Modeling PCB Interconnects as Causal Systems
Most people that have read anything about Albert Einstein have heard the word “causality” thrown around. Although causality is often talked about in terms of relativity and quantum mechanics, this idea is central in any dynamical system. PCB interconnects are causal systems, and accurately modeling a PCB interconnect requires considering specific aspects of signal behavior and the system’s response.
If you can correctly model the causal behavior of your signals, you can calculate the correct response in an interconnect. This is critical for correctly predicting interconnect impedance, impulse response, and losses. All of these aspects of signal behavior are critical in high speed interconnects, where signal bandwidths readily span into the GHz range.
Causal Systems and Causal Models
Causal systems are defined as only responding to an external stimulus after the stimulus occurs, rather than before. The standard lumped element transmission line model produces a causal signal response when the line is driven, as long as frequency domain and time domain behavior is defined properly. The critical aspect here when modeling an interconnect as having zero amplitude before t = 0. In other words, you use a Heaviside step function as a weighting function for your injected signal.
Personally, I prefer to work in the Laplace or Fourier domain as transforms for a broad range of functions are readily defined. The standard technique for accounting for causality is to determine the transfer function in either domain for an interconnect in terms of the system’s desired impulse response function or step response function. Flipping this around, if the transfer function of a system is not correctly defined, then the impulse and step responses can exhibit noncausal behavior, meaning the signal begins propagating on an interconnect before the driver switches!
The signal processing community has known about this for decades, but the PCB design community should also be aware of these potential noncausal problems in modeling interconnect behavior. The relationship between an interconnect’s causal transfer function (Fourier domain) and its impulse response function h(t) is:
This means, if you know the causal impulse response function you need for your system, you can calculate the system’s transfer function; this would be done using the convolution theorem and would yield a pair of Kramers-Kronig relations. Note that, if a transfer function for an interconnect is known, you can recover the impulse response function h(t) with an inverse Fourier transform and check to see that it is indeed causal. Incorrectly modeling the transfer function of an interconnect (i.e., if the transfer function is non-causal) will result in incorrect modeling of signal behavior. This does not normally become obvious until you are working beyond ~5 GHz signal bandwidths. This should illustrate the importance of correctly describing the impulse response of an interconnect, which requires accurate impedance calculations as a function of frequency well into the GHz range.
Causal Models for Impedance Control
Accurate impedance control at a given frequency requires accurate modeling of the dielectric constant in a causal system. If you look over a broad enough frequency range, any material will have some dispersion. A full set of data on the real and imaginary parts of the refractive index can be difficult to find over a broad range. Suppose you have access to this data, either from the literature or from measurements. In that case, you can use these values to calculate the impedance of, say, the effective dielectric constant seen by a signal travelling on a microstrip trace over a broad frequency range.
This then gives you the characteristic impedance of the trace in the presence of dielectric dispersion. Similar procedures can be used for other transmission line geometries. Whether you calculate the impedance from the effective (frequency-dependent) dielectric constant using a standard trace impedance formula (either microstrip or stripline), or if you extract the parasitics in your board as a function of frequency, you’ll be able to calculate the characteristic impedance as a function of frequency. In addition, you’ll need the losses as a function of frequency. This is more intuitive if you look at parasitics in transmission line models and calculate the frequency dependence of the parasitics. Take a look at this IEEE article for a good overview of modeling the behavior of parasitics as a function of frequency.
The alternative is to use a model that defines the frequency response of the dielectric (both the real and imaginary parts) in terms of a small number of measurements. The typical method taught in laser physics classes and in signal processing classes is to use the Kramers-Kronig relations to relate real and imaginary parts of some material property, transfer function, or the signal itself. This pair of coupled integrals is used to define the causal response of a system when the losses in the system are known, or alternatively, to calculate the losses when the causal response in the system is known. The other method is to use a standard model that defines a fitting profile for the relative permittivity; the parameters that define the fitting curve are then determined from a limited number of measurements. The wideband Debye model is widely accepted as the most accurate for modeling dispersion in PCB substrates, particularly PTFE laminates and FR4.
More on Causality
Note that we’ve only looked at the characteristic impedance of a causal single-ended transmission line. We also haven’t considered the effect of vias on losses in causal models. Because nearby transmission lines can couple inductively and capacitively, and both types of coupling are frequency dependent, the even and odd mode impedances (and thus the common and differential mode impedances) will also be functions of frequency due to dispersion in the substrate. In addition to modeling dispersion in the substrate, copper roughness at the edge of the substrate effectively increases attenuation in the trace, which is also a function of frequency due to the skin effect. The typical treatment is to rewrite the total insertion loss on an interconnect by invoking a surface roughness factor (KSR), as defined below:
I’ve discussed some aspects of copper roughness modeling in a related article. Modeling copper roughness effects as a function of frequency and the causal behavior of interconnects remains an active area of research. A related topic is the causal behavior of a PDN, as this is important for correctly describing the effects of parasitics on power integrity. As more common devices are forced to run at higher speeds, and as more designers start working in the microwave/mmWave regime, tools for properly describing interconnects as causal systems will be critical for signal integrity.
The impedance calculation and layer stack tools in Altium Designer® treat PCBs as causal systems and incorporate the wideband Debye model to calculate dispersion in your PCB. This type of utility is critical for length tuning, impedance controlled routing, and controlling losses in PCB interconnects, especially in interconnects that run at high speed/high frequencies. Altium Designer also includes an integrated set of tools for building schematics, managing components, and preparing deliverables for your manufacturer.