NEWCOM WPR3 Meeting – 6/9/04 Nonlinearity characterization and modelling Giovanni Ghione Dipartimento di Elettronica Politecnico di Torino Microwave &

NEWCOM WPR3 Meeting – 6/9/04

Nonlinearity characterization and modelling

Giovanni GhioneDipartimento di ElettronicaPolitecnico di TorinoMicrowave & RF electronics group


Agenda

A glimpse on nonlinear models Physics-based device-level models Equivalent circuit & black-box device-level models Vintage behavioral models: power series, Volterra,

envelope Advanced models: time-domain, frequency-domain,

envelope Characterization techniques (mainly loadpull…) Aknowledgements


Device models: from physical to behavioral

From: D.Root et al., IMS2004 WME-4


Physics-based nonlinear modeling

Based on the solution of transport + Poisson equations on device volume

Mainly single-device, mixed-mode intensiveOften time-domain, Harmonic Balance LS

simulation demonstrated but demanding (>10000 unknowns) order reduction techniques?

Potentially accurate, but NL operation can be a numerical killer (breakdown, direct junction conduction…)


Example: LDMOS PA simulation

From: Troyanovsky et al, SISPAD 1997


Circuit-oriented NL modelling

Equivalent circuit NL models: Extensions of DC + small signal models with NL components Ad hoc topologies for device classes: BJT, HBT, MESFETs,

HEMTs, MOS, LDMOS… Almost endless variety of topologies and component models

from the shelf, many models proprietary Empirical, semi-empirical, physics-based analytical

varieties.

Pros: numerically efficient, accurate enough for a given technology after much effort and tweaking

Cons: not a general-purpose strategy, low-frequency dispersion (memory) effect modelling difficult


NL equivalent circuit examples

Bipolar:BJT: Ebers-Moll, Gummel-PoonHBT: Modified GP, MEXTRAM…

FET:MOS: SPICE models, BSIM models…MESFET: Curtice, Statz, Materka, TOM…HEMT: Chalmers, COBRA…


Example: the Curtice MESFET model


Example: the HBT MEXTRAM model


Black-box device-level modelling

Black-box models for circuit NL components:Look-up-table, interpolatory (e.g. Root)Static Neural Network based

Global black-box (“grey-box”) device-level (?):The Nonlinear Integral Model (University of

Bologna) based on dynamic Volterra expansion + parasitic extraction

Potentially accurate, but computationally intensive


Non-quasi static effects

Device level: low-frequency dispersion due to:Trapping effects, surfaces, interfacesThermal effects

Amplifier level:Bias effect (lowpass behavior of bias tees)Thermal effect

Impact on device modelling pulsed DC and SS measurements


Pulsed IV characteristics

Investigation of the device behaviour outside the SOA region

Pulsed measurement for exploiting thermal and traps effectsDifferent QP with the same dissipated powerPoint out flaws of the fabbrication processes (e.g.

passivation faults, uncompensated deep traps)Allow the identification of the dispersive model

contributions


Pulsed IV: FET example


System-level (behavioral) NL models

Classical & textbook results:Power and Volterra series (wideband) models,

frequency or time-domainEnvelope (narrowband) static models descriptive

function

A sampler of more innovative techniques:Dynamic time-domain modelsDynamic neural network modelsDynamic f-domain models scattering functionsAdvanced envelope models


Recalling a few basics

PA single-tone testPA two-tone testPA modulated signal testIntermodulation products, ACPR…


Single-tone PA test

1 dB compression point

3rd harmonics output intercept

Output saturation power

PA


Two-tone PA test

Rationale: two-tone operation “simulates” narrowband operation on a continuous band f1 - f2

PA

CIM3


Two-tone Pin-Pout

Pin1=Pin2, dBm

Pout(f0), dBm

IMP3, dBm

1 dB

IMP3 Input Intercept Point, IIP3

IMP3 Output Intercept Point

OIP3


Modulated signal test & ACPR

fc fc+30 kHz fc+60 kHzfc-60 kHz fc-30 kHz

Pow

er

spec

tral

de

nsity

- d

Bm

/Hz

-20

0

-40

-60

-80

Main channelAdj. channel

Inputsignal

Outputsignal

Adj. channel

Spectralregrowth


Class AABC two-tone test

Fager et al, IEEE JOURNAL OF SOLID-STATE CIRCUITS, VOL. 39, NO. 1, JANUARY 2004, p. 24


Power series (PS) model Strictly speaking an IO model for a memoryless NL

system, often cascaded with a linear system with memory:

u(t)s(t) w(t)

