Advanced Nonlinear Device Characterization Utilizing New Nonlinear Vector Network Analyzer and X-parameters

Advanced Nonlinear Device Characterization Utilizing New

Nonlinear Vector Network Analyzer and X-parameters

Advanced Nonlinear Device Characterization Utilizing New

Nonlinear Vector Network Analyzer and X-parameters

presented by:

Loren Betts

Research Scientist

Presentation Outline

9 Nonlinear Vector Network Analyzer Applications (What does it do?) 9 Device Characteristics (Linear and Nonlinear)

9 NVNA Hardware

9 NVNA Measurements

Component Characterization Multi-Envelope Domain

X-Parameters

Nonlinear Vector Network Analyzer (NVNA)

ƒVector (amplitude/phase) corrected nonlinear measurements from 10 MHz to 26.5 GHz

ƒCalibrated absolute amplitude and relative phase (cross-frequency relative phase) of measured spectra traceable to standards lab

ƒ26 GHz of vector corrected bandwidth for time domain waveforms of voltages and currents of DUT

ƒMulti-Envelope domain measurements for measurement and analysis of memory effects

ƒX-parameters: Extension of Scattering parameters into the nonlinear region providing unique insight into

nonlinear DUT behavior

ƒX-parameter extraction into ADS PHD block for nonlinear simulation and design

ƒStandard PNA-X with New Nonlinear features and capability

New phase calibration standard

NVNA System Configuration

Amplitude Calibration

Vector Calibration

Phase Calibration

Phase Reference X-Parameter extraction/multi-tone source

Standard PNA-X Network Analyzer

New!! Agilent Calibrated Phase Reference New!! Agilent NVNA

Advantages:

ƒLower temperature sensitivity

ƒLower sensitivity to input power

ƒSmaller minimum tone spacing (< 1 MHz vs 600 MHz)

ƒLower frequency (< 1 MHz vs 600 MHz)

ƒMuch wider dynamic range due to available energy vs noise

Phase Reference

New!! Agilent Calibrated Phase Reference

0 10 20 30 40 50 60 70 80 90 100

0 0.2 0.4 0.6 0.8 1

F re q ue nc y in G Hz

Normalized Amplitude

Agilent’s new IC based phase reference is

superior in all aspects to existing phase reference

technologies.

NVNA Applications (What does it do?)

Time domain oscilloscope measurements with vector error correction applied

View time domain (and frequency domain) waveforms (similar to an

oscilloscope) but with vector correction applied (measurement plane at DUT terminals)

Vector corrected time domain

voltages (and currents) from

device

NVNA Applications (What does it do?)

Measure amplitude and cross-frequency phase of

frequencies to/from device with vector error correction

applied

View absolute amplitude and phase relationship between frequencies to/from a device with

vector correction applied (measurement plane at DUT terminals)

Useful to analyze/design high efficiency amplifiers such as class E/F

Can also measure frequency multipliers

Phase relationship between frequencies at output of device

Input output frequencies at device terminals 1 GHz

2 GHz

122.1 degree phase delta 1 GHz

Fundamental Frequency Source harmonics (< 60 dBc)

Stimulus to device

Output from device

Output harmonics

NVNA Applications (What does it do?)

Measurement of narrow (fast) DC pulses with vector error correction applied

View time domain (and frequency

domain) representations of narrow DC pulses with vector correction applied (measurement plane at DUT terminals)

Less than 50 ps

NVNA Applications (What does it do?)

Measurement of narrow (fast) RF pulses with vector error correction applied

View time domain (and frequency domain) representations of narrow RF pulses with vector correction applied (measurement plane at DUT terminals) Using wideband mode (resolution ~ 1/BW

~ 1/26 GHz ~ 40 ps)

Example: 10 ns pulse width at a 2 GHz carrier frequency. Limited by external source not NVNA.

Can measure down into the picosecond pulse widths

NVNA Applications (What does it do?)

Measurements of multi-tone stimulus/response with vector error correction applied

View time and frequency domain representations of a multi-tone

stimulus to/from a device with vector correction applied (measurement plane at DUT terminals)

Stimulus is 5 frequencies spaced 10 MHz apart centered at 1 GHz

measuring all spectrum to 20 GHz Measure amplitude AND PHASE of intermodulation products

Generated using external source (PSG/ESG/MXG) using NVNA and vector calibrated receiver

Input Waveform

Output Waveform

NVNA Applications (What does it do?)

