**This article is part of the TechXchange: Vector Network Analyzers.**

*Members can download this article in PDF format.*

**What you’ll learn:**

- What are the error terms in the 12-term error model?
- How are these errors related to the simpler 8-term model?
- How is 1-port calibration performed?
- How are the error terms evaluated using calibration standards?

*This is the third in a multi-part series of articles on vector network analysis. Part 1 introduced the VNA, how such instruments work, and some of their applications in the lab. Part 2 introduced S-parameter network flow diagrams and how to manipulate them to solve real-world problems.*

Because test cables experience loss and delay, vector network analyzers (VNAs) must be calibrated to make precise measurements. Those cable losses mean that phase measurements made at the VNA ports will not be the same as those made at the device under test (DUT).

If the characteristic impedances of the test cables are not precisely 50 Ω, the effective source and load match of the VNA will be slightly off, resulting in measurement ripple. In addition, the directivity error in the VNA measurement bridges must be corrected to ensure accurate reflection measurements. This short primer will quantify the various systematic errors and introduce the 12-term error-correction model, which serves as a basis for VNA calibration.

The S-parameter network diagrams, as seen in the previous article in this series, will be used to describe the various error terms.

## What is the 12-Term Error Model?

Our model starts with a perfect VNA. We create two error boxes and put them on either side of the DUT and attach the perfect VNA to the left and right sides of *Figure 1*.

Error box A is defined by its four S-parameters—e_{00}, e_{01}, e_{10}, e_{11}—and error box B by S-parameters e_{22}, e_{23}, e_{32}, and e_{33}. The DUT is defined by its S-parameters S_{11}, S_{12}, S_{21,} and S_{22}. The S-parameters measured by the perfect VNA attached to the system, S_{m,} are altered by error boxes A and B. The isolation terms e_{30} and e_{03} are usually small and may be ignored for the moment.

The measured S-parameters, S_{m}, may be expressed as in Equation 1: ** **** **

where S_{a} are the actual S-parameters of the DUT.

Unfortunately, you can’t multiply S-parameters as shown in Equation 1, but you can recast them as cascade (or transfer) parameters, and then matrix multiplication is valid. The conversion to cascade parameters is accomplished with Equations 2, 3, and 4:

where:

Then:

And we can find T_{DUT} by inverting T_{A} and T_{B}:

We then convert the cascade parameters back to S-parameters, and we will have our calibrated measurement assuming we know the eight **e** values:** **

where the T parameters are T_{DUT} from above. And:

The e values for the A and B boxes can be determined by making a few measurements of known calibration standards such as an Open, a Short, a Load, and a Thru.

But this is an 8-term error model, and the individual error terms don’t correspond well to any physical error contributor. The 8-term model can be modified into two 6-term models—one for each measurement direction—where the stimulus is emitted by Port 1 and received on Port 2 or the other way around. The error terms in this model relate to actual physical phenomena.

Referring to *Figure 2*, for a forward 6-term model, we can normalize e_{10} to 1. In doing so, any signal that comes back through e_{01} had to pass through e_{10} first, so we change e_{01} to e_{10}e_{01}. The same is true for e_{32}, so we changed it to e_{10}e_{32}.

Lastly, we eliminated node a_{3} in the forward model because there’s no driven signal on the right side to feed it. With a_{3} gone, e_{33} and e_{23} go with it and we have *Figure 2* with six error terms. Similarly, the reverse model is shown in *Figure 3*.

Focusing on the forward model, we can name these errors based on physical causes. a_{0} is the driven node and, ignoring isolation term e_{30} for now, the input signal from a_{0} immediately sees e_{00}, which sends a signal right back to b_{0}.

This is the *directivity* error, the signal that leaks from the incident signal into the reflection port of the directional bridge. The remaining forward traveling signal then encounters S_{11} of the DUT, generating a reflection, which heads back to the source.

That reflection encounters e_{11}, the *source-match error*. The source-match error is seen right at the end of the test cable and is caused by any deviation from 50 Ω due to the cable, the connectors, and to some extent, the raw source-match error of the VNA itself. This source-match reflection is pushed back up to the top of the network diagram and reenters the DUT along with the incident signal.

The remaining S_{11} reflection, that which was not diverted by e_{11}, passes through e_{10}e_{01}, the *reflection tracking error*. The reflection tracking error is essentially the frequency response of the test cable, the connectors (twice), and the internal hardware of the VNA in the reflection path of the measurement bridge. This error will be a gentle low-pass response, as the test cable will always be more lossy at higher frequencies. These first three reflections are shown in *Figure 4*.

That part of the incident signal not reflected by the directivity error, or S_{11} of the DUT, enters the DUT and is affected by S_{21}. This modified signal then sees e_{22}, the load-match error, passes to the bottom of the diagram, and heads back toward a_{2}. The load-match error is seen at the end of the test cable and is caused by any deviation from 50 Ω due to the cable, the connectors, and the raw load match of the VNA itself.

Some of that reflected signal at a_{2} is reflected upward again by S_{22} and rejoins the incident signal. The rest passes through S_{12} of the DUT—some is reflected once again by the source-match error e_{11}, and the rest passes through the reflection tracking error e_{10}e_{01}, and into Port 1 of the VNA.

