On Transfer Functions Realizable with Active Electronic Components

Abstract

In this work, we characterize transfer functions that can be realized with standard electronic components in linearized form, e.g. those commonly used in the design of analog amplifiers (including transmission lines) in the small signal regime. We define the stability of such transfer functions in connection with scattering theory, i.e. in terms of bounded reflected power against every sufficiently large load. In the simplest model for active elements, we show that unstable transfer functions exist which have no pole in the right half-plane. Then, we introduce more realistic transfer functions for active elements which are passive at very high frequencies, and we show that they have finitely many poles in the right half-plane. Finally, in contrast to the ideal transfer functions studied before, the stability of such "realistic" transfer functions is characterized by the absence of poles in the open right half-plane and the positivity of the real part of the residues of the poles located on the imaginary axis.

This report is written in a way which is suitable to the non-specialist, and every notion is defined and analyzed from first principles.

Full text

On Transfer Functions Realizable with Active Electronic
Components
Laurent Baratchart, Sylvain Chevillard, Fabien Seyfert
To cite this version:
Laurent Baratchart, Sylvain Chevillard, Fabien Seyfert. On Transfer Functions Realizable
with Active Electronic Components. [Research Report] RR-8659, Inria Sophia Antipolis. 2014,
pp.36. <hal-01098616>
HAL Id: hal-01098616
https://hal.inria.fr/hal-01098616
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ISSN 0249-6399 ISRN INRIA/RR--8659--FR+ENG
RESEARCH
REPORT
N° 8659
December 2014
Project-Team Apics
On Transfer Functions
Realizable with Active
Electronic Components
Laurent Baratchart, Sylvain Chevillard, Fabien Seyfert

RESEARCH CENTRE
SOPHIA ANTIPOLIS – MÉDITERRANÉE
2004 route des Lucioles - BP 93
06902 Sophia Antipolis Cedex
On Transfer Functions Realizable with Active
Electronic Components
Laurent Baratchart, Sylvain Chevillard, Fabien Seyfert
Project-Team Apics
Research Report n°8659 — December 2014 — 36 pages
Abstract: In this work, we characterize transfer functions that can be realized with standard
electronic components in linearized form, e.g. those commonly used in the design of analog am-
plifiers (including transmission lines) in the small signal regime. We define the stability of such
transfer functions in connection with scattering theory, i.e. in terms of bounded reflected power
against every sufficiently large load. In the simplest model for active elements, we show that un-
stable transfer functions exist which have no pole in the right half-plane. Then, we introduce more
realistic transfer functions for active elements which are passive at very high frequencies, and we
show that they have finitely many poles in the right half-plane. Finally, in contrast to the ideal
transfer functions studied before, the stability of such “realistic” transfer functions is character-
ized by the absence of poles in the open right half-plane and the positivity of the real part of the
residues of the poles located on the imaginary axis.
This report is written in a way which is suitable to the non-specialist, and every notion is defined
and analyzed from first principles.
Key-words: Amplifier, transfer function, active components, transistor, diode, transmission line,
negative resistor, stability, Hardy spaces.

Sur les fonctions de transfert réalisables avec des
composants électriques actifs
Résumé : Dans ce travail, nous caractérisons les fonctions de transfert qui peuvent être
synthétisées avec des composants électroniques standards linéarisés, y compris des lignes de
transmission. Ce sont les composants typiquement utilisés pour la synthèse d’amplificateurs
analogiques, modélisés en régime « faible signal ». Nous définissons la stabilité de telles fonctions
de transfert en nous appuyant sur la théorie de dispersion des ondes, précisément en demandant
à ce que la puissance réfléchie contre toute charge suffisamment grande reste bornée vis-à-vis de
la fréquence. Nous montrons qu’il existe des fonctions de transfert qui sont instables mais n’ont
pas de pôle dans le demi-plan droit. Nous introduisons ensuite une modélisation plus réaliste
des fonctions de transfert des composants actifs, pour traduire l’hypothèse réaliste selon laquelle
ils deviennent passifs à très haute fréquence. Nous montrons que les circuits synthétisables avec
de tels composants réalistes ont un nombre fini de pôles dans le demi-plan droit ; en outre,
nous montrons qu’on peut caractériser la stabilité des fonctions de transfert ainsi obtenues par
l’absence de pôle dans le demi-plan droit ouvert et le fait que les résidus des pôles situés sur l’axe
imaginaire aient une partie réelle positive.
Ce rapport est écrit de telle façon qu’il soit accessible au non spécialiste et chaque notion est
définie et étudiée à partir de notions élémentaires.
Mots-clés : Amplificateur, fonction de transfert, composants actifs, transistor, diode, ligne de
transmission, résistance négative, stabilité, espaces de Hardy.

Transfer functions realizable with active electronic components 3
Contents
1 Introduction 3
2 Electronic components under consideration 4
2.1 Dipoles ......................................... 4
2.2 Transmissionlines ................................... 5
2.3 Diodes.......................................... 5
2.4 Transistors ....................................... 5
3 Structure of circuits 7
4 Partial transfer functions 9
4.1 Partial transfer from a current source . . . . . . . . . . . . . . . . . . . . . . . . 9
4.2 Partial transfer from a voltage source . . . . . . . . . . . . . . . . . . . . . . . . . 10
5 R-L-C circuits with negative resistors 11
5.1 What the inverter makes possible . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
5.2 Frequency response of RLC circuits with negative resistors . . . . . . . . . . . . . 12
6 Circuits with transmission lines 13
6.1 Using transmission lines as one-port circuits . . . . . . . . . . . . . . . . . . . . . 14
6.2 Composing one-port circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
6.3 Class of all impedances of one-port circuits . . . . . . . . . . . . . . . . . . . . . 17
7 Notion of stability 18
8 Realistic model of linearized components 22
9 A stability criterion 27
10 Appendix 1: Telegrapher’s equation 28
11 Appendix 2: Transfer functions and stability 30
Acknowledgments
The research presented in this report has been partially funded by the CNES through the
grant R&T RS 10/TG1-019. We wish to thank Juan-Mari Collantes from Universidad del País
Vasco/Euskal Herriko Unibertsitatea for his patient advice and for the long discussions we had
with him regarding the stability of electronic circuits.
1 Introduction
In this work, we characterize transfer functions that can be realized with standard electronic
components in linearized form, e.g. those commonly used in the design of analog amplifiers
(including transmission lines) in the small signal regime. We define the stability of such transfer
functions in connection with scattering theory, i.e. in terms of bounded reflected power against
every sufficiently large load. In the simplest model for active elements, we show that unstable
transfer functions exist which have no pole in the right half-plane. Then, we introduce more
RR n°8659

4L. Baratchart, S. Chevillard & F. Seyfert
realistic transfer functions for active elements which are passive at very high frequencies, and
we show that they have finitely many poles in the right half-plane. Finally, in contrast to
the ideal transfer functions studied before, the stability of such “realistic” transfer functions is
characterized by the absence of poles in the open right half-plane and the positivity of the real
part of the residues of the poles located on the imaginary axis.
This report is written in a way which is suitable to the non-specialist, and every notion is
defined and analyzed from first principles.
2 Electronic components under consideration
In this section, we give a detailed account of elementary ideal models for electronic components
that we consider, along with equations satisfied by currents and voltages at their terminals.
These equations are expressed in terms of complex impedances and admittances [2, 3], i.e. we
express the relations between Laplace transforms of these currents and voltages, see Section 11
for definitions. We denote Laplace transforms with uppercase symbols, e.g. V=V(s)is a
function of a complex variable swhich stands for the Laplace transform of the voltage v=v(t)
which is a function of the time t.
By convention, we always orient currents so that they enter electronic components.
2.1 Dipoles
Electronic dipoles that we consider in the sequel are the following:
•Ideal resistor, with a positive impedance R.
•Ideal inductor, with impedance of the form Ls (L > 0).
•Ideal capacitor, with impedance of the form 1/(Cs)(C > 0).
(a) Resistor (b) Inductor (c) Capacitor
Figure 1: Symbols for linear dipoles
Relations between the currents I1and I2entering each terminal of the dipole and the poten-
tials V1and V2at each terminal are given by
1−1 0 Z
0 0 1 1 !




V1
V2
I1
I2





= 0
0!,(1)
where Z=α,Z=α s or Z=α/s with α > 0.
Remark 1. It follows directly from Equation (1) that V1I1+V2I2=Z|I1|2=Z|I2|2. Since it
is clear that <Z(s)≥0for all components as above, an immediate consequence is that <(V1I1+
V2I2)≥0when <(s)≥0.
Inria

Transfer functions realizable with active electronic components 5
2.2 Transmission lines
Transmission lines are distributed components: they are viewed as a series of infinitesimal resis-
tors, capacitors and inductors which is usually modeled by telegrapher’s equation [9, sec. 9.7.3],
see the discussion in Section 10. All transmission lines in a circuit are assumed to share the same
ground. The latter is often implicit and is not drawn along with the symbol for a transmission
line. In Section 6, it will be convenient to materialize the current loss between terminals of a line
as resulting from a current occurring in a wire (which does not actually exist) connected to the
ground. This virtual wire is drawn with a dotted segment on Figure 2. One may ignore it, in
which case one should ignore as well the last row and the last column of the matrix in Figure 2.
The behavior of a transmission line is otherwise linear and characterized by the relations
in Figure 2. In the matrix shown there, γ=p(R+Ls)(G+Cs)is sometimes called the
propagation coefficient (note it is frequency-dependent) while z0= (R+Ls)/γ is the so-called
characteristic impedance of the line (cf. Section 10). Here R,G,Land Care nonnegative
numbers.