Linear system

with memory

Nonlinear system

without memory

( ) ( ) ( )U H S


Active device PS cascading

+eg(t)

Rg LG RG

iD = f(v*)CGS

RL

v*

VGGVDD

s(t) u(t) w(t)

FET transfer curve


PS output with multi-tone excitation

Assume a multi-tone frequency-domain excitation:

Output:


Single- and two-tone PS test

The PS approach correctly yields the small-signal harmonic and IMPn slope in small-signal, class A operation

It also gives an estimate of gain compressionThe two-tone output with equal tone power

yields:Same IMPn power for right & left-hand side linesIMPn power independent on line spacing ( can be

artificially introduced through H)


Single- and two-tone gain compression

The 2-tone (modulated signal) Pin-Pout is not exactly the same as the single-tone

While the AM-AM curve is different, the AM-PM is almost the same (Leke & Kenney, MTT-S 96, TH2B-8)

Can be shown already with a PS model, assume:

then the output power is: Single-tone Two-tone Two-tone with IMP3

y b0 b1x b2x2 b3x3

P0 b12P in 3. 0b1b3P in

2 2. 25b32P in

3


2 5. 0625b32P in

3


2 5. 625b32P in

3


Example

10 12 14 16 18 20 22 24 26 28 3028

30

32

34

36

38

40

42

44

46

48

Input power, dBm

Out

put

pow

er,

dBm

Single-tone testTwo-tone testTwo-tone including IMP3

b1=10, b

3=-1


Volterra series approach

In frequency domain, generalization of the PS approach:

Exact representation, but unsuited to true LS regime or strongly NL system due to the difficulty of characterizing high-order kernels

The time-domain version is a generalization of the impulse response

1

1 2

1 2

1 2

nn

1

( )

( )2

( , , )e

n

n

q q qn

n

Q Q QN

q qn q Q q Q q Q

j t

n q q q

ay t X X

H


Envelope modeling

The PS and Volterra models are general and wideband, i.e. they hold for any excitation often in analog RF system the excitation is DC + a narrowband modulated signal

(Complex) envelope representation of input and output signals, envelope slowly varying vs. carrier:

Static envelope model (G complex “descriptive function”):

( ) Re ( )exp( ) ( ) cos ( )

( ) Re ( )exp( ) ( ) cos ( )

c c

c c

x t x t j t x t t x t

y t y t j t y t t y t

( ) ( ) ( )y t G x t x t


AM/AM and AM/PM distortion curves

-20 -15 -10 -5 0 5 10 15 202

4

6

8

10

12

14

94

96

98

100

102

104

106

108

110

Available input power, dBm

G

arg

G


Static envelope models features

No information on harmonics and out-of-band spurs bandpass filtering implied, unsuited for circuit-level modeling

G can be identified from single-tone measurements but better fitted on two-tone measurements (see caveat on fitting function Loyka IEEE Trans. VT49, p.1982)

IM3 intrinsically symmetrical and independent on tone spacing no memory (non quasi-static) effects modeled

Poor ACPR modeling in many realistic cases, performances deteriorate increasing channel bandwidth


Some “novel” approaches

Modeling strategies have ups and downs in time, the last not necessarily the best one

Recent trends: Revival on dynamic state-variable black-box (behavioral)

models based on general system identification techniques Steady interest and progress in neural network models Progress in exploiting multi-frequency NL measurement

tools Search for better system-level envelope models, also on the

basis of classical methods revisited and revamped (e.g. Volterra)


Nonlinear Time Series (NTS) model

Idea: identify a standard state-variable model on the basis of measured input and output time series [Root et al., Agilent]:

State equation ( , )

Output equation ( , )

u

y g u

x f x

x

State equation ( , )

Output equation ( , )

u

y g u

x f x

x

"Feedback" model

( , , ,..., , , ...)y f y y y u u u

"Feedback" model

( , , ,..., , , ...)y f y y y u u u


Model identification: how?

NL model identification amounts to a nonlinear inverse scattering problem

Several theoretical methods available from dynamic system theory (Whitney embedding theorem, Takens’ theorem) which allow in principle to identify f as a smooth function

Once f is identified, the implementation in commercial simulators is straightforward

Problems: system identification in the presence of noisy data identification when the state space is large building suitable sets of I/O data providing a suitable numerical approximation to f

See D.Root et al, IMS2003, paper WE2B-2 and references


Dynamic Neural Network (DNN) model

Neural networks can provide an alternative to identify the NL dynamical system

In DNNs (see Ku et al, MTT Trans. Dec. 2002, p. 2769) the NN is trained with data sequences including the input / output and their time derivatives

Once trained the NN defines a “feedback” dynamic model and simply “is” the dynamic system

Very promising technique in terms of accuracy, CPU effectiveness and generality; easy implementation in circuit simulators.


