# Minimising dispersion errors in time-domain measurements on waveguides

07 March 2019

### Hollow metal waveguides are sometimes the preferred method for transferring energy in the form of electromagnetic waves. The main reasons are that they can handle high power levels and have very low attenuation – compared with coaxial cables, for example. In certain applications they can also be more practical because of their rigidity.

When measurements on waveguides are made with a vector network analyser (VNA), it requires user calibration or error correction at the waveguide flange, where it is also required to specifically instruct the VNA that waveguide components are being used. This is due to the fact that waveguides have properties different from those of coaxial cables and require special mathematical processing of the measured signals. Properly configuring the VNA is necessary to ensure that error coefficients and offsets are calculated correctly during calibration and that subsequent measurements are made properly and represented accurately on the screen.

A key property of waveguides is the dispersion of signals, something not seen with coaxial cables. However, not all waveguides exhibit this property, and thus situations arise where initial calibration must be carried out using dispersive waveguide components but where the subsequent measurements are performed on a waveguide that does not exhibit dispersive characteristics. In such situations, it becomes necessary to instruct the VNA that the measured medium is, in fact, non-dispersive. This is a feature supported by the Anritsu Shockline family of VNAs.

Hollow waveguides and dispersion

In a hollow metal waveguide, the wave can be imagined to travel down the guide in a “zig-zag” pattern confined to the inside of the boundary walls (Fig.1). The mathematical relationship governing wave propagation in waveguides can be derived from Maxwell’s equations, describing the interdependent behaviour of electric and magnetic fields.

Using these equations, it can be shown that the speed by which an electromagnetic wave progresses down the waveguide, known as the group velocity, is slower than the speed of light. Similarly, it can be shown that any given phase of the wave, for example the crest, will have a velocity (known as the phase velocity) which appears to travel faster than the speed of light: something that we are told is impossible! It is important to emphasise that no physical entity is actually moving; instead, we are looking at geometric points in space where the phase is constant, e.g. the wave crests. This differentiates the waveguide from other media, such as free space or a coaxial cable, where the phase and group velocities are the same.

To simplify things, we can examine a parallel plate waveguide (think of just the vertical walls of the rectangular waveguide in Fig.1). By assuming that the plates are perfect conductors and that the E-field is linearly polarised in the y-direction it can be shown that we will end up with a travelling wave in the z direction and a standing wave in the x direction (superposition of plane waves bouncing off the plates). The standing wave pattern will only exist in specific, discrete modes of an integral number of half wavelengths.

From Maxwell’s equations it is then possible to derive an equation known as the “dispersion relation”: (see above)

Now, suppose we start to reduce the frequency (?). Since all the other components of the equation are constants, the only component that can be adjusted accordingly is k_z, i.e. the wave vector component in the z-direction. For the so-called fundamental mode (m=1 half wavelength), if we reduce the frequency to the point where its wavelength is 2a, or longer, or reduce the gap between the conductive plates to be half a wavelength, or smaller, no radiation will go through the waveguide (k_z=0). The frequency at which no electromagnetic wave can pass through the waveguide is called the cut-off frequency. This is another important characteristic of waveguides.

These concepts are identical for a rectangular waveguide, and in this case the cut-off frequency depends on the dimensions of the waveguide, the material inside it and the mode, and the group and phase velocities are functions of the cut-off frequency and signal frequency. These characteristics must be integrated when calibrating and measuring waveguides using a VNA.

Time-domain measurements using a VNA

VNAs are very powerful, accurate, and flexible measuring instruments. Their basic capability is to measure scattering parameters (or S-parameters), of an RF or microwave device and display the results in the frequency domain. However, at times, frequency-domain data offers little insight into the characteristics of the measured device. All Anritsu VNAs therefore have a time-domain option, which allows simulated time domain reflectometry (TDR) measurements, where measured data in the frequency domain is transformed into time or distance domain.

In order to calibrate a VNA for making measurements of a waveguide-based device the type of media, i.e. waveguide, must be part of the calibration definition. This will integrate the inherent dispersion effects based on cut-off frequency, dielectric constant, and dimensions; parameters which the user must enter. Since phase is measured and the propagator (including phase velocity) is used to calculate line lengths and offsets, including reference plane offsets, any such length information will be incorrect if dispersion effects are not considered.

Dispersion will also always affect measurements in the time, or distance, domain. A pulse being sent down a waveguide will become spread and shifted, or “smeared”: the more so the longer it travels.

This spreading means that the peak amplitude is lower but also that the peak becomes less distinct, so that identifying the distance to a reflection peak becomes less exact. The peak will also be shifted which directly leads to an erroneous distance reading. Fig.2a shows a simple example using a 223.6mm X-band waveguide through (69.6+154mm) and a 9.75mm offset short. Both calibration and measurements were made between 8.2 and 12.4GHz.

The active (orange) trace of Fig.2a shows the results without dispersion integration while the brown memory trace shows the results with dispersion integration. The spreading of the reflected signal is clear, and one can also see that the measured power is -0.8 dB, while close to 0 dB is expected from a short (assuming lossless waveguides). The total physical length of the waveguide through is known to be 233.35mm but the measured length is 302.31mm. Fig.2b shows the same measurement with dispersion integration, where a length of 231.53mm and power of -0.16 dB are measured.

There are ways in which waveguides can be designed to minimise dispersion effects and thus act as non-dispersive media similar to unbounded free space or a coaxial cable. This is sometimes desirable, for example, in applications measuring liquid levels in tanks using radar technology, where a non-dispersive medium will simplify the implementation of the radar sensor.

Fig.2c shows a time (distance) domain measurement of a 60cm open-ended coaxial cable after a waveguide calibration has been performed. The measured distance might be expected to be slightly longer than 60cm, considering also the coaxial-to-waveguide adapter not being compensated for, but the reflection from the open-ended cable appears at a distance below 60cm. Similar smearing to that in Fig.2a also appears. Since the measured medium is assumed to be waveguide based, dispersion integration is performed, resulting in smearing of the signal being put back in and an incorrect distance reading.

By setting the Line Type as Non-Dispersive, and repeating the measurement in Fig.2c, the result shown in Fig.2d is obtained. The active trace (orange) shows the results for the non-dispersive Line Type, with the smearing of the signal removed and the distance reading more in line with expectations. This feature offers users additional flexibility for measurements on hollow waveguides with non-dispersive characteristics. The solution lies in a feature found in Anritsu Shockline VNAs that allows the user to configure the measurement medium (or “line type”) (Fig.3).