I took various computer classes in high school and always wondered why the conductors in Ethernet cables were twisted around each other. Little did I know this was a simple design method that ensured signals reached their destination while eliminating noise from other conductors in the cable. Sometimes, the best solutions to complex problems are actually the simplest.

Differential routing of conductors is not limited to Ethernet cables; it is one of the key routing topologies in many PCBs that use high speed signalling. Circuit board designers often discuss transmission line impedance in terms of single-ended traces rather than differential traces, but the differential impedance is an important specification that can be found in various signalling standards.

## What is Differential Impedance?

The differential impedance of a pair of traces is the impedance measured **between the pair **when the pair is driven differentially (signals with equal magnitude and opposite polarity). Due to the opposite polarity of the driving signals, the differential impedance is equal to double the odd mode impedance, which is just the impedance of the pair of traces when the pair is driven differentially.

The differential impedance of a transmission line arises due to coupling between two parallel traces, as shown in the image below. In this image, the capacitors in the cross-sectional view show the parasitic capacitance between the traces and the nearest ground plane. However, the two traces also have some mutual capacitance, which is shown in the top view. The two traces will be part of a closed loop and will have some inductance, so the pair will also have some mutual inductance. These two parasitics and the potential difference between the two traces determine the odd mode impedance, which is just half the differential impedance.

Coupling in a differential pair, which determines the differential impedance.

Note that the exact same model shown above applies to common mode driving, i.e., when the driving signal has the same magnitude and polarity in both traces. In this case, because the pair is driven in common mode, we can define a common mode impedance, which is just the impedance measured **between the pair** under common mode driving. The common mode impedance is half of the even mode impedance, which is the impedance of the pair of traces under common mode driving.

## Differential Impedance vs. Single-Ended Impedance

The meaning of the various impedance values is not always clear, which is complicated by the existence of seven different terms (sometimes eight, depending on who you ask!) used to describe transmission line impedance. Some novice designers have a tendency to treat each member of a differential pair as its own single-ended trace. This ignores the natural coupling that exists between multiple traces and how groups of traces are driven. As we'll see momentarily, this causes the impedance of a differential pair to be much different from the impedance of a single-ended transmission line.

The single-ended impedance of a transmission line (also called characteristic impedance) is an important value as it is used to calculate all the other impedance values. This value is the impedance of a theoretical transmission line that is infinitely far away from all other transmission lines. In other words, the single-ended trace impedance is calculated by ignoring any neighboring traces, regardless of whether they contain a propagating signal. In differential pairs, where we assume that a neighboring trace propagates a return current in the opposite direction as the signal trace, the signal in one line couples to the other through inductive and capacitive coupling.

Differential pair routing and vias on a PCB

Aside from suppressing crosstalk, coupling between differential traces actually decreases the impedance of each trace. Designers should be aware that a simple single-ended trace impedance calculator should not be used to calculate the impedance of differential traces.

For digital signals, we also need to take the frequency spectrum of the signal into account when calculating differential impedance. For the mathematicians out there, the frequency content in a digital signal can be represented as a sum of analog frequencies. This means that coupling in a differential pair that carries digital signals depends largely on the entire frequency spectrum of the digital signal.

Most of the intensity in digital signals is concentrated at frequencies below the knee frequency, which is equal to approximately one-third the inverse of the rise time. All frequencies between the operating frequency and the knee frequency will be the primary contributors that determine impedance.

## Start with Single-Ended Impedance

In case it isn't obvious from the above discussion, I'll state it here so that it is clear. Calculating the impedance of any transmission line, including differential pair impedance, requires calculating the parasitic capacitance and inductance for each trace trace. It also requires calculating the mutual capacitance and inductance for the pair. With these values, you can then calculate the differential pair impedance from a lumped circuit model.

Determining these parasitic and mutual capacitance and inductance values is not cakewalk in a complex PCB. You can typically estimate them for isolated, infinitesimally small traces in certain situations. However real differential pair impedance values are complex functions of trace geometry, and the substrate and conductor dielectric constants.

Thankfully, there are some closed-form solutions for determining single-ended impedance in various transmission line geometries. Take a look at this article for microstrip impedance formulas, and this article for stripline impedance formulas. You can also take a look at this method for determining stripline differential pair impedance values directly.

## Differential Pair Impedance Calculators

Striplines and microstrip differential pairs have different impedance values due to the presence of the substrate. Symmetric and asymmetric striplines or embedded microstrips also have different impedance values compared to a surface microstrip. The substrate dielectric and geometry modify the effective dielectric constant of the trace, which also modifies the critical delay time determining whether transmission line effects become easily noticed.

Working with many differential pair impedance calculators requires that you know the effective dielectric constant of the trace beforehand. This requires another calculator tailored to your specific geometry.

Once you have your effective dielectric constant and you’ve chosen your geometry, you are ready to start running calculations. You can play with the geometric parameters until you achieve the desired impedance level, or you can constrain the geometry and use the calculated impedance value for impedance matching in your PCB.

One drawback of many online differential impedance calculators is that they do not allow you to calculate the impedance as a function of frequency in the case of unterminated transmission lines. Some RF calculators only perform calculations at a specific frequency, usually 2.4 GHz. The S-parameters in a trace are frequency dependent, and differential pair impedance calculators should take this into account.

In the frequency domain, the impedance spectrum has a minima at specific frequencies due to resonances in unterminated lines. In the time-domain, there is a minimum at a particular oscillation period/rise time, followed by a monotonic impedance rise as the signal period/rise time increases up to the round-trip delay time. This is where powerful design software and excellent simulation tools become important for determining accurate differential pair impedance values. Take a look at Altium's latest improvements to their impedance calculation tools to see how you can easily determine differential pair impedance values.

A great piece of PCB layout software like Altium Designer® makes it easy to layout differential pairs in your next high-speed or high-frequency design. The ActiveRoute® tool, xSignals® tool, and built-in simulation tools can help you easily route differential pairs and avoid signal problems due to impedance mismatch. Talk to an Altium expert today if you’re interested in learning more about Altium .

Some designers have a tendency to treat each member of a differential pair as its own single-ended trace. This ignores the natural coupling that exists between each trace and how the pair of traces are drive. As we'll see, this causes the impedance of a differential pair to be much different from a single-ended transmission line.

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