Large-Signal Network Analysis - Jan Verspechtjanverspecht.com/pdf/lsnaieeemicrowavemagazine.pdf · Large-Signal Network Analysis Jan Verspecht ... one is the transformation between

Large-Signal Network Analysis

Jan Verspecht

Jan Verspecht bvba

Mechelstraat 17B-1745 OpwijkBelgium

email: [email protected]: http://www.janverspecht.com

IEEE Microwave Magazine, Vol. 6, Issue 4, December 2005, pp. 82-92

© 2005 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.

Volume 6 • Number 4 • December 2005

for the Microwave & Wireless Engineer

3December 2005

72

82

features

Multiport Vector NetworkAnalyzer MeasurementsHardware architectures and calibration procedures for multiportmeasurements and their impacts on flexibility and uncertainties.J. Martens, D. Judge, and J. Bigelow

Large-Signal Network AnalysisAn invitation to join the rapidly growing LSNA communitythat is improving microwave measurements and is on itsway to building a multimillion dollar business.Jan Verspecht

A Potentially Significant On-WaferHigh-Frequency MeasurementCalibration ErrorJames C. Rautio and Robert Groves

IEEE Microwave Theory and Techniques Society

Volume 6 • Number 4 • December 2005 • ISSN 1527-3342

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on the cover:© ARTVILLE

for the Microwave & Wireless Engineer

82 December 20051527-3342/05/$20.00©2005 IEEE

Jan Verspecht

According to Mike McKinley of Georgia Tech, it’s “the Holy Grail ofmicrowave instrumentation and measurements,” it’s “a new measurementscience,” according to Don DeGroot of Connected Community Networksand “an enabling path to taking nonlinear models to a higher level of fideli-ty and value,” says Larry Dunleavy of Modelithics.

I prefer to call it large-signal network analysis (LSNA). Today, LSNA is undoubtedly stillin its infancy. A small community knows about the theoretical concepts, and a happy fewhave access to LSNA hardware. This community is growing rapidly, however, and I have nodoubt that the people of this community will soon revolutionize the microwave measure-ment business as profoundly as S-parameter technology did in the 1980s. I hope with thisarticle to inspire you to join this rapidly growing LSNA community that is improvingmicrowave measurements and is on its way to building a multimillion dollar business.

BackgroundUnlike cellophane or penicillin, LSNA technology was not invented by accident or by a flash ofgenius. Instead, it resulted from first recognizing a real problem faced by microwave designers

Jan Verspecht ([email protected]) is with Jan Verspecht bvba in Steenhuffel, Belgium.

December 2005 83

and then carefully fashioning a solution for it. The realproblem I am referring to is the gap between results thatcan be generated via computer-aided design (CAD) toolsand those that can be measured with current microwaveinstrumentation.

A classic vector network analyzer (VNA) can auto-matically characterize all kinds of microwave compo-nents by measuring S-parameters. These S-parameterdata can be made directly accessible to the designers ina CAD environment, where the simulated devicebehavior accurately represents the measured devicebehavior. Alternatively, a manufactured componentcan easily be verified against design specifications bymeasuring its S-parameters. In other words, a classicVNA provides a consistency check between the actualphysical device and the mathematical abstraction thatresides inside the simulator. The result is a very effi-cient design cycle. If this works so great, why, then, dowe need LSNA?

The answer is that S-parameters are based on asuperposition principle and, therefore, can only accu-rately represent linear devices, like filters, cables, cou-plers, and connectors, whose behavior is determinedsolely by the linear Maxwell’s equations. How about allthose S-parameter measurements of semiconductoramplifiers? There is certainly more than the linearMaxwell’s equations in there. Does this mean that

S-parameter measurements of amplifiers (which allexhibit nonlinearity) are questionable?

The answer is “yes.” Such measured data is ques-tionable without a verification whether the applied sig-nal levels during the S-parameter measurement aresmall enough to justify the superposition principle.This is typically done by measuring the S-parameters attwo different power levels and by verifying that theresulting curves are identical. So, we ensure that theapplied amplifier stimulus is small enough to verifythat the superposition principle was valid when wemeasure the S-parameters of a semiconductor amplifi-er. Are we happy now?

