Vector Network Analyzer (VNA)

1. Theory

• Definition: A vector network analyzer measures the complex ratios of waves leaving and entering each port of a device, its scattering parameters, over frequency (Berkeley Nucleonics).
• Reference impedance: S-parameters are defined with the ports terminated in a reference impedance, almost always 50 $\Omega$ (Berkeley Nucleonics).
• Error correction: Systematic errors of couplers, cables and probes are described by an error model whose terms are found by measuring known standards (All About Circuits; FormFactor).
• Device parameters: Corrected and de-embedded S-parameters give transit frequency, maximum oscillation frequency and small-signal models (FormFactor).

2. Working Principle

• Signal separation: A swept source drives the device; directional couplers or bridges separate incident and reflected waves (Berkeley Nucleonics).
• Receivers: Tracking receivers downconvert to a fixed IF, and the IF bandwidth trades noise floor against speed (Berkeley Nucleonics).
• Calibration: SOLT, TRL, LRM or LRRM standards define the reference plane at the cable end or probe tips (Berkeley Nucleonics; FormFactor).
• De-embedding: Dummy open and short structures remove pad and interconnect parasitics on wafer (FormFactor).

3. Applications

• RF transistors: fT, fmax and small-signal models of HBTs, HEMTs and RF MOSFETs (FormFactor).
• Passives: On-chip inductors, capacitors and transmission lines (Passive Silicon Components pages).
• Components and interconnects: Filters, cables, packages and matching networks (Berkeley Nucleonics).

S-Parameters and Matching

Name	Formula	Related Formulae	Variables/Constants
Scattering parameters	$\left[\begin{array}{c}{b}_1\\ b_2\end{array}\right]=\left[\begin{array}{cc}{S}_{11}& S_{12}\\ S_{21}& S_{22}\end{array}\right]\left[\begin{array}{c}{a}_1\\ a_2\end{array}\right]$	$S_{11}={\frac{b_1}{a_1}|}_{a_2=0},S_{21}={\frac{b_2}{a_1}|}_{a_2=0}$	$a_i,b_i:incidentandemergingwavesatporti$ $\begin{array}{r}{S}_{ij}:scatteringparameters;Berkeley\\ Nucleonics:eachS-parameteristheratio\\ ofanemergingtoanincidentwavewith\\ theotherportterminatedinthereference\\ impedance,almostalways50\Omega ,anN-port\\ devicehavinganNbyNmatrix,S_{21}as\\ insertionlossorgain;openEMS\\ documentation:matrixform\end{array}$ $\begin{array}{r}(computed:afour-portdevicehas16\\ S-parameters)\end{array}$
Reflection, return loss and VSWR	$\Gamma =\frac{Z_L-Z_0}{Z_L+Z_0},RL=-20{\log }_{10}|S_{11}|$	$VSWR=\frac{1+|S_{11}|}{1-|S_{11}|},IL=-20{\log }_{10}|S_{21}|$	$\Gamma :reflectioncoefficient$ $Z_L:loadimpedance$ $Z_0:referenceimpedance$ $RL:returnloss$ $VSWR:voltagestanding-waveratio$ $\begin{array}{r}IL:insertionloss;BerkeleyNucleonics:\\ RLandVSWRforms,20\mathrm{dB}returnlossasa\\ commonpasslinecorrespondingtoabout\\ 1.22,Smith-chartcentreatamatchand\\ rimattotalreflection;openEMS:\Gamma =S_{11},\\ Z=Z_0(1+\Gamma )/(1-\Gamma ),invertedherefor\Gamma \\ (computed),IL=-10{\log }_{10}|S_{21}{|}^2\end{array}$ $\begin{array}{r}(computed:20\mathrm{dB}is|S_{11}|=0.1andVSWR\\ 1.222;a75\Omega loadon50\Omega gives\Gamma =0.2,\\ 14.0\mathrm{dB})\end{array}$
Dynamic range and IF bandwidth	$DR=P_{max}-P_{floor}$	$\Delta P_{floor}=10{\log }_{10}\frac{B_{IF,2}}{B_{IF,1}}$	$DR:dynamicrange$ $P_{max}:largestdrivelevel$ $P_{floor}:receivernoisefloor$ $\begin{array}{r}{B}_{IF}:intermediate-frequencybandwidth;\\ BerkeleyNucleonics:narrowingtheIF\\ bandwidthlowersthenoisefloorand\\ extendsdynamicrangeatthecostofsweep\\ speed,asaveragingdoes,withsource,\\ couplersorbridges,trackingreceivers\\ andatransferswitchmeasuringallfour\\ S-parameters;logarithmicscaling\\ (computed,noisepowerproportionalto\\ bandwidth)\end{array}$ $\begin{array}{r}(computed:10\mathrm{kHz}to100\mathrm{Hz}lowersthe\\ floorby20\mathrm{dB}andmakeseachpointabout\\ 100timesslower)\end{array}$

