Analog Behavioral Modeling for High Frequency based on parameter definition (N = number of ports) Frontiers in Analog Circuits July 2011 ... GaAs HBT MMIC • Agilent HMMC 5200 –

© Copyright Agilent Technologies 2011

Page 1

David E. RootResearch Fellow

Mihai MarcuSenior Applications Engineer

Agilent Technologies, Inc.

Analog Behavioral Modelingfor High Frequency Components and Systems

Frontiers in Analog Circuits

July 2011


Page 2

Outline

• Introduction

• Fundamental concepts in behavioral modeling

• Functional block-models– Time domain: nonlinear time-series– Frequency domain: X-parameters– Mixed methods: dynamic X-parameters

• Conclusions


July 2011


Page 3

Behavioral Models and Design Hierarchy


July 2011

Equivalent Circuit Model“Compact Model”

Device

Circuit

System

Embedding Variables

( )( ) : ( , , ..., , ..., )ny t i f v v i i

( )v t( )v t( )i t

Multivariate functions for i1, i2 Behavioral Model:

Accurate model of lower level componentfor simulation at next

highest level


Page 4

When Are Behavioral Models Needed

• Complex systems– Transistor-level models become too “expensive” (resource-intensive)

• IP-protection is required– Other models reveal the IP

• Other models are simply not available– New technologies, existing HW


July 2011


Page 5

Outline

• Introduction



• Conclusions


July 2011


Page 6

Some of the Features Needed in Behavioral Models

• … in addition to accurately describing the behavior…

• Cascadability of nonlinear systems arbitrary topology

• Hierarchical modeling


July 2011


Page 7

Some of the Features Needed in Behavioral ModelsCascadability of Models• Predict signals (magnitude and phase), including harmonics

(IMD) through chains of cascaded nonlinear components

• Linear S-parameter theory does not apply– Most previous attempts to generalize are incorrect

(power-dependent S-parameters, hot-S22)


July 2011

21

Sin(2πf0t)

Freq

Pin

ters

tage

1f0

13f0

12f0

Freq

Pin

f0Freq

Pou

t

2223f02f0f0


Page 8

Some of the Features Needed in Behavioral ModelsHierarchical Modeling

• Model the cascade directly, or

• A cascade of many models reduced to one


July 2011

Dev 1 Dev 2 Dev 1 Dev 2

Mod 1 Mod 2

Mod 1 Mod 2Composite

Model(Higher Level)


Page 9

Three Components of Behavioral Modeling

• Model formulation– Time-domain– Frequency-domain– Time-frequency-domain

• Experiment design– Stimulus to excite relevant dynamics

• Model identification– Procedure to determine model parameters

• Briefly reviewed using S-parameters


July 2011


Page 10

S-ParametersPhysical Interpretation

• Based on the traveling waves concept

• A 2-port example (extendable to arbitrary N-port)


July 2011

Incident Transmitted

Transmitted Incident

DUT

Port 1 Port 2

1 11 1 12 2

2 21 1 22 2

b S a S ab S a S a

11 1S a1b

1a

2a

2b

12 2S a

21 1S a

22 2S a


Page 11

S-Parameter Behavioral Model

• Model formulation – frequency domain

• Experiment design– One tone, low-power stimulus applied successively at each port

• Model identification:– based on parameter definition (N = number of ports)


July 2011

1 11 1 12 2

2 21 1 22 2

b S a S ab S a S a

0

, , 1,k

iij

ajk j

bS i j Na


Page 12

Model Formulation

• Time-domain - nonlinear ODEs– NLTS (Non-Linear Time Series)– Transient simulation and all other simulators

• Frequency-domain - nonlinear spectral map– X-parameters– Harmonic-Balance simulation

• Time-frequency-domain – ODE-coupled envelopes– X-parameters with memory effects dynamic X-parameters– Envelope simulation

• Formulate model-equations in forms native to simulatorFrontiers in Analog Circuits

