Technical Article

# Understanding the 12-Term Error Model and SOLT Calibration Method for VNA Measurements

March 10, 2024 by Dr. Steve Arar

## The 12-term error model is a simple, effective way of modeling systematic errors in vector network analyzer (VNA) measurements. Learn about this model and an associated error correction technique in this article.

User calibration plays a fundamental role in minimizing the systematic errors of a vector network analyzer (VNA) measurement, enabling an overall measurement performance that far surpasses the capability of the raw hardware. This article will teach us about the 12-term error model, which is among the most commonly used models in the error correction algorithms of commercial VNAs. The final portion of the article will introduce the SOLT calibration technique, an equally widespread error correction method based on the 12-term error model.

### Using Signal Flow Graphs to Develop an Error Model

By examining the generic block diagram of a VNA, we can develop an error model for the measurement system. Consider the measurement block diagram in Figure 1, which is used for measuring the DUT’s input reflection (S11) and forward transmission (S21) coefficients.

##### Figure 1. The block diagram of the measurement system. Image used courtesy of Douglas Rytting and Agilent Technologies

Figure 2 shows how we can use signal flow graph concepts to model this system.

##### Figure 2. The flow graph of the measurement system. Image used courtesy of Douglas Rytting and Agilent Technologies

Let's look at this rather elaborate graph more closely. At the center is the DUT, which is modeled by its S-parameters.

At the input and output of the DUT, we observe networks that model the test setup’s cables and connectors. In contrast to simplified, ideal models, this graph doesn’t assume that the cables are lossless and present a perfect match. LC and MC denote, respectively, the loss and match of the interconnects. The model also accounts for the limited directivity of the directional couplers within the VNA. These effects are represented by the two magenta paths in the signal flow graph.

From Figure 1, it’s easy to see that the RF signal source shouldn’t directly couple to the input of the measurement receiver (b0). With real-world hardware, however, this undesired coupling is inevitable. In the flow graph, the magenta branch with coefficient LS-b0 shows a direct coupling between the signal source (denoted by aS) and the b0 receiver.

Similarly, the block diagram in Figure 1 shows that the signal reflected from DUT’s input (b1) should not appear at the input of the a0 receiver. Again, due to the finite directivity of the real-world couplers, this unwanted coupling is unavoidable. This leakage path is accounted for by the magenta branch with label L1-a0 in the signal flow graph of Figure 2.

Figure 2 provides a comprehensive view of the system’s error terms—in addition to accounting for loss, match, and leakage error terms, it also covers some terms that reflect the receivers’ nonlinearity and noise effects. However, a calibration scheme based on this model would require measuring many known loads to determine the error terms. Most VNAs opt for simpler models that can still minimize the systematic errors. The 12-term error model, being both simple and effective, is a common choice.

### The 12-Term Error Model

The 12-term error model consists of two submodels: one for the forward-direction measurements (measurement of S11 and S21 parameters) and the other for reverse-direction measurements (S22 and S12 parameters). Figure 3 shows the submodel for the forward direction.

##### Figure 3. Forward-direction submodel used in the 12-term error model. Image used courtesy of Steve Arar

There are seven error terms in the above model. However, not all of them are independent—if we write the equations for the measured S-parameters, we find that the terms e10, e01, and e32 don’t appear anywhere on their own. Instead, e01 and e32 each form a composite term with e10. This effectively reduces the number of unknowns in the above model from seven terms to six.

The submodel used for reverse-direction measurements mirrors the above. It includes another six terms (e’33, e’30, e’22, e’11, e’23e’32, and e’23e’01) with different values, giving us a total 12 error terms for the model as a whole.

One reason the 12-term error model for S-parameter measurements is favored in the industry is that its error terms can be related to the physical and comprehensible sources of error. The errors can be divided into three categories, each of which encompasses two error terms per submodel:

1. Signal leakage.
• Directivity error (e00 and e’33).
• Isolation error, also known as crosstalk (e30 and e’30).
2. Signal reflection.
• Source match error (e11 and e’22).
• Load match error (e22 and e’11).
3. Frequency response/tracking.
• Reflection tracking error (e10e01 and e’23e’32).
• Transmission tracking error (e10e32 and e’23e’01).

Figure 4 illustrates all of these systematic error types in an example VNA measurement.

##### Figure 4. Systematic errors in an example VNA measurement. Image used courtesy of Keysight

Let’s examine these errors by type. Because the submodels are mirror images of each other, we can use “error term” to mean both the forward-direction error term and its reverse-direction equivalent during this discussion.

#### Leakage Error Terms

VNA leakage error can take the form of either directivity error or crosstalk. The directivity error, as its name suggests, is related to the finite directivity of the directional coupler within the VNA. It’s not, however, exclusively a function of the coupler’s directivity.

