Advanced Nonlinear Device Characterization Utilizing New Nonlinear Vector Network Analyzer and X-parameters presented by:
Loren Betts Research Scientist
Presentation Outline Device Characteristics (Linear and Nonlinear) NVNA Hardware (The need for phase) NVNA Error Correction NVNA Measurements Component Characterization Multi-Envelope Domain X-Parameters
Nonlinear Hierarchy from Device to System Systems Modules
n tio a riz g e t n ac deli r a Ch d mo an
Chips Circuits
Instrument Front-ends Cell Phones Military systems Etc.
Devices
Amplifiers, Mixers, Multipliers, Modulators, …. Transistors, diodes, …
Nonlinearities
y (t )
x(t )
ω1
x(t ) = Ae
2ω1
ω1 j ( w0t +φ0 )
y (t ) = a0 + b0 e
jθb0
x(t )
y (t ) = a0 + b0e
jθb0
3ω1
Ae j (ω0t +φ0 )
+ c0 e
jθc0
x(t ) 2
+c0e
jθc0
A2e j 2(ω0t +φ0 )
+ d0e
jθ d0
x(t )3
+ d0e
jθ d0
A3e j 3(ω0t +φ0 )
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.
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: Convert to time domain waveforms (eg: scope mode).
y = a0 + b0e +c0e
jθ c0
+ d0e
jθ d0
jθb0
Ae j (ω0t +φ0 )
A e 2
j 2(ω0t +φ0 )
A e 3
j 3(ω0t +φ0 )
Measure phase relationships between harmonics.
Generate model coefficients.
Etc…..
Measure frequency multipliers.
NVNA Hardware Take a standard PNA-X and add nonlinear measurement capabilities
Agilent’s N5242A premier performance microwave network analyzer offers the highest performance, plus:
2- and 4-port versions Built-in second source and internal combiner for fast, convenient measurement setups Spectrally pure sources (-60 dBc) Internal modulators and pulse generators for fast, simplified pulse measurements Flexibility and configurability Large touch screen display with intuitive user interface
Measuring Unratioed Measurements on PNA-X Unratioed Measurements – Amplitude Works fine.
a10
a20
Ever tried to measure phase across frequency on an unratioed measurement? b10
Sweep 1
b20
a11
b21
b11
a12
Sweep 2
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
Phase Shifted
Sweep 2
NVNA Hardware Configuration Generate Static Phase 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. 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. 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 One phase reference is used to maintain a static cross-frequency phase relationship. A second phase reference standard is used to calibrate the cross-frequency phase at the device plane. 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). The cross-frequency phase relationship remains static.
Mathematical Representation of Pulsed DC Signal y (t ) = ( rect pw (t )) ∗ shah
1 prf
(t )
∑n δ(t-n(1/prf)) rectpw (t) t -1/2pw
0
1/2pw
*
...
...
...
... t
-2/prf -1/prf
0
1/prf 2/prf
-1/prf
Y ( 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 )
0
1/prf
Frequency Domain Representation of Pulsed DC Signal Frequency Response Drive phase reference with a Fin frequency.
Want to stimulate DUT with 1 GHz input stimulus and measure harmonic responses at 1, 2, 3, 4, 5 GHz. Fin = 1 GHz Can practically use frequency spacings less than 1 MHz
Normalized Amplitude
Example:
0.8
0.6
0.4
0.2
0 0
10
20
30
40
50
60
70
80
90
100
Frequency in GHz 1
Normalized Amplitude
Get n*Fin at the output of the phase reference.
