This paper presents an innovative approach to extend the S-parameter definition to multiport networks having conventional single-ended and differential ports, as is the case for operational amplifiers, transformers and baluns. To give maximum generality to this technique, for example, allowing for complex -parameter reference impedances, the mathematical derivation will be carried out with the most general definition of the -parameters. The presented approach gives the same results already published for circuits with differential ports only when the required simplifications are applied

Ferrero, Andrea Pierenrico

Pirola, Marco

English

PORTO Publications Open Repository TOrino

458 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 1, JANUARY 2006
Generalized Mixed-Mode S-Parameters
Andrea Ferrero, Member, IEEE, and Marco Pirola, Member, IEEE
Abstract—This paper presents an innovative approach to ex-
tend the -parameter definition to multiport networks having
conventional single-ended and differential ports, as is the case
for operational amplifiers, transformers and baluns. To give
maximum generality to this technique, for example, allowing for
complex -parameter reference impedances, the mathematical
derivation will be carried out with the most general definition of
the -parameters. The presented approach gives the same results
already published for circuits with differential ports only when
the required simplifications are applied.
Index Terms—Differential -parameters, microwave measure-
ments, mixed-mode -parameters, multiport -parameters.
I. INTRODUCTION
THE NEED of accurate experimental characterization of RFmultiport differential circuits is dramatically increasing in
today’s communication market. Recent improvements in both
device performances and system design techniques allow for the
implementation of differential circuits and systems up to RF and
microwave frequencies. In 1995, Bockelman and Einsenstadt
[1] introduced so-called mixed-mode -parameters to extend
the classical single-ended power wave theory [2] to the differen-
tial case. Since then, the characterization through mixed-mode
-parameters has become rather common, either using tradi-
tional multiport vector network analyzers (VNAs) rather then
dedicated mixed-mode ones [3], [4].
The theory shown in [1], however very innovative, can only
be applied to circuits whose ports are all to be considered as
common differential. Therefore, it cannot be extended to hy-
brid networks having some port naturally to be seen as single
ended, and others more effectively to be considered as cou-
pled. A simple, but relevant example, is the operational ampli-
fier, where the single-ended output voltage is normally seen as
a combination of the differential and common mode input volt-
ages through the corresponding gains.
This paper generalizes the theory presented in [1] for a gen-
eral -port network sketched in Fig. 1, having an even number
of ports coupled in a common-differential way, while the re-
maining are left single ended. The relationship between
the traditional single-ended -port -parameters, and the gener-
alized mixed-mode ones presented here will be derived. It will
also be shown that the formulation presented in [1] can be seen
as a reduced case of the new one. The new formulation adopts
the pseudowave definition introduced in [5], rather than the one
Manuscript received May 3, 2005.
The authors are with the Dipartimento di Elettronica, Politecnico di Torino,
10129 Turin, Italy (e-mail: andrea.ferrero@polito.it; marco.pirola@polito.it).
Digital Object Identifier 10.1109/TMTT.2005.860497
Fig. 1. n-port network with p-ports where differential parameters are defined
and the remaining single ended.
used by Bockelman and Einsenstadt [1], therefore, enabling it to
be used with complex reference port impedances. This issue is
rather common at high frequency, and implicitly assumed each
time a VNA thru-reflect line (TRL) calibration is adopted on
dispersive media [6].
This paper is organized as follows. In Section II, the gener-
alized mixed-mode -port network is introduced, and the defi-
nition of coupled port -parameters is assigned. In Section III,
starting from the traditional single-ended -port matrix, the
new generalized mixed-mode representation is defined, while
in Section IV, an application example, proving the effectiveness
of the new description, is given. Some conclusions are drawn in
Section V.
II. -PORT MIXED-MODE CIRCUIT
Let us consider the general -port network shown in Fig. 1,
and define the single-ended voltage and current state vector for
the generic port as
(1)
where the symbol indicates the transpose operator.
The overall number of -ports is split in two different sets:
the mixed-mode port set (left-hand side of the network box in
Fig. 1) composed by the even number of -ports requiring the
differential and common mode description, and the one given
by the remaining single-ended ports (right-hand side of
the network box in Fig. 1). On the generic port pair of the
0018-9480/$20.00 © 2006 IEEE
FERRERO AND PIROLA: GENERALIZED MIXED-MODE -PARAMETERS 459
mixed-mode port set, we define the differential and common
mode voltages and currents according to [1]
(2)
and the corresponding mixed-mode voltage and current state
vector
(3)
The expressions in (2) can be written in matrix notation as
(4)
(5)
and the vector is defined stacking up the single-ended state
