05 · Advanced

De-embedding and the complete VNA-to-DUT measurement chain

Use physically characterized error boxes, ordered ABCD inversion and real matched-wave conventions to extract intrinsic device S-parameters with independent numerical validation and uncertainty.

240–310 minAdvanced

In this lesson

Distinguish VNA calibration planes from intrinsic DUT terminals.
Define electrical current signs for ABCD cascaded networks.
Derive S-to-chain conversion under identical real-positive references.
Extract a two-port using independent left and right fixture error boxes.
Measure a 5+25+5Ω cascade using real 50Ω reference ports.
Recover the intrinsic 25Ω resistor and correct S11/S21.
Compute passive power absorption at both measurement and DUT planes.
Explain topology assumptions of open/short fixture corrections.
Control ill-conditioned inversions and statistical/systematic errors.
Write a reproducible VNA-to-DUT measurement and verification protocol.

This final RF-05 chapter connects VNA calibration, mathematically correct network composition and device extraction. We use a transparent two-fixture resistor example to derive and test de-embedding rather than presenting it as subtracting a trace on a plot.

0105.1 — From calibrated VNA planes to the actual DUT terminals

The error correction of a VNA is defined at calibration reference planes, often near connectors or probe tips. A DUT inside a fixture can be separated from those planes by transmission lines, launch pads, wires and transitions. The measured network then contains both access structures and intrinsic device response; the calibration step does not automatically remove their electromagnetic effects.

De-embedding is the mathematical removal of independently characterized access networks. It requires an explicit physical topology, wave reference definitions and a placement of each interface. A matched uniform delay may be handled by a plane extension, while an unmatched distributed fixture requires network operations that account for reflections in both directions.

\begin{pmatrix}V_1\\I_1\end{pmatrix}=\mathbf T\begin{pmatrix}V_2\\-I_2\end{pmatrix}

\mathbf T_{\mathrm{meas}}=\mathbf T_L\,\mathbf T_{\mathrm{DUT}}\,\mathbf T_R

\mathbf b=\mathbf S\,\mathbf a\qquad(\text{real positive wave references})

0205.2 — ABCD chain parameters and electrical current references

For the chain representation adopted in this course, both physical port currents are defined entering the positive device terminals, and the output coordinate in the $\mathrm{ABCD}$ relation is minus $I_2$ . A series resistor R is represented by the 2x2 chain matrix with diagonal unity, off-diagonal B=R and C=0. A shunt admittance Y has lower-left C=Y.

Cascaded two-port networks multiply in the physical order from the VNA input toward the output, provided all intermediate planes and modes are defined consistently. Series and shunt networks do not generally commute. Forgetting the minus sign at the output reverses the conventions and can produce plausible but incorrect matrices and power balances.

\mathbf T_{\mathrm{series}}(R)=\begin{pmatrix}1&R\\0&1\end{pmatrix}

\mathbf T_{\mathrm{shunt}}(Y)=\begin{pmatrix}1&0\\Y&1\end{pmatrix}

\mathbf T_1\,\mathbf T_2\ne\mathbf T_2\,\mathbf T_1\quad\text{in general}

0305.3 — Converting complex S to a chain matrix

With two equal real positive reference impedances $R_0$ , the $\mathrm{ABCD}$ elements can be written explicitly in terms of $S_{11}$ , $S_{12}$ , $S_{21}$ and $S_{22}$ . Both transmission directions matter. Every conversion to chain form divides by $S_{21}$ , so a weak forward transmission or a deep stopband can magnify small receiver noise and standard uncertainty.

The formulas must not be applied unchanged to arbitrary complex or unequal port reference impedances; their appropriate power-wave or pseudo-wave definitions must first be selected. Validate dimensions: A and D are dimensionless, B is an impedance in ohms and C is an admittance in siemens. Test a reciprocal resistor as a known reference case.

Figure 5.5 — Signed electrical references of a two-port

Both I₁ and I₂ are defined ENTERING the network at their positive terminals; each voltage arrow points from minus to plus.

I₁ enters the left port pointing right, while I₂ enters the right port pointing left. V₁ and V₂ are positive on the upper terminals. The conventional output current toward an attached load equals −I₂. These arrows define references, not actual solved current flow.