1
0.8
0.6
0.4
0.2
0 0
5
10
15
Frequency in GHz
20
25
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. 1 0.5
cos(ω0t + φ0 )
1 GHz
0 -0.5 -1 0
0.1
0.2
0.3
0.4
0.5 Frequency - 1 GHz
0.6
0.7
0.8
0.9
1
1 0.5
cos(2ω0t + φ1 )
2 GHz
0 -0.5
Phase relationship between frequencies remains the same
-1 0
0.1
0.2
0.3
0.4
0.5 Frequency - 2 GHz
0.6
0.7
0.8
0.9
1
1 0.5
cos(3ω0t + φ2 )
3 GHz
0 -0.5 -1 0
0.1
0.2
0.3
0.4
0.5 Frequency - 3 GHz
0.6
0.7
0.8
0.9
1
1 0.5
cos(4ω0t + φ3 )
4 GHz
0 -0.5 -1 0
0.1
0.2
0.3
0.4
0.5 Frequency - 4 GHz
0.6
0.7
0.8
0.9
1
Example NVNA Configuration #1 Using external source for phase reference drive Source 1 Source 2 (standard)
OUT 2
OUT 1
OUT 1
J11
J10
J9
J8
OUT 2
J7
SW1
J4
SW3
Test port 1
D 35 dB
65 dB
Test port 3
J1
R2
C 35 dB
J2
SW2
R4
A 65 dB
Rear panel
SW4
R3
R1
J3
B 35 dB 65 dB
65 dB
35 dB
Test port 2
Test port 4 To port 1 or 3 Ext Source in
Calibration Phase Reference
Measurement Phase Reference
Example NVNA Configuration #2 Multi-Tone, SCMM, DC/RF, Calibrated receiver mode Source 1 OUT 1
Source 2
OUT 2
OUT 2 OUT 1
Rear panel
R3
R1
R4
A
C 35 dB
65 dB
Test port 1
R2 D
35 dB 65 dB
Test port 3
B 35 dB 65 dB
65 dB
Test port 4
35 dB
Test port 2 To port 1 or 3 Ext Source in
Calibration Phase Reference
Measurement Phase Reference
NVNA Error Correction Algorithms Generalized VNA HW
a1 → Incident voltage traveling wave
• The input and output waves from a two port device are measured. Systematic measurement hardware errors prevent accurate measurements of the device. • Calibration and error correction provide the means to get an accurate representation of the device characteristics.
b1 → Reflected voltage traveling wave Zo → Normalization term V1 → Voltage applied to port 1 of device I1 → Current applied to port 1 of device V1 = Zo [ a1 + b1 ] → Incident voltage wave + Reflected voltage wave I1 =
[ a1 − b1 ] → Incident current wave Zo
Therefore (in units of 1 a1 = [ V1 + I1Zo ] 2 Zo a10
a20
b10
b20
b1 =
− Reflected current wave
Watts),
1 [ V1 − I1Zo ] 2 Zo
Incident Power = a1
2
Reflected Power = b1
2
Therefore (in units of Volts), a11
b21
b11
a12
2
1 a1 = [ V1 + I1Zo ] 2
a Incident Power = 1 Zo
1 b1 = [ V1 − I1Zo ] 2
b1
Reflected Power =
2
Zo
NVNA Error Correction Algorithms 12 Term Error Model - A 12 term error model is often used to eliminate systematic measurement errors. Assume crosstalk negligible e110
a10 e100 b10
a11 e111
S11
e101
b11
*
b21
S 21
b20
e201 *
e11 2
S22
a12
S12
Forward Direction a11 *
e111
b10
*
e101
S11 b11
e201
b21
S 21
e11 2
S 22 S12
Reverse Direction
a12
b20 e200
e10 2
a20
NVNA Error Correction Algorithms 12 Term Error Model - Terminology - Common terminology used today
e100 = Port 1 Directivity(dp1 )
e200 = Port 2 Directivity(dp2 )
e111 = Port 1 Source Match(smp1 )
e211 = Port 2 Source Match(smp2 )
*
e11 2 = Forward Load Match(lm fwd )
*
e111 = Reverse Load Match(lmrev )
e110 e101 = Forward Reflection Tracking(rt fwd )
01 e10 2 e2 = Reverse Reflection Tracking(rt rev )
10 01* 1 2
10 01* 2 1
e e
= Forward Transmission Tracking(tt fwd )
Port 1
e e
= Reverse Transmission Tracking(ttrev )
Port 2
NVNA Error Correction Algorithms Generalized 8 Term Error Model • The 8 term model accounts for changes in the match of the source and load by either measuring all the ‘a’ and ‘b’ waves or by calculating the ‘a’ waves from match coefficients (like delta match).