vectors of port , , , and , respectively,
(6)
Choosing the -port reference impedances possibly complex
(however, with positive real parts), we can define the forward
and reverse pseudowaves at port according to [5]1
(7)
where is the reference impedance at port , is its real part,
and is a free parameter whose only constrain is to have unitary
modulus, from now on assumed to be 1. The port state vector in
term of waves can be defined using (1) as follows:
(8)
where
(9)
Considering the port pair and of the mixed-mode port set,
and introducing the reference impedances for the common and
1This is not only the most general definition of waves, but also the only con-
sistent definition with the measurement provided by network analyzers, as ex-
plained in detail in [5]. In the Appendix, the alternative power wave definition
[2], [5] will be assumed, and the modifications occurring in this formulation will
be shown.
mode differential and , respectively, the mixed-mode
pseudowaves are defined as
(10)
where and are the real parts of and , re-
spectively. From (10) and (4), the mixed-mode state vector is
(11)
where
(12)
Defining the overall wave state vector of the -port pair as
(13)
from (6) and (8), we have
(14)
Finally, from (4), (11), and (14) follows the relationship
linking the classical single-ended, and mixed-mode pseu-
dowave state vectors
(15)
where is given in (16), shown at the bottom of the following
page.
460 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 1, JANUARY 2006
Equation (15) is the generalized form of [1, eq. (25)]. In fact,
considering the port pair associated to the physical ports 1 and 2,
adopting a unique real reference impedances , and assuming
, and , (16) simplifies to
(17)
which, once replaced within (15), gives
(18)
which are the same expressions reported in [1]. It is worth noting
that, in this simplified case, the mixed-mode waves and
are only dependent on their single-ended counterparts; that
is not true in the general case. For example, if a unique normal-
ization resistance is adopted (e.g.,
), we obtain
(19)
and, therefore,
(20)
To obtain the overall generalized mixed-mode wave state
vector , we combine the vectors, relative to the
mixed-mode ports, together with the ones relative to
the single-ended ports
(21)
The original single-ended overall state vector is defined as
(22)
From (15), (21), and (22), the relationship between the two
overall states becomes
(23)
(16)
FERRERO AND PIROLA: GENERALIZED MIXED-MODE -PARAMETERS 461
where is the matrix
(24)
while is the identity matrix relative
to the single-ended ports. The description given by (23) is far
more general than the traditional one based only on differential
mode parameters; in fact, it can be extended to networks with
differential and single-ended ports. Besides, as will be shown in
Section III, it brings the relationship between the single-ended
and generalized mixed-mode -parameters to a general form.
III. GENERALIZED MIXED-MODE MATRIX
By introducing the and generalized mixed-mode wave
vectors, which combine the mixed-mode port set and the re-
maining single-ended ports
(25)
we can define the generalized mixed-mode scattering matrix
such as
(26)
The aim here is to find the relationship between the generalized
mixed-mode scattering matrix and the conventional single-
ended one that links the forward and reverse vector waves.
To achieve our goal, the vector defined in (21) must be prop-
erly reordered with a set of matrices having all null elements
apart a single “1” for each row as follows:
size
elsewhere
elsewhere
elsewhere
elsewhere
size
elsewhere
elsewhere (27)
By collecting them into a permutation matrix , we
find
(28)
In a similar way, we also introduce other two matrices
elsewhere
elsewhere (29)
and combine them into another permutation matrix to reorder
the whole single-ended wave state and obtain the swapped state
(30)
From (23), (28), and (30), we obtain the relationship between the
reordered single-ended state and the corresponding generalized
mixed-mode one ( [7])
(31)
By splitting into four squared submatrices,
(32)
we find
(33)
Finally, from (26) and (33), and by using , the direct and
inverse transformations linking to become
(34)
(35)
Equation (34) represents the generalized form of the single-
ended to mixed-mode scattering matrix transformation given
in [1, eq. (7)]. In fact, if the circuit has four common-differen-
tial ports only, and once again it is assumed and
, we obtain
(36)
where is the null matrix and (34) becomes, as expected,
.
In the special case of circuits with only classical single-
ended ports, the matrix is the identity matrix,
, , and (34), as expected, gives .
However, (34) and (35) can be used with any number of ports,
and with whatsoever, even complex, reference impedances.
The generalized matrix can be split into nine submatrices
highlighting the self- and cross-coupling between the differen-
462 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 1, JANUARY 2006
Fig. 2. Operational amplifier equivalent circuit.
tial, common, and single-ended mode as reported in the fol-
lowing expression:
(37)
where the and sub-matrix subscripts refer to the differen-
tial, common, and single-ended modes, respectively.
IV. EXAMPLE
The operational amplifier is a typical simple case of a circuit
with two input coupled ports and a single-ended output port. To