Calibrated measurements of multi-tone

stimulus/response with narrow tone spacing

View time and frequency domain representations of a multi-tone

stimulus to/from a device with vector correction applied (measurement plane at DUT terminals)

Stimulus is 64 frequencies spaced

~80 kHz apart centered at 2 GHz. The NVNA is measuring harmonics to 16 GHz (8

^th

harmonic)

Multi-tone often used to mimic more complex modulation (i.e. CDMA) by matching complementary cumulative distribution function (CCDF). Multi- tone can be measured very

accurately.

NVNA Applications (What does it do?)

Calibrated measurements of multi-tone

stimulus/response with narrow tone spacing

View time and frequency domain representations of a multi-tone

stimulus to/from a device with vector correction applied (measurement plane at DUT terminals)

Stimulus is 64 frequencies spaced

~80 kHz apart centered at 2 GHz. The

NVNA is measuring harmonics to 16

GHz (8

^th

harmonic)

NVNA Applications (What does it do?)

Measure memory effects in nonlinear devices with vector error correction applied

View and analyze dynamic memory signatures using the vector error

corrected envelope amplitude and phase at the fundamental and harmonics with a pulsed (RF/DC) stimulus

Output (b2) multi-envelope waveforms

Each harmonic has a unique time

varying envelope signature

NVNA Applications (What does it do?)

Measure modeling coefficients and other nonlinear device parameters

Measure, view and simulate actual nonlinear data from your device

Dynamic Load Line

Waveforms (‘a’ and ‘b’ waves)

X-parameters

… More

Linear and Nonlinear Device Characteristics

Nonlinear Hierarchy from Device to System

Devices

Systems

Circuits

Transistors, diodes, …

Amplifiers, Mixers, Multipliers, Modulators, ….

Chips

Characterization and modeling

Modules

Instrument Front- ends

Cell Phones

Military systems

Etc.

Nonlinearities

0

0

0

0 0

2 0

3 0

( ) ( )

( ) ( )

b

c

d

j

j

j

y t a b e x t c e x t d e x t

θ θ

θ

= +

+ +

0 0

( )

( )

^{j w t}

x t = Ae

^{+φ 0 0 0}

0 0 0

0 0 0

( )

0 0

2( )

2 0

3( )

3 0

( )

^b

c

d

j j t

j j t

j j t

y t a b e Ae c e A e d e A e

θ ω φ

θ ω φ

θ ω φ

+

+

+

⎡ ⎤

= + ⎣ ⎦

⎡ ⎤

+ ⎣ ⎦

⎡ ⎤

+ ⎣ ⎦

( )

x t y t( )

ω1 ω₁ 2ω₁ 3ω₁

The Need for Phase

Cross-Frequency Phase

Notice that each frequency component has an associated static phase shift.

Each frequency component has a phase relationship to each other.

0 0 0

0 0 0

0 0 0

( )

0 0

2( )

2 0

3( )

3 0

b

c

d

j j t

j j t

j j t

y a b e Ae c e A e

d e A e

θ ω φ

θ ω φ

θ ω φ

+

+

+

⎡ ⎤

= + ⎣ ⎦

⎡ ⎤

+ ⎣ ⎦

⎡ ⎤

+ ⎣ ⎦

Why Measure This?

If we can measure the absolute amplitude and cross-frequency phase we have knowledge of the nonlinear behavior such that we can:

ƒ Convert to time domain waveforms (eg:

scope mode).

ƒ Measure phase relationships between harmonics.

ƒ Generate model coefficients.

ƒ Measure frequency multipliers.

ƒ Etc…..

ƒ 2- and 4-port versions

ƒ Built-in second source and internal combiner for fast, convenient measurement setups

ƒ Spectrally pure sources (-60 dBc)

ƒ Internal modulators and pulse generators for fast, simplified pulse measurements

ƒ Flexibility and configurability

ƒ Large touch screen display with intuitive user interface

Agilent’s N5242A premier performance

microwave network analyzer offers the highest performance, plus:

NVNA Hardware

Take a standard PNA-X and add nonlinear measurement

capabilities

Measuring Unratioed Measurements on PNA-X

Unratioed Measurements – Amplitude Works fine.

Ever tried to measure phase across frequency on an unratioed measurement?

Sweep 1 Sweep 2

1

a1

1

b1

1

b2

1

a2 0

a1

0

b1

0

a2

0

b2

Measuring Unratioed Measurements on PNA-X

Unratioed Measurements – Phase

Phase response changes from sweep to sweep. As the LO is swept the LO phase from each frequency step from sweep to sweep is not

consistent. This prevents measurement of the cross-frequency phase of the frequency spectra.