a10
e110
e100 b10
a11 e111
e101
S11 b11
e11 2
S22 S12
e201
b21
S21
a12
b20 e200
e10 2
a20
NVNA Error Correction Algorithms Conversion of 12 Term Model to 8 Term Model • To utilize the standard vector calibration algorithms a conversion is done to generate the 8 term model from the 12 term model. a10
e110
S11
e101
b11
*
b21
S 21
e111
e100
b10
a11
*
Γ 02
e200
e11 2
S 22
e10 2
a12
S12
b20
e201
a20
Forward Direction
a10
e110 *
e100
Γ10
b10
a11 e111
*
e101
S11 b11
e201
b21
S 21
S12
Reverse Direction
e200
e11 2
S 22 a12
b20
e10 2
a20
NVNA Error Correction Algorithms (17)
Conversion Equations • The conversion relationship equations to map the 12 term coefficients to the 8 term coefficients
e100 = dp1 11 1
11* 1
e =e
e200 = dp2 rt fwd Γ10 e110e101Γ10 − = lmrev − = smp1 1 − e100Γ10 1 − dp1Γ10
e110 e101 = rt fwd
11 2
11* 2
e =e
01 0 rtrev Γ 02 e10 2 e2 Γ 2 − = lm fwd − = smp2 1 − e200Γ 02 1 − dp2 Γ 02
01 e10 2 e2 = rt rev *
e110 e201 = e110 e201 1 − e200 Γ 02 = tt fwd 1 − dp2 Γ 02 a10 0 Γ1 = 0 b1
Port 1
*
01 00 0 0 e210e101 = e10 2 e1 1 − e1 Γ1 = tt rev 1 − dp1Γ1 a20 0 Γ2 = 0 b2
Port 2
NVNA Error Correction Algorithms Conversion Equations • Instead of calculating the gamma terms we can instead directly calculate the 8 term model tracking coefficients from the 12 term coefficients.
Γ 02 =
lm fwd − smp2 rtrev + dp2 lm fwd − smp2
e110 e201 = tt fwd 1 − dp2 Γ 02
10 01 1 2
e e = tt fwd
lm fwd − smp2 1 − dp2 rtrev + dp2 lm fwd − smp2
Port 1
Γ10 =
rt fwd
lmrev − smp1 + dp1 [lmrev − smp1 ]
01 0 e10 2 e1 = tt rev 1 − dp1Γ1
lmrev − smp1 01 1 e10 e tt dp = − 2 1 rev 1 rt fwd + dp1 [lmrev − smp1 ]
Port 2
NVNA Error Correction Algorithms 8 Term Model Coefficients • We now have the 8 term model coefficients…however we need to isolate the terms to relate the amplitude and cross-frequency phase.
e100
e200
e111
e11 2
e110 e101
01 e10 2 e2
e110 e201
01 e10 2 e1
Port 1
Port 2
NVNA Error Correction Algorithms Isolating Coefficients - Amplitude • Error model of VNA port and amplitude (power sensor and meter) calibration device. This isolates the amplitude of one of the tracking coefficients. a10
e110 e111
e100 b10
a11 = a10e110 + b11e111
a11
e101
es01
bs0
e11 s b11
b10 = a10 e100 + b11e101 a11 =
01 2 1
e
1 0 11 0 10 01 00 11 b e + a1 e1 e1 − e1 e1 01 1 1 e1
=
b e + a e e − e e 0 11 1 1
0 1
10 01 1 1 1 2 1
a
00 11 1 1
2 1 1
1 2 1
→ The power meter returns the power of a = a
NVNA Error Correction Algorithms Isolating Coefficients - Phase • Error model of VNA port and phase reference (harmonic comb generator) calibration device. This isolates the phase of one of the tracking coefficients by relating the phase (cross-frequency phase) at all frequencies. a10 e110 a11
e100 b10
0 1
b = e101 =
aφ0 eφ10 e101 + a10 e100 1 − e111eφ11 + a10 e110 eφ11e101 1 − e111eφ11
e101
=
eφ11
e111 b11
eφ10
aφ0
aφ0 eφ10 e101 + a10 e100 − a10 eφ11 e100 e111 − e110 e101 1 − e111eφ11
b10 1 − e111eφ11 − a10 e100 + a10 eφ11 e100 e111 − e110 e101 aφ0 eφ10
b10 1 − e111eφ11 − a10 e100 − eφ11 e100 e111 − e110 e101 φ ( e101 ) = φ → The phase reference term aφ0 eφ10 is known 0 10 aφ eφ
NVNA Error Correction Algorithms Isolating the Rest of the Coefficients in the 8 Term Model • We now have the 8 term model coefficients…however we need to isolate the terms to relate the amplitude and crossfrequency phase.
