﻿ How Your Transmission Line Functions as a Low-Pass Filter | NWES Blog

## How Your Transmission Line Functions as a Low-Pass Filter

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Every trace in a PCB is technically a transmission line, and depending on the signal you're driving into the structures, the circuit will exhibit noticeable low-pass filter behavior. All PCB transmission lines have losses, and in most cases they are driving capacitive loads which may be terminated. Due to the complex interplay between dielectric loss, conductor loss, and load termination in a transmission line, you can observe low-pass filter behavior in S-parameter measurements for the channel.

Unlike simple RC filters or high-order filters, the low-pass filter behavior of a transmission line on a PCB can be of fractional order in some frequency ranges, or it can be very nearly first-order in other ranges. When elements like stubs are present, strong losses occur, which greatly limit channel transmission at certain frequencies, i.e., bandstop behavior. Overall, the effect of losses and the resulting low-pass filter behavior is to limit channel bandwidth. In this article, I'll discuss these points and what to do about this as a PCB designer.

## How To Analyze Transmission Line Low-Pass Filter Behavior

If you look at simple discussions of transmission lines by some designers, they will incorrectly state that transmission line impedance is not a function of frequency. I have seen this from designers with decades of experience. In reality, all transmission lines have losses, and the losses in PCB materials can be used to predict the low-pass filter behavior.

To predict the low-pass filter behavior, we need several material inputs:

• Dielectric constant
• Dissipation factor (or loss tangent)
• Conductivity of your conductor material
• A measure of roughness for the conductor material
• Magnetic losses in the PCB substrate if they exist

These are used in the standard transmission line impedance equation which I have shown in many articles:

Transmission line impedance equation and terms defined in the RLCG model.

The point in this article is not necessarily to calculate impedances or S-parameters. To see why the low-pass behavior occurs at fractional-order or first-order roll-off profiles, we just need to look at the two major loss terms. The loss terms in this instance are the dielectric loss and the conductor loss.

Loss equations in dB defined from the transmission line propagation constant. The K term is the copper roughness factor as a function of frequency.

Both terms illustrate how the materials in the PCB attenuate power transmitted along the transmission line. The dielectric term is directly proportional to frequency, thus there is greater loss at high frequency. Less signal at higher frequencies in the power spectrum will reach the receiver, thus this is equivalent to low-pass filter behavior.

### Including Roughness

If you now look at how roughness affects losses on a PCB transmission line, you will find that the roughness essentially increases the skin effect resistance. Mathematically, this means applying the following functional transform:

Skin resistance formula with copper roughness factor K.

Here we have an additional frequency dependence in the copper roughness factor (K) that, in general, causes the skin resistance term to increase monotonically. This is another factor that creates a fractional power loss dependence on frequency, thus causing the transmission line to appear to behave as a low-pass filter.

### Load Capacitance

Not all transmission lines will be driving capacitive loads. For those that are driving capacitive loads, the load itself will also contribute to first order low-pass filter behavior. This applies whether the load is terminated to a target of exactly 50 ohms, left unterminated, or if some other resistive termination value is applied.

Driving a passive load with a transmission line causes the entire transmission line transfer function, which defines the output voltage seen at the load, to depend on the value of the load capacitance (and termination resistance if present). Together, we get the following expressions for the transmission line transfer function with connected load impedance in various configurations:

All of these transfer functions have the general form of a low-pass filter. However, remember that the characteristic impedance of the transmission line is also a function of frequency due to the frequency-dependent loss factors mentioned above. This is why you can get non-ideal low-pass filter behavior when working with analytical functions for transmission line transfer functions.

## Stubs as Bandwidth-Limiting Elements

Because all transmission lines exhibit low-pass filter behavior, they inherently limit bandwidth, and the designer’s job is to extend the bandwidth as far as possible such that the channel has the minimum required bandwidth to function as intended. One element in a channel that is a severe bandwidth limiter is a stub. The most well-known type of stub on a transmission line is a via stub, but there are many structures on a transmission line that can exhibit stub-like behavior. For example, balls in the ballout, structures that make up a connector, and even excessively large pads can exhibit stub-like behavior.

When a stub is present, it will generally exhibit odd integer multiple quarter wavelength anti-resonances due to destructive interference. The result is seen in an insertion loss plot and the corresponding return loss plot as shown below.

Insertion loss (S21) plot for PCIe lanes showing the effect of stubs on a PCIe edge connector.

Stubs could be more accurately characterized as exhibiting bandstop filter behavior, except there are multiple stop bands for a single stub. Whenever a stub resonance falls within your channel bandwidth requirement, it creates an extreme case of bandwidth limiting that could cause an interface to be non-functional. Simulations are most often needed to determine whether stub resonances are significant enough to require a change in the channel design.

## Real IC Packages

A real integrated circuit package or IC substrate is much more complex. The signal does not just enter the package pin and immediately encounter a load capacitance. In reality, it interacts with another conductor which might be defined as its own transmission line. At minimum, we have pin-package inductance that must be included in the load impedance. Collectively, we have all of the following additional elements that will determine signal propagation through the package and into the semiconductor die:

• Ball out or pin out with any nearby ground or stitching vias
• Internal conductors in the package/substrate
• Micro bumps on the bottom of the semiconductor die
• Conductors on the semiconductor die

These more complex linear network models for packages and substrates become more complex as you reach progressively higher frequencies, primarily because input impedance deviations at each interface become more pronounced as frequency increases. When you get into the fastest interfaces as of 2024, such as 224G PAM-4, every element in the package and substrate design must be included to accurately determine the filtering behavior of these interconnects.

Whether you're designing high-speed PCBs for mil-aero embedded systems or a complex RF product, you should work with a design and development firm that can ensure your product will be reliable and manufacturable at scale. NWES helps aerospace OEMs, defense primes, and private companies in multiple industries design modern PCBs and create cutting-edge embedded technology, including power systems for high reliability applications and precision control systems. We've also partnered directly with EDA companies and advanced ITAR-compliant PCB manufacturers, and we'll make sure your design is fully manufacturable at scale. Contact NWES for a consultation.

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