July 2011


Page 13

Model FormulationTime-Domain

• Natural for strongly nonlinear low-order (lumped) systems


July 2011

( ) , , , , ,I t F V t V t V t I t

time time

StimulusResponse


Page 14

Model FormulationFrequency-Domain

• Natural for low distortion (sparse spectrum) & high-frequency


July 2011

,...),,( 321 AAAFB kk

frequency frequency

A1

A2A3 A4

B1

B2B3 B4

B5B6B0

Stimulus (multiple harmonics)Response


Page 15

Experiment Design

• Design the experiment such that it exercises the relevant states and dynamics of the system

• Can be achieved in both “worlds”:– Simulation– Measurement

• Conceptually the same (in measurement versus simulation), but… different challenges


July 2011


Page 16

Experiment DesignSimulation

• Exercise all required states and dynamics


July 2011

System Under Test


Page 17

Experiment DesignMeasurement

• Exercise the HW on all required states and dynamics

• Calibrated magnitude and phase for all harmonics/IMD


July 2011

Magnitude and Phase Data Acquisition

RFIC

A1k B1l B2m A2nReferenceplanes

New phase calibration standard

NVNA


Page 18

Nonlinear Vector Network Analyzer (NVNA)combine with previous page• Based on the PNA-X platform

• Measures magnitude and phase of relevant frequency components

• Essential nonlinear measurements for nonlinear models (like NLTS and X-parameters)


July 2011

PNA-XNetwork Analyzer

+

Phase Reference

+

Meas. Science Algorithms & Software

= NVNA


Page 19

Nonlinear Vector Network Analyzer (NVNA)Waveforms and Complex Spectra Measurements


July 2011

pkBpkA

Port IndexHarmonic Index

2 kA1kA

1kB 2kB

1I 2I

time time


Page 20

Outline

• Introduction



• Conclusions


July 2011


Page 21

Dynamical Systems and State-Space

• The dynamics of the nonlinear system can be assumed to be described by a system of nonlinear ODEs

• The sampled solution of the ODE, y(t), is a time-series

• The solution of the dynamical equations for state variables, u(t) is a time-parameterized trajectory in the State-Space


July 2011

( ) ( 1) ( )( ) ( ,... , , ,... )n n my t f y y x x x

( ) ( ), ( ) vector of state equ

( ) (

ation

), ( ) sca

s

lar outputy t h u t

u t f u t x t

x t


Page 22

State-Space and Time-Series

• The multi-dimensional space spanned by the state variables is known as the state-space

• Any measureable output is a projection of this trajectory versus time and it is a time-series


July 2011


Page 23

Model Identification for NLTS

• Stimulate system– Sufficiently complex stimulus

• Embed– Create auxiliary variables to represent the waveform

• Sample data– Data directly or the envelope– Difficult if multiple time-scales

• Fit– Approximate nonlinear function f(.)


July 2011


Page 24

Excitation Design

• Stimulate all relevant (observable) dynamics

• Sweep power and frequency to “cover the state-space”


July 2011

‘Two-tone’

f1 f1+ff1 f1+f

f1+f

‘Three-tone’

f1+f?

‘Modulation’ (CDMA)

f1f1+f

f2

fn

‘Multi-tone’ or ‘Multi-sine’


Page 25

Embedding

• Building-up state-space to define ODE


July 2011

i(t)

v(t)

A

Bi(t)

v(t)

i(t)

v(t)

A

B

i(t)

v(t)

v’(t)

B

A

i(t)

v(t)

v (t)

B

A( ) ( ( ))i t i v t ( ) ( ( ), ( ))i t i v t v t

X(t) Y(t)

( )

( )

( ) [ ( ), ( ),..., ( )]( ) [ ( ), ( ),..., ( )]

m

n

x t x t x t x ty t y t y t y t


Page 26

Sample Data

• Dense enough in all dimensions


July 2011

X(t) Y(t)