Sweep 1 Sweep 2

Phase Shifted

NVNA Hardware Configuration

Generate Static Phase

ƒ Since we are using a mixer based VNA the LO phase will change as we sweep frequency. This means that we cannot directly measure the

phase across frequency using unratioed (a1, b1) measurements.

ƒ Instead…ratio (a1/ref, b2/ref) against a device that has a constant phase relationship versus frequency. A harmonic phase reference

generates all the frequency spectrum simultaneously.

ƒ The harmonic phase reference frequency grid

and measurement frequency grid are the same

(although they do not have to be generally). For

example, to measure a maximum of 5 harmonics

from the device (1, 2, 3, 4, 5 GHz) you would

place phase reference frequencies at 1, 2, 3, 4, 5

GHz.

NVNA Hardware Configuration

Phase Reference

ƒ One phase reference is used to maintain a static cross-frequency phase

relationship.

ƒ A second phase reference standard is used to calibrate the cross-frequency phase at the device plane.

ƒ The phase reference generates a time domain impulse. Fourier theory

illustrates that a repetitive impulse in time generates a spectra of frequency content related to the pulse repetition frequency (PRF) and pulse width (PW).

ƒ The cross-frequency phase relationship

remains static.

Mathematical Representation of Pulsed DC Signal

) ( ))

( (

)

( t rect t shah

1

t y

pw ∗

prf

=

0 1/prf -1/prf

t

...

* ...

∑ _nδ (t-n(1/ prf))

0 1/prf 2/prf -1/prf

-2/prf

... ...

rect _pw (t)

0 1/2pw -1/2pw

t

)) (

( ) ) (

( )

( s pw sinc pw s prf shah prf s

Y = × × × × ×

)) (

( )) (

( )

( s pw sinc pw s prf shah prf s

Y = × × × × ×

) (

) (

)

( s DutyCycle sinc pw s shah prf s

Y = × × × ×

Frequency Domain Representation of Pulsed DC Signal

0 10 20 30 40 50 60 70 80 90 100

0 0.2 0.4 0.6 0.8 1

Frequency in GHz

Normalized Amplitude

0 5 10 15 20 25

0 0.2 0.4 0.6 0.8 1

Frequency in GHz

Normalized Amplitude

Frequency Response

ƒ Drive phase reference with a Fin frequency.

ƒ Get n*Fin at the output of the phase reference.

ƒ Example:

ƒ Want to stimulate DUT with 1 GHz input stimulus and measure harmonic responses at 1, 2, 3, 4, 5 GHz.

ƒ Fin = 1 GHz

ƒ Can practically use frequency

spacings less than 1 MHz

NVNA Hardware Configuration

If we were to isolate a few of the frequencies from the phase reference we would see that the phase relationship remains constant versus input drive frequency and power.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-1 -0.5 0 0.5 1

Frequency - 1 GHz

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-1 -0.5 0 0.5 1

Frequency - 2 GHz

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-1 -0.5 0 0.5 1

Frequency - 3 GHz

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-1 -0.5 0 0.5 1

Frequency - 4 GHz

Phase relationship between

frequencies remains the same

1 GHz

2 GHz

3 GHz

4 GHz

0 0

cos( ω t + φ )

0 1

cos(2 ω t + φ )

0 2

cos(3 ω t + φ )

0 3

cos(4 ω t + φ )

Example NVNA Configuration #1

Using external source for phase reference drive

Test port 3

C R3

Test port 1

R1

Source 1

OUT 1 OUT 2 Source 2 (standard)

OUT 1 OUT 2

Test port 4

R4

Test port 2

R2

35 dB 65 dB

35 dB

65 dB 65 dB 35

dB 65 dB

Rear panel

A

35 dB

D B

SW1 SW3 SW4 SW2

J11 J10 J9 J8 J7 J4 J3 J2 J1

To port 1 or 3

Calibration Phase Reference Ext Source in

Measurement Phase Reference

Example NVNA Configuration #2

Multi-Tone, SCMM, DC/RF, Calibrated receiver mode

Test port 3

C R3

Test port 1

R1

Source 1

OUT 1 OUT 2 Source 2

OUT 1

OUT 2

Test port 4

R4

Test port 2

R2

35 dB 65 dB

35 dB

65 dB 65 dB 35

dB 65 dB

Rear panel

A

35 dB

D B

To port 1 or 3

Calibration Phase Reference Ext Source in

Measurement Phase Reference

NVNA Measurements

Component Characterization

The Nonlinear VNA is a new set of firmware that runs on a

standard PNA-X. Can run the

normal PNA-X firmware and the

NVNA firmware on the PNA-X.