−1 0 z0coth(γ)z0
sinh(γ)0
0−1z0
sinh(γ)z0coth(γ) 0
0 0 1 1 1













V1
V2
I1
I2
I3







=


0
0
0



Figure 2: Symbol and relations for transmission lines
2.3 Diodes
Next we consider diodes. A commonly accepted model of the diode assumes that the relation
between the current itraversing the diode and the voltage uis given as a relation i=f(u)
where fis a non linear real-valued function. In particular, one assumes that the diode has no
inductive nor capacitive effect (fonly depends on uand not on du/dtnor di/dt). We only study
small perturbations around a polarization point (u(Q), i(Q)), hence it is legitimate to linearize
the behavior of the diode, which gives us
i−i(Q)=g·(u−u(Q)), g =df
du(u(Q)).
Hereafter we rename i−i(Q)as iand u−u(Q)as u. That is, although the variables of interest
to the linearized model of the diode are incremental rather than absolute electrical quantities,
we denote them like any other intensity or voltage for notational homogeneity. Taking Laplace
transforms we get I=g U, so the (linearized) diode appears as a standard linear dipole with
admittance g∈R. Typical in our context are tunnel diodes which behave (once correctly
polarized) as ideal negative resistors: g < 0. The symbol we use for, as well as the relations
satisfied by linearized diodes are summarized in Figure 3.
2.4 Transistors
Our circuits may also contain transistors. Specifically, we consider field-effect transistors. These
have three terminals called gate, source, and drain (denoted respectively by G,Sand D). The
behavior is usually described by a relation of the form iD=f(uGS , uDS )where fis a non-linear
RR n°8659

6L. Baratchart, S. Chevillard & F. Seyfert
1−1 0 −α
0 0 1 1 !




V1
V2
I1
I2





= 0
0!
Figure 3: Symbol and relations for the linearized diode (α > 0)
real-valued function and uGS =vG−vS,uDS =vD−vS(see Figure 4). As in the case of diodes,
this simple model assumes no inductive nor capacitive effect, as fonly depends on uGS ,uDS and
not on their time derivatives, nor on the derivative of iD. Moreover the function fis increasing
in both variables. Another feature of field-effect transistors is that no current enters the gate:
iG= 0.
Exactly the same way as for diodes, we consider only small perturbations around a polariza-
tion point (u(Q)
GS , u(Q)
DS , i(Q)
D), so we may use a linear approximation:
iD−i(Q)
D=gm(uGS −u(Q)
GS )+gd(uDS −u(Q)
DS ), gm=∂1f(u(Q)
GS , u(Q)
DS )>0and gd=∂2f(u(Q)
GS , u(Q)
DS )>0.
Altogether, renaming iD−i(Q)
Das iD,uGS −u(Q)
GS as uGS ,uDS −u(Q)
DS as uDS and taking



0 0 0 1 0 0
−gm−gdgm+gd010
0 0 0 1 1 1













VG
VD
VS
IG
ID
IS










=


0
0
0



Figure 4: Symbol and relations for the linearized transistor (gm>0and gd>0)
Laplace transforms as we did for the diode, the (linearized) transistor appears as a current
source controlled by a voltage (see Figure 5).
Figure 5: Equivalent circuit for the linearized transistor
Inria

Transfer functions realizable with active electronic components 7
3 Structure of circuits
Formally speaking, a circuit is a directed graph with labeled vertices meeting the following
constraints:
•There are two kinds of vertices: electronic components and junction nodes.
•A junction node has degree greater or equal to 2.
•An electronic component labeled as a resistor, capacitor, inductor, diode or transmission
line has exactly degree 2.
•An electronic component labeled as a transistor has exactly degree 3.
•An electronic component can only be adjacent with a junction node and reciprocally.
•The edges are oriented from junction nodes to electronic components (this definition is
non-ambiguous and applies to all edges because of the previous rule).
We number the junction nodes and the edges. Without loss of generality, we can suppose
that edges adjacent to a given electronic component are numbered consecutively (because an
edge is adjacent to one and only one such component), and that the ordering gate-drain-source
prevails in the case of transistors.
To each junction node jis associated a potential Vjand to each edge kis associated an
electric current Ik. One junction node is called ground (its potential is 0by convention and
without loss of generality, we suppose that it is numbered as vertex 1). An example of circuit is
given in Figure 6: electronic components are represented with their specific symbols introduced
in Section 2, but they should now be understood as vertices of the graph. Junction nodes are
indicated with bullets, except for the ground which is represented the usual way. For clarity, the
ground is represented at multiple places on the figure, but it should be seen as a single vertex.
Figure 6: Example of a circuit
Let us denote by X=t(V1, . . . , Vn, I1, . . . , Ip)the vector made of all potentials and currents
of a circuit. Theses quantities are related as follows.
•The ground potential is 0:V1= 0.
•For each junction node kdistinct from the ground, Kirchhoff’s first law holds:
X
edge jadjacent to k
Ij= 0.
RR n°8659

8L. Baratchart, S. Chevillard & F. Seyfert
The ground is here excluded because currents may exist between transmission lines and
the ground, though they are not figured on the graph representing the circuit.
•For each electronic component k, relations from Section 2 between potentials at junction
nodes adjacent to the component and currents entering the component must be satisfied.
Collecting all these relations in a matrix, we see that potentials and currents in the circuit
must satisfy a relation MX = 0, with Ma matrix of the form:
M=










1 0 0
0C
B1
A...
Bm










.(2)
In equation (2), blocks should be interpreted as follows.
•The first row defines the ground.
•Chas n−1rows pcolumns. Each row expresses an instance of Kirchhoff’s law.
•The Biare 2×2or 3×3blocks corresponding to the right-part (i.e. multipliers of intensities)
of the matrices describing the elementary behavior of each electronic component, as detailed
in Section 2.
•Ahas prows and ncolumns. Its elements are those of the left-part (i.e. multipliers of
voltages) of the matrices describing the elementary behavior of each electronic component.
We call Mthe behavior matrix of the circuit.
The presence of active components, namely diodes and transistors, may result in Mbeing
singular. This well-known fact is illustrated by the example given in Figure 7.
Figure 7: Non-zero currents and potentials may exist in the circuit in the absence of a source
Inria

Transfer functions realizable with active electronic components 9
The behavior of this circuit is given by the equation




























1 0 0 0 0
0 1 1 0 0 0 1 0 0 0
0 0 0 0 0 0 0 0 1 1 0
0 0 0 1 1 0 0 0 0 0
1−1 0 0 0 R3
0 0 0 0 1 1 0
010−1 0 −R2
0 0 0 0 1 1
−1 0 0 1 0 R2
0 0 0 0 1 1
0 1 −1 0 0 −R1
0 0 0 0 1 1
−1 0 1 0 0 0 R1
0 0 0 0 1 1
























































V1
V2
V3
V4
I1
I2
I3
I4
I5
I6
I7
I8
I9
I10




























= 0.
One easily sees that, for any I, the vector t(0,0, R1I, −R2I, 0,0,−I, I, −I , I , I, −I, I, −I)is
a solution, which proves that the matrix Min this example is not invertible.
Now, a non-invertible behavior matrix corresponds to a situation where, in the absence of
current or voltage source in the circuit, a non-trivial equilibrium between currents and voltages
can be established. In other words, energy transfer occurs between active and passive parts of
the circuit, without external excitation. Such a property is clearly undesirable, for it entails that
the response of the circuit to external excitations is not uniquely determined by these but also
depends on certain unobservable endogenous quantities. From the point of view of design, it
indicates that the latter should be revisited in order to suppress useless loss of energy. We will
suppose in the rest of this work that the behavior matrix Min Equation (2) is invertible.
4 Partial transfer functions
The (local) stability of a circuit is studied by observing how it responds to small perturbations.
The latter can be either a set of small current sources at junction nodes or a set of small voltage
sources at terminals of the components. Due to smallness of the hypothesized perturbations,
one considers the linearized model whose behavior, when viewed as a system whose inputs are
the perturbations and whose outputs are a set voltages or currents in the circuit, will determine
whether the latter is (locally) stable or not. We are thus led to build the transfer function of
this system (see Section 11), which is a matrix whose entries are elementary transfer functions
corresponding to a single perturbation (input) applied at some junction node (in case of a current
source perturbation) or edge (in case of a voltage source perturbation) and whose effect (output)
is observed at some node or edge. These we call partial transfer functions (or partial frequency
responses) of the circuit, reserving the name transfer function (or transfer matrix for emphasis)
for the full matrix of all partial transfer functions of the circuit. In this section we review how
to compute partial transfer functions, and in later sections we shall characterize them.
4.1 Partial transfer from a current source
Some ideal current source Iis plugged in between the ground and some junction node α.
RR n°8659

10 L. Baratchart, S. Chevillard & F. Seyfert
Plugging the current source changes Kirchhoff’s law at α: it becomes PjIj=I, where the
sum is taken over all edges jadjacent to α. The behavior of the perturbed circuit is thus given
by
MX =