The answer is, unfortunately, “no.” Although themeasured S-parameters for the amplifier are validand accurate, they still can only be used to predictthe behavior for a small signal stimulus. The mea-sured S-parameters can not predict the behavior ofthe amplifier whenever the amplitude of the stimu-lus becomes significant compared to the operatingrange of the device. Whenever the behavior deviatesfrom the superposition principle, people say that thecomponent introduces nonlinear distortion effects.These effects are usually categorized as compression,amplitude modulation–phase modulation (AM-PM)conversion, and harmonic and intermodulationproduct generation.

To the best of my knowledge, the history of LSNAtechnology started in October 1988. That month, theIEEE published the paper “High-Frequency PeriodicTime-Domain Waveform Measurement System,” bySipilä et al. [2]. The paper reported on the firstmeasurement setup that had the two necessaryingredients of LSNA technology:

• a unified approach for the frequency and thetime domain

• the availability of wave as well as a voltage-cur-rent representations.

Sipilä et al. used a two-channel 14-GHz oscillo-scope and one coupler to measure the voltage andcurrent waveforms at the gate and drain of a HFtransistor.

In 1989, Lott reported on a system based exclu-sively on a VNA test set and receiver [14]. He mea-sured the harmonics with the receiver by tuning itconsecutively to each of the harmonics. An ingeniousphase reference method is used to align the harmon-ic phases (the so-called golden diode approach).

In 1990, Kompa and Van Raay reported on asimilar setup based on a two-channel microwaveoscilloscope combined with a complete VNA test setand receiver [15]. The receiver is used to measure

the fundamental data, the oscilloscope is used forthe harmonics.

A breakthrough took place in 1992 with the com-mercial introduction of the Hewlett-Packardmicrowave transition analyzer (MTA). This two-channelreceiver has a bandwidth of 40 GHz and directly mea-sures phase and amplitude of fundamental and har-monic spectral components. It was first used byKompa and Van Raay [16]. From 1994 on, it was alsoused by Demmler, Tasker, and Schlechtweg [17];Leckey [18]; and Wei and Tkachenko [19].

In 1995, my colleagues and I started using twosynchronized MTAs as a four-channel harmonicreceiver [20]. This work was performed at the lab ofProf. Barel of the Vrije Universiteit Brussel and result-ed in the first prototype of the nonlinear networkmeasurement system, later to be called the largesignal network analyzer. A picture of such a systemis shown in Figure 1.

In 2000, Arnaud and her colleagues built an LSNAsystem based on a modified VNA receiver with load-pull and pulsed measurement capability at the IRCOMin France [21]. She uses a phase reference similar toUrs Lott. The same year, I added envelope domaincapability to our LSNA prototype instrument [22].

History

84 December 2005

It is the goal of LSNA technology to go beyond S-parameters to provide a tool set that may be used toaccurately characterize nonlinear components, such asamplifiers, under large-signal excitation. This isachieved by designing instruments that can accuratelyand completely characterize devices under realisticlarge-signal operating conditions and developing soft-ware that can accurately represent the measured non-linear device behavior in a simulator.

Theoretical FoundationsLSNA technology is empowered by a unique combina-tion of two mathematical transformations. The first one is the transformation between a traveling voltage-wave formalism and a voltage-current representation.The second one is the transformation between the timedomain and the frequency domain. Classic instrumen-tation tools do not readily combine both transforma-tions. The S-parameter data of a classic VNA, forexample, is rarely studied in a time-domain represen-tation of voltages and currents. LSNA technologyenables one to easily study the data in either domain(time or frequency) and in either representation(waves or voltage-current). As will be demonstrated

later, such a combination is necessary to get insightinto large-signal behavior of nonlinear components.

Waves Versus Voltage CurrentLet us start with the transformation between a waveformalism and a voltage-current representation. S-parameters are defined as a ratio between travelingvoltage waves, where the letter A refers to the incidentwaves and the letter B refers to the scattered waves.The use of wave quantities is native to classic VNAs,which measure the ratio of the incident and scatteredwaves. It is important to note that, under the superpo-sition assumption required for classic vector networkanalysis, the stimulus frequency will be the only fre-quency present, there will be no new frequencies (e.g.,harmonics or intermodulation distortion products) pre-sent. An LSNA instrument will measure A and B wavesat the terminals of a device under large-signal operat-ing conditions. Since a ratio is not sufficient to describenonlinear behavior, an LSNA instrument will measurethe absolute values of all wave quantities (includingharmonics and intermodulation distortion products)and not merely their ratios. On the simulator side, apower amplifier designer will often simulate and

LSNA Metrology

Building an LSNA instrument is one thing, making surethat the measurements are accurate and that thecalibration procedures are traceable to national standardsis yet another technological challenge. Inevitably, all ofthe LSNA microwave and RF hardware componentsintroduce significant linear distortions. These distortionshave to be eliminated by using calibration proceduresthat are extended versions of the calibration proceduresused for classic VNAs.

