The powerful instruments known as vector network analyzers (VNAs) are indispensable in RF and microwave applications. However, before using a VNA, it’s essential to correct for the systematic errors and imperfections of the test setup by performing a user calibration.

A previous article introduced the Short-Open-Load-Through (SOLT) method, which is one of the most common user calibration techniques. In this article, we’ll explain how the SOLT calibration method works in more detail. We’ll also discuss the non-idealities of the Open and Short standards used in real-world SOLT calibrations. A thorough understanding of these concepts will help you, the VNA user, more confidently analyze the results of your measurements.

Finding the Error Terms is Key to Success

As we discussed earlier in this series of articles, the SOLT calibration method relies on the 12-term error model. This model accounts for errors arising from finite directivity, reflection tracking, port match errors, and more. Figure 1 shows its forward and reverse submodels.

Figure 1. The 12-term error model consists of a 6-term forward submodel (a) and a 6-term reverse submodel (b). Image used courtesy of Mini-Circuits

To obtain the true S-parameters of the DUT from the raw measured values, we need to correct for all of the errors modeled above. The correction process involves finding the values of the 12 error terms and applying them to a mathematical formula. Though the math for these error corrections is relatively straightforward, determining the error terms requires accurate standards and measurements, making it a challenging task.

Some applications may pose extra challenges to finding the error terms. For example, determining the error terms of a DUT at cryogenic temperatures, at extreme power levels, or with unusual connectors can be extremely difficult. Still, once the error terms are known, the equations that need to be solved are relatively simple.

To better understand the calibration process and its requirements, let’s examine SOLT calibration more closely.

The SOLT Calibration Process

SOLT calibration uses Short, Open, Load, and Through standards to determine the error terms of the measurement system. The Load, Open, and Short standards are typically collected into a calibration kit; some kits, like the one in Figure 2, also include a Through standard.

Figure 2. The S2611 calibration kit. Image used courtesy of Copper Mountain Technologies

Let’s look back at the 12-term error model in Figure 1. To find the error terms of the forward submodel in Figure 1, we use the following three steps:

Apply one-port calibration.

Determine isolation.

Make the Through measurement.

While we’re only going to walk through the process for the forward measurements, the same three-step procedure can be applied to find the error terms of the reverse submodel. All we have to do is change which error terms we plug into the equations.

Step 1: Apply One-Port Calibration

In this step, the input reflection coefficient (ΓIN) of the forward submodel is measured for three different standards: Short, Open, and Load. The input reflection coefficient measured by the VNA is related to the actual reflection coefficient of the standard (ΓL) by the following equation:

$$Gamma_{IN} ~=~ e_{00}~+~ frac{e_{10}e_{01}Gamma_L}{1~-~e_{11}Gamma_L}$$

Equation 1.

By measuring three different values of ΓL, we obtain three independent equations, each of which contains the three unknown error terms e00, e10e01, and e11. In an ideal world, the Short, Open, and Load standards should produce ΓL values of –1, 1, and 0, respectively. Of course, we don’t live in an ideal world. We’ll discuss shortly what the reflection coefficients of real-world Shorts and Opens look like.

Steps 2 and 3: Determining Isolation and the Through Measurement

To find the leakage term (e30), we connect matched loads to both Ports 1 and 2 of the VNA and measure the S21 parameter. This is an optional step—the leakage between the ports of modern VNAs is typically negligible, so we could set the leakage term to zero without suffering major consequences.

Finally, we use a Through standard to connect Ports 1 and 2 of the VNA together. By measuring S11 and S21 parameters, we obtain two independent equations to determine the two remaining error terms (e22 and e10e32).

To summarize:

Three measurements of the one-port standards (Short, Open and Load) at each port produce a total of six independent equations.

A fully characterized Through standard provides a total of four equations—two for each measurement direction.

The two isolation terms are found by connecting matched loads to both Ports 1 and 2. This gives us another two equations.

The whole calibration process produces a total of 6 + 4 + 2 = 12 independent equations to solve for the 12 error terms in the model. It’s unlikely we’ll need to solve them ourselves, however—most VNAs have built-in software that supports SOLT calibration. We only need to connect the appropriate standards and let the VNA perform the calibration.

Often, we can assume that the Load standard is a perfect 50 Ω impedance. The delay and loss of the Through standard are typically also given. Defining the Open and Short standards can be a bit trickier, as we’ll soon see.

Defining the Open Standard

Figure 3 shows the physical construction of a female Open. The left side of the center conductor is a typical female connector configuration that uses a spring finger socket. The right side of the center conductor remains unconnected, creating an open circuit.

Figure 3. Physical construction of a female Open standard. Image used courtesy of Gregory Bonaguide and Neil Jarvis

Note that there’s a short length of transmission line between the reference plane and the actual implementation of the Open. Because the transmission line adds a delay, producing a frequency-dependent phase in the reflected signal, this standard could more accurately be called an “offset Open.” However, nearly all Open standards are actually offset Opens, so it’s usually not worth making the distinction.

A fringing capacitance (Ce) forms at the center conductor’s open end between the inner and outer conductors. To make life even more complicated, this capacitance is also frequency-dependent; it affects the reflection coefficient of the standard and cannot be ignored.

At low frequencies, a fixed capacitance value (C0) may be sufficient. For frequencies higher than a few hundred MHz, the variation of capacitance with frequency becomes much more noticeable. Most VNAs use a third order polynomial equation to describe the variation of the fringing capacitance with frequency:

$$C_{e}(f) ~=~ C_{0} ~+~ C_{1}f ~+~ C_{2}f^{2} ~+~ C_{3}f^{3}$$

Equation 2.