0
.
.
.
I
.
.
.
0









.
If Mis invertible we can therefore write











V1
.
.
.
Vn
I1
.
.
.
Ip











=M−1









0
.
.
.
I
.
.
.
0









, .
It is hence clear that, for any k,Vk=M−1
[k,β]I, where βis the index of that row in Mexpressing
Kirchhoff’s law at node α. Here, M−1
[i,j]denotes the entry at row iand column jof M−1. We see
that the voltage at any node kdepends linearly on I. The ratio between (the Laplace transform
of) the voltage Vkand Iis the partial transfer function (or frequency response) of the circuit
from current at αto voltage at node k. By Cramer’s rule
Vk= (−1)k+βMβ,k
det MI,
where Mi,j denotes the minor of Mobtained by deleting row iand column j.
Note that invertibility of the behavior matrix is necessary and sufficient for all partial transfer
functions from a current source to exist.
4.2 Partial transfer from a voltage source
An edge αis chosen, which goes from a junction node kto some component. An ideal voltage
source with Laplace transform Uis plugged between kand that component. This changes
the potential at the corresponding terminal of the component, which becomes Vk+U. The
only coefficients in Mthat require change are those in the k-th column corresponding to a
row describing the behavior of the component involved. Examination of the left-part of the
matrices described in Section 2 shows that there is only one row which is affected, and that the
corresponding coefficient in Mis a nonzero real number γ(equal to 1or −1in the case of a
resistor, an inductor, a capacitor, a diode or a transmission line, and to −gm,−gdor gm+gdin
the case of transistor). Therefore we can write for the perturbed circuit
MX =









0
.
.
.
−γ U
.
.
.
0









.
Inria

Transfer functions realizable with active electronic components 11
Denoting with βthe index of the row where −γU lies in the above equation, we get using the
same argument as before that, if matrix Mis invertible, then
Ir= (−1)r+βMβ,r
det M(−γ)U, for any r∈ {1, . . . , p}.
The ratio Ir/U is the frequency response at edge rof the circuit to the voltage source U, and again
invertibility of the behavior matrix is necessary and sufficient for all partial transfer functions
from a voltage source to exist.
Altogether, we see that the partial transfer function obtained by using a voltage source is of
the same form as the partial transfer function obtained using a current source. In the rest of this
work, without loss of generality, we only deal with the latter, that is, we favor transfer functions
of impedance type.
5 R-L-C circuits with negative resistors
As a first step towards our main results, we describe in this section the structure of partial
transfer functions of circuits that have no transmission line nor transistor in their components.
Namely, each partial frequency response of a circuit made of positive resistors, negative resistors,
along with standard (i.e. positive) capacitors and inductors, belongs to the field R(s)of rational
functions in the variable swith real coefficients, and conversely any f∈R(s)can be realized as
a partial frequency response of such a circuit. Moreover, we will see (cf. Remark 2 at the end
of the present section) that the result still holds if transistors are added to the list of admissible
components.
The above statement is essentially equivalent to a classical property of impedance (rather than
partial transfer functions) of networks comprising positive resistors, negative resistors, capacitors
and inductors [4]. The latter is formally stated as Theorem 1 in Section 5.2 to come.
Note that the converse part of the statement is concerned with a single partial transfer
function, and says nothing about synthesizing an arbitrary rational matrix as the transfer matrix
of a circuit made of resistors of arbitrary sign, capacitors and inductors. Whether this is possible
or not is still an open issue, see [4] where transformers and gyrators are added to the set of
admissible elements in order to answer the question in the positive.
We now discuss the proof. The fact that each partial frequency response belongs to R(s)is
obvious from the previous section, for this frequency response has the form α Mβ,k /det(M)with
α∈R, and Mas in equation (2). Besides, since the circuit does not contain transmission lines,
all elements of Mbelong to R(s), thus also Mβ,k/det(M)∈R(s).
The converse part is a little harder, and can be established along the classical lines of Foster
synthesis [3, thm. 5.2.1] by relaxing sign conditions therein, see also [4]. Below, we give a different
proof which lends itself better to generalization when we consider circuits with transmission lines,
as will be the case in a forthcoming section.
We make extensive use of the widget given in Figure 8, called an inverter. It is made of two
dipoles with impedance X, one with impedance −Xand one with impedance Z. Using parallel
and series composition rules, one easily sees that it is equivalent to a dipole whose impedance is
X+−X(X+Z)
−X+ (X+Z)=−X2
Z.
Of course, the inverter cannot be realized from passive devices because it is impossible to realize
both a network with impedance Xand a network with impedance −Xwith passive components.
Having negative resistors at our disposal is thus crucial at this point. Hereafter, we simply speak
of impedance of a network to mean impedance of a two-terminal network.
RR n°8659

12 L. Baratchart, S. Chevillard & F. Seyfert
Figure 8: The inverter network
5.1 What the inverter makes possible
Lemma 1. If negative resistors are allowed, and if one has a network with impedance x, it is
possible to build a network with impedance −1/x.
Proof. The inverter of Figure 8 can actually be realized with X= 1 and −X= 1 since we
have negative resistors. Using the network with impedance xfor Z, we obtain a network with
equivalent impedance −1/Z =−1/x.
Corollary 1. Having positive and negative resistors, along with standard capacitors and induc-
tors allows one to emulate negative capacitors and inductors.
Proof. Applying Lemma 1 to a capacitor of capacitance α(i.e. x= 1/(α s)) gives us a negative
inductor of inductance −α. Conversely, applying the lemma to an inductor of inductance α(i.e.
x=α s) gives us a negative capacitor of capacitance −α.
Lemma 2. If negative resistors are allowed and if one has an electrical network with impedance
xand another one with impedance −x, it is possible to build a network with impedance ±x2.
Proof. We use the inverter again, this time with X=x. This is possible because we can actually
build a network with impedance −X. For Z, we can choose as a resistor of resistance ±1
(since negative resistors are allowed). We hence obtain a network with equivalent impedance
−X2/±1 = ±x2.
Lemma 3. If negative resistors are allowed and if one has electrical networks with impedances
x,−x,yand −y, it is possible to build a network with impedance ±xy.
Proof. Composing networks for xand y(respectively −xand −y) in series, we get a network
of impedance x+y(respectively −(x+y)). Using Lemma 2, we get circuits with impedances
±(x+y)2,±x2and ±y2.
Now, composing in series (x+y)2,−x2and −y2, we get a network of equivalent impedance
(x+y)2+ (−x2) + (−y2)=2xy. The same way, we also get a circuit with impedance x2+y2+
(−(x+y)2) = −2xy.
To sum up, we just proved that, having networks with impedances ±xand ±y, one can build
networks with impedance ±2xy. Applying this result to ±2xy and a resistor of resistance ±1/4,
we get a network with impedance ±xy.
5.2 Frequency response of RLC circuits with negative resistors
Theorem 1. Let P(s)/Q(s)be an arbitrary rational function with real coefficients. Using positive
and negative resistors, capacitors and inductors, it is possible to build a network with impedance
P(s)/Q(s).
Inria