e100
e200
e111
e11 2
e110 e101
01 e10 2 e2
e110 e201
01 e10 2 e1
Terms already isolated Terms to isolate Calculation path
e100
e200
e111
e11 2
e101 e110 e101 e = 01 e1 10 1
Port 1
Isolate amplitude and crossfrequency phase using power sensor and phase reference
10 01 e e e201 = 2 102 e2 01 e10 2 e1 e = 01 e1 10 2
Port 2
NVNA Error Correction Algorithms Error Correction Matrix • We now have isolated all the error coefficients in the 8 term model and can now relate the uncorrected waves to the corrected wave of the DUT. Notice 01 each ‘R’ term is multiplied by e101 and e2 which provide the cross-frequency phase relationship between the uncorrected and corrected ‘a’ and ‘b’ waves. a10
e110
e100 b10
a11 R100 1 10 b1 = R1 a12 0 1 b2 0
a11 e111
e101
S11 b11
R101 R111
0 0
0 0
R200 R210
0 a10 0 b10 R201 a20 R211 b20
e201
b21
S21
e11 2
S22 S12
a12
b20 e200
e10 2
a20
R100 =
1 10 01 00 11 e1 e1 − e1 e1 e101
R200 =
1 10 01 00 11 e2 e2 − e2 e2 e201
R101 =
1 11 e1 e101
R201 =
1 11 e2 e201
R110 =
1 −e100 01 e1
R210 =
1 −e200 01 e2
R111 =
1 e101
R211 =
1 e201
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 1 GHz
Stimulus to device
Can also measure frequency multipliers
Fundamental Frequency Source harmonics (< 60 dBc)
Output from device 2 GHz Output harmonics
Input output frequencies at device terminals
1 GHz 122.1 degree phase delta
Phase relationship between frequencies at output of device
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) Input Waveform
Stimulus is 5 frequencies spaced 10 MHz apart centered at 1 GHz measuring all spectrum to 20 GHz Output Waveform
Measure amplitude AND PHASE of intermodulation products Generated using external source (PSG/ESG/MXG) using NVNA and vector calibrated receiver
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 (8th harmonic) Multi-tone often used to mimic more complex modulation (i.e. CDMA) by matching complementary cumulative distribution function (CCDF). Multitone 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 (8th harmonic)
Multi-Envelope Domain Memory Effects in Nonlinear Devices x(t )
y (t )
ω1
ω1 x(t ) = A1 e jθ1 e− jω1t Single frequency pulse with fixed phase and amplitude versus time
2ω1
Can measure envelope of the fundamental and harmonics with NVNA error correction applied. Use to analyze memory effects in nonlinear devices. 3ω1
Get vector corrected amplitude and phase of envelope. Use to measure and analyze memory effects in nonlinear devices.
∞
y (t ) =
∑
Bn (t ) e jφn ( t ) e− jΩnt
n =−∞
Multiple frequencies envelopes with time varying phase and amplitude
NVNA Applications (What does it do?) Measure memory effects in nonlinear devices with vector error correction applied
Output (b2) multi-envelope waveforms
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
Each harmonic has a unique time varying envelope signature
NVNA Applications (What does it do?) Measure modeling coefficients and other nonlinear device parameters
… More
X-parameters Waveforms (‘a’ and ‘b’ waves)
Dynamic Load Line
Measure, view and simulate actual nonlinear data from your device
X-Parameters:
A New Paradigm for Interoperable Measurement, Modeling, and Simulation of Nonlinear Microwave and RF Components
S-parameters: linear measurement, modeling, & simulation 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 Relate to familiar measurements (gain, loss, reflection coefficient ...) Can cascade S-parameters of multiple devices to predict system performance Can import and use S-parameter files in electronic-simulation tools (e.g. ADS) BUT: No harmonics, No distortion, No nonlinearities, … Invalid for nonlinear devices excited by large signals, despite ad hoc attempts
Linear Simulation: Matrix Multiplication
S-parameters b1 = S11a1 + S12a2 b2 = S21a1 + S22a2
Model Parameters: Simple algebra
Measure with linear VNA: Small amplitude sinusoids S 21
Incident
Transmitted
a1 b2
S 11 Reflected
DUT Port 2
Port 1
b1
S 22 Reflected a2
Transmitted
S 12
Incident
bi S ij = aj
ak = 0 k≠ j
Scattering Parameters – Linear Systems Linear Describing Parameters • Linear S-parameters by definition require that the S-parameters of the device do not change during measurement.
