03 · Advanced

SOLT, TRL and multiline TRL: selecting standards

Derive characterized SOLT corrections, Engen–Hoer TRL error-box separation, line-length degeneracy, multiline TRL and verifiable calibration uncertainty, including quantitative 0.5–2 GHz design.

240–300 minAdvanced

In this lesson

Construct a one-port three-standard calibration inverse problem.
Distinguish physical SOLT standards from ideal load/open/short models.
Explain the role of through response and two-port receiver architecture.
Recognize the two-port error boxes solved in TRL.
Derive the line-phase degeneracy condition from propagation eigenvalues.
Calculate numerical phase separations at 0.5, 1 and 2 GHz.
Select complementary line lengths for multiline TRL.
Explain impedance reference ambiguities and renormalization.
Assess standards' mode, grounding, phase and dimensional uncertainties.
Cross-validate SOLT and TRL against an independent measurement.

This chapter moves from the elementary VNA calibration error model to engineering choices of physical SOLT and TRL standards, line-length conditioning, multiline TRL, and reproducible comparison of results. We derive the mathematics rather than treating calibration as a button press.

01The inverse problem: a calibrated S-matrix from standards

A VNA receiver records complex wave ratios containing systematic measurement errors. Characterized standards offer known wave relationships at chosen planes, allowing the instrument to estimate error boxes rather than treating its raw receivers as ideal. In the one-port model developed previously, three standards determine directivity, reflection tracking and source-match coefficients by a complex linear system. The mathematics clarifies which quantities are measured and which are supplied by a standard model.

A solution is reliable only if the error-parameter inverse problem is identifiable and well conditioned. The three calibrated reflections should span sufficiently different responses. Frequency-dependent standard modeling errors, residual leakage and repeatability can give a numerical answer that is internally precise but systematically wrong. A verification device not used to fit the coefficients is essential.

m_i=E_D+\frac{E_R\Gamma_i}{1-E_S\Gamma_i}

\begin{pmatrix}1&\Gamma_1&m_1\Gamma_1\\1&\Gamma_2&m_2\Gamma_2\\1&\Gamma_3&m_3\Gamma_3\end{pmatrix}\begin{pmatrix}E_D\\t\\E_S\end{pmatrix}=\begin{pmatrix}m_1\\m_2\\m_3\end{pmatrix}

t=E_R-E_D E_S

02SOLT and the limits of ideal short, open, load and thru

In SOLT, the short, open and matched load provide distinct characterized reflection standards, while the thru standard relates both transmission directions. An ideal open has positive unity reflection, a short negative unity and a perfectly matched load zero, but physical standards contain offset delays, parasitic capacitance or inductance, conductor loss and mismatch. Their calibration-kit models are part of the calculation.

Even a seemingly well-matched $51\,\Omega$ resistor on a $50\,\Omega$ real wave reference has a nonzero reflection of $\frac{1}{101}$ , corresponding to return loss near $40.1\,\mathrm{dB}$ . Fitting it as an ideal zero-reflection load can spoil measurement of even smaller DUT reflections. Two-port error models may include source/load match, tracking, switch terms and optional isolation correction; never claim every VNA implements one universal fixed equation.

\Gamma_L=\frac{Z_L-R_0}{Z_L+R_0}

\Gamma_{51\Omega}=\frac{51-50}{51+50}=\frac{1}{101}

\mathrm{RL}=-20\log_{10}\left|\frac{1}{101}\right|\approx40.1\,\mathrm{dB}

03What TRL solves: two unknown error boxes

A Thru–Reflect–Line calibration uses a thru, a sufficiently characterized or symmetry-constrained reflect and one or more transmission lines of known geometrical relation. In implementations following Engen and Hoer, the solved error boxes account for systematic receiver paths on either side of the DUT. The line's propagation information separates their effects; the reflect and thru help resolve reference-plane and sign ambiguities.

The reflect need not in every TRL variant be an ideal known complex reflection, but method-specific assumptions must hold. Consistent mode, line impedance, return-current geometry and access launches matter, particularly with coplanar on-wafer standards. A TRL algorithm cannot repair arbitrary differences between the standards and the DUT access environment.

