# Introduction to VNA Calibration Techniques

## Learn the basics of how vector network analyzer (VNA) calibration techniques correct measurement errors.

Vector network analyzers (VNAs) are among the most precise measurement instruments available for RF and microwave applications. Modern VNAs probably measure RF power with better accuracy than any other power sensor, for example. A large part of this accuracy comes from the unique calibration techniques applicable to VNAs. These techniques allow for the correction of systematic errors in both magnitude and phase measurements.

VNA calibration theory is a widely studied topic, with hundreds of papers exploring its mathematical aspects. However, it’s often presented in a relatively complex and obscure manner. As users, what we need is a more basic understanding of various calibration techniques—both advantages and disadvantages—so that we can choose the best calibration method for any specific situation. This article aims to fill this need by providing a more approachable introduction to VNA calibration.

Before diving in too far, it should be noted that the term *calibration* for most instruments refers to factory calibration performed by the manufacturer or a service center. However, for VNAs, calibration can have an additional meaning: an error correction made by the user to eliminate the systematic errors of the whole test setup, including not only the VNA itself but also its cables, connectors, and so forth. This can sometimes cause confusion for newbies to VNA technology.

To learn about VNA calibration and the errors it can correct, let’s explore a simple measurement example: using a VNA to make reflection measurements.

### Measuring the Input Reflection of a Filter

The simplified block diagram in Figure 1 shows how a VNA measures the S-parameters of a device under test (DUT). In this case, the DUT is a low-pass filter.

**Figure 1.** VNA measurement of a low-pass filter’s S-parameters.

**Figure 1.**VNA measurement of a low-pass filter’s S-parameters.

If we measure the input reflection coefficient of the DUT, the VNA generator launches a stimulus wave toward the DUT’s input through Coupler 1. As the wave reaches the input port of the DUT, it reflects back toward the directional coupler.

The coupler separates the reflected wave and applies a portion of it to the measurement receiver of Port 1 (*Rx2*). The reference receiver of Port 1 (*Rx1*) measures the original stimulus signal. Having the phase and amplitude of the incident and reflected waves, we can now determine the DUT’s input reflection coefficient. However, this basic explanation of the VNA’s operation ignores several non-idealities that can introduce error to our measurements. Let’s take a closer look.

#### Error Due to Finite Directivity of Coupler

The first error source we’ll discuss is the finite directivity of the directional coupler. Ideally, the stimulus signal that enters Coupler 1 shouldn’t appear at all at the input of the measurement receiver. However, a real-world directional coupler leaks a bit of the incident wave to the coupled port. This is shown by the magenta leakage path in Figure 2.

**Figure 2.** Undesired signals that appear in the VNA due to leakage from the coupler (magenta) and impedance mismatch at Port 2 (red) during an input reflection measurement.

**Figure 2.**Undesired signals that appear in the VNA due to leakage from the coupler (magenta) and impedance mismatch at Port 2 (red) during an input reflection measurement.

Even if the DUT’s input is perfectly matched and there’s therefore no reflection from it, the measurement receiver still detects a non-zero power due to this leakage. The amount of leakage—and, by extension, leakage error—depends on the coupler’s directivity.

#### Error Due to Reflection From VNA’s Port 2

The red path in Figure 2 shows another source of error. This error is due to VNA’s Port 2 presenting a slightly different impedance than the ideal 50 Ω. This impedance mismatch causes the signal going out of the DUT to reflect back toward it.

If the DUT is a low-loss reciprocal device—for example, a filter—the reflected signal passes through the DUT with little attenuation and couples to the input of *Rx2*. *Rx2* can’t distinguish between the signal reflected from the DUT’s input and the undesired signal reflected from Port 2 of the VNA, which creates an error in the measurement.

#### Error Due to Reflection From VNA’s Port 1

If Port 1 of the VNA presents a slightly different impedance than the ideal 50 Ω, the mismatch can result in the power reflected from the DUT not being fully absorbed by Port 1. Consequently, multiple reflections can occur between the coupler and the DUT, leading to an additional error term. The green path in Figure 3 shows these multiple reflections.