Nonlinear VNA Measurements

Stimulus/Response Setup

The user can set up a multi- dimensional segmented sweep across input frequency and power and output response.

ƒSingle tone stimulus and harmonic response

ƒMulti-tone stimulus/response

ƒDC/RF pulse response

ƒX-parameters stimulus/response

ƒMulti-Envelope response

Nonlinear VNA Measurements

Guided Calibration Wizard – 3 Step Process (Vector, Amplitude, Phase)

User is guided through all stages of calibration. Each of the three cal stages can be refreshed. No calibration shows as a ‘red’

acknowledgement.

ƒVector calibration

ƒPhase calibration

ƒAmplitude calibration

Nonlinear VNA Measurements

Guided Calibration Wizard – Vector Calibration

The Vector calibration uses the

standard PNA-X cal wizard to

generate the associated vector

correction error model. Users can

enter their own cal kit definitions.

Nonlinear VNA Measurements

Guided Calibration Wizard – Phase Calibration

User is prompted to connect the phase reference calibration

standard. Standard is USB controlled. This calibrates the cross-frequency phase.

New!

Nonlinear VNA Measurements

Guided Calibration Wizard – Amplitude Calibration

User is prompted to connect the power sensor calibration standard.

Supports any power meters/sensors

the PNA-X FW supports today and in

the future.

Nonlinear VNA Measurements

Measurement Display

The vector corrected responses from the device can

be displayed in time domain.

Nonlinear VNA Measurements

Measurement Display – Power Sweep and Phase

Can also analyze the device performance versus power.

This example illustrates the phase of the second

harmonic from the device as

a function of input power.

Nonlinear VNA Measurements

Measurement Display

User can customize the

displayed waveforms such as the voltage or current applied to and emanating from the device.

ƒ‘a’ and ‘b’ waves versus sweep domain

ƒV and I versus sweep domain

ƒV versus I, I versus V

ƒEtc...

Dynamic Load Line (RF I/V)

Nonlinear VNA Measurements

In-fixture/De-embedding

User can move the absolute amplitude and cross-frequency phase plane by de-embedding S- parameters of the fixture/adapter.

Eg: Move amplitude and phase

plane (in-coax) into the fixture

plane.

Multi-Envelope Domain

Memory Effects in Nonlinear Devices

( )

x t y t( )

1 1

1

Single frequency pulse with fixed phase and amplitude versus time

( ) ^{j j t}

x t = A e e^{θ − ω} ω1

ω1

2ω1

3ω1

( )

Multiple frequencies envelopes with time varying phase and amplitude

( ) _n( ) ^{j n t j nt}

n

y t B t e^φ e

∞ − Ω

=−∞

=

∑

ƒ Can measure envelope of the

fundamental and harmonics with NVNA error correction applied. Use to analyze memory effects in nonlinear devices.

ƒ Get vector corrected amplitude and phase of envelope.

ƒ Use to measure and analyze memory

effects in nonlinear devices.

Multi-Envelope Domain

Envelope Domain

Measure envelope of the

fundamental and harmonics with NVNA error correction applied. Use to analyze memory effects in

nonlinear devices.

Get vector corrected amplitude and phase of envelope.

Envelope resolution down to 16.7 ns

using narrowband detection mode.

X-Parameters: A New Paradigm for

Interoperable Measurement, Modeling, and Simulation of Nonlinear

Microwave and RF Components

z

Easy to measure at high frequencies

„

measure voltage traveling waves with a (linear) vector network analyzer (VNA)

„

don't need shorts/opens which can cause devices to oscillate or self-destruct

z

Relate to familiar measurements (gain, loss, reflection coefficient ...)

z

Can cascade S-parameters of multiple devices to predict system performance

z

Can import and use S-parameter files in electronic-simulation tools (e.g. ADS)

z

BUT: No harmonics, No distortion, No nonlinearities, …

Invalid for nonlinear devices excited by large signals, despite ad hoc attempts

Incident S ₂₁ Transmitted

S₁₁

Reflected S₂₂

Reflected

Transmitted Incident

b₁ a₁

b ₂

a ₂ S12

DUT

Port 1 Port 2

S-parameters

b ¹ = S ¹¹ a ¹ + S ¹² a ² b ² = S ²¹ a ¹ + S ²² a ²

S-parameters: linear measurement, modeling, & simulation

Linear Simulation:

Matrix Multiplication

Measure with linear VNA:

Small amplitude sinusoids

Model

Parameters:

Simple algebra

k 0 i ij

j a

k j

S b

a

₌

≠

=

Scattering Parameters – Linear Systems

1 11 12 1

2 21 22 2

b S S a

b S S a

⎡ ⎤ ⎡ ⎤ ⎡ ⎤

⎢ ⎥ ⎢ = ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦ ⎣ ⎦

1 11 1 12 2

2 21 1 22 2

b S a S a b S a S a

= +

= +

Linear Describing Parameters

• Linear S-parameters by definition require that the S-parameters of the device do not change during measurement.

