An Introduction to
X-Parameters*
Thomas Comberiate
Department of Electrical and Computer Engineering
University of Illinois at Urbana-Champaign
*X-Parameters is a registered trademark of Agilent Technologies.
ECE 451: Advanced Microwave
Measurements
Scattering Parameters
• Models all linear time-invariant behavior.
• Can model time-invariant nonlinear devices in the small-signal case.
• What about the large-signal case?
ECE 451 2
1 11 1 12
2 1
2
2 1 22 2
S A S A
B S A S A
B
in out
1 2
in out
in out
in o
1
ut
2
2 2
2 2
C C
C C
C C
C C
V Z I V Z IA A
Z Z
V Z I V Z IB B
Z Z +
Vin
-
Iin
2-Port
Network+
Vout
-
Iout
ZC = characteristic
impedance of
the measurement
system
A1
A2
B2
B1
T. M. Comberiate
Nonlinear Functions
with Single-Tone Stimuli
ECE 451 3
1 1 1 2
2 1 22
( , )
( , )
B
B
F A A
F A A
large-signal
harmonics
generated
small-signal linear output
f
f
f
f
Nonlinear
Device
T. M. Comberiate
Nonlinear Functions
with Multi-Tone Stimuli
ECE 451 4
large-signal
harmonics and
intermodulations
generated
small-signal linear output
f f
f f
Nonlinear
Device
1 1 1 2
2 1 22
( , )
( , )
B
B
F A A
F A AT. M. Comberiate
Nonlinear
Devicef f
Nonlinear Functions with
Commensurate Tone Stimuli
ECE 451 5
• A set of pure tones are commensurate if all the tones
in the set are located on a frequency grid fk = kf0defined by f0, called the fundamental.
• Output tones will all land on the same frequency grid
and have a same common period.
f0 2f0 3f0 4f0 f0 2f0 3f0 4f0 5f0 6f0
1 1 1 2
2 1 22
( , )
( , )
B
B
F A A
F A AT. M. Comberiate
Nonlinear Scattering Waves
• Break incident and scattered waves into their commensurate tone components, called pseudowaves.
ECE 451 6
2B
2,1B
portharmonic
wave
pseudowave
2,2B 2,3B1, 1, 1,1 1,2 1,3 2,1 2,2 2,3
2, 1,1 1,2 1,3 2,1 2,2 2,32,
( , , ,... , , ,...)
( , , ,... , , ,...)
k
k
k
k
B
B
F A A A A A A
F A A A A A A
1 1 1 2
2 1 22
( , )
( , )
B
B
F A A
F A A
+
Vin
-
Iin
2-Port
Network+
Vout
-
Iout
ZC = characteristic
impedance of
the measurement
system
A1
A2
B2
B1
T. M. Comberiate
Cross-Frequency Phase for
Commensurate Tones
ECE 451 7
• Defined as the phase of each pseudowave when
the fundamental, A1,1, has zero phase.
• B2,3 can be related to A2,2 in magnitude and phase.
A1,1
A2,2
B2,3
T. M. Comberiate
Nonlinear Scattering Functions
• Scattered pseudowave determined by a complicated time-invariant scattering function that depends on the magnitude and phase of each incident pseudowave.
ECE 451 8
, , 1,1 1,2 1,3 2,1 2,2 2,3( , , ,..., , , ,...)p k p kF A A A A AB A
A2,1 A2,2 A2,3
B2,1 B2,2 B2,3 A1,1 A1,2 A1,3
B1,1 B1,2 B1,3
Nonlinear
Time-
Invariant
2-Port
Device
T. M. Comberiate
Time-Invariance Property of
Nonlinear Scattering Function
• Shifting all of the inputs by the same time means that different harmonic components are shifted by different phases.
ECE 451 9
2 3
, 1,1 1,2 1,3
, 1,1 1,2 1,3
( e , (e ) , (e ) ,...)
( , , ,...)(e )
j j j
p k
j k
p k
F A A A
F A A A
f0 180º phase shift
2f0 360º phase shift
3f0 540º phase shift
time delay
T. M. Comberiate
Shifting reference to
zero phase of A1,1.
Defining Phase Reference
• Can use time-invariance to separate magnitude and phase dependence of one incident pseudowave.
ECE 451 10
, , 1,1 1,2 1,3
2 3
, 1,1 1,2 1,3
( , , ,...)
