
VNA Calibration Theory Introduction
April 10, 2023Introduction
Calibration is the process of configuring a measurement device such that it provides measurement results within a known and acceptable range. All measurements are susceptible to errors. If these errors can be understood and characterized, they can be mostly removed, resulting in a more accurate answer. It is impossible to remove all errors, as there will always be a certain amount of uncertainty in any measurement. VNA calibration theory helps us understand and statistically quantify this error and state a result in a concise and comprehensive manner.
Metrology of a Measurement
A metrologically complete measurement is stated as follows:
<Measurement Value> <Units> <Uncertainty Range> <Confidence Interval>
For instance, a measurement of return loss in dB may be:
Return Loss = 12.5 dB ± 0.2 dB with 95% confidence
The true value of the measurement is between 12.3 and 12.7 dB with 95% confidence and there is 5% confidence that the measurement is outside of this range. Metrological principles allow us to calculate the range and confidence interval. There will also be a wider range for which the confidence interval is 98 or 99%, but never 100%.
Uncertainties
The uncertainty of a measurement is taken to be ± 1σ for a gaussian (normal) distribution value. Since 68% of all possible values lie in this range, the confidence interval is the mean value ±σ with 68% confidence. We may want better confidence, and thus specify a wider interval like ±3σ for 99.7% confidence. This is technically three uncertainties but might still be called the uncertainty. Confusion is eliminated as long as the confidence percentage is clearly stated. For example, the uncertainties on VNA measurements given in Copper Mountain Technologies data sheets are 3σ confidence intervals, as shown in Figure 1 below.
Figure 1 – VNA Measurement Uncertainties
Figure 2 below shows a standard (Gaussian) probability distribution function centered at zero with a standard deviation (σ) of 1.0. Probability values are found by integrating the area under the curve between sigma values. The total area under the curve from minus infinity to infinity is 1.0, or 100%, and the area from -1 to +1 sigma is 0.68, or 68%. In practice, the average value –the peak of the curve– may be centered at some non-zero value, and the standard deviation may be a value other than one.
Figure 2 – Gaussian Distribution
VNA Error Terms
An error model assumes there is a Device Under Test (DUT) with error boxes on each side, which are then measured by a perfect VNA. The 12-term error model is favored in the industry, as the terms correlate to real and understandable error sources. This model is shown below in the network flow diagrams of Figure 3 and Figure 4.
Figure 3 – 12 Term Error Model, Forward Direction
Figure 4 – 12 Term Error Model, Reverse Direction
The 12 terms are named as follows:
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
Network flow diagrams depict waves moving in directions shown by the arrows. The a0 and b0 shown in Figure 3 are not two separate electrical connections. Indeed, they are a single wire. e10 and e01 flow on the same conductor but are separated on the diagram to emphasize the direction of flow. By inspecting the diagram, you can say that signal e11 will add to signal e10 and must flow towards node a1. a0 must be an independent source unaffected by other signals, since no arrows flow toward it. All other nodes must depend on a0 and the S-parameters after it.
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 two, a sample of the reflected signal entering the port. In practice, there is some leakage of the incident signal into the reflection port.
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). However, it 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 for a probe station measurement.
Figure 5 – FormFactor Coaxial InfinityXT Probes Measuring ISS Calibration Standards
Normally, the Isolation Error is lower than the VNA receiver noise at even the lowest IF bandwidths and can be safely ignored.
These errors are determined and mostly removed by measuring a series of calibration standards with known characteristics and comparing the actual measurement with error to the expected measurement based on the standard data.
For example, the directivity error is due to a leakage path in the VNA bridge. When a signal is leaving the VNA and there is no reflection, there should be a signal on the forward port of the bridge, but none on the reverse port. Unfortunately, there will always be a small leakage signal from the forward signal, as shown in Figure 6.
Figure 6 – Leakage Signal, Directivity Error
To measure the leakage signal, terminate the port with a perfect load such that there is no real reflection. After the leakage signal is known at all frequencies, it can simply be subtracted from the reflection measurements to remove the directivity error.