S-Parameter Definition

To solve VNA’s traditionally use a forward and reverse sweep (2 port error correction).

S21

S12

S11 S₂₂

10

e1

01

e1 00

e1 e₁¹¹

0

a1

0

b1

1

a1

1

b1

01

e2

10

e2 11

e2 e₂⁰⁰

1

b2

1

a2

0

b2

0

a2

( )

x t y t( )

Scattering Parameters – Linear Systems

11 12

1 1 1 1

21 22

2 2 2 2

f r f r

f r f r

S S

b b a a

S S

b b a a

⎡ ⎤ ⎡ ⎤ ⎡ ⎤

⎢ ⎥ = ⎢ ⎥ ⎢ ⎥

⎣ ⎦

⎣ ⎦ ⎣ ⎦

1 11 12 1

2 21 22 2

b S S a

b S S a

⎡ ⎤ ⎡ ⎤ ⎡ ⎤

⎢ ⎥ ⎢ = ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦ ⎣ ⎦

Linear Describing Parameters

• If the S-parameters change when sweeping in the forward and reverse directions when performing 2 port error correction then by definition the resulting

computation of the S-parameters becomes invalid.

Hot S22

This is often why customers are

asking for Hot S22 because the match is changing versus input drive power and frequency (Nonlinear

phenomena). Hot S22 traditionally measured at a frequency slightly offset from the large input drive signal.

( )

x t y t( )

1 11 1 12 2

2 21 1 22 2

b S a S a b S a S a

= +

= +

• Measurement based, device independent, identifiable from a simple set of automated NVNA measurements

• Fully nonlinear (Magnitude and phase of distortion)

• Cascadable (correct behavior in even highly mismatched environment)

• Extremely accurate for high-frequency, distributed nonlinear devices

Have the potential to revolutionize the

Characterization, Design, and Modeling of nonlinear components and systems

H a r m o n ic B a la n c e H B 2

E q u a t io n N a m e [ 3 ] = " Z lo a d "

E q u a t io n N a m e [ 2 ] = " R F p o w e r "

E q u a t io n N a m e [ 1 ] = " R F f r e q "

U s e K r y lo v = n o O r d e r [ 1 ] = 5 F r e q [ 1 ] = R F f r e q

H A R M O N I C B A L A N C E

NVNA:

Measure device X-parms

PHD component : Simulate using X-parms

ADS:

Design using X-parms

X-parameters are the mathematically correct extension of S-parameters to large-signal conditions.

X-parameters the nonlinear Paradigm:

X-parameters come from the

Poly-Harmonic Distortion (PHD) Framework

1_k 1_k

( ,

11

,

12

, ...,

21

,

22

, ...)

B = F D C A A A A

2_k 2_k

( ,

11

,

12

, ...,

21

,

22

, ...)

B = F D C A A A A

Port Index

Harmonic (or carrier) Index

Data and Model formulation in Frequency (Envelope) Domain

Magnitude and phase enables complete time-domain input-output waveforms

A 2

A 1

B 1 B ²

Unifies

S-parameters

Load-Pull,

Time-domain

load-pull

X-parameters: Systematic Approximations to NL Mapping Trade measurement time, size, accuracy for speed, practicality

+

Linear non-analytic map

( ) ( ) *

1 1

[ X

_kj^S

( D C A , ) A

_j

+ X

_kj^T

( D C A , ) A

_j

]

∑

Incident Scattered

Multi-variate NL map

1 2 3

( , , , , ...)

B

k

D C A A A

Simpler NL map

( )

( ,

1

, 0, 0, 0, ...)

