric constants are difficult to find for most common substrates. So, we developed a technique to measure the dielectric constants. It uses a coupled line microstrip resonator. A coupled line has two modes: even and odd. We measure the resonant frequencies of both modes. From these two resonant frequencies, we determine two dielectric constants. The first pair of even/odd mode resonances occurs when the resonator is one half wavelength long. From that we determine our first pair of dielectric constants. At double the frequency, the resonator is one wavelength long, and we get another pair of resonances and another pair of dielectric constants. The coupled line resonators we usually measure are almost 25 wavelengths long.
We found that the horizontal dielectric constant of a composite board is lower, not higher, than the vertical dielectric constant. We later determined that this is because composite boards have a “butter coat” with no glass fiber. The microstrip odd-mode horizontal electric field concentrates in the surface, in the butter coat. So that is why the horizontal dielectric constant is lower.
Substrates loaded with ceramic particles are also anisotropic. When the ceramic particles are not spherical, they tend to lie down, parallel to the horizontal electric field. The horizontal dielectric constant is higher. For alumina, a ceramic made from particles of sapphire, certain manufacturing techniques leave the ceramic grains standing on end, and the vertical dielectric constant is higher. Measurements of anisotropy have yielded many surprises.
For substrates we have measured, typical differences between horizontal and vertical dielectric constants are 10 to 15%. We have published quite a few papers with lots of details on our results. We have also published all the mathematical and experimental details so duplicating our work should be straight forward. Just go to www.sonnetsoftware.com/resources/technical-references-sonnet.html, or go to IEEE Xplore and search for “Rautio”.
In conclusion, two numbers are better than one!