Unfortunately, there is no such thing as a perfect load. A $7,000, 3.5mm, 26.5 GHz calibration load might have a worst-case reflection of -30 dB which would appear on the reflection port of the bridge along with the leakage signal. Therefore, the leakage may only be corrected to this level, and reflection measurements below -30 dB are meaningless. A sliding load calibration will improve on this somewhat but will not be discussed here.
After measuring the calibration standards and mathematically determining all twelve errors, they can be mostly removed from the measurement resulting in a nearly true measurement of the DUT. It is impossible to completely eliminate the errors; they are greatly reduced but not absent. These small remaining errors are called residual errors. For instance, the raw directivity of the VNA bridge might be 18 dB before calibration. After calibration, the residual directivity can be 30 dB to 26.5 GHz with a mechanical calibration kit, or 40 dB using an Automatic Calibration Module (ACM).
What is SOLT Calibration?
SOLT calibration, or Short-Open-Load-Thru calibration, uses four well-characterized calibration standards. Each of the standards are measured in turn, as shown in Figure 7, and the results are saved in order to compute the measurement correction.
Figure 7 – SOLT Calibration
Once all standards are measured—Open-Short-Load on the ends of the test cables and Thru between the cables—the measurements, along with the known values of each standard, are used to calculate all twelve error terms. With the error terms known, a correction matrix can be determined which will mathematically correct all measurements.
The characteristics of the calibration standards are known by one of two methods. First, and most likely, they are defined by a delay between the connector and the actual standard, plus a third order polynomial representing the parasitic effects. The delay and polynomial will be the same for every calibration piece of the same model number from a vendor, so there will be some error due to manufacturing tolerances. Secondly, the calibration kit might have Touchstone data files to characterize each standard [2]. This is much more accurate but requires individual measurement of each standard and is therefore costly.
A typical entry for a calibration kit definition is shown in Figure 8.
Figure 8 – Cal Kit Definition
The delays of each piece are shown and the polynomial coefficients for the parasitic capacitance of the open and the parasitic inductance of the short are on the right-hand side of Figure 8. With this information, the VNA can calculate the true characteristics of each standard.
What is TRL Calibration?
TRL calibration or the Thru-Reflect-Line method uses transmission lines and shorts or opens for calibration standards [3]. 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 [4]. TRL will always be band-limited as compared to SOLT, which can go all the way down to zero Hz. Low frequencies 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 9.
Figure 9 – 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 and
(the determinants)
Tm is measured, TDUT is unknown. Then:
Convert the cascade parameters back to S-Parameters
Where
And you have the corrected result. The process of finding the eight error terms is beyond the scope of this application note but is explored in [1].
Conclusion:
To achieve accurate results, calibrate the VNA using a defined calibration kit prior to making a measurement. The two most common methods, SOLT and TRL, were explained here. This user calibration should be differentiated from factory calibration. For a factory calibration, the VNA is examined at a certified test laboratory and verified to be performing within acceptable operating parameters. The ISO-17025 certified laboratory at Copper Mountain Technologies is capable of performing this service. It is common to have factory calibration performed every two years, but different organizations may have different required calibration intervals.
References:
- Copper Mountain Technologies, “Introduction to the Metrology of VNA Measurement, Appendix B”, https://coppermountaintech.com/introduction-to-the-metrology-of-vna-measurement/
- Copper Mountain Technologies, “Using a Databased SOLT Calibration Kit”, https://coppermountaintech.com/using-a-databased-solt-calibration-kit/
- Copper Mountain Technologies, “TRL Calibration of a VNA”, https://coppermountaintech.com/video/trl-calibration-of-a-vna/
- B. Marks, “A Multiline Method of Network Analyzer Calibration,” IEEE Tram. Microwave Theory Tech. Vol. 39, pp. 1205-1215, Jul. 1991.
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Why is it Necessary to Calibrate a Vector Network Analyzer?
