A Guide to Transmission Line Impedance

January 12, 2022 Cadence PCB Solutions

PCB transmission line input

The signals on this ASIC might act like transmission lines

Transmission line impedance matching is a critical part of any layout. Whenever you are routing traces, there are several important points to check in order to ensure signal integrity throughout your board. Let’s take a look at which transmission line impedance you need to consider for termination.

Everything You Need to Know About Transmission Line Impedance

Before getting into the topic of determining transmission line impedance, you should read this article, which shows the different impedances used to describe real transmission lines in a PCB. Just to summarize, we have some important values in a transmission line, some of which have simple formulas that can be used analytically:

  • Characteristic impedance: This is the impedance of an isolated transmission line. In other words, this is the transmission line impedance when it is not coupled in any way to any other nearby transmission lines, such as in a differential pair.
  • Differential impedance: This is the impedance of a pair of transmission lines. It is only equal to double the characteristic impedance in certain cases. In general, it is double the odd-mode impedance, which is the value we care about for differential signaling, as it is used in high-speed PCB design.

The two most common impedances that are used in PCB design are the characteristic impedance and the differential impedance. The formulas for these values in terms of basic circuit theory are shown below:

Single-ended impedance

Odd-mode impedance

Differential impedance

Equation

Equation

Equation

The equations shown above are simplified in that they do not include losses along the transmission line, which must be included in a real transmission line. The subscript “m” values are mutual values, meaning they are the mutual capacitance and mutual inductance. This is a parasitic effect and is unavoidable, even on the most carefully designed boards. This coupling produces the even and odd mode impedance values for a transmission line, depending on how both lines are driven. It will also affect which impedance value you use for termination (see the next section).

When designing an interconnect to have a specific transmission line impedance, we worry more about the characteristic impedance, but the odd-mode impedance is generally more important in high-speed layout and routing. Most design guides will only talk about the characteristic impedance. In reality, to properly route and terminate transmission lines, we have to understand the even and odd mode impedance, and the differential impedance when dealing with high-speed interfaces.

Termination and Impedance Matching

Here is something the PCB community won’t tell you about transmission lines and their critical length—all interconnects that do not run at DC will behave as transmission lines. The issue is whether the effects of a transmission line impedance mismatch are noticeable at different frequencies or different signal rise times. This brings up the question of when it is worthwhile to match the transmission line impedance to a load. However, which transmission line impedance should be used?

As a result, designers have defined various values for a critical length below which you do not need impedance matching. As a general rule, if you are working with razor thin noise margins, then you should always match impedances between a driver, load, and source, even in electrically short transmission lines. With high-speed signals and interfaces, you should also always match transmission line impedance to the load. The correct way to do this is to look at the input impedance of the transmission line, not just the characteristic impedance.

This so-called critical length is actually quite important beyond just determining when to impedance match a transmission line to the source and load. Here is how this is quantified. If you calculate the voltage V and current I along a transmission line with length ℓ, you’ll find that the impedance seen by a signal (analog or digital) that reflects off a mismatched load depends on the length of the transmission line and its capacitive and inductive characteristics. This is shown in the equation below.

Input transmission line impedance equations

The final equation defines the lossy transmission line input impedance seen by a signal that is input to the line

If the propagation constant is known, then the input impedance can be determined for any frequency. However, as we see above, the input impedance depends on the length of the line, not just the impedances.

Long or Short Lines

The author, being something of a mathematical purist, is a proponent of taking this approach in all situations. First, simply calculate the value of γ and the characteristic impedance. Next, plug in the length and γ into the equation shown above. The impedance value you calculate is the transmission line impedance the signal sees as it reflects off the mismatched load and travels on the line.

In the limit of a very long transmission line (such as when the line length is many multiples of the wavelength), then the tanh function eventually converges to 1. In this case, the input impedance is just the transmission line’s characteristic impedance:

Input impedance long transmission line

In contrast, when the transmission line is very small compared to the wavelength (i.e., at low enough frequency), the impedance seen by a traveling signal will reduce to the load impedance because tanh(0) = 0. Note that this applies to both lossy and lossless transmission lines:

 Input impedance short transmission line

This explains why we have a critical length: when the transmission line is short enough that tanh(γℓ) ~ 0 (or tan(γℓ) ~ 0 for a lossless line), then the input signal only sees the load impedance. The source and load impedance should be matched to ensure maximum power transfer into the load and prevent signal reflection.

Special Cases for a Lossless Transmission Line

For transmission lines with sufficiently low losses (i.e., Re(γ) = 0), the tanh(x) function above must be replaced with the function jtan(x), where j is the imaginary constant. You will have certain cases where Im(γ)ℓ = mπ/2, where m is an integer. In this case, you will be evaluating tan(mπ/2) in the above equation. The result reduces to:

Input transmission line impedance equations for quarter and half wavelength multiples

These are the impedances that a signal sees as it reflects back along the transmission line. If the source, load, and transmission line are all mismatched, then there are repeated reflections along the length of the line, which leads to a stair-step response seen in digital signals or standing waves with analog signals.