Figure 5.3 — 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.

\mathrm{TRL}=\{\mathrm{Thru},\mathrm{Reflect},\mathrm{Line}\}

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

\gamma=\alpha+j\beta\quad\text{(single guided-mode model)}

04TRL eigenvalues and the ill-conditioned line lengths

For a uniform single-mode line, the propagation eigenvalues associated with opposite traveling directions scale as exp(+γ $\Delta\ell$ ) and exp(−γ $\Delta\ell$ ). When the two eigenvalues approach equality in the lossless approximation, common TRL solutions have difficulty separating waves and noise is amplified. The relevant phase condition is $\beta\Delta\ell$ near integer multiples of $\pi$ , not a mystery frequency-independent rule.

The propagation constant γ=α+j $\beta$ must be defined from the same transmission-line convention in both the thru and line models. A 90-degree phase difference is a useful design center for an introductory low-loss illustration, not a universal operating guarantee. Real algorithms, losses, phase unwrapping, and finite standard length accuracy change numerical conditioning.

\lambda_\pm=\mathrm e^{\pm\gamma\Delta\ell}

\lambda_+\approx\lambda_-\quad\Longleftrightarrow\quad\beta\Delta\ell\approx k\pi

\Delta\ell=L_{\mathrm{line}}-L_{\mathrm{thru}}

05Numerical line selection from 0.5 to 2 GHz

For a deliberately hypothetical phase velocity of $2.0\times10^8\,\mathrm{m/s}$ and a line length difference $\Delta\ell=0.05\,\mathrm m$ , $\beta\Delta\ell$ is 45 degrees at $0.5\,\mathrm{GHz}$ , 90 degrees at 1.0 GHz and 180 degrees at 2.0 GHz. The 2.0 GHz case lies near a degeneracy of the ideal single-line TRL solution; at 1.0 GHz that same difference is well separated.

A second line length difference $\Delta\ell=0.025\,\mathrm m$ produces 90 degrees at 2.0 GHz and 45 degrees at 1.0 GHz. These values follow from $\beta=\frac{2\pi f}{v_p}$ , not from choosing an instrument brand or guessing a calibration specification. The example assumes the same guided mode and phase velocity in all the standards.

Figure 3.1 — Why TRL needs more than one line length

Calculated low-loss model for phase velocity 2.0×10⁸ m/s. The 5 cm line difference reaches 180° at 2 GHz, where the single-line TRL problem becomes ill-conditioned.

Line 1: Δℓ = 5 cm · Line 2: Δℓ = 2.5 cm

— 180° eigenvalue degeneracy in the lossless model

At 2 GHz the second line gives 90° while the first gives 180°. Multiple lengths alone cannot repair inconsistent port geometry or unwanted modes.

\beta=\frac{2\pi f}{v_p}

\Delta\ell=0.05\,\mathrm m,\quad v_p=2.0\times10^8\,\mathrm{m\,s^{-1}}

\begin{aligned}\Delta\phi(0.5\,\mathrm{GHz})&=45^\circ\\\Delta\phi(1\,\mathrm{GHz})&=90^\circ\\\Delta\phi(2\,\mathrm{GHz})&=180^\circ\end{aligned}

06Multiline TRL: complementary lengths and branch selection

Multiline TRL combines measured responses of several transmission lines with different lengths to estimate calibration errors and propagation more robustly across a wider band. Lines whose phase responses are nearly degenerate at one frequency may contribute useful separation at another. A numerical solver must resolve phase branches consistently and weight standard residuals and uncertainties; averaging raw complex S-matrices is not multiline calibration.

The effective reference impedance is determined by the calibrated line and the algorithm's reference definition. The line may not have exactly 50 ohms; consequently, measurements must be properly renormalized before comparison with $50\,\Omega$ Touchstone data. Uncertainty due to line dimensions, dispersion, conductor loss, connectors and thermal drift can dominate if not propagated.