Calibration

Name	Formula	Related Formulae	Variables/Constants
One-port error model	$S_{11m}=e_{00}+\frac{e_{10}{e}_{01}{S}_{11}}{1-e_{11}{S}_{11}}$	$S_{11}=\frac{S_{11m}-e_{00}}{e_{10}{e}_{01}+e_{11}\left(S_{11m}-e_{00}\right)}$	$S_{11m}:measuredreflection$ $e_{00}:directivity$ $e_{11}:sourcematch$ $\begin{array}{r}{e}_{10}{e}_{01}:reflectiontracking;Borettietal.:\\ thismodelwithasingletrackingterm{E}_{01},\\ solvedbyshort,openandloadstandards;\\ inversion(computed);AllAboutCircuits:\\ theerrortermsandshort,openandload\\ standardsoneachport;Berkeley\\ Nucleonics:couplerdirectivitysetsthe\\ flooronmeasurablereturnloss\end{array}$ $\begin{array}{r}(computed:withuncorrected40\mathrm{dB}\\ directivityatrue20\mathrm{dB}returnlossreads\\ between19.2and20.9\mathrm{dB})\end{array}$
Two-port error terms	$12\text{terms}=2\times \left(e_{00},e_{11},e_{10}{e}_{01},e_{22},e_{10}{e}_{32},e_{30}\right)$	$10\text{terms (isolation neglected)},7\text{terms (LRRM and TRL family)}$	$e_{22}:loadmatch$ $e_{10}{e}_{32}:transmissiontracking$ $\begin{array}{r}{e}_{30}:isolation(crosstalk);AllAbout\\ Circuits:sixforwardandsixreverse\\ terms,the12-termSOLTcorrection,ten\\ unknownswhencrosstalkiszero;\\ FormFactor:SOLTasa10-termmodel\\ needingopen,shortand50\Omega loadperport\\ andathru,eLRRMasa7-termmodelwith\\ freereflectvaluesandautomatic\\ load-inductanceextraction;Berkeley\\ Nucleonics:twelveerrortermsforafull\\ two-portcalibration(seetheQ/A)\end{array}$ $(computed:12-2isolationterms=10)$
TRL line bandwidth	$20^{\circ }\le \beta l\le 160^{\circ }$	$\frac{f_{max}}{f_{min}}=\frac{160}{20}=8$	$\beta :propagationconstant$ $\begin{array}{r}l:linelengthbeyondthethru;Berkeley\\ Nucleonics:asinglelinecoversaboutan\\ 8:1span,TRLpreferredforwaveguide,\\ microstripandon-waferwork,LRMasa\\ relative;FormFactor:NISTmultilineTRL\\ astheaccuracybenchmark,needingno\\ definedreflect,withlinecapacitance\\ 1.481pF/cmtoseta50\Omega referenceatthe\\ probetips;KeysightTRLhelp:THRU-LINE\\ phasedifferencebetween20and160\\ degrees,asingleLINEusableover8:1\end{array}$ $\begin{array}{r}(computed:the20to160degreelimits\\ giveexactly8:1)\end{array}$