( ) ( )1 1 1 1 1 1

( ) ( )2 2 2 2 2 2

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

m n

m n

m np p p p p p

x t x t x t y t y t y tx t x t x t y t y t y t

x t x t x t y t y t y t

nth order derivativeStimulus Response


Page 27

Function Fit

• For nth derivative of the output

• Many function-types may be used– Polynomials– Radial basis functions– Look-up tables– …– ANNs

• Artificial neural networks (ANNs) a very good option


July 2011

( ) ( 1) ( )( ,... , , ,... )n n my f y y x x x


Page 28

Function FitArtificial Neural Networks (ANNs)

• An ANN is a parallel processor made up of simple, interconnected processing units, called neurons, with weighted connections


July 2011

baxwsvxxFI

i

K

kikkiiK

1 11 ),...,(

sigmoidweights biases

x1

...

xk


Page 29

Function Fit ANN General Properties

• Fit any nonlinear function of any number of variables (universal approximation theorem)

• Infinitely differentiable better distortion than naïve splines or low-order polynomials

• Easy to train (fit) using standard tools

• Easy to train on scattered data


July 2011

baxwsvxxFI

i

K

kikkiiK

1 11 ),...,(


Page 30

Function FitANN Training

• wki,ak,b – obtained by training


July 2011

fANN

…

… …

,ki kw aweightsbiases

y(n-1)(t) y(n-2)(t) … y(t) x(n)(t) x(n-1)(t) … x(t)

y(n)(t)

( ) ( 1) ( 2) ( ) ( 1)( ) , ,..., , , ,...,n n n n nANNy t f y t y t y t x t x t x t


Page 31

Model ImplementationAn Example

• ODE in circuit simulator


July 2011

xx x(1)

+-

x(m-1) x(m)+- f vn

+-

x(1) x(2)+-

yv2v1

+-

v2v3

+-

vn-1vn

+-

( ) ( 1) ( ), , , , ,n n my f y x x x

2

1

( )1

1 2

1

, , , , , , ,n

n n

n

mnv f v v v

v yv v

v v

x x x


Page 32

NLTS – ExampleGaAs HBT MMIC

• Agilent HMMC 5200– Series-shunt amplifier– Gain: 9.5 dB @ 1.5 GHz


July 2011

Actual Circuit Detailed circuit model


Page 33

NLTS Accuracy and SpeedSingle-Tone Stimulus


July 2011

-20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 2 4-22 6

0

20

40

60

80

100

120

140

160

-20

180

dBm(In_model[::,1],z1[::,1])

ph

ase

(Ou

t_m

od

el[:

:,1

])

dBm(In_IC[::,1],z1ic[::,1])

ph

ase

(Ou

t_IC

[::,

1])

Fundamental Phase

Time psec0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.80.0 2.0

2.0

2.5

3.0

3.5

4.0

1.5

4.5

0 200

1.5

2.0

2.5

3.0

3.5

4.0

1.0

4.5

Time psecTime Domain Waveforms

229.68 seconds

11315.67 seconds

-20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 2 4-22 6

7

8

9

10

11

12

13

6

14

dbm(In_model[::,1],z1[::,1])

dB

m(O

ut_

mo

de

l[::,

1],

z2[:

:,1

])-d

Bm

(In

_m

od

el[:

:,1

],z1

[::,

1])

dbm(In_IC[::,1],z1ic[::,1])

dB

m(O

ut_

IC[:

:,1

],z2

ic[:

:,1

])-d

Bm

(In

_IC

[::,

1],

z1ic

[::,

1])

Fundamental Gain

6

14

-22 6

dBm

dBm

180

-20

1 - 19 GHz

(2) (2)1 2 1 2 1 2( ) ( , ( ), ( ), ( ), ( ), ( ), ( ))i i iI t f I V t V t V t V t V t V t

19 neurons

-22 6dBm

NLTS modelCkt model


Page 34

WCDMA Lower ACLR Comparison:Circuit Co-Sim vs. NLTSA Model vs. Measured

0

10

20

30

40

50

60

70-15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3

Input Power (dBm)

AC

LR (d

B

Circuit Co-Sim 5MHz Lower

NLTSA Model 5 MHz Lower

Circuit Co-Sim 10 MHz Lower

NLTSA Model 10 MHz Lower

Measured Data 5 MHz Lower

Measured Data 10 MHz Lower

NLTS Accuracy and SpeedResults 3GPP WCDMA Lower ACLR


July 2011

294 sec/pt NLTS

1532 sec/pt Ckt.