( , , ,...)
p k p k
p k
k
F A A A
F A
B
A A P P P 1,1arg( )1,1
1,1
j AAP e
A
using
T. M. Comberiate
Commensurate Tones
X-Parameter Formalism
• Still difficult to characterize this nonlinear term.
• If only one incident pseudowave, A1,1, is large then the other smaller inputs can be linearized about the large-signal response of Fp,k to only A1,1.
ECE 451 11
( ) 2 3
, 1,1 1,2 1,3
, 1,1 1,2 1,3
( , , ,...)
( , , ,...)
FB
p k
k
p k
X A A P A P
F A A A P
• Define
( ) 2 3
, , 1,1 1,2 1,3( , , ,...)FB k
p k p kB X A A P A P P
T. M. Comberiate
Linearization of Fp,k about A1,1
ECE 451 12
1,1
1,1
2
, , 1,1 1,2 1,
, 1,1
,, , *
, ,*1, 1 ,
,( , ) 1
, , , ,
,0, ,0,
K k
p k p k K
k
p k
q N l Kp k p kk l k l
q l q lllq l q l
q lAq l A
B F A A P A P P
F A P
F FA P A P
A P A P
T. M. Comberiate
(S), ; ,p k q l
X (T), ; ,p k q l
X
Nonlinear Mapping
Simple Nonlinear Mapping
Nonanalytic Harmonic
Superposition
Incident Waves Scattered WavesApproximates
1-Tone X-Parameter Formalism
ECE 451 13
( ) 2 3
, , 1,1 1,2 1,3( , , ,...)FB
p k p kX A A P A PB
X
p,k
(FB)( A1,1
,0,0,...)
( ) ( ) *
, ; , , , ; , ,
S T
p k q l q l p k q l q lX A X A
A1,1Large-
Signal
frequency frequency
frequency
frequency
frequency
frequency
T. M. Comberiate
1-Tone X-Parameter Formalism
ECE 451 14
, ,
( ) ( ) ( ) *
, , , ; , , , ; , ,
1, 1 1, 1( , ) (1,1) ( , ) (1,1)
q N l K q N l KFB k S k l T k l
p k p k p k q l q l p k q l q l
q l q lq l q l
B X P X A P X A P
Simple
nonlinear mapLinear harmonic map
function of incident wave
Linear harmonic map
function of conjugate of
incident wave
( )
, ; ,
S
p k q lXoutput
port input
port
input
harmonicoutput
harmonic
• X-parameters of type FB, S, and T fully characterize the nonlinear function.
• Depend on– frequency
– large signal magnitude, |A1,1|
– DC bias
1,1
1,1
AP
A
T. M. Comberiate
ECE 451 15
X-Parameters Collapse to S-
Parameters in Small-Signal Limit, ,
( ) ( ) ( ) *
, , , ; , , , ; , ,
1, 1 1, 1( , ) (1,1) ( , ) (1,1)
q N l K q N l KFB k S k l T k l
p k p k p k q l q l p k q l q l
q l q lq l q l
B X P X A P X A P
As A1,1 shrinks, the conjugate terms and harmonic terms vanish:
( ) ( )
,1 ,1 ,1; ,1 ,1
2
q NFB S
p p p q q
q
B X P X A
Remove unnecessary harmonic index and assume 2-port:( ) ( )
1 1 1,2 2
( ) ( )
2 2 2,2 2
FB S
FB S
B X P X A
B X P X A
for small A1 and P ≡ arg(A1):( )
1 1 1 1
FB
p p pX P S A P S A
1 11 1 12 2
2 21 1 22 2
B S A
B S
S A
S A A
( )
, ; ,
S
p k q lXoutput
portinput
port
input
harmonicoutput
harmonicT. M. Comberiate
Generating X-Parameters
• Traditional Generation
– Simulated using harmonic balance.
– Measured with a nonlinear vector network
analyzer (NVNA).
ECE 451 16T. M. Comberiate
ECE 451 17
Harmonic BalanceAssume nodal voltages can be represented with Fourier series and solve for the Fourier coefficients.