Calibration Procedure BasicsBefore continuing with the calibration aspects, it is usefulto consider the hardware schematic of an LSNA (seeFigure 2). During an experiment, you want to measure thephases and amplitudes of a discrete set of spectral com-ponents at the DUT signal ports. Let us call these quanti-ties the DUT quantities and denote them by D1 and D2,where the number refers to the test port. Unfortunately,we do not have direct access to these quantities. The onlyinformation we can get are the uncalibrated measured val-ues. These are called the raw quantities and are denotedby R1 and R2. All calibration procedures are based on anerror model. The main assumption we will use for ourerror model is that there exists a linear relationshipbetween the raw measured spectral components and theactual spectral components at the DUT signal ports. Inorder for this assumption to be valid, it is necessary toavoid any nonlinear distortions in the LSNA signal receiver

itself. Programmable attenuators are used for that purpose.By limiting the peak amplitude at the input of the frequen-cy convertor to about 100 mV, a spurious free dynamicrange of 65 dB is achieved. The linear error model can bedescribed by the following matrix equation:

Note that h refers to the harmonic index of a tone. TheRF correction is described by a set of 16-element matrices(one for each harmonic). Note that eight zeros are present.This corresponds to the assumption that there is no cross-coupling between Port 1 and Port 2 of the instrument. Thegoal of the RF calibration is to determine the elements ofthis matrix. This goal is achieved in three steps: a classicalVNA calibration to determine the seven coefficients ph, qh,rh, sh, th, uh and vh; an amplitude calibration to determineKh; and a harmonic phase calibration to determine the unitlength phasors e jφh .

For a coaxial DUT, the amplitude and the harmonicphase calibration are performed by connecting a powersensor and a harmonic phase reference generator(HPR) directly to the LSNA test ports. A HPR is a sourcethat produces a calibrated harmonic spectrum of com-

aD1h

bD1h

aD2h

bD2h

=Khe jφh

1 ph 0 0qh rh 0 00 0 sh th0 0 uh Vh

aR1h

bR1h

aR2h

bR2h

. (3)

December 2005 85

analyze the voltage and current waveforms at the ter-minals of a power transistor. An LSNA instrumentmust be able to convert the measured wave (noted AB)behavior of a component into a voltage-current (notedVI) representation. A good understanding of the devicebehavior is often only possible by taking a look at bothkinds of signal representations. The relationshipsbetween the VI and the AB quantities are given by sim-ple algebraic equations:

A = V + ZcI2

B = V − ZcI2

(1)

and

V = A + B I = A − BZc

. (2)

Zc is the characteristic impedance associated with thewave formalism. By convention, most of todays instru-mentation uses a Zc equal to 50 �. These simple equa-tions are based on the famous telegraphers equation.Please be aware that other wave definitions are in use.Valuable information on the scientific foundation ofwave formalisms can be found in [1].

Time Domain Versus Frequency DomainThe second transformation relates to the time domainand the frequency domain. A power amplifier designerwill typically want to visualize voltage and currentwaveforms as they appear at the terminals of a powertransistor in the time domain. An LSNA instrumenttypically measures the spectral components of the Aand B waves. Since the A and B waves contain har-monics and intermodulation distortion (IMD) prod-ucts, in addition to the fundamental frequency, thetransformation between the time and the frequencydomain can be performed only if the amplitude of allspectral components and the cross-frequency phaserelationship between the spectral components are mea-sured. Cross-frequency means that it is not sufficient todetermine the phase relationship between spectralcomponents with equal frequencies, as it is the casewith S-parameters, but one needs to measure the phaserelationship between components that have differentfrequencies (like a fundamental and harmonics or likedifferent intermodulation products). Spectral compo-nent phase information is undoubtedly the mostimportant ingredient in going from classic approachesto an LSNA approach since it enables measurements to