y (t )
x(t )
a10
e110
e111
e100
b10
a11
e101
S11 b11
e201
b21
S21 S12
e200
e11 2
S22 a12
b20
e10 2
a20
b1 = S11a1 + S12 a2 b2 = S21a1 + S22 a2 b1 S11 b = S 2 21
S12 a1 S22 a2
S-Parameter Definition To solve VNA’s traditionally use a forward and reverse sweep (2 port error correction).
Scattering Parameters – Linear Systems 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.
b1 = S11a1 + S12 a2 b2 = S 21a1 + S 22 a2 x(t )
b1 S11 b = S 2 21
b b
f 1 f 2
b S11 = b S21 r 1 r 2
S12 a1 S 22 a2
S12 a S22 a
f 1 f 2
y (t )
Hot S22
a a r 1 r 2
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-parameters the nonlinear Paradigm: Have the potential to revolutionize the Characterization, Design, and Modeling of nonlinear components and systems X-parameters are the mathematically correct extension of S-parameters to large-signal conditions. • 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 NVNA: Measure device X-parms
PHD component : Simulate using X-parms
ADS: Design using X-parms H A R M O N IC B A L A N C E H a r m o n ic B a la n c e HB2 F r e q [1 ]= R F f r e q O r d e r[1 ]= 5 U s e K r y lo v = n o E q u a t io n N a m e [ 1 ] = " R F f r e q " 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 [ 3 ] = " Z lo a d "
X-parameters come from the Poly-Harmonic Distortion (PHD) Framework
A1
A2
B1 k = F1 k ( D C , A11 , A12 , ..., A21 , A22 , ...) B 2 k = F2 k ( D C , A11 , A12 , ..., A21 , A22 , ...) Harmonic (or carrier) Index
B1
B2
Port Index
Unifies S-parameters Load-Pull, Time-domain load-pull
Data and Model formulation in Frequency (Envelope) Domain Magnitude and phase enables complete time-domain input-output waveforms
X-parameters: Systematic Approximations to NL Mapping Trade measurement time, size, accuracy for speed, practicality
Incident
Scattered B k ( D C , A1 , A 2 , A 3 , ...) Multi-variate NL map
≈ X k( F ) ( D C , A1 , 0, 0, 0, ...) Simpler NL map
+ Linear non-analytic map (S ) (T ) * [ X ( D C , A ) A + X ( D C , A ) A ∑ kj 1 j kj 1 j ]
X-parameter Experiment Design & Identification -3f0
-2f0
-f0
DC
f0
2f0
aj2*
3f0
4f0
5f0
aj2
X i(3,T )j 2 ⋅ a*j 2 X i(3,S )j 2 ⋅ a j 2
There are two contributions at each harmonic From different orders in NL conductance of driven system
5 f 0 − f1
f 0 + f1
Be , f = X ef( F ) (| A11 |) P f + ∑ X ef( S,g)h ( A11 ) P f −h ⋅ agh + ∑ X ef(T,g)h ( A11 ) P f + h ⋅ a*gh g ,h
g ,h
P = e jϕ ( A11 )
Scattering Parameters – Nonlinear Systems X-parameters bij = X ij( F ) ( a11 ) P j +
∑ (X
(S ) ij , kl
( a11 ) P j −l ⋅ akl + X ij(T,kl) ( a11 ) P j +l ⋅ akl* )
k ,l ≠ (1,1)
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.
