Identifying the relationship between frequency, the dielectric constant, and the overall effect on media propagation.
How design can counteract unwanted dispersion.
Utilizing simulation software to better model dispersion before manufacturing.
An FR4 dielectric constant may not result in the pleasant visuals often associated with dispersion, but the same physics are still at play
Since light and electromagnetism are the same, it should come as no surprise that dispersion also plays a role in the performance of 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 traveling 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 greatly simplify the number-crunching.
What FR4 Dielectric Constant Do I Need?
FR4 Dielectric Constant Influences Wave Propagation
The dielectric constant (and thus the refractive index) of 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.
It’s valuable to distinguish between the velocity of components vs. the total group velocity of a wave:
This creates several 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 known as 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. From the perspective of a pulse or periodic signal, a transmission can be described as up-chirped or down-chirped; these terms correspond to positive and negative dispersion, respectively.
Simulating a circuit model in SPICE for FR4 dielectric constant vs. frequency
Digital Pulse and FR4 Dispersion for Frequency Matching
A digital pulse is 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 for 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. This can result in a range of performance issues: lossy signals may lose coherence during their transmission and the significant amount of energy lost to the surroundings may prematurely age board materials and components. 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, a different material that is specialized for RF applications may be more appropriate.
At appreciable speeds, the substrate becomes as much an active part of the PCB as the components
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.
Find your board’s integrity needs accurately with proper simulation models
As a top-quality circuit analysis package that is specialized for PCBs, Cadence Schematic Capture, PSpice Simulation, and OrCAD PCB Designer are all integrated to easily traverse from one domain to another. These powerful tools provide pre and post-layout for analog and mixed-signal devices and integrate with thousands of components and FR4 dielectric constants, among other materials. Cadence’s PCB Design and Analysis tools, like the Constraint Manager, support users with customizable DFM to ensure designs translate seamlessly from the layout to manufacturing floors.
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