VNA Metrology and Measurement
VNA Metrology is the science of measurement. This science helps us understand the accuracy and meaningfulness of measurements. It is possible that a measurement might consist of a simple counting of items. In this case, the value measured would be an integer such as “3” with no uncertainty whatsoever. Most measurements, however, will have some uncertainty and the measured value will be said to be somewhere in a range of values with a certain probability. This application note will examine measurements, uncertainties, and the development of measurement uncertainty from the random factors which might affect it.
What is a Measurement?
A metrologically sound measurement will consist of the following:
For example, a weight measurement might be 5.75 kg ±0.05 kg with 95% Confidence. This states that with 95% certainty, the true value is between 5.70 and 5.80 kg and there is a 5% chance that the true value is outside of that range.
In practice, it is common to simply give a value without uncertainty, but this can be dangerous if a certain accuracy is required and not delivered. When in doubt, it is best to be concise and explicit.
What are Common Units for Measurement?
The modern system of units is the “SI” system, or Systѐme Internationale, within which seven primary units are defined; the Meter, the Second, the Ampere, the Kelvin, the Mole, the Candela, and the Kilogram. All other units in use are derived from these. All seven primary units may be reproduced by observing certain physical phenomena. For instance, the meter is defined in terms of a certain number of wavelengths of a spectral emission line of Krypton-86 where once it was defined by a single physical metal prototype. Up until a few years ago, the kilogram was defined by a cylindrical artifact composed of a Platinum and Iridium alloy housed in a double bell jar and stored in a vault at the International Bureau of Weights and Measures (BIPM) in Paris. This was the last remaining primary standard unit defined by a physical artifact.
Figure 1 – The “IPK”, Former Kilogram Standard Artifact, image courtesy of BIPM, Paris
In electronics, we deal primarily with Volts, Amps, Watts and Ohms. The Ampere is primary, but all the rest are derived. For instance, the Watt is 1 Joule of energy per second, J/s. The Joule is kg*m2/s2, an amount of energy, or a force over distance Newton*m or kg*m/s2 * m. Finally, the Watt is then kg*m2/s3 in SI units, an amount of energy per second.
How Do Uncertainties Propogate?
It is common for a measurement to be affected by more than one variable, each with its own uncertainty. For instance, a Vector Network Analyzer (VNA) measurement might contain some uncertainty due to cable flexure, thermal expansion, connector repeatability, and trace noise. To find the total uncertainty from each contributor it is necessary to know the sensitivity (partial derivative) of each independent component to the measurement and its statistical variance. If any two components are dependent on each other, the covariance must also be known. In formulaic form the total variance u2 is:
Where the measurement y is some function of factors xi, , u2(xi) is a variance and u(xi,xj) is the covariance between any two components.
What is a Converge Factor?
u is a standard uncertainty which for a gaussian probability distribution means that the true value will be the measured value y ±u with 68.3% confidence. For higher confidence, the value u may be multiplied by a coverage factor, k. A k of 2 gives 95.4% confidence and 3 gives 99.7%.
What Kinds of Errors Contribute to Uncertainty?
There are two kinds of errors, random and systematic. Random errors result in random variation of a measurement, whereas systematic errors result in skew of the measurement. For instance, a random pattern of hits around the center of a target demonstrates random error while a grouping of hits off center demonstrates both random and systematic error. Adjustment or calibration of the sights on the weapon can correct systematic error, leaving only the random component. Random errors can only be reduced by controlling the environment, like clamping the weapon in a vice before firing.
Figure 2 – Random and Random + Systematic Errors
Systematic errors in VNA measurement are reduced through user calibration. Random errors may only be reduced with careful methodology and environmental control. For instance, cable flexure errors might be reduced by taping the test cables to the bench top and thermal errors might be reduced with better control of the temperature in the laboratory. We can say that random errors are related to the entropy of the system. If the system has a large number of possible entropic micro states such as degrees of cable flexure then reducing them will increase entropy and reduce random error.
What are the Systematic Errors in VNA Measurement?
Systematic errors in VNA measurement are usually characterized using a 12-term error model as in Figure 3 and Figure 4.