On-Wafer Device Extraction

Name	Formula	Related Formulae	Variables/Constants
Open-short de-embedding	$Y_{DUT}={\left[{\left(Y_m-Y_{open}\right)}^{-1}-{\left(Y_{short}-Y_{open}\right)}^{-1}\right]}^{-1}$	$G_1=\mathrm{Re}\left(Y_{11,open}\right)\ge 0(\text{physical pad})$	$\begin{array}{r}{Y}_m,Y_{open},Y_{short}:admittancematricesofthe\\ deviceandofopenandshortdummy\\ structures\end{array}$ $\begin{array}{r}{G}_1:padshuntconductance;Teppatietal.,\\ Eq.1:two-stepformula;FormFactor:\\ two-stepopen-shortde-embedding,pad-open\\ G_{1}betweenabout-0.5and1.0\mathrm{mS}upto60\\ \mathrm{GHz},SOLTgivingunphysicalnegative{G}_1\\ aboveafewgigahertzfrommismatch\\ betweenrealandtabulatedstandards(open\\ -7.2against-5.5\mathrm{fF},short5.0against\\ 6.0\mathrm{pH},load-3.3against0.1and1.0\mathrm{pH})\end{array}$ $\begin{array}{r}(computed:theopen-standarderrorof1.7\\ \mathrm{fF}is0.21\mathrm{mS}ofsusceptanceat20\mathrm{GHz})\end{array}$
Transit frequency from h21	$h_{21}=\frac{-2S_{21}}{\left(1-S_{11}\right)\left(1+S_{22}\right)+S_{12}{S}_{21}}$	$|h_{21}|\approx \frac{f_T}{f}$	$h_{21}:short-circuitcurrentgain$ $\begin{array}{r}{f}_T:transitfrequency;Teppatietal.,Eq.\\ A.4:conversion;Wangetal.:|h_{21}{|}^2\\ fallingat20dBperdecadeand{f}_{T}where|\\ h_{21}|reachesunity;FormFactor:\\ spot-frequency{f}_{T}at20\mathrm{GHz}ona14.86by\\ 0.12\mu \mathrm{m}HBT,236(SOLT),242(corrected\\ SOLT),241(mTRL)and241\mathrm{GHz}(eLRRM)\end{array}$ $\begin{array}{r}(computed:f_T=241\mathrm{GHz}means|h_{21}|=\\ 12.05,or21.6\mathrm{dB},at20\mathrm{GHz})\end{array}$
Maximum oscillation frequency from U	$U=\frac{|S_{21}/S_{12}-1{|}^2}{2\left(k|S_{21}/S_{12}|-\mathrm{Re}\left(S_{21}/S_{12}\right)\right)}$	$U\approx {\left(\frac{f_{max}}{f}\right)}^2,k=\frac{1-|S_{11}{|}^2-|S_{22}{|}^2+|\Delta {|}^2}{2|S_{12}{S}_{21}|}$	$U:Maso{n}^{\prime }sunilateralgain$ $k:Rollettstabilityfactor,\Delta =S_{11}{S}_{22}-S_{12}{S}_{21}$ $\begin{array}{r}{f}_{max}:maximumoscillationfrequency;Teppati\\ etal.,Eqs.A.1andA.2:theseforms;\\ Wangetal.:Ufallingat20dBper\\ decade,f_{max}fromU;FormFactor:f_{max}from\\ Maso{n}^{\prime }sgain,329(SOLT),302(corrected\\ SOLT),297(mTRL)and295\mathrm{GHz}(eLRRM),\\ SOLTabout10percenthighbecauseof\\ probemisplacement\end{array}$ $\begin{array}{r}(computed:f_{max}=297\mathrm{GHz}meansU=220,or\\ 23.4\mathrm{dB},at20\mathrm{GHz};329against297is\\ 10.8percenthigh)\end{array}$

Important Questions

Ques: What does a vector network analyzer measure?

The complex scattering parameters of a device, the amplitude and phase of waves it reflects and transmits. A swept source drives one port, part of its output is tapped as a reference, and directional couplers or bridges separate the incident and reflected waves; tracking receivers downconvert them to a fixed intermediate frequency for comparison (Berkeley Nucleonics). A transfer switch routes the source to either port, so one sweep gives all four S-parameters of a two-port, and an N-port has $N^2$, 16 for four ports (computed). Each parameter is an emerging-to-incident wave ratio with the other port terminated in the reference impedance, almost always 50 $\Omega$. $S_{21}$ is insertion loss for a cable or filter and gain for an amplifier. Reference: Berkeley Nucleonics, ‘Vector Network Analysis’, chapter 6 of an online RF and microwave guide (fetch-tool summary).

