# Modeling FR4 Dielectric Constant vs. Frequency

April 22, 2019

If you’ve ever played with a prism, you’re already familiar with dispersion, even if you don’t know it yet. This important optical effect is also important in high speed and high frequency PCBs, where different signals travel at different speeds in a trace.

Just like any other material, FR4 has dispersion that affects travelling pulses and waves in PCB traces. The physics that describes dispersion is well-known and can be used to develop analytical models for signal behavior in a PCB, but working with a simulation package can make your life much easier.

## How FR4 Dispersion Affects Analog and Digital Signals

For those who may not remember their engineering or physics classes, the dielectric constant (and thus the refractive index) in a material is a function of a traveling electromagnetic wave’s oscillation frequency. This is the reason that a prism can be used to split white light into the colors of the rainbow. Similarly, the absorbance of an electromagnetic wave is also a function of the electromagnetic wave’s frequency.

This creates a number of effects in PCBs on FR4 that are especially important in high speed or high frequency applications. The variation in FR4 dielectric constant vs. frequency is called dispersion, which causes different frequency components in an electrical pulse in a PCB trace to travel with different velocities. With positive dispersion (the dielectric constant increases with frequency), higher frequency components arrive at a load later than lower frequency components, and vice versa for negative dispersion.

## Digital Pulse and FR4 Dispersion for Frequency Matching

A digital pulse is really just a superposition of analog waves, and dispersion affects each of these frequency components slightly differently. FR4 happens to have negative dispersion in terms of signal propagation speed, but placing a laminate with positive dispersion on the substrate can compensate signal distortion and reduce losses.

The majority of the frequency spectrum in a digital pulse (about 75%) is concentrated between the switching frequency and the knee frequency. The knee frequency is approximately one-third of the reciprocal of the signal rise time. A decent approximation is to account for dispersion only at the switching frequency, but this approximation is only acceptable for low to moderate dispersion.

The loss tangent for FR4 also varies with frequency, increasing rapidly up to about 100 KHz and then increasing steadily up to about 100 GHz. Thus, attenuation is larger at higher frequencies, but the stretching induced in digital pulses is less severe. Stretching is more important at lower frequencies and data rates, which affects the trace length mismatch tolerance.

With analog signals, PCB traces on FR4 tend to have higher losses than other PCB materials that are specialized for applications involving analog signals in the GHz range. For this reason, FR4 boards used in high speed/high frequency applications should include high speed laminates that reduce loss and compensate for the inherent negative dispersion of FR4. Alternatively, you should use a different material that is specialized for RF applications.

## Modeling Dispersion in FR4

Taking account of dispersion in a circuit model for a transmission line is done on a per-unit-length basis. In other words, the important parameters in modeling a transmission line are the series resistance and series inductance of the conductor, shunt conductance of the dielectric, and capacitance between the conductor and its return path. The important piece here is to take account of changes in the shunt conductivity and the dielectric constant as frequency changes.

The conductivity of a material is divided into a static component and a frequency-dependent component, where the latter component is proportional to the dielectric loss and the frequency. Meanwhile, the dielectric constant is inherently a function of frequency due to excitation of surface charge or dipolar oscillations at lower frequencies, or the excitation of lattice vibrations and electronic transitions at high frequencies.

In terms of constructing a circuit model for a board on FR4, the total capacitance and shunt conductance must be determined at the signal frequency of interest in FR4. These values must be included in a circuit model for a trace on an FR4 board when modeling the behavior of a circuit. The calculations involved are elementary, but getting the values wrong will cause your model to produce results that do not match reality.

While you could certainly analyze transmission lines in each portion of your board using the Telegrapher’s equations, you can also use a SPICE-based circuit simulator. You’ll need to include the correct values for shunt conductance and capacitance for your FR4 substrate at the frequency of interest.

Alternatively, because you have already determined the electrical properties of FR4 at the relevant frequency, you can include the correct values in a 3D field solver. This gives you the ability to examine radiated fields that can create signal integrity issues throughout your device or in multi-board designs.

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