ponents. The power sensor and HPR can be consid-ered additional connectorized calibration elements,just like a load, a short, or an open calibration ele-ment. Note that, unlike the HPR calibration, the powercalibration is performed by applying one tone at atime since the power sensor is not frequency selec-tive. The HPR and power sensor measurements can-not directly be done, however, for an on-wafer mea-surement since both the HPR and the power sensorhave a coaxial output connector. The solution to thisproblem is based on the use of the reciprocity princi-ple between the probe tip and the generator inputconnector of the test set. The harmonic phase refer-ence generator and the power sensor are connectedto this coaxial input port, and the final measured char-acteristic can be transformed to the probe tip basedon the fact that the test set is reciprocal. This transfor-mation does require an additional load-open-short cal-ibration at the generator input connector. During all ofthe above measurements, the probe tips are connect-ed to a line calibration element.

The Harmonic Phase Reference GeneratorAn HPR generates fundamental harmonics with a sta-ble and very well characterized phase relationship.The main components of the HPR are a power ampli-fier, a step-recovery diode, and a pulse-sharpeningnonlinear transmission line (together with cables and

padding attenuators). Developing a traceable calibra-tion procedure for the HPR was the most challengingaspect of the LSNA calibration.

The HPR is characterized by using a broadband sam-pling oscilloscope. A discrete Fourier transformationreturns the phase relationship between all of the relevantharmonics (up to 50 GHz). Note that the HPR needs tobe characterized across a whole range of fundamentalfrequencies with enough resolution to allow interpolation.Our HPR covered one octave of fundamental frequen-cies, from 600 up to 1,200 MHz. The HPR is a stabledevice, such that it is sufficient to recalibrate it with anoscilloscope measurement about once every year.

The accuracy of the HPR characterization is, in turn,determined by the accuracy of the sampling oscillo-scope measurement. The sampling oscilloscope intro-duces linear distortions that can be characterized bythe so-called nose-to-nose calibration procedure,which was invented by Rush at Hewlett-Packard in1989 [4]. I performed extensive research on the accu-racy of the nose-to-nose calibration procedure as partof my Ph.D. work and later, at the National Institute ofStandards and Technology (NIST), Remley and hercolleagues [23] performed it. The research at the NISTresulted in a very accurate harmonic phase standardbased on a broadband (110 GHz) electro-optic sam-pler setup [5].

86 December 2005

be represented in the time as well as frequency domain.Because of this fact, time domain gurus often refer to anLSNA instrument as a “calibrated oscilloscope withvoltage and current probe.”

Envelope DomainNext to the pure time and frequency domain, manyengineers represent their signal as a spectrum thatvaries over time. This kind of representation is useful ifthe signals are modulated carriers, as is the case formost wireless telecommunication applications. Signalsresulting from both analog and digital modulationschemes can be represented as so called complexenvelopes. A complex envelope is a complex time func-tion where the real part represents the in-phase compo-nent and the imaginary component represents thequadrature component of a modulated carrier. The in-phase component is usually called I, and the quadra-ture component is usually called Q. This IQ or enveloperepresentation is not only used by engineers but is alsoused inside advanced envelope domain simulators.The concept is also native to instrumentation like the

vector signal analyzers (VSAs) and digital modulationsynthesizers. An LSNA instrument is able to constructthese envelope representations of the measured A and B waves and can, as such, also perform typical VSAmeasurements.

LSNA HardwareThe combination of the simple yet powerful conceptsof the wave versus voltage-current formalism and thetime to frequency domain transformation are not pre-sent with a single classic instrument like a VNA, VSA,oscilloscope, or spectrum analyzer. In many cases,many separate measurements are being performedusing several classic microwave instruments. A VNA isused to characterize the small-signal matching charac-teristics, a VSA is used to characterize the distortion ofthe complex envelope of a signal, a spectrum analyzeris used to take a look at the harmonic power, and anoscilloscope is used to look at time-domain wave-forms. In contrast to classic instrumentation, an LSNAinstrument provides one measurement tool that mea-sures all of the aforementioned characteristics in a con-sistent and accurate way. In the following, I explain thearchitecture of the large-signal network analyzerinstrument as my colleagues and I developed it whileworking for the Hewlett-Packard company (later tobecome Agilent Technologies). A picture of such a sys-tem is shown in Figure 1.