Ques: How are return loss and VSWR related?

Both are functions of the reflection magnitude. Return loss is $-20{\log }_{10}|S_{11}|$ and VSWR is $(1+|S_{11}|)/(1-|S_{11}|)$ (Berkeley Nucleonics). A 20 $\mathrm{dB}$ return loss means $|S_{11}|=0.1$, so 1 per cent of the incident power is reflected and the VSWR is 1.222, the 1.22 Berkeley Nucleonics quotes; it calls 20 $\mathrm{dB}$ a common pass line (computed). A 75 $\Omega$ load on a 50 $\Omega$ system reflects $\Gamma =(75-50)/(75+50)=0.2$, from openEMS’s $Z=Z_0(1+\Gamma )/(1-\Gamma )$ inverted, a return loss of 14.0 $\mathrm{dB}$ (computed). On the Smith chart a perfect match sits at the centre and total reflection on the rim. Larger return loss means a better match; in this convention return loss is positive, while $S_{11}$ in decibels is negative. Reference: Berkeley Nucleonics, ‘Vector Network Analysis’, chapter 6 of an online RF and microwave guide (fetch-tool summary); openEMS documentation, ‘Calculate S-parameters’: [b1; b2] = S [a1; a2], Gamma = S11, Z = Z0(1 + Gamma)/(1 - Gamma), return loss -10 log10(|S11|^2), insertion loss -10 log10(|S21|^2) (fetch-tool summary).

Ques: Why must a VNA be calibrated, and what limits accuracy?

Because the couplers, cables and probes add their own reflections and losses. For one port, the measured reflection is $e_{00}+e_{10}{e}_{01}{S}_{11}/(1-e_{11}{S}_{11})$, with directivity $e_{00}$, source match $e_{11}$ and reflection tracking $e_{10}{e}_{01}$ (Boretti et al., who write the tracking term as a single $E_{01}$); measuring a short, an open and a load gives three equations for the three unknowns (All About Circuits). Without correction, coupler directivity limits the return loss that can be measured (Berkeley Nucleonics): with 40 $\mathrm{dB}$ directivity a true 20 $\mathrm{dB}$ match can read anywhere from 19.2 to 20.9 $\mathrm{dB}$ (computed). After calibration, accuracy depends on how well the standards are known and how repeatably cables and probes are positioned (FormFactor). Reference: E. Boretti, N. Pompeo, E. Silva, K. Torokhtii and A. Alimenti, ‘Design and Repeatability Assessment of an RF MEMS Switching Platform for Automated Microwave SOL Calibration’, arXiv:2607.06747: Gamma_m = E00 + E01 Gamma/(1 - E11 Gamma), directivity, reflection tracking and source match, SOL with short, open and load (fetch-tool summary); S. Arar, ‘Understanding the 12-Term Error Model and SOLT Calibration Method for VNA Measurements’, All About Circuits, 10 March 2024 (fetch-tool summary); Berkeley Nucleonics, ‘Vector Network Analysis’, chapter 6 of an online RF and microwave guide (fetch-tool summary); FormFactor, ‘Calibration Tools: Achieving Consistent Parameter Extraction for Advanced RF Devices’, application note CALTOOL-AN-0612 (2020) (fetch-tool summary).

Ques: Does SOLT use a 12-term or a 10-term error model?