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If the VNA isn’t calibrated before measurement, then the reference plane will be at the VNA connectors, and any phase or delay will include the excess delay and phase shift of the test cables. Additionally, there may be ripple in the transmission measurements (s12 and s21) due to error term constructive and destructive interference.

What is an Automatic Calibration Module?
January 6, 2023
What is an ACM? An Automatic Calibration Module, or ACM, is used to perform user calibration on a Copper Mountain Technologies Vector Network Analyzer (VNA). This is the calibration which is performed prior to making a series of measurements, not the one performed on an annual or semi-annual basis at a calibration laboratory. The ACM presents a series of calibration standards to the VNA – that is, one or more Opens, Shorts, Loads, and a Thru – to complete a SOLT (Short/Open/Load/Thru) full 2-port calibration, or a SOL 1-port calibration. A simplified block diagram is shown in Figure 1. Figure 1 - ACM Block Diagram A mechanical calibration kit contains individual Open, Short, Load, and Thru calibration pieces, which must be attached to the VNA one after another. A calibration kit definition file informs the VNA of what reflection coefficient each piece should exhibit at each frequency, and calibration is calculated by comparing measured values to actual values. In contrast, the Short, Open, Load, and Thru in the ACM have been precisely characterized at the CMT factory and the results are stored in its non-volatile memory. This is much more accurate than the typical polynomial curve fit definition of the mechanical kit. The ACM completes the calibration process with a single connection to the test cables, automatically cycling through the internal calibration standards. How is it Used? A USB cable must be connected between the ACM and the PC running the VNA software to use the ACM. The ACM is then attached to the test cables and either 1-port, 2-port, or 2-port one-path calibration methods are chosen under the Calibration/Autocal menu. The ACM may be attached to the test cables in either direction, with Port A on the Port 1 test cable, or with Port B when the orientation option is set to AUTO. The operating frequency range of the ACM is printed on the label. When using an ACM, make sure that the VNA frequency range is within the range printed on the ACM. Leave the thermo compensation function ON, as this will provide more accurate calibration when ambient conditions change. Figure 2 - 8 GHz 2-Port ACM Allow the calibration to complete, and then COR should appear in the lower-left corner of the VNA UI screen. The calibration type should be added at the end of the trace definition in the upper left corner – [F1], [F2], and [OP] for full 1-port, 2-port, and one-path 2-port calibration respectively. For 4-port calibration, the ACM4509 AND ACM4520 are 9 and 20 GHz 4-port models respectively. They are capable of performing 1-, 2-, 3-, or 4-port calibration as needed. The Confidence Check feature, which may be enabled in the Automatic Calibration menu, helps verify the integrity of the test cables. After pressing the Confidence Check button in the User Interface, an approximately 20 dB attenuator is connected internally between the A and B ports of the ACM and a precision factory measurement of that same attenuator is loaded into the active trace memory. The memory trace will normally be covered by the active measurement, so it is best to subtract the live measurement from the memory trace by navigating to Display>Memory>Data Math>Data / Memory. Set the scale to a low value, such as 0.05 dB/Div, for a magnitude reading and then move the ACM from side to side to flex the test cables slightly. There should be very little movement of the displayed difference. If wild movement is seen, then the test cables are either loose or defective. Defective cables should be discarded immediately. Figure 3 - ACM Connection for 1-Port Calibration, Load Unused Port Figure 4 - ACM Connection for 2-Port Calibration Why Use an ACM? The measurement accuracy of the VNA depends on the quality of the user calibration. You can best understand the quality of calibration by examining the residual errors which remain after the calibration is performed. The 12-term error model is generally used to characterize these errors – six terms for forward measurements, such as S11 and S21, and six for the reverse, S22 and S12. These errors are depicted in the S-parameter network diagrams shown below. Figure 5 - 12 Term Error Model, Forward and Reverse The actual Device Under Test (DUT) S-parameters are shown in the grey box, and the error vectors appear on either side of it. These errors are: Table 1 - 12 Term Errors After calibration, these errors are greatly reduced but are not zero. The remaining errors are called residual errors. For instance, the raw directivity of the directional bridge