LSNA Hardware Architecture OverviewThe hardware architecture of the LSNA, illustrated inFigure 2, is relatively simple. Four couplers are used forsensing the spectral components of the incident andscattered voltage waves at both device-under-test(DUT) signal ports. The sensed signals are attenuatedto an acceptable level before being sent to the inputchannels of a four-channel broadband RF–IF converter.This RF-IF converter operates based on the harmonicsampling principle (to be described in more detail inthe next section) and converts all of the spectral compo-nents coherently to a lower frequency copy (below4 MHz). The resulting IF signals are digitized by a setof four high-performance analog-to-digital converters(ADCs). A computer does all the processing needed totransform the calibrated data into one of the three for-mats (AB or VI, time, frequency, or envelope domain).

The primary specifications of the LSNA prototype,as it was developed by our research group, are as fol-lows: the calibrated RF frequency range equals 600 MHz to 50 GHz, the maximum RF power equals 10 W, and the maximum bandwidth of the modulatedsignal equals 8 MHz. The repetition frequency of themodulation is typically a few kilohertz. Note that syn-thesizers and tuners for the signal generation can be(and usually are) added externally. They are not con-sidered part of the LSNA. Although not shown here forreasons of simplicity, dc bias circuitry is also present.Figure 2. LSNA architecture.

Tuner

Attenuators

...

10 MHz A-to-D

RF-IF Converter

DUT Quantities

Raw Quantities

haR1

haD2

hbD2haD1

hbD1

Computer R1hb R2

ha R2hb

Figure 1. Picture of the LSNA at IRCOM (France).

December 2005 87

Harmonic SamplingThe four-channel RF–IF sampling frequency converter,illustrated in Figure 3, is the core of the LSNA. It isbased on broadband sampler technology. One digitalsynthesizer drives four sampler switches at a rate closeto 20 MHz. This rate is called the local oscillator (LO) fre-quency ( fLO). Each time the switch closes for a durationof about 10 ps, it samples a little bit of charge and sendsit to a low-pass filter. When the synthesizer frequencyis properly chosen, a low-frequency copy of all inputspectral components at the output of the filter is found.

The harmonic sampling idea is illustrated inFigure 4. It corresponds to the case where one has afundamental tone and harmonics. Assume that weneed to measure a 1-GHz fundamental together withits second and third harmonic. We choose a LO signalhaving a frequency of 19.98 MHz. The 50th harmonicof the LO will have a frequency of 999 MHz. This com-ponent mixes with the fundamental and results in a 1-MHz mixing product at the output of the converter.The 100th harmonic of the LO has a frequency of1998 MHz, mixes with the second harmonic at 2 GHz,and results in a 2-MHz mixing product. The 150th har-monic of the LO has a frequency of 2,997 MHz, mixeswith the third harmonic at 3 GHz, and results in a 3-MHz mixing product. The final result at the output ofthe converter is a 1-MHz fundamental together withits second and third harmonic. This is actually a low-frequency copy of the high-frequency RF signal. Thissignal is then digitized, and the values of the spectralcomponents are extracted by applying a discreteFourier transformation.

Although based on the same principles, the processfor a modulated signal is more involved. After theRF–IF conversion, all harmonics and modulationtones can be found back in the IF channel, ready fordigitizing and processing. The technique is describedin detail in [3].

ApplicationsThe following gives an overview of the many applica-tions based on the LSNA concepts. Practical examplesare presented covering a whole spectrum of applica-tions, including transistor characterization, loadpullwaveform engineering, frequency-domain black-boxmodeling, state-space black-box modeling techniques,active digital-signal integrity measurements, and spec-tral regrowth measurements.

Waveform MeasurementsAs explained above, the LSNA measures both theamplitude and the phase of all significant harmonics ofboth the incident and the scattered traveling voltagewaves. Applying an inverse Fourier transform resultsin corresponding time-domain waveforms. The time-domain traveling voltage waves can be transformedinto a current and a voltage waveform. In the time-

domain current-voltage visualization mode, the LSNAemulates a broadband oscilloscope having calibratedvoltage and current probes. Such information can bevery valuable to get a unique insight into issues (e.g.,transistor reliability) that are often related to hard non-linear phenomena.