Measured

Model: - 40 neurons, cascadable

- generated from sinusoidal signals only

- works in Transient, HB, Envelope

Simulation: 3 GHz (courtesy Greg Jue)Measured: 2.4 GHz


Page 35

Outline

• Introduction



• Conclusions


July 2011


Page 36

X-Parameters

• Similar to S-parameters, but for nonlinear behavior

• The mathematically correct superset of S-parameters

• Applicable to:– Both large- and small-signal– Both linear and nonlinear systems

• All pieces of the puzzle work together:– Measure– Model– Simulate


July 2011


Page 37

X-Parameters

• Cascadable models – correctly account for mismatch

• Model identification from a simple set of measurements

• Very accurate


July 2011


Page 38

X-ParametersFundamental Concepts

• Spectral map of complex large input phasors to large complex output phasors


July 2011

2A1A

1B 2B1 1 11 12 21 22

2 2 11 12 21 22

( , , , ..., , , ...)( , , , ..., , , ...)

k k

k k

B F D C A A A AB F D C A A A A

Port Index Harmonic Index


Page 39

X-Parameters for Multi-Harmonic StimulusHarmonics Are Large-Signals

• Response is a nonlinear function of stimulus

• Behavior can be described by X-parameters

• Phase-coefficient comes from time-invariance:– Output of delayed input is the delayed output


July 2011

, , 11 12 21 22, , , . , ,p k p kB F D C A A A A

( )

( ), ,

1,1 1, 2,

1,1 1, 2,

, 1, , , , 2, , , 1,

, 1, , , , 2, , , 1,

1, , 1,

k kq k k

k kq k k

FDCRp p

FB kp k p k

DCR X

B

DCS q N A A P k K A P k K

DCS q N A A P k K A P k K

p N k K

X P

1,1 j phase AP e


Page 40

• If all stimuli are small except for the fundamental simplify the general nonlinear spectral mapping by spectral linearization

• X(S) and X(T) terms synthetic load-pull

X-Parameters for Single-Tone StimulusHarmonics Are Small-Signals


July 2011

( )1,1 ,, ; ,

11

, 1

( ), 1,1.

( ) *1,1 ,, ; ,

11

, 1 1,1 ,

q Nl K

S k lq lp k q l

ql

q

FB kp k p k

q Nl K

T k lq lp k q

l

lql

q l

X A P AB X A P X A P A

Mismatch terms linear in Aq,l

Perfectly matched response

Mismatch terms linear in A*q,l


Page 41

Practical Applications of X-Parameters

• Black-box description holds for arbitrary systems (amplifiers, mixers, receivers, transmitters, etc.)

• Supports multi-variable swept conditions:– bias– RF-power– frequency– load-pull– additional user-defined variables


July 2011


Page 42

Single-Tone Amplifier


July 2011

CktXpar


Page 43

Mismatch Effects in a Nonlinear Cascade


July 2011

CascadedX-parameter models

Cascaded PA circuits

Mismatch

Mismatch


Page 44

Mismatch Effects in a Nonlinear CascadeResults


July 2011

Fundamental

2nd Harmonic

3rd Harmonic

Overlaid Results

Model (Red)Vs.