**Figure 3.** The green path shows the multiple reflections that can occur between the DUT and Port 1 of the VNA.

**Figure 3.**The green path shows the multiple reflections that can occur between the DUT and Port 1 of the VNA.

In the above discussion, the errors originate from the non-idealities of the VNA—limited directivity of the coupler and the mismatch of the test ports. However, the overall error is also dependent upon the cables and connectors used in the test setup.

Even if the cables provide a perfect 50 Ω impedance, the length of the cable between Port 1 and the DUT’s input determines the length of the green path in Figure 3. This in turn affects the phase of the corresponding error term. The loss of the cables can also affect the amplitude of the error signal.

As we just saw, the measurement error depends on a variety of factors—the VNA, the cables and connectors used in the test setup, and the properties of the DUT all play a role. We now have three error terms, each corresponding to a signal path:

- The directivity error (magenta, Figure 2).
- The Port 2 reflection error (red, Figure 2).
- The Port 1 reflection error (green, Figure 3).

We can refer to these error terms as *x*_{1}, *x*_{2}, and *x*_{3}, respectively. To better understand the measurement error and get a sense of these error terms’ significance, let's work through an example using the VNA we’ve been examining and some typical values.

### How Large Are These Error Terms?

Assume the following:

- The DUT is a low-pass filter with a 1 dB insertion loss (
*L*= 1) and a 20 dB return loss (_{filter}*RL*= 20)._{filter} - The directivity of Coupler 1 is 30 dB (
*D*= 30). - The VNA port return loss is 25 dB (
*RL*= 25 dB)._{port}

Given these values, how large are *x*_{1}, *x*_{2}, and *x*_{3}?

Let’s start by finding *x*_{1}, the directivity error. For simplicity’s sake, we’ll ignore the main line loss of the coupler.

#### Calculating the Directivity Error

First, let’s examine the desired signal. The stimulus signal travels through the coupler and reflects from the DUT’s input to show up at the input of the measurement receiver (*Rx2*). This path has a total loss of:

$$L_{total}~=~C~+~RL_{filter}$$

**Equation 1.**

**Equation 1.**

where *C* is the coupling factor of Coupler 1.

In a previous article, we explored how limited directivity affects power measurements. As you may recall from that discussion, *C *is one of three factors used to characterize directional couplers:

- The isolation factor (
*I*). - The coupling factor (
*C*). - The directivity factor (
*D*).

These factors are related by the following equation:

$$I~=~C~+~D$$

**Equation 2.**

**Equation 2.**

The signal that couples through the magenta path experiences a loss equal to the isolation factor of the coupler. We therefore know that the power of the undesired signal from the magenta path is *C* + *D* decibels below the power of the stimulus signal. Figure 3 compares the power terms associated with the three signals:

*P*, the power of the incident (stimulus) signal._{i}*P*, the power of the desired component reflected from the DUT’s input._{d}*P*_{c}_{1}, the power of the undesired component coupled through the magenta path.

**Figure 4.** The relative power levels of the incident signal (P_{i}*), the desired signal (*P_{d}*), and an undesired signal (*P_{c1}*).*

**Figure 4.**The relative power levels of the incident signal (

The difference (in decibels) between the power of the desired signal and the power of the undesired signal is given by:

$$P_{d}~-~P_{c1}~=~D~-~RL_{filter}$$

**Equation 3.**

**Equation 3.**

For our example, *D* is given as 30 dB and *RL _{filter}* as 20 dB.

*P*

_{c}_{1}is therefore 10 dB lower than

*P*. If we consider the voltage quantities, we can calculate the error term as follows:

_{d}$$x_1~=~10^{-(D~-~RL_{filter})/20}~=~10^{-10/20}~=~0.32$$

**Equation 4.**

**Equation 4.**

The amplitude of the undesired voltage is a factor of 0.32 less than the desired voltage. Note that this error term depends on the directivity of the coupler and the return loss of the DUT.