properly study this circuit, we adopt the simplified model shown
in Fig. 2. The reference impedance for the two input ports is
assumed unique , however, different from the one of the
output port . With this assumption, the 3 3 single-ended
matrix has the following elements:
(38)
As expected from the classical single-ended approach, the in-
formation regarding the differential and common mode amplifi-
cation is all together embedded within the - and -param-
eters, and the common and differential contributions cannot be
easily recognized and separated. Using the above results, it is
simple to transform into the corresponding new mixed-mode
matrix. In this particular case, the sub-matrices defined in (37)
are simply the elements of , and are given by
(39)
The differential and common mode amplifications are separated
as highlighted by (39) that show the absence of cross-mode
components at the operational amplifier output. Furthermore,
the elements and only depend on and , respec-
tively, showing in the most straightforward way the separated
contributions of the input differential and common mode to the
single-ended output port. Proceeding with some simplifications:
matched output port and a common reference impedance for all
ports , we find
(40)
Finally, assuming the traditional mixed-mode reference imped-
ances , , we find
(41)
This expression points out once more the effectiveness of the
proposed approach, compared with the traditional one based
on the classical single-ended -parameters, which, in the same
conditions, would have given
(42)
FERRERO AND PIROLA: GENERALIZED MIXED-MODE -PARAMETERS 463
(44)
V. CONCLUSIONS
This paper has presented an innovative approach that intro-
duces the generalized mixed-mode -parameters starting from
the single-ended ones. These novel parameters allows for a con-
sistent representation of networks with differential and single-
ended ports with real or complex reference impedances. The
new approach is fully consistent with the classical mixed-mode
-parameter theory when the required simplifications are ap-
plied. The method effectiveness, together with its capability to
immediately identify the common and differential mode effects,
has been proven on a test circuit having both classical single-
ended and mixed-mode ports.
APPENDIX
Alternative definitions to (7) and (10) have been proposed by
[2]
(43)
where is the normalization impedance and is its real part.
Despite these definitions have been demonstrated to be incom-
patible with the experimental data in presence of complex nor-
malization impedances [5] (e.g., when TRL calibration on dis-
persive media is adopted), they are rather popular, and reduce
to (7) for real normalization impedances. The only variations
on the formulation concern the matrix that have to be eval-
uated according to (43), and becomes (44), shown at the top of
this page.
REFERENCES
[1] D. Bockelman and W. R. Einsenstadt, “Combined differential and
common-mode scattering parameters: Theory and simulation,” IEEE
Trans. Microw. Theory Tech., vol. 43, no. 7, pp. 1530–1539, Jul. 1995.
[2] K. Kurokawa, “Power waves and scattering matrix,” IEEE Trans. Mi-
crow. Theory Tech., vol. MTT-13, no. 3, pp. 194–202, Mar. 1965.
[3] D. Bockelman and W. R. Einsenstadt, “Pure-mode network analyzer for
on-wafer measurements of mixed-modeS-parameters of differential cir-
cuits,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 7, pp. 1071–1077,
Jul. 1997.
[4] A. Ferrero, U. Pisani, and K. Kerwin, “A new implementation of a mul-
tiport automatic network analyzer,” IEEE Trans. Microw. Theory Tech.,
vol. 40, no. 11, pp. 2078–2085, Nov. 1992.
[5] R. Marks and D. Williams, “A general waveguide circuit theory,” J. Res.
Nat. Inst. Standards Technol., vol. 97, pp. 533–561, Sep. 1992.
[6] , “Characteristic impedance determination using propagation con-
stant measurement,” IEEE Microw. Guided Wave Lett., vol. 1, no. 6, pp.
141–143, Jun. 1991.
[7] G. Golub and C. V. Loan, Matrix Computations, 2nd ed. Baltimore,
MD: The John Hopkins Univ. Press, 1989.
Andrea Ferrero (S’86–M’88) was born in Novara,
Italy, on November 7, 1962. He received the Elec-
tronic Engineering degree and Ph.D. degree in elec-
tronics from the Politecnico di Torino, Turin, Italy, in
1987 and 1992, respectively.
In 1988, he joined Aeritalia as a Microwave Con-
sultant. In 1991, he was a Summer Student with the
Microwave Technology Division, Hewlett-Packard
Company, Santa Rosa, CA. In 1995, he was a Guest
Researcher with the Electrical Engineering Depart-
ment, Ecole Polytechnique de Montrèal, Montrèal,
QC, Canada. In 1998, he became an Associate Professor of electronic mea-
surements with the Politecnico di Torino. His main research activities are in the
area of microwave measurements techniques, calibration, and modeling.
Marco Pirola (M’97) was born in Velezzo Lomel-
lina, Italy, in 1963. He received the Laurea degree in
electronic engineering and Ph.D. degree from the Po-
litecnico di Torino, Turin, Italy, in 1987 and 1992 re-
spectively.
In 1992 and 1994, he was a Visiting Re-
searcher with the Microwave Technology Division,
Hewlett-Packard Company, Santa Rosa, CA.
Since 1992, he has been with the Dipartimento
di Elettronica, Politecnico di Torino, initially as
a Researcher and, since 2000, as an Associate
Professor. His research concerns the simulation, modeling, and measurements
of microwave devices and systems.