in the VNA might be 18 dB, but after calibration with an ACM, it might be 46. The Importance of Residual Directivity The directional bridge in the VNA has four ports: an input, an output, a forward sample port, and a reverse sample port. When a forward traveling signal passes through the bridge, the forward port should have an attenuated version of this signal exiting it, assuming 20 dB coupling attenuation for example. A reverse traveling signal should cause a 20 dB down signal to appear on the reverse sample port. Ideally, there would be no signal on the reverse sample port when there is only a forward traveling wave passing through the bridge. As no bridge is ideal, a leakage signal will be present as shown in Figure 6. If the normal coupling is 20 dB and the leakage signal is 35 dB, then we say the bridge has 15 dB of directivity. To understand the importance of the quality of the calibration kit, consider the following: to improve the directivity, you would have the VNA produce a signal and terminate it by attaching the calibration load to the output port. Theoretically, there would be only a forward traveling signal leaving the VNA and no reflection from the load, hence no reverse traveling signal. The leakage signal could then be measured on the reverse sample port and simply subtracted from all subsequent measurements. The residual (corrected) directivity would be infinite! Unfortunately, the assumption that there is no reflection from the calibration load is incorrect. A very expensive (~$6k) 26 GHz, 3.5mm calibration load will have worst case return loss of about 30 dB, so there will be a reflected signal, 30 dB down entering the port. This will show on the reverse sample port. The residual directivity of the bridge after calibration will therefore be no better than 30 dB. Figure 6 - Bridge with Leakage Signal Because the ACM has a stable, temperature-compensated, data-based calibration load, the residual directivity after calibration is 46 dB, much better than 30 dB. Residual Errors and Reflection Uncertainty The other error terms are corrected to the much smaller residual values based on the quality of the calibration standards within the ACM. The uncertainty of VNA measurements depends on the values of these residual errors. A typical reflection uncertainty chart after ACM calibration is shown in Figure 7. Figure 7 - Reflection Uncertainty The floor for reflection measurement accuracy is determined by the residual directivity error. The quality of the calibration load is very important. To put it another way, 1-port reflection measurements are not noise limited, but rather interference limited by the residual directivity. The reflection uncertainty curve above is only valid for calibration with an ACM. If using a mechanical kit with a 30 dB return loss calibration load instead, the above curve would be shifted 16 dB to the right and the uncertainty for a -20 dB reflection measurement would be about ± 3 dB instead of ± 0.6. From EUROMET [1], the equation for the approximate uncertainty of a reflection measurement in terms of the residual errors is (in linear terms): Where Sii is the measured reflection T is the Residual Reflection Tracking M is the Residual Source Match L is the Residual Load Match And R accounts for random factors such as connector repeatability The Load match term, L, only applies for reflection measurements of a DUT, which is terminated on the other side by the opposite port of the VNA and insertion loss through the DUT is low (S12 = S21≈1). Residual reflection tracking and source match dominate for high reflection, and residual directivity dominates for low reflection measurements. Residual directivity, D, is approximately equal to the calibration load uncertainty; 30 dB for the expensive mechanical kit and 46 dB for the ACM. Residual source match is approximately equal to the weighted square root of the sum of the squares of the calibration load uncertainty and the angular phase error (radians) of the Open and Short calibration standards. Residual load match is approximately equal to the residual source match for a 2-port calibration with unknown thru. The uncertainties of the calibration standards determine the residual errors and how those residual errors affect the final reflection measurement uncertainty. A more thorough treatment of VNA metrology is given in [2]. Residual Errors and Transmission Uncertainty A typical transmission measurement uncertainty chart after ACM calibration (CMT model S5065) is shown in Figure 8. Figure 8 - Transmission Uncertainty, Model S5065 For transmission measurements, the approximate measurement uncertainty is given by: Where: T is the Residual Transmission Tracking error M is a residual error due to DUT mismatch I is the residual Isolation error R includes a number of random factors but is dominated by the noise floor for small reflections Residual transmission tracking is proportional to the product: Where: µ1 