DNN result example


F-domain dynamic behavioral models

The availability of Large-signal Network Analyzers (LSNA) have fostered the development of generalizations of the scattering parameter approach:


Describing (scattering) functions

NL relationship between power wave harmonics in LS steady state (ij port & harmonics index) [Verspecht, IMS2003]:


Relationship with S parameters Describing functions reduce to multifrequency S-parameters for a linear device (lowercase used for PW):

however, simplifications can be made (scattering functions model) if a11 is the only “large” component superposition can be applied to the other terms.

b11 S11 1a11 S12 1a21

b21 S21 1a11 S22 1a21

b1N S11 Na1N S12 Na2N

b2N S21 Na1N S22 Na2N

b11 F11a11,a21, a1N,a2N

b21 F21a11,a21, a1N,a2N

b1N F1Na11,a21, a1N,a2N

b2N F2Na11,a21, a1N,a2N


Frequency superposition

aj,kN aj,k

N exp ik arga1,1

bj,kN bj,k

N exp ik arga1,1

a1,1N |a1,1 |

Normalization:


Scattering function model

Introducing phase normalized variables one has the relationship [Verspecht, IMS2003]:

b1,1N S11,11a1,1

N a1,1N S12,11a1,1

N a2,1N S12,11

a1,1N a2,1

N j 1,2

k 1

S1j,1ka1,1N aj,k

N S1j,1k a1,1

N aj,kN

b2,1N S21,11a1,1

N a1,1N S22,11a1,1

N a2,1N S22,11

a1,1N a2,1

N j 1,2

k 1

S2j,1ka1,1N aj,k

N S2j,1k a1,1

N aj,kN

b1,NN S11,N1a1,1

N a1,1N S12,N1a1,1

N a2,1N S12,N1

a1,1N a2,1

N j 1,2

k 1

S1j,Nka1,1N aj,k

N S1j,Nk a1,1

N aj,kN

b2,NN S21,N1a1,1

N a1,1N S22,N1a1,1

N a2,1N S22,N1

a1,1N a2,1

N j 1,2

k 1

S2j,Nka1,1N aj,k

N S2j,Nk a1,1

N aj,kN


Scattering functions features

Also called large-signal scattering parametersDirectly measurable through a VNAEffective in providing a model for a HB

environment and for strongly nonlinear components

Can be used at a circuit level, providing interaction with higher harmonics; not an envelope model


Envelope LS scattering parameters

Two-port extension of descriptive function concept, same features and limitations:

1 11 1 2 1 12 1 2 2

2 21 1 2 1 22 1 2 2

( ) ( ) exp( ) , ( ) ( ) exp( )

( ) ( , ) ( ) ( , ) ( )

( ) ( , ) ( ) ( , ) ( )

i i c i i ca t a t j t b t b t j t

b t S a a a t S a a a t

b t S a a a t S a a a t


Envelope models

Envelope models consider (narrowband) modulated signal “time varying spectrum” signals

Model purpose: relating input and output signal envelopes Well suited to envelope circuit simulation techniques


Limitations of static envelope models

IMD simmetry & independence on tone spacing Both properties are not observed in practice owing to low-

frequency dispersion (memory) effects thermal, trap related, bias related (Pollard et al, MTTS-96, paper TH2B-5):


Improving static models: simple solutions

Add a state-variable Z dependence (temperature, bias) [Asbeck IMS2002, p.135]; Z in turn depends (linearly or not) on the input variable:

( ) ( ) , ( ) ( )y t G x t Z t x t


High-frequency dispersion

While low frequency (long memory) effects arise due to heating etc., also high-frequency (short memory) phenomena can arise leading to high-frequency dispersion

This amount to an output sensitivity when the modulation bandwidth increases e.g. in next generation systems

General (usually, but not only) Volterra-based approaches have been suggested to overcome the static limitation


Examples of low- and high-frequency dispersion

LDMOS amplifier, from Ngoya et al., BMAS 2003


More general approaches

In general, the descriptive function can be turned into a descriptive functional:

Volterra-based solutions, with slight variations: Derivation from Dynamic Volterra Series [Ngoya et al MTT-