Transfer functions realizable with active electronic components 13
Proof. Using Corollary 1 together with Lemma 3, we see by an elementary induction that we
can build networks with impedances ±αskfor any k∈Z. Composing them in series allows us
to realize ±P(s)and ±Q(s). Now, using Lemma 1, we can realize ±1/Q(s), and finally, using
Lemma 3 again, we get ±P(s)/Q(s).
Let finally P(s)and Q(s)be two polynomials such that 06≡ Q6=P. To establish the result
announced at the beginning of this section, it remains to observe in view of Theorem 1 that the
circuit shown in Figure 9, where we set R(s) = P(s)/Q(s), is realizable with positive and negative
resistors, inductors, capacitors. Indeed, it consists of a series of two elements with impedance
1and R(s)respectively, with both ports connected to the mass. Now, one easily checks that
the partial frequency response to a current source plugged at the junction node with output the
voltage at that node is exactly P(s)/Q(s). This achieves the proof.
Figure 9: A circuit with partial frequency response R(s)
Remark 2. Circuits made of positive and negative resistors, capacitors, inductors and linearized
transistors have exactly the same class of partial frequency responses as those without transistors.
Indeed, elements in the matrix describing the behavior of transistors (see Figure 4) also belong
to R(s), so the frequency response of circuits with transistors in turn lies in R(s). Since all
functions from R(s)are already realizable without transistors, this remains a fortiori true when
transistors are al lowed.
6 Circuits with transmission lines
Below, we generalize the result of the previous section to the case where circuits consist of all
elements listed in Section 2. More precisely, let Ebe the smallest field containing R(s)as well
as all functions of the form γ(s) sinh(γ(s)) and cosh(γ(s)), where γ(s) = p(a+bs)(c+ds)for
some real numbers a, b, c, d ≥0. The determination of the square root involved in the expression
for γis irrelevant: choosing a determination or its negative defines the same functions because
cosh is even and sinh is odd. Another way of defining Eis, e.g.,
E={(f1+· ·· +fn)/(g1+· ·· +gm),with f1, . . . , fn, g1, . . . , gm∈ A} where
A=(α sk
n
Y
i=1
γi(s) sinh(γi(s))
m
Y
i=n+1
cosh(γi(s)))α∈R, k∈N,
γi(s)=√(ai+bis)(ci+dis)
where ai, bi, ci, di≥0
.(3)
The main result of this section is that the class of functions realizable as partial transfer functions
of circuits made of elements listed in Section 2, namely positive and negative resistors, capacitors,
inductors, linearized transistors and transmission lines, is exactly E.
We follow the same approach as in the previous section. Again, proving that each partial
frequency response belongs to Eis very easy: all entries of the matrix Mgiven in Equation (2)
belong to E, which is a field, thus any partial frequency response (which is proportional to the
ratio of a minor of Mand its determinant) also belongs to E.
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14 L. Baratchart, S. Chevillard & F. Seyfert
We now show the converse. For this, we use a trick: we only consider networks with one
available terminal, for which a relation of the form V=Z I is satisfied, where Vis the potential
of the terminal and Iis the current entering the network through the terminal. We call such a
network a one-port circuit and we call Zthe impedance of the circuit.
Figure 10: Representation of a one-port circuit
Of course such circuits form a strict, and in fact very particular subset of all circuits one can
built from positive or negative resistors, inductors, capacitors and transmission lines. One could
think a priori that partial frequency responses realizable in this way are only a small subset of
all partial frequency responses arising from more general topologies. As we will see this is not
the case, as all functions of Ecan be realized as impedances of one-port circuits already. Now, if
R(s)is an element of E, so is R(s)/(1 −R(s)), and we can use again the the circuit of Figure 9.
Indeed, it can be synthesized with a one-port circuit of impedance R(s)/(1 −R(s)) and a resistor
of resistance 1. One easily checks that the partial frequency response obtained by plugging a
current source at the junction node and looking at the voltage at that same node is again R(s).
6.1 Using transmission lines as one-port circuits
The reason why we limit our study to one-port circuits is the following. Whereas dipoles are easy
to compose in series or parallel and this composition results in nice algebraic combinations of
their impedance functions, it is not so for transmission lines. Their behavior, recalled in Figure 2,
cannot be reduced to a single scalar relation, and one fundamentally needs two linear relations
to express it. Thus, when composing lines with other elements in a circuit, one is led to multiply
2×2matrices from which it is not easy to keep track of the algebraic structure of the resulting
elements.
In contrast, using a transmission line as a one-port circuit by forcing either the current or
the potential at one of its terminals simplifies the relations and allows us to retain a single linear
equation to represent its behavior in the form of the impedance of a one-port circuit. This we
see from the following two lemmas.
Lemma 4. Let a,b,c, and dbe four nonnegative numbers. Let us set γ(s) = p(a+bs)(c+ds),
where the determination of the square root is arbitrary. It is possible to realize a one-port circuit
with impedance (a+bs)/γ(s)·coth(γ(s)).
Figure 11: One-port circuit with impedance (a+bs)/γ(s)·coth(γ(s))
Inria

Transfer functions realizable with active electronic components 15
Proof. Let us consider a transmission line with characteristics R=a,L=b,G=c, and C=d.
If the second terminal of the line is left open, I2is forced to 0. Using the relations given in
Figure 2, we get
V1=a+bs
γ(s)coth(γ(s)) I1+a+bs
γ(s) sinh(γ(s)) I2=a+bs
γ(s)coth(γ(s)) I1
Lemma 5. Let a,b,c, and dbe four nonnegative numbers. Let us set γ(s)as in previous lemma.
It is possible to realize a one-port circuit with impedance γ(s)/(a+bs)·tanh(γ(s)).
Figure 12: One-port circuit with impedance γ(s)/(a+bs)·tanh(γ(s)).
Proof. Let us consider a transmission line with characteristics R=c,L=d,G=a, and C=b.
Remark that γ(s)is still equal to p(a+bs)(c+ds). This time, we connect the second terminal
directly to the ground. This forces V2= 0. Using the second relation given in Figure 2, we get
c+ds
γ(s) sinh(γ(s)) I1+c+ds
γ(s)coth(γ(s)) I2=V2= 0,
and hence I2=−I1/cosh(γ(s)).
The other relation is
V1=c+ds
γ(s)coth(γ(s)) I1+c+ds
γ(s) sinh(γ(s)) I2=c+ds
γ(s)cosh(γ(s))
sinh(γ(s)) −1
sinh(γ(s)) cosh(γ(s)) I1.
Since cosh(γ(s))2−1 = sinh(γ(s))2, the expression simplifies to V1= (c+ds)/γ(s)·tanh(γ(s)) I1.
We conclude by remarking that (c+ds)/γ(s) = γ(s)/(a+bs).
Remark 3. Let us consider a two-ports network with impedance R(s). Connecting one of the
terminals of the network to the ground, we get a one-port circuit. The potential Vat the remain-
ing terminal is the voltage between both terminals of the network, since the ground has potential 0.
By definition, V=R(s)I, hence the one-port circuit so obtained also has impedance R(s).
6.2 Composing one-port circuits
Composing one-port circuits with other electrical elements is a little more complicated than
composing dipoles, since only one terminal remains available. Still, we can compose one-ports
circuits in parallel or compose a one port-circuit in series with a dipole: this indeed works as
expected which is shown in the following two lemmas.
Lemma 6 (Parallel composition of one-port circuits).If one has a one-port circuit with impedance
xand a one-port circuit with impedance y, it is possible to build a one-port circuit, with impedance
xy/(x+y).
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16 L. Baratchart, S. Chevillard & F. Seyfert
Figure 13: Composition of one-port circuits in parallel
Proof. We simply connect both circuits in parallel. Then I=Ix+Iyby Kirchhoff’s law and
V=Ux=Uy. Moreover, by hypothesis Ux=x Ixand Uy=y Iy. We deduce that
I=Ux
x+Uy
y=1
x+1
yV=y+x
xy V,
hence the result.
Lemma 7 (Series composition of a one-port circuit and a dipole).If one has a one-port circuit
with impedance x, it is possible to build a one-port circuit with impedance x+R(s), for any
rational function Rwith real coefficients.
Figure 14: Composition of a one-port circuit and a dipole in series
Proof. Let us consider a dipole of impedance R(s), as given by Theorem 1 and let us connect it
in series with the one-port circuit of impedance x.Then I=Ixbecause the current is preserved
from one terminal of the dipole to the other. Moreover UR=R(s)Iand Ux=x Ixby hypothesis.
Therefore, V=Ux+UR= (x+R(s)) I.
There is obviously a problem to compose one-port circuits in series: by construction, one-
port circuits have only one terminal available, which prevents us from chaining them to add
their impedances. Yet, we remark that Lemma 1 still holds if “network” is now understood as
“one-port circuit”. Indeed, according to Lemmas 6 and 7, the inverter of Figure 8 is realizable
with resistors X= 1 and −X=−1and a one-port circuit of impedance Z=x. This is sufficient
to emulate composition in series:
Lemma 8. If one has a one-port circuit with impedance xand another one with impedance y,
it is possible to build a one-port circuit with impedance x+y.
Inria

Transfer functions realizable with active electronic components 17
Proof. Using Lemma 1 it is possible to build one-port circuits with impedances −1/x and −1/y.
Hence, using Lemma 6 we get a circuit with impedance
(−1/x)(−1/y)
(−1/x)+(−1/y)=−1
y+x.
Using Lemma 1 again, we get a one-port circuit with impedance x+y.
Now that composition in series has been proved to be possible, we see that Lemmas 2 and 3
hold for one-port circuits as well, and we shall be able to proceed much in the same way as we
did to establish Theorem 1.
6.3 Class of all impedances of one-port circuits
Lemma 9. Let a,b,c, and dbe four nonnegative numbers. Let us set γ(s) = p(a+bs)(c+ds),
where the determination of the square root is arbitrary. It is possible to realize one-port circuits
with impedances −(a+bs)/γ(s)·coth(γ(s)) and −γ(s)/(a+bs)·tanh(γ(s)).
Proof. According to Lemma 5 (respectively, Lemma 4) there is a one-port circuit with impedance
γ(s)/(a+bs)·tanh(γ(s)) (respectively (a+bs)/γ(s)·coth(γ(s))). Using Lemma 1 we get a one-port
circuit of impedance −(a+bs)/γ(s)·coth(γ(s)) (respectively −γ(s)/(a+bs)·tanh(γ(s))).
Lemma 10. Let a,b,c, and dbe four nonnegative numbers. Let us set γ(s)as in previous
lemma. It is possible to realize one-port circuits with impedances ±cosh(γ(s))2.
Proof. According to Lemmas 5 and 9, there are one-port circuits with impedances ±γ(s)/(a+
bs)·tanh(γ(s)). Using Lemma 2, we can hence build one-port circuits with impedances
±γ(s)2
(a+bs)2tanh(γ(s))2=±c+ds
a+bs tanh(γ(s))2.
Now, by Remark 3 and Theorem 1, there are one-port circuits with impedances ±(a+bs)/(c+ds).
Therefore, using Lemma 3, we can build one-port circuits with impedances ±tanh(γ(s))2. Using
Lemma 7 we get impedances ±(tanh(γ(s))2−1), and finally, using Lemma 1, we get impedances
±1
1−tanh(γ(s))2=±cosh(γ(s))2.
Theorem 2. Any element of the set Edefined in Equation (3) can be realized as the impedance
of a one-port circuit.
Proof. It is enough to prove that, for any nonnegative numbers a,b,c, and d, one can realize
±γ(s) sinh(γ(s)and ±cosh(γ(s)), where γ(s)is defined as in Lemma 9. Indeed, this is sufficient
to show that any element of Ais realizable (using Remark 3 and Lemma 3). Then, Lemma 8
allows us to realize any sum of elements of A, and we conclude using Lemmas 1 and 3.
Let us consider arbitrary nonnegative numbers a,b,c, and d. We define γ(s)as before:
γ(s) = p(a+bs)(c+ds). We set a0=a/2,b0=b/2,c0=c/2and d0=d/2and γ0(s) =
p(a0+b0s)(c0+d0s). We obviously have γ(s) = ±2γ0(s)(the determinations of the square roots
in the definitions of γand γ0are not necessarily the same). According to Lemma 10, there are one-
port circuits with impedances ±cosh(γ0(s))2. Now, using the identity cosh(2x) = 2cosh(x)2−1,
RR n°8659