#### Calculating the Port 2 Reflection Error

Next, let’s consider the undesired component that travels through the red path in Figure 2. This signal:

- Passes through the DUT, resulting in a loss of
*L*._{filter} - Reflects from the VNA’s port, resulting in a loss of
*RL*._{port} - Passes through the DUT again as it travels toward the coupler, once again suffering a loss of
*L*._{filter} - Passes through the coupler to appear at the input of the receiver
*Rx2*, having undergone an attenuation equal to the coupling factor.

The total loss of this path is therefore:

$$L_{total}~=~2L_{filter}~+~RL_{port}~+~C$$

**Equation 5.**

**Equation 5.**

This is illustrated in Figure 5, where the power of this undesired component is labeled as *P _{c}*

_{2}.

**Figure 5.** The relative power levels of the incident signal (P_{i}*), the desired signal (*P_{d}*), and an undesired signal (*P_{c2}*).*

**Figure 5.**The relative power levels of the incident signal (

The difference between *P _{d}* and

*P*

_{c}_{2}is given by:

$$P_{d}~-~P_{c2}~=~2L_{filter}~+~RL_{port}~-~RL_{filter}~=~2~\times~1~+25~-20~=~7~\text{dB}$$

**Equation 6.**

**Equation 6.**

The power of this undesired component is therefore 7 dB lower than the power of the desired component. We can now find *x*_{2} the same way we found *x*_{1}:

$$x_2~=~10^{-(2 L_{filter}~+~RL_{port}~-~RL_{filter})/20}~=~10^{-7/20}~=~0.45$$

**Equation 7.**

**Equation 7.**

The amplitude of the undesired voltage is a factor of 0.45 less than that of the desired voltage. The error term depends on both the VNA and DUT parameters.

#### Calculating the Port 1 Reflection Error

Finally, we calculate the error term associated with the green path in Figure 3. This signal does the following:

- Reflects from the DUT’s input, suffering a loss of
*RL*._{filter} - Bounces off the VNA’s Port 1, for a loss of
*RL*._{port} - Reflects from the DUT’s input one more time, suffering a loss of
*RL*again._{filter} - Passes through the coupler, adding a loss of
*C*, before appearing at the input of*Rx2*.

The total loss of this path is therefore:

$$L_{total}~+~2RL_{filter}~+~RL_{port}~+~C$$

**Equation 8.**

**Equation 8.**

The relationship between the incident power, the desired power, and the power of the undesired component (*P _{c}*

_{3}) can be seen in Figure 6.

**Figure 6.** The relative power levels of the incident signal (P_{i}*), the desired signal (*P_{d}*), and our final undesired signal (*P_{c3}*).*

**Figure 6.**The relative power levels of the incident signal (

The difference between *P _{d}* and

*P*

_{c}_{3}can be expressed as:

$$P_{d}~-~P_{c3}~=~RL_{filter}~+~RL_{port}~=~20~+~25~=~45~\text{dB}$$

**Equation 9.**

**Equation 9.**

*P _{c}*

_{3}is 45 dB lower than the desired component. If we consider the voltage quantities, the amplitude of the undesired voltage is a factor of 0.006 less than the desired voltage:

$$x_3~=~10^{-(RL_{filter}~+~RL_{port})/20}~=~10^{-45/20}~=~0.006$$

**Equation 10.**

**Equation 10.**

Because this error term results from multiple reflections, its magnitude drops quickly, especially when both the DUT and test port present a relatively matched impedance. Note that *x*_{3}, like *x*_{1} and *x*_{2}, is also dependent upon the properties of the VNA and DUT.

#### The Measurement Uncertainty Range

We now have the relative amplitudes of all three undesired components with respect to the desired component. If we assume that the desired signal has an amplitude of unity, the three undesired signals have amplitudes of 0.32, 0.45, and 0.006. By adding and subtracting these three signals to and from the desired signal, we can find the worst-case measurement uncertainty range.