(M’97) was born in Velezzo Lomellina, Italy,

(S’86–M’88) was born in

[6] , “Characteristic impedance determination using propagation constant measurement,”

A new implementation of a multiport automatic network analyzer,”

and 1994, he was a Visiting Researcher with the Microwave Technology Division, Hewlett-Packard Company,

Combined differential and common-mode scattering parameters: Theory and simulation,”

he has been with the Dipartimento di Elettronica, Politecnico di Torino, initially as a Researcher and, since 2000, as an Associate Professor. His research concerns the simulation, modeling, and measurements of microwave devices and systems.

he joined Aeritalia as a Microwave Consultant. In

Power waves and scattering matrix,”

Pure-mode network analyzer for on-wafermeasurementsofmixed-modeS-parametersofdifferentialcircuits,”IEEETrans.Microw.TheoryTech.,vol.45,no.7,pp.1071–1077,

Generalized mixed-mode S-parameters

A. Ferrero

M. Pirola

Crossref

FERRERO, ANDREA PIERENRICO

PIROLA, Marco

PORTO@iris (Publications Open Repository TOrino - Politecnico di Torino)

05 August 2020POLITECNICO DI TORINORepository ISTITUZIONALEGeneralized mixed-mode S-parameters / A.FERRERO; PIROLA M.. - In: IEEE TRANSACTIONS ON MICROWAVETHEORY AND TECHNIQUES. - ISSN 0018-9480. - STAMPA. - 54:1(2006), pp. 458-463.OriginalGeneralized mixed-mode S-parametersPublisher:PublishedDOI:10.1109/TMTT.2005.860497Terms of use:openAccessPublisher copyright(Article begins on next page)This article is made available under terms and conditions as specified in the  corresponding bibliographic description inthe repositoryAvailability:This version is available at: 11583/1433496 since:458 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 1, JANUARY 2006Generalized Mixed-Mode S-ParametersAndrea Ferrero, Member, IEEE, and Marco Pirola, Member, IEEEAbstract—This paper presents an innovative approach to ex-tend the -parameter definition to multiport networks havingconventional single-ended and differential ports, as is the casefor operational amplifiers, transformers and baluns. To givemaximum generality to this technique, for example, allowing forcomplex -parameter reference impedances, the mathematicalderivation will be carried out with the most general definition ofthe -parameters. The presented approach gives the same resultsalready published for circuits with differential ports only whenthe required simplifications are applied.Index Terms—Differential -parameters, microwave measure-ments, mixed-mode -parameters, multiport -parameters.I. INTRODUCTIONTHE NEED of accurate experimental characterization of RFmultiport differential circuits is dramatically increasing intoday’s communication market. Recent improvements in bothdevice performances and system design techniques allow for theimplementation of differential circuits and systems up to RF andmicrowave frequencies. In 1995, Bockelman and Einsenstadt[1] introduced so-called mixed-mode -parameters to extendthe classical single-ended power wave theory [2] to the differen-tial case. Since then, the characterization through mixed-mode-parameters has become rather common, either using tradi-tional multiport vector network analyzers (VNAs) rather thendedicated mixed-mode ones [3], [4].The theory shown in [1], however very innovative, can onlybe applied to circuits whose ports are all to be considered ascommon differential. Therefore, it cannot be extended to hy-brid networks having some port naturally to be seen as singleended, and others more effectively to be considered as cou-pled. A simple, but relevant example, is the operational ampli-fier, where the single-ended output voltage is normally seen asa combination of the differential and common mode input volt-ages through the corresponding gains.This paper generalizes the theory presented in [1] for a gen-eral -port network sketched in Fig. 1, having an even numberof ports coupled in a common-differential way, while the re-maining are left single ended. The relationship betweenthe traditional single-ended -port -parameters, and the gener-alized mixed-mode ones presented here will be derived. It willalso be shown that the formulation presented in [1] can be seenas a reduced case of the new one. The new formulation