18 L. Baratchart, S. Chevillard & F. Seyfert
we see that cosh(γ0(s))2=1
2cosh(±γ(s))+ 1
2.Using Lemma 7, and since cosh is an even function,
we get impedances ±1
2cosh(γ(s)), and by addition we can get rid of the factor 1/2.
Furthermore, since we have impedances ±1
2cosh(γ(s)) and ±γ(s)
a+bs tanh(γ(s)), we get
±γ(s)
2(a+bs)sinh(γ(s))
by Lemma 3. Using Remark 3 and Theorem 1, there are one-port circuits with impedances ±2(a+
bs). Thus, by Lemma 3 again, we can build one-port circuit with impedance ±γ(s) sinh γ(s), as
desired.
Remark 4. Note that each element of Eis a meromorphic function on C, for branchpoints like
−ai/bi, and −ci/diin Equation (3) (cf. page 13), which are a priori of order 2, are in fact
artificial by evenness of s7→ γ(s) sinh γ(s)and s7→ cosh γ(s).
7 Notion of stability
It is natural to say that a circuit is (locally) stable with respect to small current perturbations
if every partial transfer function is stable, and similarly for voltages. In other words, stability
should refer to the collection of all partial transfer functions (i.e. to the transfer matrix). In
practice, however, the computational complexity involved with plugging current sources at every
junction node (or voltage sources at every edge) and checking their effect on each current or
voltage in the circuit often lies beyond computational and experimental capabilities. Typically,
one is content with checking stability on a couple of well-chosen partial transfer functions. We
do not lean on the issue of how to pick those, but we shall discuss stability of a single partial
transfer function from a current source at a node to a voltage at a node.
The standard definition of stability for a linear dynamical system is that it maps input signals
of finite energy (i.e. of bounded L2-norm, both in time and frequency domain since the Fourier
transform is an isometry) to output signals of finite energy. This type of stability is denoted as
BIBO, that stands for Bounded Input Bounded Output. Equivalently, a linear dynamical system
is stable if its transfer function belongs to the Hardy space H∞,i.e. if it is holomorphic and
bounded in the right half-plane (see Section 11). This definition is not satisfactory here, for
it would term unstable such simple passive components as pure inductors (i.e. Z=Ls), pure
capacitors (i.e. Z= 1/(Cs)), or ideal transmission lines (those for which when R=G= 0 in
Section 2.2). To circumvent this difficulty, we first argue that no current source is ever ideal: it
always has internal resistance, that in the near to ideal case can be considered as a very large
resistor connecting the ground to the node where we plug the current source. Next, we take a hint
from scattering theory: if Zis the partial transfer function and R > 0is the load of the current
source, then (R−Z)/(R+Z)is the transfer-function from the incoming (maximum available)
power wave to the reflected power wave. In other words, |(R−Z(iω))/(R+Z(iω))|2is that fraction
of the maximum power that the current source can supply to the system which bounces back, at
frequency ω, and if Iis the intensity of the current then (1 − |(R−Z(iω))/(R+Z(iω)|2)RI2/4
is the power actually dissipated by Z(see [8]).
When the circuit is passive, then <Z≥0hence the fraction of reflected power is less than 1,
that is to say (R−Z)/(R+Z)lies in H∞and its supremum norm is at most 1. This indicates
that the system does not generate energy. When the circuit is active, which is the case when it
contains diodes or transistors, the reflected fraction of the incoming power can be greater than
1 at some or all frequencies, which means that the system generates energy at those frequencies.
For instance, an amplifier is expected to magnify the signal it receives. Of course, the necessary
Inria

Transfer functions realizable with active electronic components 19
power supply to do this has to come from an external source, used to generate voltage at terminals
of the primary circuits of diodes and transistors. Thus, even if it has norm greater than unity,
(R−Z)/(R+Z)should still lie in H∞to prevent instabilities, namely, working rates for which
the energy demand to these primary circuits becomes infinite. In view of this, it seems natural to
say that Zis stable if (R−Z)/(R+Z)∈ H∞. However, we do not want the degree of stability of
Zto depend on the actual value of the load, which leads us to the make the following definition
which we could not locate in the literature.
Definition 1. Let Zbe a partial frequency response of a circuit. We say that Zis stable if there
exists R0>0and M > 0such that
∀R > R0,R−Z
R+Z∈ H∞and 



R−Z
R+Z


H∞≤M. (4)
The same definition applies to the impedance of a two-ports.
In other words, our definition states that a circuit is stable, if in terms of power waves it is
of BIBO type, and this for all near to perfect feeding current sources (i.e R≥R0). Definition 1
of stability is more general than the BIBO condition on Z:
Lemma 11. If Z∈ H∞(i.e. if Zis stable in the usual sense), then Zis also stable in the sense
of Definition 1.
Proof. Since Z∈ H∞it holds that |Z(s)|< M for some Mand all swith <s > 0. If we set
R0=M+ 1, then |Z(s) + R| ≥ 1for all R≥R0, so that |(R−Z(s))/(R+Z(s))|is bounded
above by R+M, implying that it lies in H∞.
Note that the converse of Lemma 11 is not true, e.g. ideal inductors and capacitors becomes
stable with Definition (4), although they are not themselves in H∞.
As pointed out in Section 5, circuits made of positive resistors, negative resistors, capacitors
and inductors have rational partial transfer functions, in which case Definition 1 simply says
that the rational function (R−Z)/(R+Z)has no pole in the closed right half-plane for all R
large enough, including at infinity. In fact, this function cannot have poles at infinity anyway
(i.e. it is proper), because either Zhas a pole there and then (R−Z)/(R+Z)(∞) = −1, or else
Z(∞) = a∈Cin which case (R−Z)/(R+Z)has finite value at infinity for each R > |a|. Thus,
for such circuits, stability simply means that (R−Z)/(R+Z)has no no pole at finite distance
in the closed right half-plane, whenever Ris large enough. This familiar criterion for stability
no longer holds when lines are present in the circuit, as follows from the example below. Set
f(s) = stanh(s)−1
s+ 1 and Z(s) = 2f(s)
f(s)+2.(5)
We will show that for all Rlarge enough, s7→ (R−Z(s))/(R+Z(s)) is defined and finite on
{<s≥0}but does not belong to H∞. This will provide us with an example of a transfer function
which is unstable in the sense of Definition 1, yet has no poles in the right half-plane. We proceed
via the following steps.
•First, the function Zis realizable since it obviously belongs to the set Edefined in Equa-
tion (3). In fact, one can check that the circuit shown in Figure 15 has partial transfer
function Zfrom a current source between the ground and the black bullet (bottom-left of
the circuit) to the voltage at that bullet.
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20 L. Baratchart, S. Chevillard & F. Seyfert
Figure 15: Realization of a function that is unstable, though it has no pole in the right half-plane.
•Next, we remark that the equation f(s) = −awhere a∈[1,+∞[has no solution sin the
right half-plane. Indeed,
f(s) = −a⇐⇒ tanh(s) + 1
s+ 1 =1−a
s.
We observe that tanh(s) + 1/(s+ 1) has a positive real part whenever <(s)≥0though
(1 −a)/s has a non-positive real part. Thus they cannot be equal.
•Consequently, for all R > 2, the function (R−Z)/(R+Z)is finite at every point of the
closed right half-plane. Indeed, since
R−Z
R+Z=R−2
R+ 2 ·f(s) + 2R
R−2
f(s) + 2R
R+2
,(6)
it follows from the previous item that the denominator cannot vanish for <s≥0and,
though fmay have poles there, the function (R−Z)/(R+Z)is analytically continued at
them with value (R−2)/(R+ 2).
•To recap, we just showed that for any R > 2the function (R−Z)/(R+Z)is holomorphic
in {<s≥0}. Still, we claim that it does not belong to H∞. Indeed, put for simplicity
α= 2R/(R+ 2) and consider the sequence of points sk=i(kπ +α/(kπ)). Then
sktanh(sk) = −kπ +α
kπ tan α
kπ =−α+◦1
k,
Inria