S Digest 2000] Nonlinear Impulse Response Transient (NIRT) envelope

model [Soury et al. MTT-S Digest 2002 paper WE2E-1] Extracting memory effects from modified Volterra series

[Filicori et al., IEEE CAS-49, p.1118 and IEEE Instr. & Meas. V.53 p.341]

( ) ( )y t x t


Dynamic Volterra in a nutshell

1st step: from the conventional Volterra series extract a modified series in the instantaneous deviations x(t)-x(t-); truncate the series to the first term; one has:

1 ˆ( ) ( , ) exp( )2

ˆ ( , ) ( , ) ( ,

( ) ( ) ( )

( ) ( )

)

)

(

( 0)

DCy H jx t x ty t t d

H H

X

x t x t x tH

DC response

small-signal response

line

arity

frequency

am

plit

ude

memory

ss regime

DC (LF) regime

Volterra

Dynamic Volterra


Dynamic Volterra – cntd.

2nd step: introduce an envelope representation of input and output into the dynamic Volterra series; one has:

* *

*

/ 2

1/ 2

/ 2

2/ 2

/ 2

1/ 2

2

*

*

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

1 ˆ( , ) ( , , ) exp( )2

1 ˆ ( , , ) exp( )2

1 ˆ ( , ) e( ) ( ) ( ) ( )

(

xp( )2

1 ˆ ( , ) ex2

) ( )

( )BW

DC BW

BW

BW

BW

BW

y H j t d

H j t

x t x t x t x t X

x d

G

t x t X

x H j tt x t

y t

x t X

x t

d

H X

/ 2

/ ( )2p( 2 )x t

BW

BWj t d

AM/AM – AM/PM


Dynamic Volterra – cntd.

3rd step: identify the AM/AM and AM/PM response from two-tone (one-tone?) measurements; identify the two transfer functions with two-tone measurements vs. tone spacing and tone amplitude

Comments: the Dynamic Volterra Envelope approach still has problems when long-memory effects with highly nonlinear features are present; further modifications are suggested in Soury et al. MTT-S 2003 p.795


Example

from Ngoya et al., BMAS 2003


Nonlinear Dynamic Measurements

Amplifiers and two port devices 50 Ohm fixed impedance systems

Spectrum Analyzer basedPower Meter based

Load Pull systemsFundamental Load PullHarmonic Load PullWaveform Load Pull


Spectrum Analyzer and PWM Based

1- Pout measurement

2- IM3, ACPR measurement

3- Gain measurement


Load pull – Source pull

Load-pull procedure characterization of a device performance as a function of the load reflection coefficient, in particular the output power

Source pull same when changing the source reflection coefficient


Class A Load-Pull theory (Cripps)

-0.8 -0.6 -0.4-0.2 0 0.2 0.4 0.6 0.8

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Re(L)

Im(L)

-1 dB-2 dB

-3 dB

-4 dB

-5 dB

PRF,M

|Z'L|<RLo |Z'L|>RLo


Basics of Load Pull

Example of Load Pull data

Output Power [dBm] @ 1dB gain compression

Power Added Efficiency (PAE) [%] @ 2dB gain compression


Comments on load pull contours

Ideally the loadpull measurement indicates the “maximum power” or “saturation power” for each load

In practice the power sweep is stopped up to a certain compression value (e.g. 1 or 2 dB compression point)

Points having the same output power (curves in red) do not usually have the same gain

2 dB gain compression constant output power curves

Constant power curves

Measured loads


Load Pull Systems

Power meter or scalar analyzer-basedonly scalar information on DUT performanceseconomic

Vector receiver (VNA)vector and more complete information on DUT

performanceshigh accuracy, thanks to vector calibrationexpensive

Time Domain Receiver (MTA)Waveform capabilitiesComplexity, high cost


Passive Load Pull Systems I

Passive loadsMechanical tunersElectronic tuners (PIN diode-based)

PowerMeter

PowerSensor

PowerSensor

Passive tuners

S L


Passive Load Pull Systems II

FeaturesSingle or double slug tunersHigh repeatability of tuner positionsPre-characterization with a network