In Figure 5, we plot the voltage and current timedomain waveforms as they appear at the gate anddrain of a microwave field-effect transistor (FET).These measurements are performed while applying anexcitation signal of 1 GHz at the gate. The signal ampli-tude is increased until we see a so-called breakdowncurrent. It shows up as a negative peak (20 mA) for thegate current and as an equal amplitude positive peak at

Figure 3. Sampling frequency convertor.

Figure 4. Sampling a fundamental tone with harmonics.

1

2

3

4

1

2

3

4

RF IF

fLO

Freq. (GHz)1 2 3

50 fLO 100 fLO 150 fLO

Freq. (MHz)1 2 3

RF

IF

fLO =19.98 MHz = (1GHz-1MHz)/50

Unlike cellophane or penicillin, LSNAtechnology was not invented byaccident or by a flash of genius.

88 December 2005

the drain. Using the LSNA, we can actually witness thebreakdown current that flows from the drain towardsthe gate. This kind of operating condition deterioratesthe transistor and is a typical cause of transistor failure.To our knowledge, these measurements are the firstever performed of breakdown current under large-sig-nal RF excitation [6].

Another example is the usage of a high-electronmobility transistor (HEMT) transistor in a resistivemixer configuration [7]. The schematic of the mixer isshown in Figure 6. A large LO signal is driving the gate.This causes the drain of the transistor to behave as atime-dependent switch. The switch is toggled at therate of the LO (in our example, 3 GHz). Any incidentvoltage wave at the drain (in our example, 4 GHz)experiences a fast time-varying reflection coefficient(toggling at a 3 GHz rate). The voltage wave reflectedfrom the drain will, thus, be equal to the incident wavemultiplied by the time-varying reflection coefficient.The scattered wave contains the primary mixing prod-ucts (1 and 7 GHz) of the LO and the incident RF sig-nal. The working principle is nicely demonstrated bylooking at the measured time-domain representation ofthe voltage waves. Figure 7 illustrates how the incidentwave (RF) and reflected wave (IF) are in phase or inopposite phase, depending on the instantaneous ampli-tude of the LO. For a high LO, the RF experiences a lowdrain impedance and behaves like a short correspond-ing to a reflection coefficient close to −1 (reflection inopposite phase); for a low LO, the RF experiences ahigh drain impedance and behaves like an open corre-sponding to a reflection coefficient close to +1 (reflec-tion in phase). Note that a 1- and a 7-GHz componentin the resulting reflected wave (IF) can be clearly dis-tinguished. For clarity, the amplitude of the LO signalon Figure 7 has been divided by two.

Another interesting application is the calibrated on-wafer measurement of high-speed digital signals [8].The classic technique for performing such measure-ments makes use of a microwave oscilloscope. The

Figure 5. Breakdown currents.

5.8Vds (V)

Ids (mA)Igs (mA)

Vgs (V)

5.6

5.4

5.2

5.0

4.8

15

10

5

0

0 1 2

0 1 2

0

−8

0

−5

−10

−15

−6

−4

−2

1 2

0 1 2

(All Time Scales in ns)

1 GHz

IBreakdown

Figure 6. Resistive mixer schematic.

Figure 7. Resistive mixer time domain waveforms.

(No Drain Bias Applied)

LO3 GHz

RF4 GHz

IF1 GHz7 GHz

• • •

LO/2

RF

IF

0 0.2 0.4 0.6

Time (ns)

0.8 1

0.15

Vol

tage

Wav

es (

V)

0.1

0.05

0

−0.05−0.1

−0.15

December 2005 89

cables, connectors, and probes that are used to connectthe oscilloscope to the wafer introduce significant dis-tortions in the measurement of the eye-diagrams ofdigital signals. Performing similar measurements withan LSNA results in fully error-corrected waveforms allthe way up to the tip of the probe. In Figure 8, it is clearthat the oscilloscope data lacks certain details like offsetand overshoot when compared to thecalibrated LSNA data. This is due to dis-persion of the oscilloscope cables as wellas to the presence of jitter on the time-base of the oscilloscope. Both of thesedistortions are also present with theLSNA frequency converter but are com-pensated with VNA-like accuracy bythe LSNA calibration procedure.

