Math & Units

De-Embedding (Math)

Q: Why must S-parameters be converted to T-parameters or ABCD before de-embedding?

S-parameters reference incident and reflected power waves and do not cascade by simple matrix multiplication, because the output of one stage is not the input of the next in the wave sense. T-parameters (transfer scattering parameters) and ABCD parameters do cascade: the overall matrix of two networks in series equals the product of their individual matrices. To remove a fixture half from a measurement, you convert the measured S-matrix to T-parameters, then left-multiply by the inverse of the input fixture's T-matrix and right-multiply by the inverse of the output fixture's T-matrix. The result is converted back to S-parameters at the DUT reference plane. Skipping the conversion and trying to subtract S-parameters directly produces meaningless results because reflections and transmission are not additive across cascaded two-ports.

Q: What is the difference between de-embedding and port extension?

Port extension applies only a phase rotation (and optionally a loss term) to the measured S-parameters, modeling the fixture as an ideal lossless transmission line of a given electrical length. It corrects the phase reference plane but cannot remove mismatch reflections, frequency-dependent loss, or coupling. Full de-embedding uses a complete two-port (or multi-port) error network characterized at every frequency point, so it removes magnitude error, phase error, and the multiple internal reflections that bounce between the fixture and the DUT. Port extension is a quick first-order fix suitable for short, well-matched launches; de-embedding is required when the fixture has appreciable loss, impedance discontinuities, or when you need accurate S11 and S22 from a poorly matched device.

Q: How do you obtain the error network matrix for de-embedding?

There are three common routes. First, direct measurement: build a fixture with a known standard (a thru, a short, or a calibrated load) at the DUT plane and back out the fixture two-port from the measured response. Second, calibration-based methods such as TRL (thru-reflect-line) place the reference plane at the DUT by measuring three on-fixture standards, so the error terms are solved as part of the VNA calibration rather than as a separate de-embedding step. Third, electromagnetic simulation: model the launch, pad, or interconnect in a 3D field solver, export its S-parameters, and use that simulated network as the error box. The simulation route is the basis of the related EM de-embedding workflow and is preferred when physical standards are impractical, such as on-wafer probe pads at millimeter-wave frequencies.

/dee-em-BED-ing math/

Mathematically peeling a test fixture, probe pad, or interconnect off a measurement is the job of de-embedding. The measured S-parameters are first converted into a cascadable form (T-parameters or the ABCD matrix), the characterized error network is inverted, and the inverse is multiplied with the measured data so the reference plane lands exactly at the device under test. Unlike a simple port extension that only rotates phase, the full matrix approach removes loss, mismatch reflections, and the internal bounces between the test fixture and the DUT, recovering true device behavior up through millimeter-wave bands.

Category: Math & Units

Core operation: T-matrix inversion

Reference plane: Shifted to DUT

The Matrix Algebra Behind Fixture Removal

Every two-port measured on a vector network analyzer is really the cascade of three networks: the input fixture half (launch, trace, probe pad), the device under test, and the output fixture half. The fixture halves add their own loss, phase, and mismatch to whatever the DUT does, so the raw S-parameters describe the whole sandwich rather than the device alone. De-embedding is the bookkeeping that undoes this cascade. The mathematical obstacle is that S-parameters relate power waves and therefore do not multiply when networks are placed in series; the transmission (T) and ABCD representations were created precisely because they do.

Once the measured S-matrix is converted to T-parameters, the cascade becomes a clean matrix product: T_meas = T_A × T_DUT × T_B, where T_A and T_B are the input and output error networks. Solving for the device is then a left and right inversion: T_DUT = T_A⁻¹ × T_meas × T_B⁻¹. The same logic works in ABCD form for series-connected two-ports, which is convenient for analytic fixture models built from transmission-line segments and lumped pads. After the inversion the result is converted back to S-parameters, now referenced to the DUT plane.

Accuracy hinges entirely on how well the error networks are known. If T_A and T_B are derived from measurement of real standards or from a 3D field solver, the de-embedded result is faithful; if they are crude guesses, the inversion can amplify error rather than remove it, especially near the fixture's resonances where the matrix becomes ill-conditioned.