Both descriptions appear. All About Circuits and Berkeley Nucleonics describe full two-port SOLT correction with twelve error terms, six forward and six reverse, while FormFactor’s application note describes SOLT as a 10-term model. They agree once isolation is considered: All About Circuits notes that setting crosstalk to zero reduces the unknowns from 12 to 10. So 12 terms is the general model and 10 terms the common practical form, which is the more likely meaning of FormFactor’s statement (my reading). Keysight’s TRL help, by contrast, says TRL determines the full 2-port 12-term error coefficients, so the term count describes the error model solved, not the standards used (my reading). Reference: S. Arar, ‘Understanding the 12-Term Error Model and SOLT Calibration Method for VNA Measurements’, All About Circuits, 10 March 2024 (fetch-tool summary); Berkeley Nucleonics, ‘Vector Network Analysis’, chapter 6 of an online RF and microwave guide (fetch-tool summary); FormFactor, ‘Calibration Tools: Achieving Consistent Parameter Extraction for Advanced RF Devices’, application note CALTOOL-AN-0612 (2020) (fetch-tool summary); Keysight Technologies, VNA Help, ‘TRL Calibration’: THRU-LINE phase difference greater than 20 and less than 160 degrees, a single LINE usable over an 8:1 frequency range, TRL determining the 2-port 12-term error coefficients (fetch-tool summary).

Ques: Which on-wafer calibration is best, SOLT, TRL or LRRM?

FormFactor’s comparison favours multiline TRL and enhanced LRRM. SOLT needs open, short and 50 $\Omega$ load standards whose values must be known; on its test substrate, the tabulated values differed from the extracted ones (open -7.2 against -5.5 $\mathrm{fF}$, short 5.0 against 6.0 $\mathrm{pH}$, load -3.3 against 0.1 to 1.0 $\mathrm{pH}$), and probe misplacement added errors. NIST multiline TRL needs no defined reflect, uses several lines, and is the accuracy benchmark, each line covering about an 8:1 band (Berkeley Nucleonics; Keysight’s 20 to 160 degree rule, computed). Enhanced LRRM uses free reflect values and extracts the load inductance, avoiding probe-distance readjustment. FormFactor found both advanced methods better than SOLT already at about 20 $\mathrm{GHz}$ and recommends eLRRM as accurate and easy to implement. Reference: FormFactor, ‘Calibration Tools: Achieving Consistent Parameter Extraction for Advanced RF Devices’, application note CALTOOL-AN-0612 (2020) (fetch-tool summary); Berkeley Nucleonics, ‘Vector Network Analysis’, chapter 6 of an online RF and microwave guide (fetch-tool summary); Keysight Technologies, VNA Help, ‘TRL Calibration’: THRU-LINE phase difference greater than 20 and less than 160 degrees, a single LINE usable over an 8:1 frequency range, TRL determining the 2-port 12-term error coefficients (fetch-tool summary).

Ques: Why is de-embedding needed, and how is it checked?

Because calibration moves the reference plane only to the probe tips, while the device sits behind pads and interconnects. Open and short dummy structures model these parasitics, and the two-step formula $Y_{DUT}=[(Y_m-Y_{open}{)}^{-1}-(Y_{short}-Y_{open}{)}^{-1}{]}^{-1}$ removes them (Teppati et al., Eq. 1). FormFactor checks calibration quality with the pad-open shunt conductance, which must stay positive: multiline TRL and eLRRM gave the expected behaviour up to 60 $\mathrm{GHz}$, while SOLT gave a conductance that fell with frequency and turned negative above a few gigahertz, physically impossible for a passive pad. It traced this to the mismatch between real and tabulated standards; the 1.7 $\mathrm{fF}$ open error alone is 0.21 $\mathrm{mS}$ of susceptance at 20 $\mathrm{GHz}$ (computed). Reference: V. Teppati et al., ‘Accuracy of Microwave Transistor fT and fMAX Extractions’, IEEE Trans. Electron Devices 61, 984-990, 2014: Z_DUT,os = (Y_DUT - Y_O)^-1 - (Y_S - Y_O)^-1, k, U and H21 from S-parameters (Eqs. 1, A.1, A.2, A.4) (fetch-tool summary); FormFactor, ‘Calibration Tools: Achieving Consistent Parameter Extraction for Advanced RF Devices’, application note CALTOOL-AN-0612 (2020) (fetch-tool summary).

Ques: How are fT and fmax extracted, and how sensitive are they?