analyzer, no real time load measurementsHigh power handling


Passive Load Pull Systems III

Slab LineMotors

DUT

Tuners


Passive Load Pull Limits

DrawbacksLoad reflection coefficient limited in magnitude

by tuner and test-set lossesThis is true especially for harmonic tuning

higher frequency optimum load on the edge of the Smith Chart

Pre-Matching using tuners or networksTo reach higher gamma while characterizing

highly mismatched transistors


Pre-Matching

LLOSS

LLOSS L

Tuners

Networks


Real Time VNA based Load Pull

Vector network analyzer-based system

VECTOR INFO

NORMAL VNA CAL

LOSSES

TUNABLELOADS

TUNABLELOADS


Real Time MTA based Load Pull

Transition Analyzer based system

VECTORAND TD INFOREF SIGNAL

TD CAL REQUIRED

TUNABLELOADS

TUNABLELOADS


Active Load

Active loop technique

a

b = a·C·A·exp(j)·G

G

exp(j)A

C


Harmonic Load Pull

Controlling the Load/Source condition at harmonic frequencies

Wave-shaping techniques at microwave frequencies

Great complexity of the system but potential improvement of the performance


Passive harmonic Load Pull

A Tuner for each harmonicComplexEasy to change frequencyMore harmonic load control

Harmonic Resonators within the slugOnly Phase control of the loadDifficult to change frequency

f0

2f0

FundamentalHarmonic


Active Harmonic Load Pull

Politecnico di Torino implementation


Four Loop Harmonic System

VNA

Switching Unit

Couplers

Amplifier

Loop Unit

DUT and Probe


M TA

Sweeper

Ch1Ch2

Couplers Bias TCouplersBias T

Sw itch A

Sw itch B

Sw itc h C

DUT

Co

up

lers

Co

up

lers

IF S w itch A

IF S w itch B

Load Sw itch

SourceSw itch

F1, F2

TRIPLEXER

ES G

2nd 3rd

Load P ullSource

PhaseAlign

Source P ullSource

ScopeCh1Ch2

Bias TD.C.

ES GES G

TRIPLEXER

F1

2nd 3rd

ES G

F2ES G

Bias TD.C.

Low Freq S ource

Source P ull Load P ull

RF & BB Load Pull System

BB Frequency Test Set

RF Frequency Test Set

Exploit BB Load Pull: wide band analysis


Load Pull and PA Design

Classical PA design information like:Power SweepOptimum Loads

Load/Source Map based designActive Real Time System Additional info

Gamma InAM/PM conversion

Harmonic Load conditionsTime Domain Info



Data set example


Power Sweep and More

Power Sweep @ Best Load for Pout

-70.00

-60.00

-50.00

-40.00

-30.00

-20.00

-10.00

0.00

10.00

20.00

30.00

27.5526.7525.4423.6021.6019.5817.7115.9614.3112.74

Pav (dBm)

dB

/ d

Bm

40.00

42.00

44.00

46.00

48.00

50.00

52.00

54.00

56.00

58.00

60.00

Pout

Gain

IM3L

IM3R

AM/PM

Eff

GammaL= 0.41 , 167 Frequency= 18 GHz

1dB Compression

1dB compression PointPout=26.29 dBmGain= 9.72 dBIM3R= -18.34 dBcIM3L=-18.50 dBcEff=48.07%



OUTPUT POWER @ 1 dB GAIN COMPRESSION

POWER GAIN@ 1 dB GAIN

COMPRESSION

COMBINING LP MAP INFORMATIONTO OPTIMIZE POWER PERFORMANCES

26dBm

12 dB


PAE @ 1 dB GAIN

COMPRESSION

C/I 3 LEFT @ POUT = 24 dBm

COMBINING LP MAP INFORMATION TO OPTIMIZE LINEARITY PERFORMANCES

50% -28 dBm



Harmonic LP Example

PAE

2nd Harmonic Load Plane

f: 3.6 GHz


TD Harmonic Source Pull

0 2 4 6 8 10 12 140

0.020.040.060.080.1

0.120.140.160.180.2

Vds, V

Ids,

A

PAE=65%PAE =55%PAE =51%

0.65 88 0.54 65

0.21 149

SII harm

mag phase

Instantaneous working point for different harmonic Gamma S

Fundamental Freq: 1 GHzGamma L fixed at1 GHz and at 2 GHz


TD Harmonic Source Pull

0

0.04

0.08

0.12

0.16

0.2

0 0.4 0.8 1.2 1.6 20

2

4

6

8

10

12

time, ns

Ids,

A

Vds

, V

PAE=65%


Acknowledgements

The presentation includes work from many colleagues from the Microwave & RF Group:Prof. Andrea FerreroProf. Marco PirolaDr. Simona DonatiDr. Laura TeppatiDr. Vittorio Camarchia

Documents

NEWCOM WPR3 Meeting – 6/9/04 Nonlinearity characterization and modelling Giovanni Ghione Dipartimento di Elettronica Politecnico di Torino Microwave &