The coefficients C0, C1, C2, and C3 depend on the specific Open standard’s geometry and material compositions. The coefficients should be in the appropriate units for the final value to have units of farad. For example, if C0 is in femtofarads, then C1 should be in fF/Hz, C2 should be in fF/Hz2, and so on.

Figure 4 shows the parameters of a typical Open standard as they would be specified in one of Keysight’s VNAs.

Figure 4. Parameters of a typical Open standard. Image used courtesy of Keysight

As you can see, the parameters of the transmission line—delay, loss, and characteristic impedance—are specified along with the coefficients of the fringing capacitance. For some calibration kit models, the same third-order polynomials and delays are used to describe the calibration standards. The kit manufacturers rely on precision manufacturing and machining to enable this. Even so, some error will always persist.

Another way of defining the calibration standards is to use a database of reflection vs. frequency measurements from a very accurately calibrated VNA. The database method is much more accurate than the polynomial method, but also much costlier.

The Open Standard on a Smith Chart

An ideal open circuit is located on the circumference of a Smith chart at a single point with a phase angle of zero. However, if we measure the reflection coefficient of an Open standard for a given frequency range, what we get is an arc—not a dot. We can see this in Figure 5, which shows the measured reflection coefficient of the S2611 calibration kit’s Open standard.

Figure 5. Smith chart showing the measured reflection coefficient of the S2611 kit’s Open standard. Image used courtesy of Copper Mountain Technologies

The measured reflection coefficient appears as an arc. It starts at a phase angle of zero when frequency is low, then moves clockwise as frequency increases. This is due to two factors:

The fringing capacitance of the Open.

The short transmission line that appears before the actual open circuit.

Defining the Short Standard

Figure 6 shows the physical construction of a female Short. The center conductor is shorted to the outer conductor on the right side of the diagram.

Figure 6. Physical construction of a female Short standard. Image used courtesy of Gregory Bonaguide and Neil Jarvis

As with the Open standard, there’s a short length of transmission line before the standard’s actual implementation. The standard is therefore an offset Short. Also as with the Open, this is true of nearly all Shorts—we’re only making the distinction here to explain why the standard’s reflected signal experiences a frequency-dependent change in phase.

An inductance (Le) is created at the location of the Short. Like the fringing capacitance we discussed in the previous section, this inductance is frequency-dependent. We might be able to ignore Le at low frequencies and for large connector sizes (≥7 mm). At higher frequencies and for small (≤3.5 mm) connectors, we need at least a third-order polynomial to describe the variation of the inductance with frequency:

$$L_{e}(f)~=~L_{0} ~+~ L_{1}f ~+~ L_{2}f^{2} ~+~ L_{3}f^{3}$$

Equation 3.

Figure 7 shows some typical values for a Short standard’s parameters.

Figure 7. Parameters of a typical Short standard. Image used courtesy of Keysight

The Short Standard on a Smith Chart

On a Smith chart, the measured reflection coefficient of a Short appears as an arc that starts at a phase angle of 180 degrees at low frequencies and moves clockwise as the frequency is increased. This is due to the Short’s parasitic inductance and the length of transmission line that makes it an offset Short. Figure 8 shows the measured reflection coefficient of the S2611 calibration kit’s Short.

Figure 8. Smith chart showing the measured reflection coefficient of the S2611 kit’s Short standard. Image used courtesy of Copper Mountain Technologies

Measuring Calibration Standards

Let’s say we’re using an Open and a Short standard in our user calibration. If we use the VNA after it’s calibrated to measure these standards’ reflection coefficients, will we still see arcs on the Smith chart?

In a word, yes. Most real Opens and Shorts are actually offset Opens and offset Shorts, so their responses correspond to an arc on the Smith chart rather than a single point. For more information on why, please refer to Examples 4 and 5 in “Learn by Example—Using an Impedance Smith Chart.”

The calibration process doesn’t change this. It only removes imperfections from the test setup and determines the correct error terms to map the standard’s measured response to the one expected from the third-order polynomial description. In fact, the VNA will adjust the result to be consistent with its polynomial description even if the standard is slightly damaged in some way and doesn’t produce the characteristics specified by its manufacturer.

For that reason, you should verify the result of your completed calibration by measuring an Open or Short standard that doesn’t come from the kit you used in the calibration process. This process produces the error terms based on the measured response of the standards—if we re-measure a standard from the same kit, we might get a false sense of confidence that the calibration is correct. The VNA has already been adjusted to match the characteristics of that standard.

By using a different standard, we can see how well the VNA measures a device that wasn’t part of the calibration process. This lets us find any errors or inconsistencies, such as incorrect standard definitions or loose connections, that might have occurred during the calibration.

Wrapping Up

In this article, we focused on SOLT calibration—both the calibration method itself and the imperfections of its Open and Short standards. Though the SOLT method is one of the most common ways to calibrate a VNA, it’s by no means the only one. Other methods, such as TRL (Through-Reflect-Line) and LRM (Line-Reflect-Match) calibration, also exist. Each calibration method has its own advantages and disadvantages, depending on:

The type and frequency range of the DUT.

The availability and quality of the standards.

The desired accuracy and speed of the calibration.

This article concludes my series on VNAs and VNA calibration. I hope it has helped you gain a basic understanding of the relevant concepts, which you can use to explore other calibration methods should you desire to do so.

The preceding articles in this series are listed below in order of publication:

Introduction to the Directional Coupler for RF Applications

Understanding RF Power Measurement Errors in Directional Couplers

Understanding the Inner Workings of Vector Network Analyzers

Understanding the Significance of Dynamic Range and Spurious-Free Dynamic Range

How to Estimate and Enhance the Dynamic Range of a Vector Network Analyzer

Introduction to VNA Calibration Techniques

Understanding the Limits of VNA Calibration

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

Featured image used courtesy of Adobe Stock