A last application of waveform mea-surements is power amplifier design byapplying waveform engineering [9].With waveform engineering, differentharmonic impedances at the DUT signalports are applied, and specific perfor-mance parameters, like power-added-efficiency (PAE), are optimized by look-ing at the voltage and current wave-forms. This is illustrated in Figure 9.Traditionally, current and voltage wave-forms are not readily available frommeasurements, so the method is usually

applied only in a simulator. The problem with usingsimulators is that the accuracy of the final result willdepend heavily on the quality of the large-signal tran-sistor model.

Connecting a harmonic tuner to the test set of theLSNA makes it possible to apply the method in real life,without the need of a simulator or transistor model.

Figure 8. Measuring digital signals.

Oscilloscope Data

LSNA Data

See Overshoot

See Ripple

See Offest

See Pattern-Dependent Jitter

0 25 50 100 125 15075Time (ps)

Calibrated Eye Measurement on Wafer

Figure 9. Harmonic loadpull voltage and current waveforms.

Vgs

(V

)

−10−9−8−7−6−5−4−3−2−10

0 0.2 0.4 0.6

Time (ns)

Gate Voltage

0.8 1 1.2

12

Vds

(V

)

10

8

6

4

2

00 0.2 0.4 0.6

Time (ns)

0.8 1 1.2

Drain Voltage

0.04

0.03

I gs

(Am

ps)

0.02

0.01

0

−0.03

−0.02

−0.01

−0.040 0.2 0.4 0.6

Time (ns)

Gate Current

0.8 1 1.2

0.08

I ds

(Am

ps)

0.06

0.04

0.02

0

−0.02

−0.040 0.2 0.4 0.6

Time (ns)

Drain Current

0.8 1 1.2

90 December 2005

Using the combination of an LSNA with a tuner andusing the waveform engineering technique, Nébus etal. could demonstrate an increase of the PAE of a metalsemiconductor FET (MESFET) Class F amplifier from50% to a record 84% [9].

Transistor ModelingNext, we will show how the LSNA data can be used inadvanced physical transistor modeling. By physical

models, we refer to electrical circuit schematics con-taining nonlinear capacitors, voltage-controlled cur-rent sources, and resistors. All nonlinear elements aretypically defined by means of analytical functions,which are often derived from the physical description

of the transistor (geometry and semiconductordoping profile). These models can have more than100 parameters that need to be determined.Examples of such models are BSIM3, Chalmers,and Materka. The approach of optimizing existingphysical models using the LSNA is pretty straight-

forward [10]. First, a set of LSNA experiments is per-formed that covers the operating range of the transis-tor. This data is imported into a simulator. The mea-sured incident voltage waves are applied to the modelin the simulator. The model parameters are thenfound by tuning them such that the differencebetween the measured and modeled scattered voltagewaves is minimized. Note that one can often use built-in optimizers for this purpose. As with all nonlinearoptimizations, it is necessary to have reasonable start-ing values (these can be given by a simplified versionof the classical approaches).

A recent and upcoming transistor-modelingapproach is based on state-space models [11]. Thesemodels are black-box time-domain models that are typ-ically used in physics for describing all kinds of non-linear dynamical processes. A first microwave transis-tor model of this kind is the so-called Root-model. It isa simple version of the more general state-space func-tion model. It was originally invented by David Root.For a Root-model, the model parameter extraction is

Figure 11. Measured time domain waveforms.

−0.6−0.4

−0.2

00.2

(V)

0.40.6

−1

−2

0

0 0.1 0.2 0.3

Transmitted Signal (b2)

Incident Signal (a1)

0.4

Normalized Time

0.5

0.40.30.20.10

Normalized Time

0.5

1

2

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LSNA technology is empowered by a unique combination of twomathematical transformations.