A=\frac{(1+S_{11})(1-S_{22})+S_{12}S_{21}}{2S_{21}}

B=R_0\,\frac{(1+S_{11})(1+S_{22})-S_{12}S_{21}}{2S_{21}}

\begin{aligned}C&=\frac{(1-S_{11})(1-S_{22})-S_{12}S_{21}}{2R_0S_{21}}\\D&=\frac{(1-S_{11})(1+S_{22})+S_{12}S_{21}}{2S_{21}}\end{aligned}

0405.4 — Error-box de-embedding by matrix inversion

Let $\mathbf T_L$ and $\mathbf T_R$ be independently determined input and output fixture chain matrices, and $\mathbf T_{\mathrm{meas}}$ the calibrated measured cascade. Under the single-mode, linear two-port hypothesis $\mathbf T_{\mathrm{meas}}=\mathbf T_L\mathbf T_{\mathrm{DUT}}\mathbf T_R$ . If both error boxes are known and invertible, the intrinsic network follows as $\mathbf T_{\mathrm{DUT}}=\mathbf T_L^{-1}\mathbf T_{\mathrm{meas}}\mathbf T_R^{-1}$ .

An algebraic inverse is not sufficient for a reliable result when the fixture model is inaccurate or its condition number is large. Errors amplify under inversion, and one scalar insertion-loss offset cannot reproduce the multiple-reflection corrections generated by two general fixture matrices. De-embedding is also distinct from renormalization to a new reference impedance.

\boxed{\mathbf T_{\mathrm{DUT}}=\mathbf T_L^{-1}\,\mathbf T_{\mathrm{meas}}\,\mathbf T_R^{-1}}

\det\mathbf T_L\ne0,\qquad\det\mathbf T_R\ne0

\operatorname{cond}(\mathbf T_L),\operatorname{cond}(\mathbf T_R)\;\text{must be controlled}

0505.5 — Numerical example: 5 Ω + 25 Ω + 5 Ω

Consider the physical input fixture as a $5\,\Omega$ series resistor, the DUT as a $25\,\Omega$ series resistor, and the output fixture as another $5\,\Omega$ resistor. With identical $50\,\Omega$ wave references, the three chain matrices multiply to a network with a $35\,\Omega$ total series resistance. Its directly observed $S_{11}$ = $\frac7{27}$ ≈0.259259 and forward $S_{21}$ = $\frac{20}{27}$ ≈0.740741.

After multiplying by the two inverse fixture matrices, the extracted DUT chain B parameter is $25\,\Omega$ , and its $S_{11}$ =.2 and $S_{21}$ =.8. For single-sided matched incident excitation at the intrinsic reference planes, the DUT absorbed fraction is $1-0.2^2-0.8^2=0.32$ . This 32% must not be confused with total cascade dissipation under the external excitation.

\mathbf T_L=\mathbf T_R=\begin{pmatrix}1&5\,\Omega\\0&1\end{pmatrix}

\mathbf T_{\mathrm{meas}}=\begin{pmatrix}1&35\,\Omega\\0&1\end{pmatrix}

\mathbf T_{\mathrm{DUT}}=\begin{pmatrix}1&25\,\Omega\\0&1\end{pmatrix}

0605.6 — Open/short subtraction and physical topology

A measured open structure can characterize parasitic shunt admittance when the layout genuinely permits a parallel equivalent; a short structure can help estimate residual series impedance under a justified circuit reduction. The familiar open-then-short correction is not an arbitrary matrix subtraction valid for any distributed two-port.

The exact subtraction depends on which networks are in series or parallel at the chosen planes, and whether substrate leakage, coupling, radiation or additional modes violate that picture. An incorrectly modeled open standard can remove a real device capacitance or create unphysical negative dissipation. Verify with dedicated empty, thru and known-resistor patterns where possible.

\begin{aligned}S_{11}^{\mathrm{meas}}&=\frac{35}{100+35}=\frac7{27}\\S_{21}^{\mathrm{meas}}&=\frac{100}{135}=\frac{20}{27}\end{aligned}

\begin{aligned}S_{11}^{\mathrm{DUT}}&=\frac{25}{100+25}=0.2\\S_{21}^{\mathrm{DUT}}&=\frac{100}{125}=0.8\end{aligned}

A_{\mathrm{DUT}}=1-|S_{11}|^2-|S_{21}|^2=0.32

0705.7 — Stability, noise and covariance after de-embedding

Both chain-matrix inversion and the conversion from S to T can be ill-conditioned, particularly if $S_{21}$ approaches zero, the fixture matrices are nearly singular, or the measurement contains an unmodeled mode. A dramatic recovered resonance can therefore arise from numerical instability rather than a physical plasmon or cavity mode.