Transfer functions realizable with active electronic components 21
from which we deduce the following identities:







f(sk) + α=−1
sk+ 1 +◦1
k=−1
ikπ +◦1
k
R−2
R+2 f(sk) + α=1−R−2
R+2 α+◦(1) = 8R/(R+ 2)2+◦(1) .
From Equation (6), we obtain
R−Z
R+Z=
R−2
R+2 f+α
f+α,
and so, its value at point skis −8ikπR/(R+2)2+◦(k). Hence, we did exhibit (for any R > 2)
a sequence of points on the imaginary axis along which (R−Z)/(R+Z)is unbounded.
This shows it cannot lie in H∞(cf. Section 11).
Remark: (R−Z)/(R+Z)does not belong either to the space H2nor to any Hp(see definition
in Section 11). To see this, remark that
f(s) = s2tanh(s) + stanh(s)−1
s+ 1 ,
hence, setting β= (R−2)/(R+ 2), we have
R−Z
R+Z=βf (s) + α
f(s) + α=β(s2tanh(s) + stanh(s)−1) + α(s+ 1)
s2tanh(s) + stanh(s)−1 + α(s+ 1) .
Now, setting M= 1/tanh(s)we can rewrite the previous expression as
R−Z
R+Z=β(1 + 1/s −M/s2) + αM/s +αM/s2
1 + (1 + αM)/s + (α−1)M/s2.
Whenever R > 2, we have 1< α < 2and 0< β < 1. Let us consider s=i(kπ +t)where kis an
integer such that |k|π≥6+4/β and t∈[π/4,3π/4]. Then, |s| ≥ 6+4/β and |M| ≤ 1. From
this we deduce that

R−Z
R+Z≥
β1−1
6+4/β −1
(6+4/β)2−2
6+4/β −2
(6+4/β)2
1 + 3
6+4/β +1
(6+4/β)2
=29β2+ 30β+ 8
55β2+ 60β+ 16 β≥β
2.
In conclusion, we obtained a positive lower bound of |(R−Z)/(R+Z)|valid on any interval of
the form [iπ(k+ 1/4),iπ(k+ 3/4)] (k∈Zlarge enough): this shows that (R−Z)/(R+Z)does
not belong to Lpof the imaginary axis, so it belongs to no Hardy space.
A pending issue. The example just constructed has pathological behavior because it has a
sequence of poles located in the open left half-plane but asymptotically close to the imaginary
axis at large frequencies. One may ask if a partial transfer function can be unstable (with respect
to Definition 1) when the set of poles is at strictly positive distance from the axis. Symmetrically,
if a partial transfer function has poles arbitrarily close to the axis, is it necessarily unstable? It
is not known to the authors whether stability, in the sense of Definition 1, can be described
solely in terms of poles of the partial transfer function. In this connection, we mention that the
behavior of poles of rational functions in the variable sand in real powers of exp(s)(the subset
RR n°8659

22 L. Baratchart, S. Chevillard & F. Seyfert
of Eattainable with lossless lines, i.e. those for which a=c= 0) is well-known [10]: they are
asymptotic either to vertical lines or to curves separating exponentially fast from the imaginary
axis. The situation with lossy lines has apparently not attracted much attention, and may be
more complicated.
Until now, we considered linearized diodes as pure negative resistors and linearized transistors
as pure current sources controlled by voltages. Such ideal models are simple and usually lead
to good approximation at working frequencies, therefore they are widely used in simulation
and design. However, the example of Figure 15 shows that checking stability does not reduce
then to verify that poles have strictly negative real part. This somewhat contradicts common
engineering practice, and puts a question mark on the use of ideal linearized models in connection
with stability.
Actually, ideal models are somewhat unrealistic: even if it does not show at working fre-
quencies, no active component has gain at all frequencies for there are always small resistive,
capacitive and inductive effects in a physical device. As we will see in the next section, tak-
ing them into account restricts considerably the class of transfer functions realizable with such
circuits.
8 Realistic model of linearized components
So far, we modeled a diode as i=f(u), where fis a non-linear real-valued function. To
take into account small resistive, capacitive or inductive effects, we should rather postulate that
i=f(u, du/dt, di/dt)where fhas very small derivatives with respect to the second and third
variables.
Simple, “realistic” models of a diode can be given along these lines, e.g. those given in
Figure 16a and Figure 16b.
(a) With inductive effect and high resistance (e.g., of the
air around the diode)
(b) With capacitive effect and small resistance (e.g.,
of the wire)
Figure 16: Two realistic models of a linearized diode
Inria

Transfer functions realizable with active electronic components 23
Of course, physical reality is much more complex and it is pointless to attempt at giving a
complete and accurate model of a linearized diode at all frequencies. Besides, different types of
diodes would require different descriptions and it would be cumbersome to distinguish between
them, while the differences are completely negligible in practice in the range of frequencies where
the diodes are intended to be use. Instead, we put forward the paradigm that “what happens at
very large frequencies is unimportant beyond passivity”, and we shall use a somewhat abstract
definition to accommodate various cases occurring in practice.
Definition 2. A realistic linearized diode is a dipole with complex impedance Zhaving the
following characteristics:
•Z(s)is a rational function with real coefficients.
•When |s|< ω0, it holds that Z(s) = −R+(s)where is a function whose modulus is
negligible compared to other quantities in the circuit.
•When |s|> ω1and <(s)≥0, we have <(Z(s)) ≥0.
In this definition, 0< ω0< ω1as well as Rare positive real numbers.
The hypothesis “Zis a rational function with real coefficients” simply states that a realistic
negative resistor may, in principle, be described as a combination of a pure negative resistor and
standard passive linear dipoles. In particular both models proposed in Figure 16 comply with
Definition 2, but many other models would also meet our requirements.
Remark 5. The same argument as in Remark 1 (cf. page 4) shows that, if V1,V2,I1and I2
denote the potentials and currents at both terminals of a realistic linearized diode, then
V1I1+V2I2=Z|I1|2=Z|I2|2.
Therefore, when |s|> ω1and <(s)≥0, we have that <(V1I1+V2I2)≥0.
Transistors can in turn be modeled in a “realistic” way, to account for the fact that they do not
provide gain at very high frequencies. For instance, the model presented in Figure 17 corresponds
to what is called the intrinsic model of the linearized transistor and displays capacitive effects
appearing at the junctions between semiconductors. As in the case of a diode, reality is still
more complex and involves both inductive and capacitive effects that we do not try to model
since they are irrelevant to our discussion.
Figure 17: Intrinsic model of a linearized transistor
This leads us to the following definition, in the spirit of Definition 2:
RR n°8659

24 L. Baratchart, S. Chevillard & F. Seyfert
Definition 3. A realistic linearized transistor is an electronic component with three terminals
such that voltages and currents, at these terminals, satisfy relations of the fol lowing form:



−Y(s) 0 Y(s) 1 0 0
−Ym(s)−Yd(s)Ym(s) + Yd(s)010
0 0 0 1 1 1













VG
VD
VS
IG
ID
IS










=


0
0
0


,
where Y,Ymand Ydare complex admittances meeting the fol lowing requirements:
•There exist positive real numbers τ1, . . . , τpsuch that Y,Ymand Ydare rational functions
in the variables s,exp(−τ1s),. . . ,exp(−τps)) having real coefficients.
•For |s|< ω0it holds that Y(s) = (s),Ym(s) = gm+m(s)and Yd(s) = gd+d(s), where
,mand dare functions whose moduli are negligible compared to other quantities in the
circuit.
•The function |Ym(s)|tends to 0as |s|tends to infinity, subject to <(s)≥0. Moreover,
whenever |s|is large enough and <(s)≥0, it holds that <(Y(s)) > α and <(Yd(s)) > β
with αand βtwo positive constants.
Corollary 2. For a realistic linearized transistor, it holds whenever |s|is large enough with
<(s)≥0that
<(VGIG+VDID+VSIS)≥0
Proof. From Definition 3, we see that, on the one hand, IG+ID+IS= 0, and the other hand,
IG
ID!= Y(s) 0
Ym(s)Yd(s)! VG−VS
VD−VS!=M V.
Now, VGIG+VDID+VSIS= (VG−VS)IG+(VD−VS)ID+ (VS−VS)IS, since IG+ID+IS= 0.
Thus,
VGIG+VDID+VSIS=tV M V .
We remark that tV M V is a 1×1matrix, so it is equal to its transpose tV M?V. Consequently,
<(VGIG+VDID+VSIS) = 1
2tV M?V+tV M V =1
2
tV(M?+M)V.
The hypotheses of Definition 3 ensure that the matrix M+M?is Hermitian positive definite
when |s|is large enough and <(s)≥0, so <(VGIG+VDID+VSIS)≥0for such s.
It is immediate from Definitions 2 and 3 that partial frequency responses of circuits containing
positive resistors, inductors, capacitors, transmission lines, and realistic linearized diodes and
transistors are elements of the class Epresented in Section 6. This comes from the fact that
such a frequency response is still obtained by inverting a matrix whose entries are members of E.
However, not every function in Ecan be realized with a realistic circuit, in particular the function
ffrom the ideal circuit in Figure 15 cannot. Indeed, this a consequence of the following theorem.
Inria