The power measured by the measurement receiver *Rx2* can be 20log(1 + 0.32 + 0.45 + 0.006) = 4.99 dB higher or 20log(1 – 0.32 – 0.45 – 0.006) = –13 dB lower than the ideal value. This is an unacceptably large uncertainty, but one we can significantly reduce by making some adjustments and applying VNA calibration techniques.

### Reducing Measurement Uncertainty

Part of the large measurement uncertainty in the above example stems from the DUT being a low-loss, reciprocal device. Note that the return loss of the filter (20 dB) and the VNA port (25 dB) are comparable. The undesired signal that reflects from the VNA’s Port 2 therefore has a power comparable with our desired signal, except that the undesired component experiences twice the attenuation of the filter.

Because the filter has a relatively small insertion loss (1 dB) in its pass-band, the undesired term isn’t significantly suppressed by the filter. To attenuate this error component, we can disconnect the output of the filter from Port 2 of the VNA and terminate the output of the filter in a well-matched load. We can also reduce the mismatch uncertainty by inserting a high-quality attenuator between the output of the filter and Port 2 of the VNA.

Let’s assume that, by applying one of these two techniques, we can reduce the signal reflected from the VNA’s Port 2 to a negligible level. In that case, the power measured by *Rx2* can be 20log(1 + 0.32 + 0.006) = 2.45 dB higher or 20log(1 – 0.32 – 0.006) = –3.43 dB lower than the ideal value.

This is still a considerable amount of uncertainty—enough, in production testing, to potentially make a filter that actually meets the specifications fail the tests, or a filter that actually *doesn’t* meet the specifications pass. Fortunately, as we’ll see below, VNA calibration techniques allow us to increase the accuracy further.

### One-Port Calibration

To model the imperfections of the VNA and its test cables, let’s assume that an error network (or error box) defined by its unknown S-parameters is placed between the VNA and the DUT’s input. This is illustrated in Figure 7.

**Figure 7.** We can use this error box to account for the non-idealities of the VNA.

**Figure 7.**We can use this error box to account for the non-idealities of the VNA.

Since the non-idealities are accounted for through the error box, we can assume that the VNA is ideal. The input reflection coefficient (Γ_{in}) measured by the ideal VNA is related to the actual load reflection (Γ_{L}) by the following equation:

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

**Equation 11.**

**Equation 11.**

We have a total of four unknowns in this equation: *e*_{00}, *e*_{01}, *e*_{10}, and *e*_{11}. However, we can reduce the number of unknowns to three by interpreting the term *e*_{10}*e*_{01} as a single parameter. Each of these three error terms is associated with a physical source of system error—*e*_{00} is related to the effective directivity of the system, *e*_{11} represents the source match error, and *e*_{10}*e*_{01} is the reflection tracking error.

To determine these unknown parameters, we measure three known terminations. In the context of a VNA, these terminations are referred to as calibration standards. Once we measure the Open, Short, and (matched) Load calibration standards, we can solve equations for each of the three unknowns (*e*_{00}, *e*_{10}*e*_{01}, and *e*_{11}).

With these error terms no longer unknown, we can use the measured value of Γ_{in} to determine Γ_{L} and the load termination impedance. Since the error signal components are added to the desired signal vectorially, we need to know both the magnitude and phase information of the error terms. The VNA can then correct such systematic errors using mathematical methods.

Though the error can’t be eliminated completely, calibration techniques can still significantly reduce the measurement uncertainty—for example, applying calibration techniques might improve the directivity of the system from about 30 dB to 45 dB. In the next article of this series, we’ll learn about some errors that are more difficult to correct in this way.

This article is Part 6 of a series on vector network analyzers. In order of publication, the articles in this series are:

- 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
- Understanding RF Calibration Using Short, Open, Load, and Through Terminations

*All images used courtesy of Steve Arar*

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