\Delta\phi_i(f)=\beta(f)\Delta\ell_i

\max_i\bigl|\sin\Delta\phi_i(f)\bigr|>0\quad\text{(necessary nondegeneracy indicator)}

\Delta\ell_1=0.05\,\mathrm m,\quad\Delta\ell_2=0.025\,\mathrm m

07Reference impedance, fixture geometry and differential uncertainty

Calibration and reference-impedance renormalization are distinct operations. Renormalization changes how the same physical two-port is represented by reference waves; de-embedding removes a modeled network cascade from measurement data. For correctly defined real wave references, transform S through the underlying physical impedance matrix, accounting for unequal port references and the invertibility/conditioning of the needed matrices.

On-wafer probes, PCB launches and coax transitions must preserve compatible port modes, grounds and return-current topology. A correct algebraic TRL solution can still be inaccurate if a line standard couples an additional mode or if the reflect standard has inconsistent metal/ground geometry. Differential measurements also need a consistent mixed-mode treatment and uncertainty budget.

\boldsymbol\zeta=\mathbf R^{-1/2}\mathbf Z\mathbf R^{-1/2}

\mathbf S=(\boldsymbol\zeta-\mathbb I)(\boldsymbol\zeta+\mathbb I)^{-1}

\text{TRL line reference}\ne\text{automatically }50\,\Omega

08Reproducible SOLT-versus-TRL verification

To compare SOLT and TRL, place both solutions at physically comparable planes and express results with the same wave reference. Measure an independent verification load, thru or circuit not used to determine the calibration coefficients. Compare the complex reflection and transmission, not merely plotted magnitudes or gain in decibels.

Archive the instrument and firmware, actual calibration-kit standard definitions, probe pitch and geometry, frequency grid, source power, IF bandwidth, averaging, cable motion, environmental conditions and the measured complex Touchstone files. Repeat calibrations and connector landings and report residual model deviations with uncertainty. Disagreement identifies assumptions to investigate, not automatic evidence of a new DUT effect.

\Delta\mathbf y\approx\mathbf J\,\Delta\mathbf x

\boldsymbol\Sigma_y\approx\mathbf J\boldsymbol\Sigma_x\mathbf J^\dagger

\text{validation standard}\ne\text{calibration fitting standards}

SOLT, TRL and multiline calibration problems

Formulate the one-port three-standard error model as a linear 3×3 inverse problem, explaining why independent standards matter.
At R0= $50\,\Omega$ , compute $\Gamma$ , reflected-power fraction and return loss for a $51\,\Omega$ resistive calibration load.
Explain why a physical open with parasitic capacitance cannot have $\Gamma=+1$ at all RF frequencies.
State which information SOLT derives from reflection and thru standards, and why two-port calibration depends on receiver architecture.
Starting from propagation eigenvalues, derive the low-loss TRL degeneracy condition $\beta\Delta\ell\approx k\pi$ .
For vp= $2.0\times10^8\,\mathrm{m/s}$ and $\Delta\ell=0.05\,\mathrm m$ , compute Δφ at 0.5, 1 and $2\,\mathrm{GHz}$ .
Choose a second TRL line-length difference giving 90 degrees at $2\,\mathrm{GHz}$ ; evaluate it also at $1\,\mathrm{GHz}$ .
Explain what must be done if a TRL calibration defines the wave reference through a line with impedance not equal to $50\,\Omega$ .
Diagnose a coplanar TRL error caused by inconsistent ground return or an unwanted additional mode.
Design a SOLT/TRL cross-check with an independent verification network and an uncertainty record.

Worked solutions

1. For each standard i, write $m_i=E_D+\frac{E_R\Gamma_i}{1-E_S\Gamma_i}$ , set $t=E_R-E_D E_S$ and multiply by the denominator. The resulting row $\begin{pmatrix}1&\Gamma_i&m_i\Gamma_i\end{pmatrix}$ multiplies column $\begin{pmatrix}E_D\\t\\E_S\end{pmatrix}$ to produce m_i. Three linearly independent rows solve the three complex unknowns; then $E_R=t+E_D E_S$ . Close standard responses produce an ill-conditioned matrix even if formal inversion exists.

