# Transmission Line Impedance Measurement: Even vs. Odd Mode Zachariah Peterson
|  Created: April 29, 2020  |  Updated: September 25, 2020 I still work on research in lasers, and pump-probe measurements are one way to examine how the electromagnetic field interacts with charge carriers in optical materials. With electronics, you won’t have to do anything as complex as a pump-probe technique for a transmission line impedance measurement. You’ll still need to examine how signals travel down a transmission line, and how they interact with the dielectric and load component.

That’s just for single-ended signals. What about coupled transmission lines? Even-mode and odd-mode transmission lines couple capacitively and inductively, causing signals on the two lines to see impedance values that do not match the characteristic impedance. When designing interconnect solutions for ultra-high speed and high frequency boards, you’ll inevitably need to gather impedance measurements for your proposed designs. Here are the tools you need in order to measure even and odd mode transmission line impedance, and how they relate to other fundamental measurements in digital systems.

## Transmission Line Impedance Measurement Techniques

Impedance can be measured in the frequency domain and in the time domain (normally referring to measurements from TDR data). Time-dependent impedance may seem odd and related to a transient effect that can appear due to dispersion, where the signal takes time to reach equilibrium as it travels back-and-forth on a transmission line. In reality, you can only infer the impedance from a time-domain impulse response measurement, which will be inherently bandlimited.

In my opinion, the easiest transmission line impedance measurement is gathered in the frequency domain from S-parameters, which can be converted into Z-parameters (this gives the line's self impedance and coupling impedance with respect to nearby lines/conductors). The alternative is to look in the time domain by sourcing a finite impulse or by sourcing a digital signal, which will give a transfer function for the line that can be converted back to a characteristic impedance. There are a few important reasons for this:

1. Impedance is not constant throughout the signal bandwidth, i.e., it varies with frequency, and the variation is not a simple inverse square root dependence. Anyone familiar with the RCLG model should be aware of this.
2. The effective dielectric constant is also not constant throughout the signal bandwidth. This creates different deviations from ideal impedance at different frequencies, which can be difficult to determine from time-domain data.

The two ideal transmission line impedance measurement techniques are time-domain reflectometry (and the related time-domain transmission measurement) and S-parameter measurements in the frequency domain. With some simple techniques and a VNA, you can measure the single-ended characteristic impedance of an isolated transmission line, and the differential impedance of a differential pair. A spectrum analyzer, VNA, and oscilloscope are all important for taking a transmission line impedance measurement.

## Single-ended Characteristic Impedance

### Time-Domain Reflectometry (TDR)

TDR measurements are useful for inspecting optical fibers, the same technique can be for a transmission line impedance measurement. This involves sending an impulse down a channel and measuring the time required for a signal to reflect off of an imposed impedance discontinuity. For a transmission line impedance measurement, this requires placing an element with a known impedance at the far end of the line. A related measurement is time-domain transmission (TDT), where the transmitted signal is measured.

This time-domain measurement reveals the phase shift due to reflection (either 0° or 180°) and the level of the reflected/transmitted signal. From this data, you can calculate the transmission line’s impedance from the complex reflection coefficient using the formula below: Complex reflection coefficient between a transmission line and the source/load. For reflection off the source end, Z0 is the source impedance, and ZL is the transmission line characteristic impedance. For reflection off the load end, Z0 is the transmission line characteristic impedance, and ZL is the source impedance.

Here, this assumes that the source is perfectly matched to the transmission line, which is an unknown quantity. In a real measurement, there is reflection at the source end and the load end, giving you two possible reflection coefficient measurements. Most TDR instruments (i.e., a VNA) can perform this calculation directly.

Signal distortion also occurs as the signal travels down the transmission line. The reflection coefficient you measure in the time domain is effectively an average by simply comparing signal levels. Either you need to convert your time-domain signal level measurements to the frequency domain with an FFT, or you need to determine the reflection coefficient in the frequency domain directly. The latter is more accurate and can be done with an S-parameter measurement.

### S-parameter Measurement

An S-parameter measurement treats a transmission line as a 2-port network, and the incoming/outgoing voltage and current are measured. This type of measurement can be easily configured with a VNA. Rather than go into all of the math behind this, I’ll refer you to any advanced electronics textbook, or you can take a look at this PDF to see how to convert Z-parameters or a characteristic impedance value to S-parameters. The important point here is that the reflection coefficient at each end of the line can be calculated from the S11 coefficient, which can then be converted back to the transmission line impedance as a function of frequency.

Note that a VNA is an invaluable piece of equipment to keep in your lab. These units can provide S-parameter to impedance parameter calculations automatically, and they can provide a time-domain reflectometry measurement. You can also extract the electrical length with these measurements.

## Even and Odd Mode Impedance for Coupled Lines

When examining coupled transmission lines for common-mode or differential driving, you either have to source two separate TDR/TDT signals on the two lines simultaneously, or you have to measure the even/odd mode impedances. The even mode impedance is simply the impedance of a single line when the two lines are driven in common mode. This is quite simple with a VNA as you can directly measure the S-parameters in the frequency domain and then convert this to an impedance.

The same procedure applies for odd mode impedance, where the coupled lines are driven in differential mode. After you calculate the even and odd-mode impedances, simply calculate the differential and common impedance as shown below.

Note that, when we are dealing with coupled lines, the characteristic impedance is not so important anymore. The important values are the even-mode and differential impedance values. In an ideal situation, the even-mode impedance will be nearly equal to the characteristic impedance, and the differential impedance will be nearly double characteristic impedance.

## Compare Your Measurements Against EM Field Simulations

Whenever you’re designing and measuring interconnects for advanced applications, you should compare your results to data from an integrated EM field solver. Routing tools which include an EM field solver can account for parasitics in a real layout, and can help you identify any sources of impedance deviation along an interconnect. If you want to learn more about working with transmission line measurements and calculations, take a look at these articles:

The layer stack design and routing tools in Altium Designer® include an integrated electromagnetic field solver. This type of tool is ideal for controlled impedance routing and simulating signal integrity in your interconnects. The layer stack manager also gives you access to a range of exotic stackup materials, including important electrical properties. 