Circuit (Blue)


Page 45

Systems with Frequency ConversionMixer (2-Tone: RF, LO)


July 2011

1.0E9

2.0E9

3.0E9

4.0E9

5.0E9

6.0E9

7.0E9

8.0E9

9.0E9

1.0E10

0.0

1.1E10

-160

-140

-120

-100

-80

-60

-40

-180

-20

freq, Hz

dB

m(v

ou

t_ck

t)

m3

freq+(0.05 GHz)

dB

m(v

ou

t_m

dl)

m4

m3indep(m3)=plot_vs(dBm(vout_ckt), freq)=-38.914

2.500E8

m4indep(m4)=plot_vs(dBm(vout_mdl), freq+(0.05 GHz))=-38.914

3.000E8

1.0E9

2.0E9

3.0E9

4.0E9

5.0E9

6.0E9

7.0E9

8.0E9

9.0E9

1.0E10

0.0

1.1E10

-150

-100

-50

0

50

100

150

-200

200

freq, Hz

ph

ase

(vo

ut_

ckt)

m1

freq+(0.05 GHz)

ph

ase

(vo

ut_

md

l)

m2

m1indep(m1)=plot_vs(phase(vout_ckt), freq)=-50.105

2.500E8

m2indep(m2)=plot_vs(phase(vout_mdl), freq+(0.05 GHz))=-50.105

3.000E8

CktXpar

v out_mdlv out_ckt

HarmonicBalanceHB1

Order[2]=5Order[1]=5Freq[2]=2 GHzFreq[1]=1.75 GHzMaxOrder=5

HARMONIC BALANCE

P_1TonePORT3

Freq=2 GHzP=polar(dbmtow(-50),0)Z=50 OhmNum=3 P_1Tone

PORT4

Freq=1.75 GHzP=polar(dbmtow(-5),0)Z=50 OhmNum=4

V_DCSRC2Vdc=5 V

X4PXNP1File="a0_Mixer.ds"

4

1 2

3 Ref

RR2R=50 Ohm

RR1R=50 Ohm

P_1TonePORT1


PORT2


V_DCSRC1Vdc=5 V

x_2_GilCellMixX1

IF_portRF_port

LO_port

Bias_port


Page 46

Systems with Frequency Conversion Mixer Sub-System - Spectrum Measurement Setup

• RFpwr = -50

• LOpwr = -5 dBm


July 2011

v out_mdl

v out_ckt

X4PXNP1File="d0_MixerBPF.ds"

4

1 2

3 Ref

RR2R=50 Ohm

RR1R=50 Ohm

BPF_Cheby shevBPF1

N=15Ripple=0.5 dBBWpass=100 MHzFcenter=250 MHz

V_DCSRC2Vdc=5 V

HarmonicBalanceHB1

Order[2]=5Order[1]=5Freq[2]=2 GHzFreq[1]=1.75 GHzMaxOrder=5

HARMONIC BALANCE

P_1TonePORT1


PORT2


V_DCSRC1Vdc=5 V

x_2_GilCellMixX1

IF_portRF_port

LO_port

Bias_port

P_1TonePORT4


P_1TonePORT3

Freq=2 GHzP=polar(dbmtow(-50),0)Z=50 OhmNum=3


Page 47

Systems with Frequency ConversionMixer Sub-System – Results: Output Spectrum

• small intentional display offset – to distinguish traces


July 2011

1.0E9

2.0E9

3.0E9

4.0E9

5.0E9

6.0E9

7.0E9

8.0E9

9.0E9

1.0E10

0.0

1.1E10

-800

-600

-400

-200

-1000

0

freq, Hz

dB

m(v

ou

t_ck

t)

freq+(0.05 GHz)

dB

m(v

ou

t_m

dl)

1.0E9

2.0E9

3.0E9

4.0E9

5.0E9

6.0E9

7.0E9

8.0E9

9.0E9

1.0E10

0.0

1.1E10

-100

0

100

-200

200

freq, Hz

ph

ase

(vo

ut_

ckt)

freq+(0.05 GHz)

ph

ase

(vo

ut_

md

l)

