Frequency-Dependent Material Properties

Estimated reading time: 2 minutes

At this year's DesignCon, I presented a paper on the topic of causal models for materials. Whenever you go from a frequency domain description of a function into the time domain, you have to worry about causality.

Causality means a response can't happen before the stimulus. This is the essence of cause and effect. The problem is that sine waves, and arbitrary combinations of sine waves in the time domain, are not usually causal. They extend back to minus infinity in time.

To make combinations of sine waves appear causal and be useful to describe time domain responses of real systems, we need to conspire the real and imaginary parts of the frequency domain response so all the waves at t < 0 cancel out, leaving just stuff that happens after t = 0.

Any dielectric properties, described by the real part of the dielectric constant, Dk, and the ratio of the imaginary to the real, Df, must be causal, and so must have this relationship between their real and imaginary parts. This is described by the Kramers-Kronig relationship.

This means that "to know the real part of the dielectric constant is to know the imaginary part." If we make the assumption that the real part decreases with the log of the frequency, then we can build a simple analytical expression for the dissipation factor Df.

We find that if the Dk drops with the log of the frequency, the Df will be roughly proportional to the slope of Dk and log F. This means the higher the dissipation factor, the more the dispersion.

It's the dispersion that causes increased rise time degradation, above and beyond what we expect from just the losses. If your simulator does not include this frequency-dependent dielectric constant, you will under estimate the rise time and actual performance may be worse than you predict.

If you care about high-speed serial links and use FR-4-like materials, you should care about frequency-dependent material properties.

You can download a copy of the paper I wrote with my partners which was presented at DesignCon.

I am turning the DesignCon presentation into a video lecture. This will be posted on my Web site in the next few weeks. Stay tuned!

Eric Bogatin    , the "signal integrity evangelist," is the founder of Bogatin Enterprises and a longtime design instructor. He will be teaching his "High-Speed Design For Signal Integrity" workshop in Shanghai during CPCA on March 17. Click here to register.