December 2005 91

based on bias-dependent S-parameter measurements.In a state-space function approach, one assumes thatthe port currents of a device are a general nonlinearfunction of a limited set of state variables. The statevariables are the port voltages, the first- and higher-order derivatives of the port voltages, and the first- andhigher-order derivatives of the port currents. One canidentify the relevant set of state variables and the func-tions themselves by performing sufficient LSNA exper-iments and by processing the gathered data. The state-space models are technology independent and can beapplied to all nonlinear microwave circuits as long asthey do not contain distributed elements. Examples aresingle transistors or small RFICs. As soon as distributedelements are present, the set of relevant state-spacevariables becomes excessively large (in theory, an infi-nite number) and the state-space functions can nolonger be practically identified. One of the main chal-lenges is the experiment design. In order to have agood fit to the state-space functions, it is necessary tohave a dense coverage of the state-space over a mean-ingful range. An original solution to this was devel-oped by Schreurs [12]. She simultaneously appliedexcitation tones having different frequencies to the gateand the drain of a microwave transistor. The two fre-quencies used were 4.2 and 4.8 GHz. Note that both fre-quencies are integer multiples of 600 MHz, whichimplies that a single-frequency grid of 600 MHz is suf-ficient for the LSNA data acquisition. Figure 10 illus-trates the resulting coverage of the state-space corre-sponding to two measurements (two different bias set-tings are applied). The two-tone experiment allowsdense coverage of the gate and drain voltage state-space over a meaningful range. It might be interestingto note that the two-tone technique was actuallyinvented several decades ago by mechanical engineersdealing with the characterization of nonlinear mechan-ical structures.

Scattering FunctionsWhile the LSNA can improvemeasurement capability forthe traditional measurementsdescribed here, I have nodoubt that the breakthroughof LSNA technology will comefrom the ability to measurescattering functions [13], [24],[25]. This is a black-box, fre-quency-domain modelingtechnique. The approach is anextension of S-parameterstowards large-signal behavior.Ideally, one connects a DUT toan LSNA instrument, and amodel is automatically ex-tracted that accurately de-

scribes all kinds of nonlinear behavior, such as ampli-tude and phase of harmonics, compression character-istics, AM-PM, spectral regrowth, amplitude-depen-dent input, and output match. A simple measurement

example is given in Figure 11. It shows the measuredincident and transmitted wave at the input and,respectively, output terminals of an RFIC. The appliedsignal was carefully designed to have characteristicsthat are very similar to the signals that are to be usedfor the final application [in this case, a code division,multiple-access (CDMA) signal with a 1.9-GHz carri-er]. Note the use of the normalized time scale. In orderto visualize the distortion throughout the envelope,the envelope time scale and the actual RF carrier timescale differ. This is done to avoid that the RF oscilla-tions would look like a black blur of ink. The RF sig-nal would oscillate about 1,000 times before therewould be any noticeable change in the envelope. Theratio between the modulation period and the RF car-rier period is artificially lowered for the purpose ofvisualization. A black-box frequency-domain behav-ioral model was then extracted based on the measureddata. Figure 12 shows the spectral regrowth that ispredicted by the model and the spectral regrowth as itwas actually measured. As you can see, the corre-spondence between the modelled and the measuredvalue is very good.

The real beauty of the approach is that it provides muchmore than just plots of the aforementioned characteristics.The scattering functions can be used in a CAD environ-ment and describe the interdependency between the non-linear characteristics. As with S-parameters, the scattering

Figure 12. Modeled and measured spectral regrowth.

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An LSNA instrument must be ableto convert the measured wave

behavior of a component into avoltage-current representation.

92 December 2005

functions approach works both ways. It not only allowsfor a really accurate automatically extracted CADmodel, but it also allows you to experimentally verifythe large-signal behavior of a component once it hasbeen designed. A nice characteristic of the scatteringfunctions is that they reduce to classic S-parameters forsmall input amplitudes. As such, the measurement capa-bility of an LSNA instrument equipped with the meansto measure the scattering functions performs a supersetof the measurements that are possible with a classicVNA. As such, LSNA instruments will gradually replaceall VNAs that are used today to characterize semicon-ductor devices, easily a multimillion dollar business.

ConclusionsThe dream of accurate and complete large-signal char-acterization of components under realistic operatingconditions is made real. The only limit to the scope ofapplications is the imagination of the R&D people whohave access to this measurement capability. LSNAinstrumentation and scattering functions are powerfulconcepts. The associated technology is emerging andwill revolutionize the way microwave semiconductorcomponents will be characterized in the future.

AcknowledgmentsI would like to thank Larry Dunleavy of Modelithicsfor reviewing the manuscript and the ARFTG organi-zation (www.arftg.org) for the continued support to theLSNA community. The copyright of the figures andpart of the text is with Agilent Technologies (1998), thefigures and text are used with permission.

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