Figure 3 – 12-Term Forward Error Model
Figure 4 – 12-Term Reverse Error Model
The 14 error contributions form 12 designated errors in the Forward and Reverse directions (e10-e01 and e23-e32 are combined).
e00, e33’ Directivity Error
e10e01, e23’e32’ Reflection Tracking Error
e11, e22’ Source Match Error
e10e32, e23’e01’ Transmission Tracking Error
e22, e11’ Load Match Error
e30, e03’ Isolation Error
The Directivity Error, e00, is due to the leakage in the VNA directional bridge. The two ports on the bridge should produce two pure signals. One, a sample of the incident signal leaving the port, and the other, a sample of the reflected signal entering the port. In practice, there is some leakage of the incident signal into the reflection port and some leakage of the reflected signal into the incident sample port. The incident and reflection ports are coupled off of the main signal line, perhaps 20 dB down. If the leakage signal is 40 dB down, we say that the raw directivity is 20 dB, the difference. Clearly, if the leakage were at 20 dB, there would be no directivity at all.
Reflection Tracking, e10e01 accounts for the indignities suffered by an incident signal leaving the port, passing through cables and connectors, e10, and being reflected by the DUT, to go through them once again, e01.
Source Match Error, e11, accounts for the complex error in the source impedance of the incident signal as it appears at the input of the DUT. Even if the source impedance of the stimulus of the VNA is a perfect 50 Ohms, small variations in the characteristic impedance of cables and connectors will alter it somewhat.
Transmission Tracking, e10e32, is similar to Reflection Tracking and includes the first part of it, e10, but also includes the errors in the path to the other VNA port, e32.
Load Match Error, e22, is the load impedance error from 50 Ohms as seen at the output of the DUT and includes impedance variations caused by the output cable and connectors and any small load impedance error through the bridge of the VNA itself.
Isolation Error, e30, accounts for any signal which bypasses the DUT entirely. Either through leakage within the VNA itself, which is uncommon, or from the electromagnetic coupling between the two DUT connections, such as that experienced between the probes within a probe station measurement.
Figure 5 – FormFactor Coaxial InfinityXT Probes Measuring ISS Calibration Standards
Normally the Isolation Error is lower than the receiver noise at even the lowest IF bandwidths and can be safely ignored.
The 12-Term error model is useful as each term represents a real physical source of error and is therefore appealing. Other models, such as the 8-Term model, might also be used, but a better understanding of calibration and residual errors is best based on the more intuitive 12-Term model.
What Does Calibration Do?
The process of calibration characterizes the error terms and reduces them as much as possible. A full 2-Port calibration includes all twelve error terms, a full 1-Port need only include three terms: directivity, source match and reflection tracking. One-Path 2-Port calibration includes only forward directivity, forward source match, forward reflection tracking and forward transmission tracking errors. Load match is not considered and none of the reverse errors are included. Response Open, Short and Thru calibrations include only a few of the error terms and the results are unsurprisingly less precise.
Calibration characterizes some number of these errors and reduces them. They are never completely eliminated, and the remaining errors retain the same names preceded by the word “residual”. For instance, the raw, uncorrected directivity might be 18 dB and after correction the residual directivity might be 45 dB.
It is possible to de-embed the errors on each side using them to convert measured S-Parameters to actual S-parameters of the Device Under Test (DUT). The calculation is lengthy and not particularly enlightening and is not included here.
Different calibration methods and different calibration standard kits will produce different levels of accuracy. The uncertainty of the load standard in a Short, Open, Load and Thru (SOLT) calibration kit is responsible for most of the residual error. Use of a sliding load or a data-based kit will provide considerable improvement. TRL calibration can be very accurate indeed since it doesn’t require a load standard, but TRL is only practical over a limited bandwidth.
How is Calibration Accomplished?
Calibration standards with known characteristics are measured by the VNA. A comparison of actual to measured results in a 2-port calibration allows for the determination of all twelve error terms. One common calibration method is SOLT or Short, Open, Load and Thru calibration. A known Short, Open, Load, and Thru are applied in turn to the VNA for measurement as shown in Figure 6.
Upon completion of calibration, the errors due to cables and connectors are mostly removed and the reference plane is moved to the ends of the test cables. The reference plane is the location of zero insertion loss, phase and delay.
Figure 6 – SOLT Calibration
Other calibration methods such as Thru, Reflect and Line, (TRL) might be used.
TRL calibration or the Thru-Reflect-Line method uses transmission lines and shorts or opens for calibration standards. TRL does not require a Load standard, which is the weakest link in the SOLT method. Three standards are employed for TRL calibration three standards:
- a Thru, which can be zero length if the connectors on the ends of the test cables mate directly, or a short length of coaxial line.