F

X

k

D C A

≈

X-parameter Experiment Design & Identification

DC f ₀ 2f ₀ 3f ₀ 4f ₀ 5f ₀ -f ₀

-2f ₀ -3f ₀

, ,

( )

11

( ) ( )

, 1 1

, ,

1

*

(| |) ( ) (

1

)

ef gh ef gh

ef

S f

F f

e f

T f h

gh g

h gh

g h h

B = X A P + ∑ X A P

⁻

⋅ a + ∑ X A P

⁺

⋅ a ^{P = e}

^jϕ(A11)

There are two contributions at each harmonic

From different orders in NL conductance of driven system

a _j2 a _j2 ^*

0 1

5 f − f

2

( ) *

3, 2 _j T

i j

X ⋅ a

0 1

f + f

²

( )

3, 2 _j S

i j

X ⋅ a

Scattering Parameters – Nonlinear Systems X-parameters

Definitions

• i = output port index

• j = output frequency index

• k = input port index

• l = input frequency index

Description

• The X-parameters provide a

mapping of the input and output frequencies to one another.

(

^*

)

, (1,1)

( ) ( ) ( )

11 , 11 , 11

( )

^j

( )

^{j l}

( )

^{j l}

ij kl kl

k l

F S T

ij ij kl ij kl

b X a P X a P

⁻

a X a P

⁺

a

≠

= + ∑ ⋅ + ⋅

X-Parameters Collapse to S-Parameters in Linear Systems

Definitions

• i = output port index

• j = output frequency index

• k = input port index

• l = input frequency index

1 11 1 12 2

2 21 1 22 2

b S a S a b S a S a

= +

= +

For small |a₁₁| (linear), X^T terms go to 0.

Cross-frequency terms also go to 0

( )

( ) (

, ( 1

1,

) ,

1)

(

1

)

F j S j l

ij kl

l j k

ij j kl

l

b X a P X

i

P

⁻

a

=≠

= + ∑ ⋅

Consider fundamental frequency (j = 1).

Harmonic index is no longer needed.

( )

)

1

( ( )

(

11

)

F S

i ik

i k

k

P a

a

b X X

≠

= + ∑ ⋅

Assume 2 port (i and k = 1 ->

( ) ( )

1 11 1

1 2

( ) ( )

2 11 2

2

2 2 2

( ) ( )

F S

F S

b P a

b a a

a P

X X

X X

= + ⋅

= + ⋅

1 11 1 12 2

2 21 1 22 2

b S a P S a b S a P S a

= +

= +

For small |a₁₁| (linear), X^F terms go to S_i1·|a₁₁|,

and X^S terms are equal to linear S parameters

(

^*

)

, (1,1)

( ) ( ) ( )

11 , 11 , 11

( )

^j

( )

^{j l}

( )

^{j l}

ij kl kl

k l

F S T

ij ij kl ij kl

b X a P X a P

⁻

a X a P

⁺

a

≠

= + ∑ ⋅ + ⋅

By definition, ¹

1

P a

= a

Example: Fundamental Component

11

( )

21,11 11 21

| | 0

( )

S

X A

A

s

→

→

11

( )

21,21 11

| | 0

( ) 0

T

X A

A

→

→

-60 -25 -20 -15 -10 -5 0 5 10

|A

₁₁

| (dBm)

-40 40 20

0 -20 dB

( ) 21,21

X

S

( ) 21,21

X

T ( )

21,11

X

S

Reduces to (linear) S-parameters in the appropriate limit

( ) ( )

2 1 1 1

( ) 2 1, 2 1

2 *

2 1

(

1 1

) X

^F

( )

^S, 2 1

(

1 1

)

2 1 ^T 2 1

(

1 1

)

2 1

B A = A P + X A A + X A P A

11

( )

21,21 11 22

| | 0

( )

S

X A

A

s

→

→

( ) 21,21

( ) ( )

21,11 11 1

2 *

21

(

11

)

^S

( )

1 ^S

(

11

)

21 21,21^T

(

11

)

21

B A = X A A + X A A + X A P A

( ) ( ) ( ) *

11 , 11 , 11

( ) ( ) ( )

F m S m n T m n

pm pm pm qn qn pm qn qn

B = X A P + X A P ⁻ A + X A P ⁺ A X-parameter Experiment Design & Identification

Ideal Experiment Design

Perform 3 independent experiments with fixed A

₁₁

using orthogonal phases of A

₂₁

E.g. functions for B _pm (port p, harmonic m)

( ) ( ) ( )

(1) ( ) ( ) (1) ( ) (1)

11 , 11 , 11

F m S m n T m n

pm pm pm qn qn pm qn qn

B =X A P +X A P ⁻ A +X A P ⁺ A

( ) ( ) ( )