X-Parameters Collapse to S-Parameters in Linear Systems By definition, P =
bij = X ij( F ) ( a11 ) P j +
∑ (X
(S ) ij , kl
a1 a1
( a11 ) P j −l ⋅ akl + X ij(T,kl) ( a11 ) P j +l ⋅ akl* )
k ,l ≠ (1,1)
For small |a11| (linear), XT terms go to 0. Cross-frequency terms also go to 0
bij = X ij( F ) ( a11 ) P j +
l= j k ,l ≠ (1,1)
b1 = S11a1 + S12 a2 b2 = S21a1 + S22 a2 Definitions • i = output port index • j = output frequency index • k = input port index • l = input frequency index
∑ (X
b1 = S11 a1 P + S12 a2 b2 = S21 a1 P + S22 a2 For small |a11| (linear), XF terms go to Si1·|a11|, and XS terms are equal to linear S parameters
(S ) ij , kl
P j −l ⋅ akl )
Consider fundamental frequency (j = 1). Harmonic index is no longer needed.
bi = X i( F ) ( a11 ) P + ∑ ( X ik( S ) ⋅ ak ) k ≠1
Assume 2 port (i and k = 1 -> 2)
b1 = X 1( F ) ( a11 ) P + X 1(2S ) ⋅ a2 b2 = X 2( F ) ( a11 ) P + X 2(2S ) ⋅ a2
Example: Fundamental Component (S ) (T ) 2 * B 21 ( A11 ) = X 2( 1F ) ( A11 ) P + X 21 ( A ) A + X ( A ) P A , 21 11 21 11 21 2 1, 21
(S ) (S ) (T ) B21 ( A11 ) = X 21,11 ( A11 ) A11 + X 21,21 ( A11 ) A21 + X 21,21 ( A11 ) P 2 A2*1
dB
40
(S ) X 21,11
(S ) X 21,11 ( A11 ) → s21 | A11 | → 0
20 (S ) X 21,21
0
(S ) X 21,21 ( A11 ) → s22 | A11 |→ 0
-20 -40 -60
-25
X
-20 -15 -10
(T ) X 21,21 ( A11 ) → 0
(T ) 21,21
|A11| (dBm)
-5
| A11 |→ 0
0
5
10
Reduces to (linear) S-parameters in the appropriate limit
X-parameter Experiment Design & Identification Ideal Experiment Design E.g. functions for Bpm (port p, harmonic m) (F ) (S ) m−n (T ) m+n * B pm = X pm ( A11 ) P m + X pm ( A ) P A + X ( A ) P Aqn , qn 11 qn pm , qn 11 Perform 3 independent experiments with fixed A11 using orthogonal phases of A21
output Bpm
input Aqn
(0) (F) Bpm = X pm ( A11 ) Pm
Im
Im (1) ( F) Bpm = Xpm ( A11 ) Pm + Xpm(S),qn ( A11 ) Pm−n Aqn(1) + X(pmT),qn ( A11 ) Pm+n Aqn(1)*
Re Re
(F ) m (S ) m − n (2) (2)* B (2) Aqn + X (pmT ),qn ( A11 ) P m + n Aqn pm = X pm ( A11 ) P + X pm , qn ( A11 ) P
X-parameter Application Flow Automated DUT X-params measured on NVNA
Simulator
Application creates data-specific instance Compiled PHD Component simulates using data
NVNA
PHD instances v1 I_Probe i1
R R11 R=50 Ohm DC_Block DC_Block1
MCA_ZFL_11AD Connector MCA_ZFL_11AD_1 X1 fundamental_1=fundamental
v2 I_Probe i2 MCA_ZX60_2522 MCA_ZX60_2522_1 fundamental_1=fundamental
R R1 R=25 Ohm
DC_Block DC_Block2
V_1Tone SRC14 V=polar(2*A11N,0) V Freq=fundamental
MDIF File
Data-specific simulatable instance Compiled PHD component
Harmonic Bal: Simulation and Design 76
25
75 20 74 15
.
73
.
DUT
Φ Ref
magnitude & phase of harmonics, frequency dep. and mismatch
72
10
71 5
70 69
0 -30
Envelope:
-28
-26
-24
-22
-20
-18
-16
-14
-12
-10
-8
-6
-30
-4
-28 -26 -24 -22 -20 -18
-16
-14
-12
-10
-8
-6
-4
-16
-14
-12
-10
-8
-6
-4
-16
-14
-12
-10
-8
-6
-4
.
. 10
-115
0
-120
-10
.
-125
-20
-130
-30
20
-135
-40
10
-140 -30
-28
-26
-24
-22
-20
-18
-16
-14
-12
-10
-8
-6
-4
-30 -28 -26
-24
-22
-20
-18
-30
-24
-22
-20
-18
.
0
-20
10
14
0
12
-10
10
-20
8
-30
.
-10 .
.
