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Transmission line notes.

Waveguides and Cavity Resonators.

Microwaves for satellite communications.

Scattering parameters, scattering matrix, s-parameters.

Uncertainties in measurements

Keeping a laboratory notebook

You have 18 hours lab time on this experiment; this should be sufficient. You may need extra time to write up.

If you are a local UoS user, you can view, down-load, and print a postscript version of the SMITH chart.

You can amend the source code with an editor to add your own title and to plot points.

There is a previous report Xerox copy in the lab; a backup of this report is held by the lab steward Dave Fishlock.

Much of the experiment requires some facility with making vernier and micrometer measurements accurately; allowing for backlash in adjusting screws; learning how to use unfamiliar equipment against the clock; analysis of sources of error; and getting to grips with unfamiliar concepts and theory. There is some help elsewhere on these web pages, including a book list having core and extended sections.

Although microwave measurements in practice are made automatically nowadays by network analyser, there is no substitute for the laborious slotted line measurements introduced here for getting a real feel for how wave measurements are made and how various components perform. One can observe the physical wavelength directly, and one realises that VSWR measurements are a very sensitive indicator of impedance mismatch. One also comes to understand how all the behaviour is governed by the ratio of the wavelength to the size of the devices, apparatus, transmission line, and the tolerances in manufacturing.

Scaling one's thought process from the 10GHz frequency to say 100GHz where the free space wavelength is 3mm, one realises the constraints of manufacturing tolerance on making reliable microwave systems at mm wave frequencies, for an error of 1/100 of a wavelength in the size of a component makes a measurable difference to the performance, and at 100GHz this is a distance of 30 microns.

Counters and pre-scalers for direct frequency measurement in terms of a quartz crystal reference oscillator are often used at lower frequencies, but they give up currently at frequencies above about 10GHz. An alternative is to measure the wavelength of microwaves and calculate the frequency from the relationship (frequency) times (wavelength) = wave velocity. Of course, the direct frequency counter will give a far more accurate indication of frequency. For many purposes the 1% accuracy of a wavelength measurement suffices. A resonant cavity made from waveguide with a sliding short can be used to measure frequency to a precision and potential accuracy of 1/Q of the cavity, where Q is the quality factor often in the range 1000-10,000 for practical cavities.

Precision is governed by the fineness of graduations on a scale, or the "tolerance" with which a reading can be made. For example, on an ordinary plastic ruler the graduations may be 1/2mm at their finest, and this represents the limiting precision.

Accuracy is governed by whether the graduations on the scale have been correctly drawn with respect to the original standard. For example, our plastic ruler may have been put into boiling water and stretched by 1 part in 20. The measurements on this ruler may be precise to 1/2mm, but in a 10 cm measurement they will be inaccurate by 10/20 cm or 5mm, ten times as much.

In a cavity wavemeter, the precision is set by the cavity Q factor which sets the width of the resonance. The accuracy depends on the calibration, or even how the scale has been forced by previous users winding down the micrometer against the end stop...

Wavelength is measured by means of signal strength sampling probes which are moved in the direction of wave propagation by means of a sliding carriage and vernier distance scale. The signal strength varies because of interference between forward and backward propagating waves; this gives rise to a standing wave pattern with minima spaced 1/2 wavelength.

At a frequency of 10 GHz the wavelength in free space is 3 cm. Half a wavelength is 15mm and a vernier scale may measure this to a precision of 1/20mm. The expected precision of measurement is therefore 1 part in 300 or about 0.33%

The location of a maximum is less precise than the location of a minimum; the indicating signal strength meter can be set to have a gain such that the null is very sharply determined. In practice one would average the position of two points of equal signal strength either side of the null; and one would also average the readings taken with the carriage moving in positive and negative directions to eliminate backlash errors.

Multiple readings with error averaging can reduce the random errors by a further factor of 3 for a run of 10 measurements.

The 10 GHz microwave signal in the waveguide is "chopped" by the PIN modulator at a frequency of 1kHz (audio) and the square wave which does this is provided by the bench power supply.

The detector diodes in the mounts on the wavemeter and slotted line rectify and filter this 10GHz AM signal and return a 1kHz square wave which you can observe directly on the oscilloscope. They are actually being used as "envelope detectors" as is the detector diode in your AM radio.

The VSWR indicator is a 1kHz tuned audio amplifier with 70dB dynamic range at least, and a calibrated attenuator sets its gain. The meter measures the size of the audio signal at 1kHz.

Since the detectors are "square law" their output voltage is proportional to the square of the microwave signal voltage. Regarded as a linear meter then, the VSWR indicator gives a deflection proportional to the POWER of the microwave signal (V*V/Zo). That is the reason for the curious calibration on the VSWR scales.

Half scale deflection on the VSWR meter therefore represents a microwave voltage of 1/sqrt(2) or 0.707 of that corresponding to full scale deflection.

Moreover, the VSWR meter is calibrated "backwards" in that one sets the voltage maximum at full scale deflection, then reads the VSWR from the voltage minimum. Thus the calibration point at half scale deflection is actually 1/0.707 or 1.414 VSWR. Check this. At 1/10 of full scale deflection the VSWR calibration point is sqrt(10) or 3.16. At this point one increases the gain by a factor of 10 with the main attenuator adjustment, and reads the VSWR scale from 3.16 to 10 on the other half of the VSWR scale. Get a demonstrator to show you how if this isn't yet clear.