adoptsthe pseudowave definition introduced in [5], rather than the oneManuscript received May 3, 2005.The authors are with the Dipartimento di Elettronica, Politecnico di Torino,10129 Turin, Italy (e-mail: andrea.ferrero@polito.it; marco.pirola@polito.it).Digital Object Identifier 10.1109/TMTT.2005.860497Fig. 1. n-port network with p-ports where differential parameters are definedand the remaining single ended.used by Bockelman and Einsenstadt [1], therefore, enabling it tobe used with complex reference port impedances. This issue israther common at high frequency, and implicitly assumed eachtime a VNA thru-reflect line (TRL) calibration is adopted ondispersive media [6].This paper is organized as follows. In Section II, the gener-alized mixed-mode -port network is introduced, and the defi-nition of coupled port -parameters is assigned. In Section III,starting from the traditional single-ended -port matrix, thenew generalized mixed-mode representation is defined, whilein Section IV, an application example, proving the effectivenessof the new description, is given. Some conclusions are drawn inSection V.II. -PORT MIXED-MODE CIRCUITLet us consider the general -port network shown in Fig. 1,and define the single-ended voltage and current state vector forthe generic port as(1)where the symbol indicates the transpose operator.The overall number of -ports is split in two different sets:the mixed-mode port set (left-hand side of the network box inFig. 1) composed by the even number of -ports requiring thedifferential and common mode description, and the one givenby the remaining single-ended ports (right-hand side ofthe network box in Fig. 1). On the generic port pair of the0018-9480/$20.00 © 2006 IEEEFERRERO AND PIROLA: GENERALIZED MIXED-MODE -PARAMETERS 459mixed-mode port set, we define the differential and commonmode voltages and currents according to [1](2)and the corresponding mixed-mode voltage and current statevector(3)The expressions in (2) can be written in matrix notation as(4)(5)and the vector is defined stacking up the single-ended statevectors of port , , , and , respectively,(6)Choosing the -port reference impedances possibly complex(however, with positive real parts), we can define the forwardand reverse pseudowaves at port according to [5]1(7)where is the reference impedance at port , is its real part,and is a free parameter whose only constrain is to have unitarymodulus, from now on assumed to be 1. The port state vector interm of waves can be defined using (1) as follows:(8)where(9)Considering the port pair and of the mixed-mode port set,and introducing the reference impedances for the common and1This is not only the most general definition of waves, but also the only con-sistent definition with the measurement provided by network analyzers, as ex-plained in detail in [5]. In the Appendix, the alternative power wave definition[2], [5] will be assumed, and the modifications occurring in this formulation willbe shown.mode differential and , respectively, the mixed-modepseudowaves are defined as(10)where and are the real parts of and , re-spectively. From (10) and (4), the mixed-mode state vector is(11)where(12)Defining the overall wave state vector of the -port pair as(13)from (6) and (8), we have(14)Finally, from (4), (11), and (14) follows the relationshiplinking the classical single-ended, and mixed-mode pseu-dowave state vectors(15)where is given in (16), shown at the bottom of the followingpage.460 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 1, JANUARY 2006Equation (15) is the generalized form of [1, eq. (25)]. In fact,considering the port pair associated to the physical ports 1 and 2,adopting a unique real reference impedances , and assuming, and , (16) simplifies to(17)which, once replaced within (15), gives(18)which are the same expressions reported in [1]. It is worth notingthat, in this simplified case, the mixed-mode waves andare only dependent on their single-ended counterparts; thatis not true in the general case. For example, if a unique normal-ization resistance is adopted (e.g.,), we obtain(19)and, therefore,(20)To obtain the overall generalized mixed-mode wave statevector , we combine the vectors, relative to themixed-mode ports, together