Measure X-Parameters
Accurately simulate NB multi-tone or complex stimulus
-40
-30
2
-60
0
-70
-15
-10
-5
0
5
10
15
20
-2 -30
-28
-26
-24
-22
-20
-18
-16
.
.
6 4
-50
-40
-14
-12
-10
-8
-6
-4
-28
-26
.
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 ZFL-AD11+ Source
11dB gain, 3dBm max output power
R R11 R=50 Ohm DC_Block DC_Block1 V_1Tone SRC14 V=polar(2*A11N,0) V Freq=fundamental
v1 I_Probe i1
Connector 80 ps delay
ZX60-2522M-S+ 23.5dB gain, 18dBm max output power
v2 MCA_ZFL_11AD Connector MCA_ZFL_11AD_1 X1 fundamental_1=fundamental
I_Probe i2
MCA_ZX60_2522 MCA_ZX60_2522_1 fundamental_1=fundamental
DC_Block DC_Block2
Load
R R1 R=25 Ohm
Amplifier Component Models from individual X-parameter measurements
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 (b2NoST[::,2]-b2ref[::,2])/b2ref[::,2]*100 (b2NoT[::,2]-b2ref[::,2])/b2ref[::,2]*100 (b2[::,2]-b2ref[::,2])/b2ref[::,2]*100
Fundamental Phase unwrap(phase(b2NoST[::,1])) unwrap(phase(b2NoT[::,1])) unwrap(phase(b2ref[::,1])) unwrap(phase(b2[::,1]))
76
74
72
70
68
66 -30
-28
-26
-24
-22
-20
-18
-16
Pinc Pincref
-14
-12
-10
-8
-6
-4
Second Harmonic % Error 100
80
60
40
20
0 -30
-28
-26
-24
-22
-20
-18
-16
Pinc
-14
-12
-10
-8
-6
-4
Large-mismatch capability (load-pull) Full Nonlinear Dependence on both A1 & A2 [13]
B ik = X
(F ) ik
Terms linear in the ( A11 , A21 , 0, 0, ...) +remaining components
Mismatched Loads: 0 ื |ˁ| ื 1
Fundamental Phase
Fundamental Gain 10.5
160
10.0
155
9.5 150
9.0
145
8.5 8.0
140
7.5 135
7.0 130
6.5 -30
-25
-20
-15
-10
-5
0
5
10
Pinc
-30
-25
-20
-15
-10
-5
0
Pinc
Fundamental frequency: 4 GHz PHD Behavioral Model (solid blue)
Circuit Model (red points)
5
10
Large-mismatch capability (load-pull) Mismatched Loads: 0 ื |ˁ| ื 1 Magnitude
Phase
20
-100 -120
0
-140
Second Harmonic
-160
-20
-180 -40
-200 -220
-60
-240 -80
-260 -30
-25
-20
-15
-10
-5
0
5
10
-30
-25
-20
-15
Pinc
-10
-5
0
5
10
-5
0
5
10
Pinc
20
60
0
40 20
-20
0
Third Harmonic
-40 -20 -60
-40
-80
-60
-100
-80
-120
-100
-30
-25
-20
-15
-10
-5
0
5
10
Pinc
-30
-25
-20
-15
-10
Pinc
Fundamental frequency: 4 GHz PHD Behavioral Model (solid blue)
Circuit Model (red points)
Large-mismatch capability (load-pull) Mismatched Loads: 0 ื |ˁ| ื 1
Dynamic Load-lines (green for matched case) 0.08
0.06
0.04
0.02
Port 2 Current
0.00
Port 2 Voltage
0
1
2
3
4
6
5
6
0.08
5 0.06
4 3
0.04
2 0.02
1 0
0.00
0
100
200
300
400
500
time, psec
0
100
200
300
400
time, psec
Fundamental frequency: 4 GHz PHD Behavioral Model (solid blue)
Circuit Model (red points)
500
PHD-Simulated vs Load-Pull Measured Contours & Waveforms from Load-dependent X-parameters (WJ transistor) Red: Load-pull data
d (blue) and Measured (red) Efficiency C
Blue: PHD model simulated Fundamental Gamma = 0.383+j*0.31 m9 m10
WJ Transistor
Measured & Simulated Waveforms 0.15 0.10
5
0.05 0 0.00 -5
-0.05 -0.10
-10 0.0
0.2
0.4
0.6
time, nsec
0.8
1.0
SimulatedCurrent, A MeasuredCurrent
Power Delivered
SimulatedVoltage, V MeasuredVoltage
10
m1 m2
Efficiency