Transfer functions realizable with active electronic components 25
Theorem 3. Let a circuit consists of resistors, inductors, capacitors, transmission lines and
realistic linearized diodes and transistors (cf. Definitions 2 and 3). Assume as always that the
behavior matrix of the circuit is invertible. To each partial frequency response Z(s)of the circuit,
there exists K > 0such that, whenever <(s)≥0and |s| ≥ K, we have <(Z(s)) ≥0.
In order to prove this, recall Tellegen’s theorem[3, thm. 2.14.1] expressing the conservation
of power inside a circuit. For our purpose, it is convenient to state it in graph-theoretic form:
Theorem 4 (Tellegen).Let Gbe an oriented graph whose vertices are numbered from 1to n
and edges from 1to m. Suppose that each vertex iis assigned a weight Vi∈Cand that each edge
jis assigned a weight Ij∈C. Denote by in(i)the set of incoming edges and by out(i)the set of
outgoing edges at vertex i. Moreover, for each edge k, we put head(k)for the vertex kis pointing
to and tail(k)for the vertex koriginates from. Subsequently, we set U(k) = Vtail(k)−Vhead(k).
If Gsatisfies the junction rule
∀i∈ {1,...n},X
j∈in(i)
Ij=X
j∈out(i)
Ij,
then the following relation holds:
X
j
(Vtail(j)−Vhead(j))Ij= 0.(7)
Proof. Let us define the incidence matrix of G, say M= (Mij ), where





Mij = 1 if i=tail(j),or, equivalently, if j∈out(i),
Mij =−1if i=head(j),or, equivalently, if j∈in(i),
Mij = 0 otherwise.
The junction rule means that MI = 0. Moreover, if we let Uindicate the vector with components
U(k), we get by definition tU=tV M. Consequently, tUI =tV MI = 0 which is (7).
Remark 6. If the junction rule is satisfied at all vertices but one, then it is satisfied at all
vertices. To see it, observe that (1, . . . , 1) M= (0, . . . , 0) because every column jof Mcon-
tains exactly once the value 1and once the value −1(at rows corresponding, respectively, to
the initial vertex and the final vertex of edge j). If the junction rule is satisfied at al l ver-
tices but one (we assume without loss of generality that it is vertex n), then the vector MI
looks like t(0, . . . , 0, α). Therefore, α= (1, . . . , 1) (M I). Finally, since (1, . . . , 1) (MI) =
((1, . . . , 1)M)I= (0, . . . , 0) I= 0, we conclude that αis indeed zero.
We can now prove Theorem 3.
Proof of Theorem 3. We consider the situation described in Section 4.1. Thus, the circuit gets
excited at some junction node αby a current source i(t). Because the behavior matrix is in-
vertible, voltages and currents are well defined in response to the excitation. We denote by I
the Laplace transform of t7→ i(t)and by I1, . . . , Ipand V1, . . . , Vnthe Laplace transforms of
currents and voltages in the circuit. We further fix the value of sso that <(s)≥0.
Define a graph Gto be the circuit with one edge added between the ground and each trans-
mission line (the virtual wire described in Figure 2), and we also add an edge between the ground
and the junction node α. We set the weight of a junction node kto be Vk(s). The weight of each
node corresponding to an electronic component is defined as being 0. The weight of each edge j
is Ij(s)(for the virtual wire of a transmission line, it is thus −(Ij0(s) + Ij00(s)) according to the
RR n°8659

26 L. Baratchart, S. Chevillard & F. Seyfert
last line of the matrix in Figure 2). The weight of the edge between the ground and node αis
defined as I(s).
We first observe that Gsatisfies the junction rule. Indeed, for each vertex of Gcorresponding
to a junction node of the circuit (except maybe the ground), this follows from Kirchhoff’s law; at
nodes that are electronic components, this is true by examination of the last line of the behavior
matrix of each component (cf. Section 2). The junction rule is hence satisfied at all vertices
of Gexcept maybe for the ground. But from Remark 6, we see that the junction rule is then
necessarily satisfied at all vertices including the ground. We can thus apply Tellegen’s theorem.
By construction, all edges but the one between the ground and node αare oriented from a
junction node to an electronic component. Since the weight of an electronic components is 0,
each term (Vtail(j)−Vhead(j))Ijin Equation (7) becomes Vtail(j)(s)Ij(s)in the present case.
Moreover, since the ground has voltage 0, the term corresponding to the edge between the
ground and node αis (0 −Vα(s)) I(s). Since by definition Vα(s) = Z(s)I(s), Tellegen’s theorem,
applied to the graph G, gives us
X
j
Vtail(j)(s)Ij(s) = Z(s)|I(s)|2.(8)
Now, since each edge in the circuit is adjacent to exactly one electronic component, we can
rewrite the left hand side of (8) as a sum over all electronic components ito obtain
X
iX
junction node β
adjacent to i
Vβ(s)Iβ→i(s) = Z(s)|I(s)|2,
where Iβ→i(s)denotes the weight of the edge between node βand electronic component i.
Applying Remark 1, Remark 5, Corollary 2 or Lemma 12 (cf. Section 10) according whether i
is a passive dipole, a realistic linearized diode, a realistic linearized transistor or a transmission
line, we see whenever <(s)≥0and |s|is large enough that, for each i,
<
X
β
Vβ(s)Iβ→i(s)
≥0.
Hence, the left hand side of Equation (8) has positive real part provided that <(s)≥0and |s|
is large enough, and so does Z(s).
Below we draw two important consequences of Theorem 3. They point at a remarkable
difference between the case of ideal linearized elements described in Section 6, and the case of
realistic linearized elements.
Corollary 3. Let Zbe a partial frequency response of a circuit made of transmission lines,
resistors, capacitors, inductors, and realistic linearized diodes and transistors. For <(s)≥0and
|s|large enough, it holds for all R≥0that |(R−Z(s))/(R+Z(s))| ≤ 1.
Proof. Let Kbe as in Theorem 3. Then, for |s|> K and <(s)≥0, we have as soon as R≥0
that |<(R−Z(s))|≤<(R+Z(s)), and it is otherwise clear that =(R−Z(s)) = −=(R+Z(s)),
hence |R−Z(s)| ≤ |R+Z(s)|, as desired.
Corollary 4. Assumptions and notations being as in Theorem 3, it holds for each R≥0that
the meromorphic function (R−Z)/(R+Z)has finitely many poles in the closed right half plane.
Inria

Transfer functions realizable with active electronic components 27
Proof. From the previous corollary we see that the poles of (R−Z)/(R+Z)in the closed right
half plane have to lie in a disk of radius Kcentered at 0. Since (R−Z)/(R+Z)∈ E, it is a
meromorphic function on Cby Remark 4. Hence poles cannot accumulate, therefore they are
finite in number.
9 A stability criterion
Below we set up a criterion for stability, in the sense of Definition 1, of a partial frequency
response of a circuit made of transmission lines, resistors, capacitors, inductors, and realistic
linearized diodes and transistors. It turns out to be similar to the classical passivity criterion [5,
ch.6] , except that the condition “residues of imaginary poles should be positive” gets relaxed
into “residues of imaginary poles should have positive real part”.
Theorem 5. Let Zbe a partial frequency response of a circuit made of transmission lines,
resistors, capacitors, inductors, and realistic linearized diodes and transistors. Then, Zis stable
in the sense of Definition 1 if and only if it has no pole in the open right half plane, while each
pole it may have on the imaginary axis is simple and has a residue with strictly positive real part.
Proof. From Corollary 3 we know there is K > 0such that |(R−Z(s))/(R+Z(s))| ≤ 1as soon
as <(s)≥0and |s|> K, therefore the stability of Zin the sense of Definition 1 is equivalent to
the boundedness of (R−Z)/(R+Z)for |s| ≤ Kand <s > 0, uniformly with respect to large R.
Let ζbe a pole of Zwith |ζ| ≤ K, and m≥1be its multiplicity. In a neighborhood of ζ, we
can write
Z(s) = am(s−ζ)−m+· ·· +a1(s−ζ)−1+F(s), am6= 0,(9)
where F(s)is holomorphic and bounded in some disk D(ζ , a) = {s:|s−ζ|< a}. By (9), Zis
holomorphic from D(ζ, a)into the Riemann sphere, and as such it is an open map, meaning that
the image of an open set is open. In particular, the image of D(ζ , a)under Zis a neighborhood
of ∞, meaning that it contains {s:|s|> A}for some A.
If <ζ > 0, we can pick aso small that D(ζ, a)is included in the open right half plane, and
by what precedes Ztakes every negative value of sufficiently large modulus in D(ζ , a). Hence
Zcannot be stable, because (R−Z)/(R+Z)will have a pole nearby ζfor R > 0large enough
(note that R−Zcannot vanish at a point where Z=−R).
Assume now that <ζ= 0. To say that |(R−Z)/(R+Z)| ≤ Mfor some M > 0is equivalent
to say that |1 + R/Z| ≥ ε1for some ε1>0. Now, if s∈D(ζ , a), then by (9)
1 + R
Z(s)=am+am−1(s−ζ) + · ·· +a1(s−ζ)m−1+ (R+F(s))(s−ζ)m
am+am−1(s−ζ) + · ·· +a1(s−ζ)m−1+F(s)(s−ζ)m.(10)
If m≥2, then to each R > 0large enough and each ε > 0, one can find sR,ε ∈D(ζ, a), with
<sR,ε >0, such that
|(R+F(ζ))(sR,ε −ζ)m+am|< ε. (11)
Indeed, the image of D(a, ζ)under the map s7→ (s−ζ)mis D(0, am)if m≥3, and D(0, a2)
deprived from the negative real axis if m= 2. Moreover, it is clear that sR,ε has to converge
to ζwhen R→ ∞, uniformly with respect to ε. Therefore, if we let {Rk}be a sequence
tending to +∞, we readily see from (10) that |1 + R/Z(sRk,1/k)|goes to zero as k→ ∞ so that
(R−Z)/(R+Z)cannot be uniformly bounded with respect to Rin D(ζ, a)⊂ {s:<s > 0}.
Hence Zis unstable.
Finally if m= 1 and Ris large enough, it is easily checked that there exists sR,ε ∈D(ζ, a)
satisfying <sR,ε >0and such that (11) holds for arbitrary small εif and only if <a1≤0. This
establishes the only if part of Theorem 5.
RR n°8659