(2) ( ) ( ) (2) ( ) (2)*

11 , 11 , 11

F m S m n T m n

pm pm pm qn qn pm qn qn

B =X A P +X A P ⁻ A +X A P ⁺ A

input A _qn output B _pm Im

Re Re

Im ^{( )}

(0) ( )

11

F m

pm pm

B =X A P

X-parameter Application Flow

MDIF File

NVNA

DUT

^{Φ Ref}

Measure X-Parameters

Automated DUT X-params measured on NVNA

Application creates data-specific instance Simulator

v2 v1

Connector

X1 MCA_ZX60_2522

MCA_ZX60_2522_1 fundamental_1=fundamental MCA_ZFL_11AD

MCA_ZFL_11AD_1 fundamental_1=fundamental

R R1 R=25 Ohm V_1Tone

SRC14 Freq=fundamental V=polar(2*A11N,0) V R R11 R=50 OhmDC_Block

DC_Block1 DC_Block

DC_Block2 I_Probe i2 I_Probe

i1

PHD instances

Data-specific simulatable instance

Compiled PHD Component simulates using data

-28 -26 -24 -22 -20 -18 -16 -14 -12 -10-8-6

-30 -4

-30 -20 -10 0

-40 10

.

.

-28 -26 -24 -22 -20 -18 -16 -14 -12 -10-8-6

-30 -4

-60 -50 -40 -30 -20 -10 0

-70 10

.

.

-28 -26 -24 -22 -20 -18 -16 -14 -12 -10-8-6

-30 -4

5 10 15 20

0 25

.

.

-28 -26 -24 -22 -20 -18 -16 -14 -12 -10-8-6

-30 -4

70 71 72 73 74 75

69 76

.

.

-28 -26 -24 -22 -20 -18 -16 -14 -12 -10-8-6

-30 -4

-135 -130 -125 -120

-140 -115

-28 -26 -24 -22 -20 -18 -16 -14 -12 -10-8-6

-30 -4

0 2 4 6 8 10 12

-2 14

.

.

-10 -5 0 5 10 15

-15 20

-30 -20 -10 0 10

-40 20

.

.

Simulation and Design

Compiled PHD component

Harmonic Bal:

magnitude & phase of harmonics, frequency dep.

and mismatch

Envelope:

Accurately simulate NB multi-tone or complex stimulus

Measurement on the NVNA

Switching from general measurements to X-parameter measurements is as simple as selecting “Enable X-parameters”

Measuring X-parameters for 5 harmonics at 5 fundamental frequencies with 15 power points each (75 operating points) can take less than 5 minutes

DC bias information can be measured using external instruments

(controlled by the NVNA) and included in the data

PHD Design Kit

The MDIF file containing measured X-parameters is imported into ADS by the PHD Design Kit, creating a component that can be used in

Harmonic Balance or Envelope simulations.

Measurement-based nonlinear design with X-parameters

v2 v1

Connector X1

MCA_ZX60_2522 MCA_ZX60_2522_1 fundamental_1=fundamental MCA_ZFL_11AD

MCA_ZFL_11AD_1 fundamental_1=fundamental

R R1 R=25 Ohm

V_1Tone SRC14

Freq=fundamental V=polar(2*A11N,0) V R

R11

R=50 Ohm DC_Block DC_Block1

DC_Block DC_Block2 I_Probe

i2 I_Probe

i1

Amplifier Component Models from individual X-parameter measurements

ZFL-AD11+

11dB gain, 3dBm max output power Source

Connector 80 ps delay

ZX60-2522M-S+

23.5dB gain, 18dBm max output power

Load

Results

Cascaded Simulation vs. Measurement

Red: Cascade Measurement

Blue: Cascaded X-parameter Simulation

Light Green: Cascaded Simulation, No X

^(T)

terms Dark Green: Cascaded Models, No X

^(S)

or X

^(T)

terms

-28 -26 -24 -22 -20 -18 -16 -14 -12 -10 -8 -6

-30 -4

68 70 72 74

66 76

Pinc

unwrap(phase(b2[::,1]))

Pincref

unwrap(phase(b2ref[::,1]))unwrap(phase(b2NoT[::,1]))unwrap(phase(b2NoST[::,1]))

Fundamental Phase Second Harmonic % Error

-28 -26 -24 -22 -20 -18 -16 -14 -12 -10 -8 -6

-30 -4

20 40 60 80

0 100

Pinc

(b2[::,2]-b2ref[::,2])/b2ref[::,2]*10(b2NoT[::,2]-b2ref[::,2])/b2ref[::,2]*(b2NoST[::,2]-b2ref[::,2])/b2ref[::,2]

Large-mismatch capability (load-pull)

Full Nonlinear Dependence on both A1 & A2 [13]

Fundamental Gain Fundamental Phase

-25 -20 -15 -10 -5 0 5

-30 10

7.0 7.5 8.0 8.5 9.0 9.5 10.0

6.5 10.5

Pinc

-25 -20 -15 -10 -5 0 5

-30 10

135 140 145 150 155

130 160

Pinc

Mismatched Loads: 0 ≤ |Γ| ≤ 1

Fundamental frequency: 4 GHz

PHD Behavioral Model (solid blue) Circuit Model (red points) ( )

1 1 2 1

( , , 0, 0, ...)