Amplifier with bias; standard compliant wideband modulation source; Parametric sweep of A) source power (dBm), & B) electrical distance (degrees) from output of amp to 2nd harmonic notch filter [fundamental = 1GHz] ENVELOPE
PARAMETER SWEEP ParamSweep Sweep_pwr SweepVar="Pavs" Start=-8 Stop=0 Step=1
Var Eqn
VAR SourcePwr Pavs=-2 _dBm
V_DC VDC Vdc=VDC
Envelope Env1 Freq[1]=RFfreq Order[1]=3 Stop=tstop Step=tstep
I_Probe IDC
IS-95 CDMA Source
TLIN TL1 Z=50.0 Ohm E=T_line F=2 GHz
PARAMETER SWEEP ParamSweep Sweep_Tline SweepVar="T_line" Start=90 Stop=360 Step=30 Var VAR Eqn
VAR2 T_line=360
Harmonic Filter
Vin PtRF_CDMA_IS95_REV SRC6 F0=RFfreq Power=dbmtow(Pavs) Z=50 Ohm Num=1
Vout DC_Block DC_Block2
ZX60_2522M_DC_PHD ZX60_2522M_DC_PHD_1 fundamental_1=_freq1
s_harmfilter X1
DC_Block DC_Block1
Term Term1 Num=1 Z=50 Ohm
Gain in dB +21 to +24 in 0.5 dB steps (g_contour) constant GAIN contours of T-line length -vs- Pavs src pwr 360 340
Eqn gmin=21 Eqn gmax=24 Eqn gstp=0.5
level=22.000, number=3
320 300 280
Tline length to filter
260 240
level=21.000, level=21.500, number=2 number=2 level=22.000, number=2 level=22.500, number=2
220 200
level=24.000, number=1 level=23.500, number=1 level=23.000, number=1
180 160 140 120 level=22.500, number=1
100 level=22.000, number=1
level=21.500, number=1level=21.000, number=1
80 -8.0
-7.5
-7.0
-6.5
-6.0
-5.5
-5.0
-4.5
-4.0 Pavs
-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
Power delivered (dBm) +14 to +23 in 1 dB steps (P_contour) constant Channel Power contours of T-line length -vs- Pavs src pwr 360
Eqn pmin=14
340
Eqn pmax=23 Eqn pstp=1
320 300
level=19.000, number=2
280
Tline length to filter
260 240 220 level=23.000, number=1
200
level=22.000, number=1 level=21.000, number=1
180
level=20.000, number=1
160
level=19.000, number=1
140 120 100 level=14.000, number=1 level=15.000, number=1 level=16.000, number=1 level=17.000, number=1
80 -7.5
-7.0
-6.5
-6.0
-5.5
-5.0
-4.5
-4.0
-3.5
Pavs
-3.0
-2.5
-2.0
level=18.000, number=1
-1.5
-1.0
-0.5
0.0
Power Added Efficiency (%) 6% to 36% in 2% steps (PAE_contour) constant (percent) PAE contours of T-line length vs Pavs src power 360
Eqn pae_min=6
340
Eqn pae_max=36
320
Eqn pae_stp=2
300
level=22.000, number=2 level=20.000, number=2
280
level=18.000, number=3
Tline length to filter
260 240 220 200
level=36.000, level=34.000, level=32.000, level=30.000, level=28.000, level=26.000, level=24.000,
180
number=1 number=1 number=1 number=1 number=1 number=1 number=1
level=22.000, number=1
160
level=20.000, number=1 level=18.000, number=2
140
level=16.000, number=2
120 100
level=6.000, number=1 level=8.000, number=1 level=10.000, number=1 level=12.000, number=1 level=14.000, number=1 level=16.000, number=1 level=18.000, number=1
80 -8.0
-7.5
-7.0
-6.5
-6.0
-5.5
-5.0
-4.5
-4.0 Pavs
-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
Summary X-parameters are a mathematically correct superset of Sparameters 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 Nonlinear Modeling
Nonlinear Simulation Customer Applications
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