1 2
1 1
1 2
1 2
2
, , ,
0 0 0
( ) Ren
n n
n
n
K K Kj k f k f t
k k k
k k k
v t V e
Initial Guess of
Node Voltages
Simulation
CompleteIFFT Nonlinear
Voltages to
Time Domain
Separate Linear
and Nonlinear
Component
Voltages
Calculate Linear
Currents in the
Frequency Domain
from Linear
Voltages
Calculate
Nonlinear
Currents in Time
Domain
FFT Nonlinear
Currents to
Frequency
Domain
KCL Error =
abs(Linear
Currents +
Nonlinear
Currents)
Linear
Components
Nonlinear
Components
KCL Error <
threshold?
Update Node Voltages
No
Yes
T. M. Comberiate
Generating X-Parameters with
Harmonic Balance• Need to set proper values for:
– Frequency range
– Fundamental power
– DC bias
• X-parameter measurements are unidirectional because of large-signal fundamental |A1,1| on one port.
• Different types of X-parameter ports:
ECE 451 18T. M. Comberiate
Source Load Bias
X-Parameter Generation
Example
ECE 451 19T. M. Comberiate
Nonlinear Vector Network
Analyzer (NVNA)
ECE 451 20T. M. Comberiate
PNA-X
• Four ports.
• Two filtered
microwave
sources.
• Microwave
combiner.
Amplitude Calibration
• Necessary for any nonlinear measurement
because linear property of homogeneity does
not apply.
• Measures power and is controlled via GPIB.
ECE 451 21T. M. Comberiate
Phase Calibration
ECE 451 22T. M. Comberiate
• Enables cross-frequency phase measurement.
• Takes frequency input from external microwave source.
Vector Calibration
• Can use ECal.
• Based on eight-term error model.
• Works for forward, reverse, and combined stimuli.
ECE 451 23T. M. Comberiate
Large-Signal X-Parameter
Extraction• Apply large-signal stimulus A1,1 without
any small-signal stimulus.
• Measure the response at all ports and
harmonics of interest.
• term is the measured response to
the large-signal stimulus at port p and
harmonic k
ECE 451 24T. M. Comberiate
(FB)
,p kX
Offset-Phase Small-Signal
X-Parameter Extraction• Apply large-signal stimulus A1,1 and one
small-signal stimulus Aq,l at zero phase.
• Measure the response at all ports and harmonics of interest.
• Apply large-signal stimulus A1,1 and one small-signal stimulus Aq,l at 90º phase.
• Measure the response at all ports and harmonics of interest.
• Use both measurements to extract
ECE 451 25T. M. Comberiate
(S) (T)
, ; , , ; , and .p k q l p k q lX X
Using X-Parameters
• Traditional Uses
– Modeling mixers and amplifiers in steady-
state simulations for RF systems.
– Can be used to determine nonlinear figures of
merit.
• 1-dB Compression Point
• AM/AM and AM/PM
• Third Order Intercept
ECE 451 26T. M. Comberiate
Cascading S-Parameter Blocks
ECE 451 27
[T](T)
=
[T](1)
[T](2)
A1(1)
A2(2)
B2(2)
B1(1)
[S](T)
A1(1)
A2(2)
B2(2)
B1(1)[S]
(1)A1
(1)
A2(1)
B2(1)
B1(1) [S]
(2)A1
(2)
A2(2)
B2(2)
B1(2)
[T](1)
A1(1)
A2(1)
B2(1)
B1(1) [T]
(2)A1
(2)
A2(2)
B2(2)
B1(2)
=
=
Can disregard circuit
behavior at internal node.[T] = transfer scattering parameters.
T. M. Comberiate
Cascading X-Parameter Blocks
ECE 451 28
These equations at
the internal node
must always be
satisfied:
B1,k(1) = A1,k
(2),
A1,k(2) = B2,k
(1)
for all values of k.
[X](1)
[X](2)
[X](T)
A2,k(1)
B2,k(1)
A1,k(1)
B1,k(1)
A1,k(2)
B1,k(2)
A2,k(2)
B2,k(2)
A2,k(2)
B2,k(2)
A1,k(1)
B1,k(1)
=
=
T. M. Comberiate
Using X-Parameters in
Simulation
ECE 451 29T. M. Comberiate
Can construct entire receiver chains made of S- and X-parameter blocks.
ECE 451 30
X-Parameter Extensions
• Multiple Large Signals
• DC Components of Scattered Waves
– DC current port bias: X(Z)p,k
– DC voltage port bias: X(Y)p,k
• Memory Effects
T. M. Comberiate