F

ik ik

B = X A A + Terms linear in the

remaining components

Large-mismatch capability (load-pull)

Magnitude Phase

Second Harmonic

Third Harmonic

-25 -20 -15 -10 -5 0 5

-30 10

-60 -40 -20 0

-80 20

Pinc

-25 -20 -15 -10 -5 0 5

-30 10

-240 -220 -200 -180 -160 -140 -120

-260 -100

Pinc

-25 -20 -15 -10 -5 0 5

-30 10

-80 -60 -40 -20 0 20 40

-100 60

Pinc

-25 -20 -15 -10 -5 0 5

-30 10

-100 -80 -60 -40 -20 0

-120 20

Pinc

Mismatched Loads: 0 ≤ |Γ| ≤ 1

Fundamental frequency: 4 GHz

PHD Behavioral Model (solid blue) Circuit Model (red points)

Large-mismatch capability (load-pull)

100 200 300 400

0 500

1 2 3 4 5

0 6

time, psec

100 200 300 400

0 500

0.02 0.04 0.06

0.00 0.08

time, psec

1 2 3 4 5

0 6

0.02 0.04 0.06

0.00 0.08

Port 2 Voltage Port 2 Current

Dynamic Load-lines (green for matched case)

Mismatched Loads: 0 ≤ |Γ| ≤ 1

Fundamental frequency: 4 GHz

PHD Behavioral Model (solid blue) Circuit Model (red points)

PHD-Simulated vs Load-Pull Measured Contours & Waveforms from Load-dependent X-parameters (WJ transistor)

m1 m2

( ) ( ) y

m9 m10

Power Delivered

Efficiency Red: Load-pull data

Blue: PHD model simulated

0.2 0.4 0.6 0.8

0.0 1.0

-5 0 5

-10 10

-0.05 0.00 0.05 0.10

-0.10 0.15

time, nsec

MeasuredVoltage MeasuredCurrent SimulatedCurrent, A

SimulatedVoltage, V

Measured & Simulated Waveforms

Fundamental Gamma = 0.383+j*0.31

WJ Transistor

Load-Dependent Measured X-parameters ^[16]

Summary

X-parameters are a mathematically correct superset of S-

parameters for nonlinear devices under large-signal conditions

– Rigorously derived from general PHD theory; flexible, practical, powerful

X-parameters can be accurately measured by automated set of experiments on the new Agilent NVNA instrument

Together with the PHD component, measured X-parameters can be used in ADS to design nonlinear circuits

All pieces of the puzzle are available and they fit together!

Nonlinear

Measurements

Customer Applications

Nonlinear Modeling

Nonlinear

Simulation

Summary

New Nonlinear Vector Network Analyzer based on a standard PNA-X

New phase calibration standard

Vector (amplitude/phase) corrected nonlinear measurements from 10 MHz to 26.5 GHz

Calibrated absolute amplitude and relative phase (cross-frequency relative phase) of measured spectra traceable to standards lab

26 GHz of vector corrected bandwidth for time domain waveforms of voltages and currents of DUT Multi-Envelope domain measurements for

measurement and analysis of memory effects

X-parameters: Extension of Scattering parameters into the nonlinear region providing unique insight into nonlinear DUT behavior

X-parameter extraction into ADS PHD block for

nonlinear simulation and design

Configuration Details

Description

N5242A - 400 10 MHz to 26.5 GHz 4-port

Opt 419: 4-port attenuators & bias tees

Opt 423: Source switching & combining network Opt 080: Frequency Offset Mode

Opt 510: Nonlinear component characterization measurements (requires 400,419,080)

Opt 514: Nonlinear X-parameters

(requires 423, opt 510 & ext. source) Opt 518: Nonlinear pulse envelope domain

(requires opt 510,021,025) Accessories:

U9391C Phase reference (2 units required) Power meter & sensor or USB power sensor 3.5mm calibration standards

External source for X-parameters