We won’t learn about the other parameters that can affect it in this article, but I recommend Joel Dunsmore’s “Handbook of Microwave Component Measurements: With Advanced VNA Techniques” if you want to know more. This term can introduce significant error into reflection measurements.

Isolation error, also known as crosstalk, models the finite isolation between the test ports. In other words, it accounts for any signal that bypasses the DUT entirely. These signals can introduce error into transmission measurements.

This form of leakage error can occur within the VNA itself, though it’s uncommon in modern VNAs. More often, crosstalk takes the form of electromagnetic coupling between the two DUT connections—for example, between the probes of a probe station measurement system like the one in Figure 5.

##### Figure 5. A probe station measurement system. Image used courtesy of Adobe Stock

Note that crosstalk correction for probe station measurement systems can be very sensitive to positional changes. If a probe is moved only a small distance away from the crosstalk calibration position, the crosstalk correction vectors may reinforce—or even worsen—the crosstalk problem.

Because the isolation between the ports of a modern VNA is normally greater than the noise floor of the system, the crosstalk can’t be adequately characterized. For that reason, it’s typically set to zero. By ignoring the crosstalk term in both the forward and reverse submodels, we can reduce the number of unknowns in the model from 12 to 10.

#### Signal Reflection Terms

Signal reflection errors are related to the imperfect impedance match of the VNA’s ports. The source match error accounts for the impedance mismatch of the port that supplies the stimulus signal, while the load match error reflects the mismatch of the VNA port connected to the DUT’s output.

Using different error terms for each port means that the port match depends on whether or not the test port is supplying the stimulus. This is necessary because activating the signal source of a test port changes its configuration and hence its impedance match.

#### Tracking Errors

Frequency response errors, also known as tracking errors, affect both transmission and reflection measurements. As well as representing the relative losses of the signal paths in a given measurement, these error terms reflect the difference in the frequency response of the receivers associated with the measurement. We call these terms tracking errors because they indicate how well the various receivers in the network analyzer track one another across a frequency sweep.

The reflection tracking error term accounts for the frequency response error that occurs as the incident signal does the following:

1. Leaves the VNA’s port.
2. Travels through the cables and connectors.
3. Reflects from the DUT’s input.
4. Travels through the cables one more time back toward the VNA.
5. Is finally detected by the measurement receiver of the VNA.

Similarly, the transmission tracking error takes into account the relative losses and phase shifts experienced by the incident signal as it travels from the source test port to the load test port.

Both the transmission and reflection tracking errors are represented by composite error terms. To understand them, we should note that S-parameter measurements are ratios. If we refer back to the basic diagram in Figure 1, for example, the input reflection coefficient is obtained by dividing the output of the b0 receiver by that of the a0 receiver. If the frequency responses of the two signal paths aren’t exactly the same, an error will be introduced in the measured reflection coefficient.

Now that we’ve discussed all of the error terms in our model, let’s talk about how to correct them.

### 12-Term Error Correction

Figure 6 shows the final 6-term submodels for the forward (a) and reverse (b) directions.

##### Figure 6. 6-term forward submodel (a) and 6-term reverse submodel (b). Image used courtesy of Mini-Circuits

To correct for the measurement errors, we need to find the values for all of the error terms above. One common method is SOLT calibration, so named because it uses Short, Open, Load, and Through standards. Each of the standards is measured in turn during the SOLT calibration process (Figure 7).

##### Figure 7. SOLT calibration. Image used courtesy of Copper Mountain Technologies

To perform the SOLT calibration, each port of the VNA is calibrated separately by connecting it to a short circuit, an open circuit, and a matched load. This constitutes a one-port calibration of the two ports. The two ports are then connected together by the Through standard, which provides a known transmission coefficient. Since there are a total of 12 error terms, the two-port calibration is often referred to as 12-term error correction.

Finally, the results are analyzed to determine the error terms of the model. Once we’ve found the error terms, we can mathematically correct for the errors in our measurements. Most VNAs have built-in software that supports the SOLT calibration method—you won’t need to apply the equations yourself.

If you’re interested, you can find the equations relating the error terms and measured S-parameters to the DUT’s S-parameters in this Mini-Circuits application note. Though the equations are relatively lengthy, the basic concept behind them is straightforward.

### Up Next

SOLT calibration is a convenient and reliable way to calibrate most VNAs, but it does have limitations—it requires high-quality standards, for example. In addition to going into greater detail regarding the steps of SOLT calibration, the next and final article will discuss the non-idealities of the Open and Short standards used by this calibration in the real world.