with the ones relative tothe single-ended ports(21)The original single-ended overall state vector is defined as(22)From (15), (21), and (22), the relationship between the twooverall states becomes(23)(16)FERRERO AND PIROLA: GENERALIZED MIXED-MODE -PARAMETERS 461where is the matrix(24)while is the identity matrix relativeto the single-ended ports. The description given by (23) is farmore general than the traditional one based only on differentialmode parameters; in fact, it can be extended to networks withdifferential and single-ended ports. Besides, as will be shown inSection III, it brings the relationship between the single-endedand generalized mixed-mode -parameters to a general form.III. GENERALIZED MIXED-MODE MATRIXBy introducing the and generalized mixed-mode wavevectors, which combine the mixed-mode port set and the re-maining single-ended ports(25)we can define the generalized mixed-mode scattering matrixsuch as(26)The aim here is to find the relationship between the generalizedmixed-mode scattering matrix and the conventional single-ended one that links the forward and reverse vector waves.To achieve our goal, the vector defined in (21) must be prop-erly reordered with a set of matrices having all null elementsapart a single “1” for each row as follows:sizeelsewhereelsewhereelsewhereelsewheresizeelsewhereelsewhere (27)By collecting them into a permutation matrix , wefind(28)In a similar way, we also introduce other two matriceselsewhereelsewhere (29)and combine them into another permutation matrix to reorderthe whole single-ended wave state and obtain the swapped state(30)From (23), (28), and (30), we obtain the relationship between thereordered single-ended state and the corresponding generalizedmixed-mode one ( [7])(31)By splitting into four squared submatrices,(32)we find(33)Finally, from (26) and (33), and by using , the direct andinverse transformations linking to become(34)(35)Equation (34) represents the generalized form of the single-ended to mixed-mode scattering matrix transformation givenin [1, eq. (7)]. In fact, if the circuit has four common-differen-tial ports only, and once again it is assumed and, we obtain(36)where is the null matrix and (34) becomes, as expected,.In the special case of circuits with only classical single-ended ports, the matrix is the identity matrix,, , and (34), as expected, gives .However, (34) and (35) can be used with any number of ports,and with whatsoever, even complex, reference impedances.The generalized matrix can be split into nine submatriceshighlighting the self- and cross-coupling between the differen-462 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 1, JANUARY 2006Fig. 2. Operational amplifier equivalent circuit.tial, common, and single-ended mode as reported in the fol-lowing expression:(37)where the and sub-matrix subscripts refer to the differen-tial, common, and single-ended modes, respectively.IV. EXAMPLEThe operational amplifier is a typical simple case of a circuitwith two input coupled ports and a single-ended output port. Toproperly study this circuit, we adopt the simplified model shownin Fig. 2. The reference impedance for the two input ports isassumed unique , however, different from the one of theoutput port . With this assumption, the 3 3 single-endedmatrix has the following elements:(38)As expected from the classical single-ended approach, the in-formation regarding the differential and common mode amplifi-cation is all together embedded within the - and -param-eters, and the common and differential contributions cannot beeasily recognized and separated. Using the above results, it issimple to transform into the corresponding new mixed-modematrix. In this particular case, the sub-matrices defined in (37)are simply the elements of , and are given by(39)The differential and common mode amplifications are separatedas highlighted by (39) that show the absence of cross-modecomponents at the operational amplifier output. Furthermore,the elements and only depend on and , respec-tively, showing in the most straightforward way the separatedcontributions of the input differential and common mode to thesingle-ended output port. Proceeding with some simplifications:matched output port and a common reference