From the current gain and Mason’s unilateral gain. Converting S-parameters to $h_{21}$ and using $|h_{21}|\approx f_T/f$ at a spot frequency gives the transit frequency, and $U\approx (f_{max}/f{)}^2$ gives the maximum oscillation frequency (Teppati et al. give $H_{21}$ and U from S-parameters; Wang et al. show both falling at $20\mathrm{dB}$ per decade). On FormFactor’s HBT at 20 $\mathrm{GHz}$, $f_T$ was 236 $\mathrm{GHz}$ with SOLT and 241 $\mathrm{GHz}$ with mTRL and eLRRM, about 3 per cent apart, but $f_{max}$ was 329 $\mathrm{GHz}$ with SOLT against 297 and 295 $\mathrm{GHz}$, 10.8 per cent high (computed); correcting the SOLT standards gave 302 $\mathrm{GHz}$. A gain of 21.6 $\mathrm{dB}$ in $h_{21}$ and 23.4 $\mathrm{dB}$ in U at 20 $\mathrm{GHz}$ correspond to 241 and 297 $\mathrm{GHz}$ (computed). fmax is more sensitive because U depends on the small reverse transmission. Reference: V. Teppati et al., ‘Accuracy of Microwave Transistor fT and fMAX Extractions’, IEEE Trans. Electron Devices 61, 984-990, 2014: Z_DUT,os = (Y_DUT - Y_O)^-1 - (Y_S - Y_O)^-1, k, U and H21 from S-parameters (Eqs. 1, A.1, A.2, A.4) (fetch-tool summary); H. Wang et al., ‘Black Phosphorus Radio-Frequency Transistors’, arXiv:1410.3880: |h21|^2 and U follow 20 dB/dec slopes, fT where |h21| reaches unity, fmax from U or MSG/MAG (fetch-tool summary); FormFactor, ‘Calibration Tools: Achieving Consistent Parameter Extraction for Advanced RF Devices’, application note CALTOOL-AN-0612 (2020) (fetch-tool summary).

References

1. Berkeley Nucleonics, “Vector Network Analysis,” chapter 6 in an online RF and microwave guide. Available: https://www.berkeleynucleonics.com/books/rf-and-microwave/html/06-vector-network-analysis.html
2. S. Arar, “Understanding the 12-Term Error Model and SOLT Calibration Method for VNA Measurements,” All About Circuits, Mar. 10, 2024. Available: https://www.allaboutcircuits.com/technical-articles/understanding-the-12-term-error-model-and-solt-calibration-method-for-vna-measurements/
3. FormFactor, “Calibration Tools: Achieving Consistent Parameter Extraction for Advanced RF Devices,” application note CALTOOL-AN-0612, 2020. Available: https://www.psirep.com/system/files/achieving_consistent_parameter_extraction_for_advanced_rf_devices_-_calparaextraction-an.pdf
4. openEMS documentation, “Calculate S-parameters.” Available: https://openems.readthedocs.io/en/latest/_sources/concepts/postproc/sparams.rst.txt
5. E. Boretti, N. Pompeo, E. Silva, K. Torokhtii and A. Alimenti, “Design and repeatability assessment of an RF MEMS switching platform for automated microwave SOL calibration,” arXiv:2607.06747. Available: https://arxiv.org/abs/2607.06747
6. Keysight Technologies, “TRL Calibration,” PNA VNA Help. Available: https://helpfiles.keysight.com/csg/N52xxB/S3_Cals/TRL_Calibration.htm
7. V. Teppati, S. Tirelli, R. Lovblom, R. Fluckiger, M. Alexandrova and C. R. Bolognesi, “Accuracy of microwave transistor fT and fMAX extractions,” IEEE Trans. Electron Devices, vol. 61, pp. 984-990, 2014. Available: https://iris.polito.it/retrieve/handle/11583/2536694/62156/TEDTeppati2014_Review2_CLEAN.pdf
8. H. Wang et al., “Black phosphorus radio-frequency transistors,” arXiv:1410.3880. Available: https://arxiv.org/abs/1410.3880

Related pages in this vault: SMU and Probe Station, LCR Meter and C-V, AC Circuits and Impedance, Spreading Resistance Profiler (SRP), Four-Point Probe, Charge Pumping, On-Chip Transmission Lines, On-Chip Transformers and Baluns, RF and Non-Quasi-Static Effects, Planar Spiral Inductors, Langmuir Probe.