For a transparent uncertainty treatment, collect real and imaginary components of the relevant measured quantities into a real vector and propagate their covariance by the local Jacobian. Also vary the fixture models, calibration standards, contact resistance, probe landing and physical plane lengths to estimate systematic sensitivity. Repeating the same smooth curve does not prove the fixture model.

\mathbf Y_{\mathrm{DUT}}=\mathbf Y_{\mathrm{meas}}-\mathbf Y_{\mathrm{open}}\quad\text{only for the assumed parallel topology}

\mathbf Z_{\mathrm{DUT}}=\mathbf Z_{\mathrm{corrected}}-\mathbf Z_{\mathrm{series}}\quad\text{only for the assumed series topology}

\mathbf T(S)\;\text{can be ill-conditioned as }S_{21}\to0

0805.8 — A complete and reproducible VNA-to-DUT protocol

Choose physically realizable reference planes, characterize the SOLT or $\mathbf T_R$ L standards, calibrate the VNA and measure an independent verification network. Save complex Touchstone traces and their version, data order, port wave references and instrument settings. Measure the two fixture error boxes or appropriate calibration structures with the same modal topology as the DUT.

Perform the matrix extraction only within the frequency band where the standards and fixture inverses remain stable. Convert back to S, check passivity and expected limiting cases, compare with known resistor/through devices and repeat connections to obtain variability. Report the intrinsic S curves alongside the raw cascade and uncertainty, and keep the code and models that generate each plotted figure.

\mathbf x=\begin{pmatrix}\operatorname{Re}S_{11}\\\operatorname{Im}S_{11}\\\operatorname{Re}S_{21}\\\operatorname{Im}S_{21}\end{pmatrix}

\boldsymbol\Sigma_y\simeq\mathbf J\,\boldsymbol\Sigma_x\,\mathbf J^{\mathsf T}

\text{calibration}\ne\text{renormalization}\ne\text{de-embedding}

De-embedding exercises and worked solutions

Draw the two port currents and explain why the $\mathrm{ABCD}$ output current coordinate is − $I_2$ .
Derive the $\mathrm{ABCD}$ matrix of a series resistor R using Kirchhoff's laws.
Reproduce S-to- $\mathrm{ABCD}$ conversion for equal real $R_0$ , and explain the singularity at $S_{21}$ =0.
Show how to isolate $\mathbf T_{\mathrm{DUT}}$ from $\mathbf T_{\mathrm{meas}}=\mathbf T_L\mathbf T_{\mathrm{DUT}}\mathbf T_R$ and identify the conditions on fixtures.
Compute external $S_{11}$ and $S_{21}$ for a $5\,\Omega$ / $25\,\Omega$ / $5\,\Omega$ series cascade with $R_0$ = $50\,\Omega$ .
Apply the two inverse $5\,\Omega$ error boxes and recover the intrinsic $25\,\Omega$ resistance.
Compute the absorbed-power fraction for the corrected $25\,\Omega$ DUT and distinguish it from the external cascade.
Explain why subtracting fixture insertion loss expressed in dB cannot represent general multi-reflection de-embedding.
Describe a case where an open/short correction or $\mathrm{ABCD}$ inversion fails despite a plausible numerical answer.
Draft a full calibrated RF measurement procedure from standards and Touchstone to verified intrinsic DUT parameters.

Worked solutions

1. Define $I_1$ and $I_2$ as positive when entering their corresponding positive terminals. If a current physically flows from left to right through a two-port, it enters port 1 but leaves port 2, so its chain-coordinate current on the output side is − $I_2$ . Hence $\begin{pmatrix}V_1\\I_1\end{pmatrix}=\mathbf T\begin{pmatrix}V_2\\-I_2\end{pmatrix}$ , consistent with identical currents through a series resistor.