S to T Conversion and the Inversion Step

The conversion from a two-port S-matrix to T-parameters is exact and reversible, which is what makes the round trip lossless in principle. The danger is numerical: when the determinant of an error-network T-matrix approaches zero (a deep notch or a long, lossy line), the inverse blows up small measurement noise into large de-embedded artifacts. Practitioners limit the frequency span, use symmetric fixture halves where possible, and cross-check against a calibration method such as TRL that solves the error terms directly.

De-Embedding Versus Calibration Reference Planes

De-embedding and VNA calibration both move the reference plane, but at different stages. Calibration corrects the analyzer's systematic errors up to the cable ends or probe tips; de-embedding then removes whatever fixture remains between that calibrated plane and the device. A TRL calibration can do both at once by placing standards on the fixture itself, which is why on-wafer and millimeter-wave labs often prefer it over a separate post-measurement de-embedding pass.

Cascade in T-parameters:
[T_meas] = [T_A] × [T_DUT] × [T_B]

De-embedding (solve for the device):
[T_DUT] = [T_A]⁻¹ × [T_meas] × [T_B]⁻¹

Simple port-extension (phase only):
S₂₁^corr = S₂₁^meas × e^+jβℓ, βℓ = 2πfℓ/v_p

ABCD cascade (series two-ports):
[ABCD]_total = [ABCD]_A × [ABCD]_DUT × [ABCD]_B

Where T = transfer scattering matrix, β = phase constant, ℓ = fixture length, v_p = phase velocity. The inverse matrices [T_A]⁻¹ and [T_B]⁻¹ must be computed at every frequency point because the error networks are dispersive.

Method Comparison

Method	Removes Loss	Removes Mismatch	Error Network Source	Typical Use	Limitation
Port extension	Approx. (loss term optional)	No	Assumed ideal line	Short, matched launches	Phase-only, ignores reflections
S-to-T matrix de-embed	Yes	Yes	Measured or modeled two-port	Connectorized fixtures, pads	Ill-conditioned at notches
ABCD de-embed	Yes	Yes	Analytic line + pad model	PCB traces, lumped pads	Series two-port assumption
TRL calibration	Yes	Yes	On-fixture thru/reflect/line	On-wafer, mmWave	Needs precise line standards
EM-simulated de-embed	Yes	Yes	3D field solver export	Probe pads, no physical std.	Only as good as the model

Common Questions

Frequently Asked Questions

Why must S-parameters be converted to T-parameters or ABCD before de-embedding?

S-parameters reference power waves and do not cascade by matrix multiplication, so the output of one stage is not the input of the next in the wave sense. T-parameters and ABCD parameters do cascade: networks in series multiply. To strip a fixture half, convert the measured S-matrix to T, left-multiply by [T_A]⁻¹, right-multiply by [T_B]⁻¹, then convert back. Subtracting S-parameters directly is meaningless because reflection and transmission are not additive across cascaded two-ports.

What is the difference between de-embedding and port extension?

Port extension applies only a phase rotation (with an optional loss term), modeling the fixture as an ideal lossless line of fixed electrical length. It cannot remove mismatch reflections or frequency-dependent loss. Full de-embedding uses a complete error two-port characterized at every frequency, removing magnitude error, phase error, and the internal reflections bouncing between fixture and DUT. Port extension suits short, matched launches; de-embedding is required for lossy fixtures or accurate S11/S22 from a mismatched device.

How do you obtain the error network matrix for de-embedding?

Three routes: direct measurement of a known standard (thru, short, or load) at the DUT plane to back out the fixture two-port; calibration methods such as TRL that solve the error terms while placing the reference plane at the DUT; or 3D electromagnetic simulation of the launch or pad, exporting its S-parameters as the error box. The simulation route underpins the EM de-embedding workflow and is preferred for on-wafer probe pads at millimeter-wave frequencies where physical standards are impractical.

Related Terms

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