2. The real-load reflection is $\Gamma=\frac{51-50}{51+50}=\frac1{101}\approx0.00990099$ . The reflected-power fraction is $|\Gamma|^2=\frac1{10201}\approx9.80\times10^{-5}$ , or approximately $0.0098\%$ . Return loss is $-20\log_{10}\!\left(\frac1{101}\right)=20\log_{10}(101)\approx40.086\,\mathrm{dB}$ . Treating this load as $\Gamma=0$ can bias a high-return-loss measurement.

3. At the physical open plane an open-ended connector can exhibit fringe capacitance C and propagation delay. A capacitive termination has $Z=\frac1{j\omega C}$ under the declared exp(+jωt) convention, and $\Gamma=\frac{Z-R_0}{Z+R_0}$ is frequency dependent. The ideal value +1 is recovered in an appropriate low-frequency or zero-capacitance limit, not automatically at all RF bands.

4. The short, open and load supply calibrated reflections which constrain directivity, source-match and reflection tracking. The thru supplies a characterized transfer between ports to constrain transmission-path behavior. Additional switch, load-match and isolation corrections vary with VNA hardware and measurement sequence, so the exact coefficient count must be documented for the real test set.

5. A line-length difference $\Delta\ell$ gives forward/reverse eigenvalue factors $\lambda_\pm=\mathrm e^{\pm\gamma\Delta\ell}$ . For $\gamma\approx j\beta$ , equality requires $\mathrm e^{2j\beta\Delta\ell}=1$ , hence $2\beta\Delta\ell=2k\pi$ and $\beta\Delta\ell=k\pi$ . At those phases the eigenvalues merge and noise can make separation ill-conditioned. Attenuation changes exact equality but not the underlying near-degeneracy risk.

6. With $\beta\Delta\ell=\frac{2\pi f\Delta\ell}{v_p}$ = $2\pi$ f×0.05/(2.0×10⁸), the phases are $\frac\pi4=45^\circ$ at $0.5\,\mathrm{GHz}$ , $\frac\pi2=90^\circ$ at $1\,\mathrm{GHz}$ and $\pi=180^\circ$ at $2\,\mathrm{GHz}$ . Thus the same line difference is useful around the center of this example but approaches an ill-conditioned condition at $2\,\mathrm{GHz}$ .

7. For the desired $90^\circ$ at $2\,\mathrm{GHz}$ , solve $\beta\Delta\ell$ = $\frac\pi2$ to obtain $\Delta\ell=\frac{v_p}{4f}=\frac{2.0\times10^8}{4(2.0\times10^9)}=0.025\,\mathrm m$ . At $1\,\mathrm{GHz}$ , the phase is half as large, $45^\circ$ . A pair of $0.05\,\mathrm m$ and $0.025\,\mathrm m$ differences provides complementary phase separations at 1 and $2\,\mathrm{GHz}$ in this simplified single-mode model.

8. A TRL calibration may establish a wave reference linked to its characteristic line impedance $Z_c$ , which is not necessarily $50\,\Omega$ . Convert calibrated scattering data to a compatible underlying Z representation using the actual reference, then normalize to the intended $50\,\Omega$ waves with a suitable matrix transformation. A simple adjustment of scalar S21 dB values is not a correct renormalization.

9. If the TRL standards and DUT have different ground-return paths or an extra coupled/radiating mode, they no longer share the same error-box model. A visually plausible mathematical solution can hide this mismatch. Inspect port layout, conductor/ground symmetry, phase consistency and independent verification response; redesign the standards or use a multimode model as appropriate.

10. Document the SOLT and TRL calibration planes, standard geometry/models, frequency band, wave reference, probe connections, source power, IF bandwidth and repeatability. Measure a device not used during either calibration, compare the complex S11/S21 after consistent plane and reference corrections, repeat landings and estimate both random and systematic uncertainty. Preserve raw and calibrated files.

Next we will study Touchstone data, complex uncertainty and reference-plane reporting, then complete a reproducible VNA-to-DUT de-embedding chain.