Mag

Phase

CktXpar


Page 48

Amplifier Load-Pull


July 2011

Pd

el_

ckt_

con

tou

rs_

pP

de

l_m

dl_

con

tou

rs_

p

Pdel_dBm

PA

E_

ckt_

con

tou

rs_

pP

AE

_m

dl_

con

tou

rs_

p

PAE

PA

E_

ckt_

con

tou

rs_

pP

AE

_m

dl_

con

tou

rs_

p

PAE

Pd

el_

ckt_

con

tou

rs_

pP

de

l_m

dl_

con

tou

rs_

p

Pdel_dBm

CktXpar

50 Ω X-parameters

Load-dependent X-parameters

-18 -16 -14 -12 -10 -8 -6 -4 -2 0 2 4-20 6

0

5

10

15

-5

20

Power in (dBm)

Pow

er o

ut (

dBm

)


Page 49

Load-Dependent Waveforms27 Ω Validation of Measurement-Based 50 Ω Model


July 2011

v1

100 200 300 400 500 6000 700

-0.5

0.0

0.5

-1.0

1.0

time, psec

v2

100 200 300 400 500 6000 700

-1.0

-0.5

0.0

0.5

-1.5

1.0

time, psec

100 200 300 400 500 6000 700

-0.005

0.000

0.005

-0.010

0.010

time, psec

i1

100 200 300 400 500 6000 700

-0.02

0.00

0.02

0.04

-0.04

0.05

time, psec

i2

Measurement-Based X-parameter Model Independent NVNA Data


Page 50

Rough Comparison of Methods and Applicability

NLTS• Works in TA, HB, Envelope

• Excellent for strongly nonlinear, but lumped (low-order ODE) systems

• Training non-algorithmic

• Experiment design not fully solved

• Not as robust for convergence

• Scales well with complexity

• Great gains in simulation speed

X-Parameters• Frequency-domain natural for highly

nonlinear, distributed, broad-band circuits

• Experiment design completely solved

• Highly automated model identification

• Works in HB and Envelope

• Very robust for convergence

• Always accurate if sampled densely

• Complexity increases rapidly for multiple tones


July 2011


Page 51

Outline

• Introduction



• Conclusions


July 2011


Page 52

Why Mixed-Methods

• Applies to systems under large-signal modulated drive

• Long-term memory effects are present

• Time-varying spectra for all inputs, outputs and state-variables

• Perfectly suited for Envelope simulations

• Well matched for data from NVNA


July 2011


Page 53

Envelope Domain for Long-Term Memory

• Xh(t) – a set of time-varying complex-waveforms (amplitude and phase) at each harmonic-index, h

• Modeling problem: map input envelopes to output envelopes


July 2011

Freq. (GHz)1 2 3 4DC

Time-varying spectrum

02

0

( ) Re ( )H

j h f th

h

x t X t e

B2[1](t) – time-domain envelope


Page 54

Merging Frequency and Time Domains

• The static spectral mapping becomes a differential equation in envelope domain


July 2011

11 12 21 22( , , ..., , , ...)pk pkB f A A A A

1 1 110 10 11 11 20 21 2,..., , , ,..., , ,..., ,..., ,...,n m

pk pk pk pk kB f B t B t A t A t A t A t A t A t A t

Envelope index

Order of time derivative

21 21 20 11

22011 21

,

,

B t f B t A t

dB tg A t B t

dt

Example:

Port index


Page 55

Envelope Model: Amplifier with Self-Heating


July 2011

Pulsed RF signal at 1GHz: Thermal Time Const. 10usec

10 20 30 400 50

0.1

0.2

0.3

0.0

0.4

Fundamental Input

time, usec time, usec10 20 30 400 50

1

2

3

0

4

Fundamental Output

10 20 30 400 50

0.01

0.02

0.03

0.00

0.04

10

20

30

0

40

Third Harmonic Output Mag & Phase

time, usec


Page 56


July 2011

Dynamic X-parameterPNA-X NVNA

Dynamic Compression Characteristic

Out

put (

Vp)

Input (Vp)

Dynamic X-Parameters - Memory Effects [12]

Power Sweep @ 19.2 kHz Tone Spacing


Page 57

Conclusions

• Time, frequency and mixed domain methods:– NLTS could become practical soon– X-parameters are mature, both measurement and simulation– Dynamic X-parameters proven for memory

• Commercial availability– NVNA – based on PNA-X HW platform– X-parameters available in circuit and system simulators (ADS, Genesys,

SystemVue)

• Great opportunities for applications– Specifications of active components by X-parameters– Better understanding of nonlinear behavior new design practices– Nonlinear stability, matching, etc.


