The 12-Term Error Model: What VNA Calibration Corrects

Why a VNA needs error correction

A vector network analyzer (VNA) does not measure your device directly. It measures the device through its own couplers, switches, cables and connectors. Each of these adds small, unwanted reflections, leakage and gain or loss variations.

Calibration is the process that separates the device from the instrument. The VNA measures standards whose true behavior is known, compares what it sees with what it should see, and solves for a set of error terms. Those terms are then used to correct every later measurement.

The device is described by its S-parameters, which relate the waves at its two ports.

The raw numbers the VNA reports are the device's S-parameters mixed with the errors of the measurement path. The error model says exactly how they are mixed, so the mixing can be undone.

Systematic and random errors

The error model only covers one class of error.

• Systematic errors are repeatable. The same setup gives the same error each time. Directivity, source match, load match, tracking and isolation are all systematic. Calibration removes them.
• Random errors change from one measurement to the next. Noise and connector repeatability are the standard examples. Calibration cannot remove them.

This split decides what you can fix with software and what you must fix at the bench. A good calibration leaves the random errors untouched, so tightening connectors the same way each time still matters after you calibrate.

The six forward error terms

With port 1 driving, six terms describe the instrument.

• Directivity (E_D). The coupler should respond only to the wave coming back from the device. Directivity is its ability to separate the forward wave from the reflected wave. With finite directivity, part of the forward wave leaks into the reflection channel. The result is a ripple on reflection measurements, because the leakage adds to the true reflection with a phase that changes with frequency.
• Source match (E_S). The source port is not a perfect termination. A wave reflected from the device partly reflects again at the source and goes back to the device. This changes the signal that reaches the device.
• Reflection tracking (E_R). The frequency response of the reflection path, covering both the forward and the return trip through the coupler and cable. It scales the measured reflection in magnitude and phase.
• Load match (E_L). The receiving port is not a perfect termination either. Energy that arrives through the device partly reflects at port 2 and returns into the device.
• Transmission tracking (E_T). The frequency response of the transmission path from port 1 to port 2. It scales the measured transmission.
• Isolation (E_X). Signal that reaches the port 2 receiver without passing through the device. This is crosstalk, and it limits the measurement of high-loss devices.

The reverse direction has the same six terms, with port 2 driving.

From one-port to twelve terms

A reflection measurement at one port involves only three terms: directivity, source match and reflection tracking. The measured reflection coefficient is

where Γ is the true reflection coefficient of the load and Γ_m is what the VNA reads. Solving for the true value gives the correction:

Read the numerator first. Subtracting E_D removes the directivity leakage. The denominator then undoes the tracking and the source-match re-reflection.

For a two-port device, each direction needs its own three reflection terms plus load match, transmission tracking and isolation. That gives 6 forward and 6 reverse, so 12 in total. The correction of one S-parameter generally uses all four measured S-parameters. A device that reflects at port 2 changes what port 1 sees, and the load-match term accounts for that coupling.

How known standards solve for the terms

Each standard gives the VNA an equation with a known answer. The idea is simple: unknowns need equations, and each known standard supplies one.

For the reflection terms, take three standards with known reflection coefficients.

• A matched load has Γ = 0. In the equation above, Γ_m = E_D. The VNA reads directivity directly.
• A short and an open have large, known reflection coefficients of different phase. Each gives one equation in E_R and E_S, and once E_D is known, two equations solve for the two remaining unknowns.

For the transmission terms, a through connection between the ports gives a known transmission, which solves for transmission tracking and load match. Measuring with a load on each port and no device between them reveals the leakage, which gives isolation. Repeating this in the reverse direction completes the 12 terms.

The standards must be known well. Any error in a standard's definition goes straight into the error terms, and from there into every corrected result.

Worked example: directivity ripple

Take a device with a true return loss of 30 dB, measured on a VNA whose directivity is 30 dB. These values are assumed for illustration.

A return loss of 30 dB means |Γ| = 10^(−30/20) = 0.0316. A directivity of 30 dB means the leakage term has magnitude |E_D| = 0.0316 as well.

The measured reflection is roughly the sum of the true reflection and the leakage. Their relative phase changes across frequency, so the sum swings between two limits:

• Worst case, in phase: |Γ_m| = 0.0316 + 0.0316 = 0.0632, which is a return loss of −20 log₁₀(0.0632) = 24.0 dB.
• Best case, opposite phase: |Γ_m| = 0.0316 − 0.0316 = 0, which reads as a very high return loss.

The trace therefore ripples from about 24 dB up to a very large value, while the true value is a steady 30 dB. The error is about 6 dB on the low side. This first-order sum ignores source match, which is a small additional effect at this level. Calibration measures E_D with the load and subtracts it, so the ripple disappears.

In plain words · for students and beginners

VNA calibration in plain words

An everyday picture

Think of a bathroom scale that always reads 2 kg too high and also wobbles a little if you stand off-center. Before you weigh anything, you put on a known 10 kg weight and a known 20 kg weight. Now you know how the scale is wrong, and you can fix every later reading.

A VNA calibration does the same thing. The "known weights" are called standards. They are small parts, such as a short, an open and a load, whose true behavior is known.

A returning signal is much like an echo.

The VNA measures echoes from your device, but the instrument also makes tiny echoes of its own. Calibration finds out how big the instrument's own echoes are.

The picture breaks down in one place. A scale's error is one number. A VNA has 12 error terms, and each one changes with frequency.

The same example in simple steps

1. Your device really reflects a small amount: a return loss of 30 dB. (Return loss is how far below the input the reflected signal is. Bigger means a better match.)
2. The VNA's coupler is imperfect. Its directivity is 30 dB, so it leaks a signal of the same size as the true echo into the reading.
3. The leak and the true echo add up. Depending on frequency, they help each other or cancel.
4. When they help, the VNA reads 24.0 dB instead of 30 dB. The trace wobbles up and down.
5. During calibration, the VNA measures a load, which reflects nothing. Whatever it reads is the leak. It stores that number.
6. From then on, it subtracts the leak from every reading. The wobble is gone.

Mistakes beginners make

• Expecting calibration to remove noise. Calibration only removes errors that repeat. Random noise and connector wobble stay.
• Trusting an old calibration after moving a cable. The cable is part of the measurement path. If it changes, the stored corrections are no longer right.
• Using damaged or dirty standards. The VNA believes the standards are perfect. If they are not, the error is built into every result.
• Doing a one-port calibration for a two-port measurement. One-port needs three terms. A two-port device needs all 12 terms to correct transmission as well.

Remember

• A full two-port calibration solves 12 error terms: 6 forward and 6 reverse.
• Known standards give the VNA equations it can solve, so repeatable errors can be measured and removed.
• Calibration cannot remove random errors such as noise and connector repeatability.

Related pages on RF View

• S11, S21, S12 and S22 Explained for a Two-Port