2. For a series resistor the same current flows through the device and voltage drops by $V_1-V_2=RI_1$ , while $I_1=-I_2$ under the entering-current convention. Thus $V_1=V_2+R(-I_2)$ and $I_1=-I_2$ . The matrix is $\begin{pmatrix}1&R\\0&1\end{pmatrix}$ , with B in ohms; two such networks add their B entries.

3. For equal real $R_0$ , one formula is A= $\frac{(1+S_{11})(1-S_{22})+S_{12}S_{21}}{2S_{21}}$ , while B= $R_0\frac{(1+S_{11})(1+S_{22})-S_{12}S_{21}}{2S_{21}}$ . Companion expressions yield C and D with the correct units. $S_{21}$ occurs in every denominator, so near a stop-band transmission zero the calculated T becomes very sensitive to noise.

4. Given the cascade $\mathbf T_{\mathrm{meas}}=\mathbf T_L\mathbf T_{\mathrm{DUT}}\mathbf T_R$ , left-multiply by $\mathbf T_L$ ^-1 and right-multiply by $\mathbf T_R$ ^-1, preserving matrix order. The result is $\mathbf T_{\mathrm{DUT}}=\mathbf T_L^{-1}\mathbf T_{\mathrm{meas}}\mathbf T_R^{-1}$ . Both fixture matrices must be known at compatible physical planes and modes, invertible and sufficiently well conditioned; otherwise the algebra can amplify errors.

5. The three series resistors total $5+25+5=35\,\Omega$ . For a series R under real $50\,\Omega$ references, $S_{11}=\frac{R}{2R_0+R}$ and $S_{21}=\frac{2R_0}{2R_0+R}$ . Thus $S_{11}=\frac{35}{135}=\frac7{27}\approx0.259259$ and $S_{21}=\frac{100}{135}=\frac{20}{27}\approx0.740741$ , with $S_{22}$ = $S_{11}$ and $S_{12}=S_{21}$ by the reciprocal symmetric series model.

6. The fixture matrices are both $\begin{pmatrix}1&5\,\Omega\\0&1\end{pmatrix}$ . Their inverses have B=− $5\,\Omega$ . Multiplying the measured $\begin{pmatrix}1&35\,\Omega\\0&1\end{pmatrix}$ between these inverses gives $\begin{pmatrix}1&25\,\Omega\\0&1\end{pmatrix}$ exactly. Converting back to S under $R_0$ = $50\,\Omega$ yields $S_{11}=S_{22}=\frac{25}{125}=0.2$ and $S_{21}=S_{12}=\frac{100}{125}=0.8$ .

7. With a matched external load at the DUT reference plane and only port 1 excitation, the reflected power fraction is $|S_{11}|^2=0.04$ and the transmitted fraction is $|S_{21}|^2=0.64$ . The remainder 1−.04−.64=.32 is absorbed by the DUT series resistance. In the uncorrected full cascade, the analogous fraction is $1-\left(\frac7{27}\right)^2-\left(\frac{20}{27}\right)^2=\frac{280}{729}\approx0.3841$ and includes fixture dissipation.

8. The decibel insertion loss describes one scattering response under specific terminations. Cascading two unmatched networks changes waves and standing fields at their interface, so transmission amplitudes do not generally multiply as independent scalar loss terms. The exact two-port cascade is a matrix multiplication; de-embedding requires the ordered inverse of measured fixture networks.

9. At $S_{21}$ ≈0, each equal-reference S-to-T expression divides by a very small number and can explode under tiny receiver noise. An open/short subtraction also fails if a parasitic network is distributed, introduces multiple modes, or cannot be represented as the assumed parallel and series elements. Independent standards, condition-number checks and topology validation are required.

10. Define calibrated wave planes and port sign convention, then document kit models and instrument sweep settings. Calibrate using SOLT/ $\mathbf T_R$ L, verify with an independent standard and export complex versioned Touchstone data. Measure both compatible fixture boxes, compute the ordered $\mathrm{ABCD}$ inverse, convert the intrinsic response back to S, confirm passive power balance and limit cases, repeat contacts, and archive uncertainty and source models.

RF-05 is complete in scope: combine these measurement and error-box methods in a reproducible VNA measurement report with checked mathematical assumptions.