impedance for allports , we find(40)Finally, assuming the traditional mixed-mode reference imped-ances , , we find(41)This expression points out once more the effectiveness of theproposed approach, compared with the traditional one basedon the classical single-ended -parameters, which, in the sameconditions, would have given(42)FERRERO AND PIROLA: GENERALIZED MIXED-MODE -PARAMETERS 463(44)V. CONCLUSIONSThis paper has presented an innovative approach that intro-duces the generalized mixed-mode -parameters starting fromthe single-ended ones. These novel parameters allows for a con-sistent representation of networks with differential and single-ended ports with real or complex reference impedances. Thenew approach is fully consistent with the classical mixed-mode-parameter theory when the required simplifications are ap-plied. The method effectiveness, together with its capability toimmediately identify the common and differential mode effects,has been proven on a test circuit having both classical single-ended and mixed-mode ports.APPENDIXAlternative definitions to (7) and (10) have been proposed by[2](43)where is the normalization impedance and is its real part.Despite these definitions have been demonstrated to be incom-patible with the experimental data in presence of complex nor-malization impedances [5] (e.g., when TRL calibration on dis-persive media is adopted), they are rather popular, and reduceto (7) for real normalization impedances. The only variationson the formulation concern the matrix that have to be eval-uated according to (43), and becomes (44), shown at the top ofthis page.REFERENCES[1] D. Bockelman and W. R. Einsenstadt, “Combined differential andcommon-mode scattering parameters: Theory and simulation,” IEEETrans. Microw. Theory Tech., vol. 43, no. 7, pp. 1530–1539, Jul. 1995.[2] K. Kurokawa, “Power waves and scattering matrix,” IEEE Trans. Mi-crow. Theory Tech., vol. MTT-13, no. 3, pp. 194–202, Mar. 1965.[3] D. Bockelman and W. R. Einsenstadt, “Pure-mode network analyzer foron-wafer measurements of mixed-modeS-parameters of differential cir-cuits,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 7, pp. 1071–1077,Jul. 1997.[4] A. Ferrero, U. Pisani, and K. Kerwin, “A new implementation of a mul-tiport automatic network analyzer,” IEEE Trans. Microw. Theory Tech.,vol. 40, no. 11, pp. 2078–2085, Nov. 1992.[5] R. Marks and D. Williams, “A general waveguide circuit theory,” J. Res.Nat. Inst. Standards Technol., vol. 97, pp. 533–561, Sep. 1992.[6] , “Characteristic impedance determination using propagation con-stant measurement,” IEEE Microw. Guided Wave Lett., vol. 1, no. 6, pp.141–143, Jun. 1991.[7] G. Golub and C. V. Loan, Matrix Computations, 2nd ed. Baltimore,MD: The John Hopkins Univ. Press, 1989.Andrea Ferrero (S’86–M’88) was born in Novara,Italy, on November 7, 1962. He received the Elec-tronic Engineering degree and Ph.D. degree in elec-tronics from the Politecnico di Torino, Turin, Italy, in1987 and 1992, respectively.In 1988, he joined Aeritalia as a Microwave Con-sultant. In 1991, he was a Summer Student with theMicrowave Technology Division, Hewlett-PackardCompany, Santa Rosa, CA. In 1995, he was a GuestResearcher with the Electrical Engineering Depart-ment, Ecole Polytechnique de Montrèal, Montrèal,QC, Canada. In 1998, he became an Associate Professor of electronic mea-surements with the Politecnico di Torino. His main research activities are in thearea of microwave measurements techniques, calibration, and modeling.Marco Pirola (M’97) was born in Velezzo Lomel-lina, Italy, in 1963. He received the Laurea degree inelectronic engineering and Ph.D. degree from the Po-litecnico di Torino, Turin, Italy, in 1987 and 1992 re-spectively.In 1992 and 1994, he was a Visiting Re-searcher with the Microwave Technology Division,Hewlett-Packard Company, Santa Rosa, CA.Since 1992, he has been with the Dipartimentodi Elettronica, Politecnico di Torino, initially asa Researcher and, since 2000, as an AssociateProfessor. His research concerns the simulation, modeling, and measurementsof microwave devices and systems.

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Generalized mixed-mode S-parameters

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