Finally, the remaining incident signal not reflected by the load match passes through e_{10}e_{32}, the *transmission tracking error*. This error is the frequency response of the Port 2 test cable, the connectors, and the internal hardware of the VNA behind Port 2. It looks like a gentle low-pass response, much like the reflection tracking error.

These reflections are shown in *Figure 5*. The reflections that occur in the reverse model are identical in nature and needn’t be shown again.

## How are the 12 Error Terms Found?

The first step to the full 2-port correction is to perform a full 1-port correction on each side. This means finding e_{00}, e_{11}, and e_{10}e_{01} on the left side and e_{33}’, e_{22}’, and e_{23}e_{32}’ on the right, where the primed error terms are for the reverse direction.

First, we place a load with reflection coefficient Γ_{L} after a network defined by its S-parameters as in *Figure 6*.

_{in}, is given by Mason’s Rule:

^{1}

This is an equation with three unknowns, as we don’t need to know e_{10} and e_{01} separately. We need only apply three known Γ_{L} loads and measure three Γ_{in} values to find them. The three Γ_{L} values may be Open, Short, and Load, or +1, −1, and 0, if we had perfect calibration standards. Note that if a perfect Load is applied, the fraction will be zero and the input reflection coefficient will be simply e_{00}, the directivity error. This is an important result, because it highlights the significance of this error in the total 1-port measurement error.

There’s a very convenient matrix operation to find e_{00}, e_{10}e_{01}, and e_{11}. Form two matrices **C** and **V**:^{2}

where Γ_{a1}, Γ_{a2}, and Γ_{a3} are the three __actual__ load values, which must be known. These are usually the Open, Short, and Load calibration standards. Γ_{m1}, Γ_{m2}, and Γ_{m3} are the three measured input reflection coefficients with the three known loads applied.

Note that the three applied loads might also be three total reflections with different delays, whereby they’re widely separated on the Smith Chart, such as 0°, 120°, and 240°. Any three calibration standards may be used if they remain far apart on the Smith Chart over the frequencies of calibration. Open, Short, and Load are the simplest because the Open and Short will maintain a 180° difference on the circumference if they have equal internal delays, and the 50-Ω load is at (or near) the center.

Next, we calculate matrix **E**:

where H is the Hermitian transpose operator, or the matrix transposed with its entries conjugated.

From **E**, we can find the three error terms:

Note that (C^{H} * C)^{−1} * C^{H} is a least-squares calculation. If our measurements are a little noisy, we can improve our results by making more known measurements and add more rows to C and V. Matrix E will still have three values in the end, and the results will be somewhat better in the face of slightly noisy measurements.

The last three error terms can be found now that the error terms on the left-hand side are known. e_{22} and e_{10}e_{32} are found mathematically and e_{30}, the isolation, is found by a last measurement with no connection between the two ports.

We’ll find e_{10}e_{32} and e_{22} on the right-hand side of the forward model. First, the test cables are terminated, and measuring S_{21} gives isolation term e_{30} directly. Then with the two ports—or test cables—connected directly together:

where:

and:

This process is repeated in the reverse direction to obtain the six primed error terms of the reverse model. With all 12 error terms, we can now convert measured S-parameters into actual S-parameters of the DUT. The equations for this are given in Reference 4.

The assumption that the two cables are connected directly together greatly simplified these last two calculations. If using a Thru with some loss and delay instead, it’s easier to revert to the 8-term error model and use the “unknown Thru” or SOLR calibration calculation.^{5} This method doesn’t require precise knowledge of either the loss or the delay of the Thru calibration standard.

## Conclusion

We’ve seen here how the measurement system can be modeled as a perfect VNA with error boxes on either side and how those errors can be related to real-world phenomena. We provided an example of the 1-port calibration calculation and then the final calculations to achieve a 2-port calibration.

With these error terms characterized, do we now have a perfect measurement? No, we will still have residual errors. These residual errors may be traced back to several factors, not the least of which are the uncertainties of the calibration kit standards. This will be a topic for the next article in the series.

## Acknowledgments

The great majority of the work on VNA calibration was done by Douglas Rytting,^{3} and this paper draws extensively from his published work. The intention of this article is to elaborate on that work and make it more accessible to those without extensive VNA experience.

**Read more articles in the TechXchange: Vector Network Analyzers.**

## References

1. Mason, Samuel, “Feedback Theory – Further Properties of Signal Flow Graphs”, Proceedings of the IRE (Volume: 44, Issue: 7, July 1956).

2. Walker, Brian, “One-Port VNA Calibration: A Look Under the Hood,” Microwaves & RF, Jan 11, 2021.

3. Rytting, Douglas, “Network Analyzer Error Models and Calibration Methods,” University of Illinois, Urbana-Champaign.

4. Walker, Brian, “Introduction to the Metrology of VNA Measurement,” Appendix A, May 31, 2022.

5. Walker, Brian, “VNA Master Class: Full 2-Port VNA Calibration Math Explained,” Webinar.