is Residual Source Match L2 is Raw Load Match µ2 is Residual Load Match L1 is Raw Source Match Raw source and load match are properties of the VNA hardware, and residual source and load match are approximately equal to the uncertainty of the calibration load standard. The raw source and load match of the VNA should be reasonable, but the two products above are dominated by the small residual values. For a well-matched DUT, the M term may be ignored, and the port-to-port isolation of the VNA is normally below thermal noise in 10 Hz bandwidth, so it can also be ignored. The R term is mostly trace noise due to proximity of the measurement to the receiver noise floor. The uncertainties of the calibration standards affect uncertainty of transmission measurements over a broad range from about 0 to -60 dB. Below -60 dB, random noise introduced by the receiver noise floor dominates, and above 0 dB, receiver compression introduces errors. Receiver compression errors may be avoided by reducing the stimulus power for amplifier measurements. Again, the quality of the calibration kit is important for precise transmission measurements. Adding Adapters to the ACM Normally, you would use an ACM that matches the connectors of the test cables. For instance, an ACM with female-to-female N connectors would be selected for use with test cables having male N connectors. Sometimes, adapters are added to the ACM to allow for calibration using test cables with non-matching connectors. User Characterization can be performed on the ACM to allow for this. As an example, use test cables that have 3.5mm male connectors and an ACM that has female N connectors. Place the N Male to 3.5mm female adaptors on the ACM and torque them appropriately. Calibrate the ends of the test cables using a 3.5mm calibration kit, another ACM, or a good mechanical kit. Make sure the frequency range is appropriate for the adapters and set the number of points to 1601. Attach the test cables to the ACM with adapters using proper torque. In the VNA User Interface under Calibration/AutoCal, select Characterization and change it from “factory” to one of three user characterizations. Press the Characterize AutoCal Module button, allow the calibration to complete, then fill out the information on the dialogue screen which will automatically appear. After User Characterization, the adapters should be left on the ACM. If they need to be removed and then re-attached, the user characterization should be repeated. Conclusion The accuracy of the VNA measurement greatly depends on the quality of the calibration kit in use, especially for 1-port reflection measurements. Because of its thermally compensated, data-based internal calibration standards, the ACM delivers consistently superior results as compared to calibration using even a very high-grade mechanical calibration kit. In addition, the Confidence Check feature provides a fast and easy method to verify the robustness of the calibration and test set-up. For precision VNA measurement, an ACM is required. 2 and 4-Port ACM modules can be found on the CMT website here. References [1] EURAMET, “Guidelines on the Evaluation of Vector Network Analyzers”, Euramet cg-12 Version 2.0, 3/2011. [2] Brian Walker, Copper Mountain Technologies, “Introduction to the Metrology of VNA Measurement”, May 31st, 2022.

Power Calibration with Copper Mountain Technologies VNAs - Quick Start Guide
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Performing amplifier, mixer, or other sensitive DUT measurements accurately and correctly depends on proper setup and appropriate calibration. Power calibration using power sensors at the DUT calibration plane ensures precise results.

TRL Calibration
July 8, 2019
Thru-Reflect-Line (TRL) calibration has a number of advantages over the Short-Open-Load-Thru (SOLT) method often used in VNA calibration. Read the app note to learn more about the TRL calibration method.

Calibration Types and Considerations
April 19, 2018
One of the most frequently asked questions we receive at Copper Mountain Technologies’ sales and support departments goes something like this: “What about calibration?” It’s an unfortunate reality that in the English language, Calibration has two completely distinct definitions. The first relates to checking out the instrument periodically to make sure it is operating within its specifications. “Performance test” is the procedure by which the analyzer performance is verified, typically annually. The second meaning is to do with Measurement or User calibration, a collection of techniques by which measurement accuracy is maximized and made to exclude elements of the system from those measurements (such as cables, adapters and the like). In this application note, we discuss both meanings of calibration as related to Copper Mountain Technologies’ Vector Network Analyzers (VNAs). First, we describe Annual Calibration and then later we discuss measurement calibration.