Note that the gain dB scales and the attenuator on the VSWR indicator correspond to POWER of the microwave signal, not to POWER of the 1kHz audio input.

A visit to your favourite microwave book shows that a measurement of the standing wave ratio alone is sufficient to determine the magnitude, or modulus, of the complex reflection coefficient. In turn this gives the return loss from a load directly. The standing wave ratio may be measured directly using a travelling signal strength probe in a slotted line. The slot in waveguide is cut so that it does not cut any of the current flow in the inside surface of the guide wall. It therefore does not disturb the field pattern and does not radiate and contribute to the loss. In the X band waveguide slotted lines in our lab, there is a ferrite fringing collar which additionally confines the energy to the guide.

To determine the phase of the reflection coefficient we need to find out the position of a standing wave minimum with respect to a "reference plane". The procedure is as follows:-

First, measure the guide wavelength, and record it with its associated accuracy estimate.

Second, find the position of a standing wave minimum for the load being measured, in terms of the arbitrary scale graduations of the vernier scale.

Third, replace the load with a short to establish a reference plane at the load position, and measure the closest minimum (which will be a deep null) in terms of the arbitrary scale graduations of the vernier scale. Express the distance between the measurement for the load and the short as a fraction of a guide wavelength, and note if the short measurement has moved "towards the generator" or "towards the load". The distance will always be less than 1/4 guide wavelength towards the nearest minimum.

Fourth, locate the r > 1 line on the SMITH chart and set your dividers so that they are on the centre of the chart at one end, and on the measured VSWR at the other along the r > 1 line. (That is, if VSWR = 1.7, find the value r = 1.7).

Fifth, locate the short circuit point on the SMITH chart at which r = 0, and x = 0, and count round towards the generator or load the fraction of a guide wavelength determined by the position of the minimum.

Well done. If you plot the point out from the centre of the SMITH chart a distance "VSWR" and round as indicated you will be able to read off the normalised load impedance in terms of the line or guide characteristic impedance. The fraction of distance out from centre to rim of the SMITH chart represents the modulus of the reflection coefficient [mod(gamma)] and the angle round from the r>1 line in degrees represents the phase angle of the reflection coefficient [arg(gamma)].

These demonstration benches introduce the novice student to the essentials of the behaviour of microwaves in the laboratory. The wavelength is convenient at the operating frequency in X band (8-12 GHz approx) The waveguide used is WG90, so called because the principal waveguide dimension is 0.900 inches, 900 "thou" or "mils" depending on whether you are using British or American parlance. The guide wavelength at 10 GHz in WG90 is notionally 3.98 cm (the free space wavelength is 3 cm) so the standing wave pattern repeats at a distance of about 2 cm.

The benches include an attenuator, and an isolator. Both of these help to stop the reflected power from reaching the oscillator and pulling the frequency of the cavity and Gunn diode off tune when the load impedance is varied.

There is a dual directional coupler, arranged as a pair of crossed waveguides, which samples some of the forward wave power and couples it to a calibrated cavity wavemeter for measuring the oscillator frequency. Taken together with a measurement of guide wavelength, we have then two independent checks on the oscillator frequency. There is also a PIN modulator which chops the 10GHz signal at a frequency of 1KHz square wave.

The guide wavelength is an important property to be measured, and should not be changed during the course of a series of measurements. A half guide wavelength (about 2 cm) represents a plot of once round the SMITH chart. As remarked, we can determine the position of a minimum to about 1/20mm precision, or about 1 degree of angle around the chart. That represents 0.00125 lambda error in the phase plot on the SMITH chart.

A network analyser makes measurements of complex reflection coefficients on 2-port microwave networks. In addition, it can make measurements of the complex amplitude ratio between the outgoing wave on one port and the incoming wave on the other. There are thus four possible complex amplitude ratios which can be measured. If we designate the two ports 1 and 2 respectively, these ratios may be written s11 s12 s21 s22. These are the four "s-parameters" or "scattering parameters" for the network. Together they may be assembled into a matrix called the "s-matrix" or "scattering matrix".

The network analyser works on a different principle to the slotted line. It forms sums and differences of the port currents and voltages, by using a cunning bridge arrangement. The phase angles are found by using synchronous detection having in-phase and quadrature components. From the measured voltage and currents it determines the incoming and outgoing wave amplitudes. As we recall from elsewhere in the notes, V+ = (V + ZoI)/2 and V- = (V - ZoI)/2.

Network analysers can be automated and controlled by computer, and make measurements at a series of different frequencies derived from a computer controlled master oscillator. They then plot the s-parameters against frequency, either on a SMITH chart or directly.

The important experimental technique to the use of a network analyser lies in the calibration procedure. It is usual to present the analyser with known scattering events, from matched terminations and short circuits at known places. It can then adjust its presentation of s-parameters for imperfections in the transmission lines connecting the analyser to the network, so that the user never has to consider the errors directly providing he/she can trust the calibration procedure. It is even possible to calibrate out the effects of intervening transmission components, such as chip packages, and measure the "bare" s-parameters of a chip at reference planes on-chip.

Copyright D.Jefferies 1996, 2004.

D.Jefferies email 13th March 2004.