- A Reflect standard, which can be any total reflection like a short or open.
- A Line, which is a quarter wavelength longer than the Line at the center frequency of calibration, fc. The Line must be pristine, with precise characteristic impedance and extremely low reflection. Loss in the Line is acceptable. Alternatively, the Line may be data-based if these conditions can’t be met.
TRL is valid over the frequency range where the Line is 20 to 160º longer than the Thru, centered at 90º at fc. If the useable range is f1 to f2 then:
For a broader frequency range, multiple Lines can be used for multi-line TRL calibration. However, if it is implemented, TRL will always be band-limited as compared to SOLT, which can go all the way down to nearly zero Hz. Low frequencies will require long Lines for TRL, which may not be practical.
The Reflect standard is usually a Short, since it is easiest to implement. A delay between the standard connector and the actual short will result in phase variation over frequency. The phase of the Reflect must be known within ±λ/4 so the delay is usually specified in the TRL calibration kit definition. This requirement comes from the solution of a square root in the correction calculation. The sign of the root is ambiguous without the known approximate phase of the Reflect, and if this sign is chosen incorrectly, subsequent corrected measurements will exhibit an erroneous 180º phase shift.
The best Residual Directivity that can be achieved with SOLT is about 47 dB, due to the uncertainty of the Load standard. However, if a mechanically precise Line with excellent characteristic impedance is used for TRL, this can be improved to as high as 60 dB, allowing accurate VNA reflection measurements to much lower levels.
TRL correction comes from the solution of an 8-term error correction model which does not correlate directly with physical sources of errors. The network flow diagram is shown below in Figure 7.
Figure 7 – 8-Term Calibration Model
The process of correction amounts to solving for all eight error terms, converting to Cascade Parameters which may be multiplied directly (unlike S-parameters), solving for the inverse, and multiplying left and right sides to leave only the DUT.
Starting with measured S parameters with error, Sm:
Sm = [Sa] [SDUT] [Sb]
Convert to equivalent cascade parameter measurement Tm:
Where (the determinants)
Tm is measured, TDUT is unknown. Then:
Convert the cascade parameters back to S-Parameters
The process of finding the eight error terms is explored further in the application note titled “Introduction to the Metrology of VNA Measurement” in Appendix B.
How Does the Calibration Kit Affect Residual Errors?
The uncertainty of the Load standard in a calibration kit is simply equal to its worst-case return loss over the frequency range of use. That might be 30 dB for a high quality 3.5 mm, 26.5 GHz load standard. If we want to correct the directivity error of the VNA bridge, we can simply terminate the VNA output with a perfect load such that no reflection returns and measure the leakage signal from the reflection port. With the leakage perfectly characterized, we can then subtract it from all subsequent reflection measurements and the residual directivity will be infinite!
Unfortunately, the Load standard is not a perfect load. In fact, it will produce at worst a 30 dB reflection back into the VNA which will be seen on the reflection port of the bridge. This unwanted reflection cannot be distinguished from the directivity leakage so the leakage may only be characterized to that 30 dB level. This sets an absolute floor for reflection measurement accuracy; with infinite uncertainty at this level. Because of this reflection uncertainty will be ±3.3 dB at 20 dB and ±1.0 dB for 10 dB reflection measurements.
Where ΔdB is the difference between the residual directivity and the reflection measurement, -10 and -20 for the two examples above.
An Automatic Calibration Module (ACM) performs much better than this since its load standard is data-based. There will still be some reflection from the load, but the amount of reflection is known to the VNA and can be taken into account. Attainable residual directivity is 47 dB for most ACM models.
The uncertainty of the Open and Short standards, particularly the uncertainty of the tiny delay between the connector reference plane and the actual Short or Open within the standard contribute immediately to the uncertainty of reflection phase measurement and the slight phase error of the calculated correction results in ripples in the magnitude of both reflection and transmission reflection measurements. These are the residual reflection tracking error and residual transmission tracking errors.
In conclusion, random errors are minimized by controlling the environment as much as possible and systematic errors are reduced through the process of user calibration. Some residual measurement uncertainty remains after calibration, and this can be mostly attributed to the uncertainties of the calibration kit. Use of a precision calibration kit or an ACM is highly recommended to attain the most accurate results.