Matching the transmission line’s characteristic impedance and the load prevents reflection at the load end, and the input impedance will just be the characteristic impedance. In this case, there are no reflections at the load, but you do not have maximum power transfer down the line if the source is unmatched. If you go a step further and match the source to the characteristic impedance, you now ensure maximum power transfer across the line.

Analog vs. Digital Transmission Lines

We should make a distinction here regarding the value of γ. In distinguishing the effects of transmission for different types of signals, think of transmission lines (either isolated or coupled) as filters with some transfer function. With an analog signal oscillating at a single frequency, γ is just the complex wavenumber multiplied by the effective dielectric constant (pi divided by half the wavelength inside the trace) plus the attenuation per unit length along the line. However, you might remember from discussions of dispersion, that the dielectric constant and the characteristic impedance depend on frequency.

This affects signals in different ways. For analog and digital transmission lines, we need to analyze transmission lines in the following ways:

  • Analog signals: Generally, we only care about one specific frequency. For modulated signals, we typically only design to the carrier frequency.
  • Digital signals: With digital signals, we have to design so that the characteristic impedance (or differential impedance) and termination are determined at a very high limiting frequency. The limited frequency of interest is usually the Nyquist frequency for the receiver or some limit determined from the rise time.

For frequency-modulated analog signals, the characteristic impedance of a transmission line has a constant value throughout the signal’s frequency spectrum as long as the relevant frequency range is high enough. At lower frequencies and with amplitude modulated signals, this may not be the case, and the other relevant impedance values will depend on frequency and driving mode, i.e., they will have some associated spectrum.

Considering amplitude modulation in transmission line impedance matching

Transmission line impedance with modulated signals only concerns the carrier frequency

With digital signals, one must remember that the source and load impedances are not consistent at all frequencies. The relevant bandwidth to consider for transmission line impedance matching is the range extending from the pulse repetition rate up to some very high frequency. For high-speed receivers, this is usually the Nyquist frequency. If you’re worried about oversampling the digital signal or measuring it, then you need to use at least 0.5/(rise time). As long as the component’s bandwidth is flat within this range of frequencies, then you can consider a single value for your PCB routing design rules and transmission line impedance matching.

A Note on Ringing

If there is some ringing on a transmission line due to signal reflections, it will be a function of the trace length relative to the length the signal travels during its rise time. For slowly rising signals, where the signal reaches the end of the transmission line before the signal rises to full strength, the signal will still ring even with perfect transmission line impedance matching. However, the strength of ringing may be so small as to be unnoticeable when compared to the noise margin as long as the transmission line’s impedance spectrum is flat up to the signal’s bandwidth limit. This is discussed beautifully by Dr. Howard Johnson. This is one reason why some designers encourage routing traces with the shortest possible length.

In essence, the ringing amplitude depends on the voltage difference between the two ends of the line. If the signal rises faster, the voltage difference across the line will be larger, leading to a larger ringing amplitude for a given line length. In order to compensate for ringing, you need to shift the resonance frequency of the transmission line to much higher than the knee frequency while ensuring impedance matching to eliminate reflection or you need to critically damp the response with a series resistor.

When in Doubt, Check the Datasheet

Many components are designed to specific signaling standards and have specified input and output impedance values. These components are designed to operate with a specific transmission line impedance, where the impedance is normally specified in terms of the characteristic--even or common (for parallel data transfer, for older standards) or differential impedance (for high-speed differential standards). These signaling standards will also specify the line lengths you should use for different applications and how components that use different standards (or no standard at all) should be terminated in order to ensure compatibility.

Pin arrangement on through hole connector

Check the datasheet to determine the impedance you should use for termination

If you are not working with a specific signaling standard, you will have some work ahead of you to ensure compatibility when connecting components. DRCs are generally not designed to account for potential intermixing between components with different signaling standards, and you’ll need to check component datasheets to see exactly how components should be matched to the transmission line impedance.

Transmission line impedance matching is a critical part of ensuring signal integrity, and you can ensure your interconnects are designed properly when you use the right PCB design and analysis software package. Allegro PCB Designer and Cadence’s full suite of analysis tools make it easy to determine the various transmission line impedance values and perform important signal integrity simulations in your circuits.

If you’re looking to learn more about how Cadence has the solution for you, talk to us and our team of experts.

About the Author

Cadence PCB solutions is a complete front to back design tool to enable fast and efficient product creation. Cadence enables users accurately shorten design cycles to hand off to manufacturing through modern, IPC-2581 industry standard.

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