July 2011


Page 58

References

[1] J. Wood, D. E. Root, N. B. Tufillaro, “A behavioral modeling approach to nonlinear model-order reduction for RF/microwave ICs and systems,” IEEE Transactions on Microwave Theory and Techniques, Vol. 52, Issue 9, Part 2, Sept. 2004 pp. 2274-2284

[2] D. E. Root, J. Wood, and N. Tufillaro, “New Techniques for Non-Linear Behavioral Modeling of Microwave/RF ICs from Simulation and Nonlinear Microwave Measurements,” in 40th ACM/IEEE Design Automation Conference Proceedings, Anaheim, CA, USA, June 2003, pp. 85-90 (videotaped)

[3] D. E. Root, “Nonlinear Analog Behavioral Modeling of Microwave Devices and Circuits,” IEEE Microwave Theory and Techniques Distinguished Microwave Lecture (now available from MTT-S Speaker’s Bureau)

[4] Agilent HMMC-5200 DC-20 GHz HBT Series-Shunt Amplifier, Data Sheet, August 2002.

[5] J. Verspecht, M. Vanden Bossche, F. Verbeyst, “Characterizing Components under Large Signal Excitation: Defining Sensible `Large Signal S-Parameters'?!,” in 49th IEEE ARFTG Conference Dig., Denver, CO, USA, June 1997, pp. 109-117.

[6] D. E. Root, J. Verspecht, D. Sharrit, J. Wood, and A. Cognata, “Broad-Band Poly-Harmonic Distortion (PHD) Behavioral Models from Fast Automated Simulations and Large-Signal Vectorial Network Measurements”, IEEE Transactions on Microwave Theory and Techniques Vol. 53. No. 11, November, 2005 pp. 3656-3664

[7] J. Wood, D. E. Root, editors, Fundamentals of Nonlinear Behavioral Modeling for RF and Microwave Design, 1st ed. Norwood, MA, USA, Artech House, 2005.

[8] D. E. Root, D. Sharrit, J. Verspecht, “Nonlinear Behavioral Models with Memory: Formulation, Identification, and Implementation,” 2006 IEEE MTT-S International Microwave Symposium Workshop (WSL) on Memory Effects in Power Amplifiers

[9] Blockley et al 2005 IEEE MTT-S International Microwave Symposium Digest, Long Beach, CA, USA, June 2005.

[10] Soury et al 2005 IEEE International Microwave Symposium Digest pp. 975-978

[11] J. Verspecht and D. E. Root, “Poly-Harmonic Distortion Modeling,” in IEEE Microwave Theory and Techniques Microwave Magazine, June, 2006.

[12] J. Verspecht, J. Horn, L. Betts, D. Gunyan, R. Pollard, C. Gillease, and D. E. Root, “Extension of X-parameters to include long-term dynamic memory effects,” IEEE MTT-S International Microwave Symposium Digest, 2009. pp 741-744, June, 2009

[13] J. Horn, L. Betts, C. Gillease, D. Gunyan, J. Xu, J. Verspecht, and D. E. Root, “X-Parameters: Extending S-parameters for Measurement, Modeling, and Simulation of Nonlinear Components,” 2008 IEEE Power Amplifier Symposium, Orlando, FL


July 2011

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Analog Behavioral Modeling for High Frequency based on parameter definition (N = number of ports) Frontiers in Analog Circuits July 2011 ... GaAs HBT MMIC • Agilent HMMC 5200 –