28 L. Baratchart, S. Chevillard & F. Seyfert
Conversely, suppose that Z(s)has only pure imaginary poles for <s≥0and |s| ≤ K, which
are simple and whose residue has strictly positive real part. Let ζbe such a pole and a1its
residue. We just saw that sR,ε as in (11) does not exist for large Rand small ε, therefore there
is ε0>0such that |(R+F(ζ))(s−ζ) + a1| ≥ ε0for s∈D(ζ, a)and all Rlarge enough. Then,
by inspection of (10), we see that |1 + R/Z(s)| ≥ ε0/(2|a1|)as soon as Ris large enough and
s−ζis small enough, <s > 0. That is to say, there is r0>0and R0>0such that |s−ζ|< r0,
<s > 0, ad R > R0together imply |(R−Z(s))/(R+Z(s))| ≤ Mfor some Mindependent of R
and ssatisfying the preceding conditions.
Because there are only finitely many poles, the preceding argument shows that we can choose
R0>0and M > 0so large, and r0>0so small, that |(R−Z(s))/(R+Z(s))| ≤ Mas soon as
slies in the right half plane but closer than r0to one of the poles. But if <s > 0,|s| ≤ K, and
the distance from sto one of the poles is bigger than r0, then Z(s)is bounded independently
of sand increasing R0if necessary we may assume that still |(R−Z(s))/(R+Z(s))| ≤ Mfor
such sand R > R0(note that (R−Z(s))/(R+Z(s)) tends to 1as Rtends to +∞while Z(s)
remains bounded). This shows that Zis stable, as announced.
10 Appendix 1: Telegrapher’s equation
A transmission line is commonly modeled as a succession of infinitesimal capacitors, resistors
and inductors whose impedances do not depend on their position on the line (see Figure 18. In
the figure, Gdenotes the conductance of the resistor).
Figure 18: Model of a transmission line (the line goes from x= 0 to x=`and is a succession of
the same LR-GC infinitesimal elements)
This model leads to what is known as Telegrapher’s equation: expressing the relation between
currents and voltages at positions xand x+dx, one sees that





V(x+dx)−V(x) = −(R+Ls)I(x)dx
V(x+dx) = 1
(G+Cs)dx(I(x)−I(x+dx)).
From these local equations, we deduce a system of first order equations:







∂V
∂x =−(R+Ls)I
∂I
∂x =−(G+C s)V
=⇒






∂2V
∂x2=γ2V
∂2I
∂x2=γ2I ,
(12)
Inria

Transfer functions realizable with active electronic components 29
where γis one of the complex square roots of (R+Ls)(G+Cs).
We set z0= (R+Ls)/γ, so that we can write R+Ls =γ z0and G+Cs =γ
z0
.
Since R,G,Land Cdo not depend on x, Equation (12) leads to the following explicit
solutions: 


V(x) = Aexp(γ x) + Bexp(−γ x)
I(x) = Dexp(γ x) + Eexp(−γ x).
(13)
The transmission line has a given length `and we want to express the relations between I(0),
V(0),I(`)and V(`). For this purpose, we simply need to express the constants A,B,Dand E
in function of I(0) and V(0).
On the one hand, V(0) = A+Band I(0) = D+Eby letting x= 0 in Equation (13). On
the other hand, differentiating Equation (13) at point x= 0, we get







∂V
∂x (0) = γ(A−B)
∂I
∂x (0) = γ(D−E).
(14)
Now, using Equation (12) at x= 0, we see that ∂V
∂x (0) = −γ z0I(0) and ∂I
∂x (0) = −γ
z0
V(0),
thus B−A=z0I(0) and E−D=V(0)/z0. This gives us the values of A,B,Dand E:















A=1
2(V(0) −z0I(0))
B=1
2(V(0) + z0I(0))
D=1
2(I(0) −V(0)/z0)
E=1
2(I(0) + V(0)/z0).
(15)
Finally, putting these constants in Equation (13), at point x=`, and collecting the terms in
I(0) and in V(0), we obtain:





V(`) = cosh(γ `)V(0) −z0sinh(γ `)I(0)
I(`) = −sinh(γ `)
z0
V(0) + cosh(γ `)I(0).
(16)
The second line of Equation (16) gives us
V(0) = z0coth(γ `)I(0) −z0
sinh(γ `)I(`),
which we inject in the first line of the system, leading to
V(`) = z0
sinh(γ `)I(0) −z0coth(γ `)I(`).
Remark that multiplying R,G,Land Cby some constant αdoes not change the value of
z0, but has the effect of multiplying γby α. Therefore, the relations between the currents and
voltages at the terminals of a line of length `with characteristics R,G,Land Care the same
that the relations at the terminals of a line of length 1with characteristics R/`,G/`,L/` and
RR n°8659

30 L. Baratchart, S. Chevillard & F. Seyfert
C/`. In conclusion, from a theoretical viewpoint, the length of the line can be arbitrary and is
not worth mentioning. We arbitrarily set it to 1in this report.
Note that, by convention, all currents are oriented so as to enter electronic components. In
Figure 2 on page 5, we hence have I1=I(0), but I2=−I(`).
Lemma 12. When <(s)≥0, we have <(V1¯
I1+V2¯
I2)≥0.
Proof. We remark that
V1¯
I1+V2¯
I2=V(0) I(0) −V(`)I(`)
=−Z`
0∂V
∂x (ξ)I(ξ) + V(ξ)∂¯
I
∂x (ξ)dξ.
Replacing ∂V /∂x and ∂I/∂x by their values in function of Iand V, as given by Equation (12),
we obtain
V1¯
I1+V2¯
I2=−Z`
0−(R+Ls)|I(ξ)|2−(G+Cs)|V(ξ)|2dξ.
Finally, we get that
<(V1¯
I1+V2¯
I2) = Z`
0
(R+L<(s)) |I(ξ)|2+ (G+C<(¯s)) |V(ξ)|2dξ.
The expression under the integral symbol being positive for <(s)≥0, the integral itself is positive
when <(s)≥0.
11 Appendix 2: Transfer functions and stability
In this section, we discuss transfer functions in connection with stability and harmonic response,
a thorough account of which seems hard to find in the literature.
On the real line R, we let L1(R),L2(R)and L∞(R)indicate respectively the spaces of
complex-valued summable, square summable, and essentially bounded measurable functions with
respective norms
kfkL1(R)=Z+∞
−∞ |f(t)|dt, kfkL2(R)=Z+∞
−∞ |f(t)|2dt1/2
(17)
and
kfkL∞(R)= sup A≥0 : m{x∈R;|f(x)|> A}>0,(18)
where m(E)stands for Lebesgue measure of a set E. We put L1
loc(R)(resp. L2
loc(R),L∞
loc(R))
for spaces of functions fsuch that χEf∈L1(R)(resp. L2(R),L∞(R)) whenever Eis a bounded
measurable subset of R, where χEstands for the characteristic function of E(which is 1 on E
and 0elsewhere). We shall have to deal with corresponding spaces when Rgets replaced by the
imaginary axis iR, or by the positive semi-axis R+= [0,∞). We then write L1(iR),L1(R+),
and so on.
In order to study even fairly common systems1, one cannot work entirely with functions and
it is convenient to use the distributional formalism as follows. Let D(R)be the space of smooth
(i.e. C∞) functions with compact support on R; recall that the support of ϕ, abbreviated as
1For instance transmission lines like in the previous Appendix.
Inria

Transfer functions realizable with active electronic components 31
supp ϕ, is the closure of those t∈Rwith ϕ(t)6= 0. A distribution [12, ch. 6] is a form (i.e. a
complex-valued linear map) Φon D(R)which is continuous in the sense that to each compact
K⊂R, there is a constant Cand an integer n≥0for which |hΦ, ϕi| ≤ Ckϕ(k)kL∞(R)for all
0≤k≤n, whenever ϕ∈ D(R)is supported on K; here and below, brackets indicate the action
of distributions and superscript (k)stands for the k-th derivative. Each f∈L1
loc(R)identifies
with a distribution upon setting hf, ϕi=RRfϕ. Distributions can be multiplied by smooth
functions: if ψ∈C∞(R), then ψΦis the distribution given by hψΦ, ϕi=hΦ, ψϕi. They can be
differentiated as well, the derivative of Φbeing given by hΦ(1), ϕi=−hΦ, ϕ(1) i. Also, if Φis a
distribution and ψlies in D(R), the convolution Φ∗ψis the function Φ∗ψ(t) = hΦ, ψ(t−.)i;
here, the dot in the argument stands for a dummy variable. The support supp Φof a distribution
Φis the complement of the largest open set Ωfor which hΦ, ϕi= 0 whenever supp ϕ⊂Ω.
Let us think of t∈Ras being time. A linear dynamical system is a linear map u7→ ysending
each ubelonging to a certain function space of the variable t(the admissible inputs) to a function
yof t(the output corresponding to u), which is causal (i.e. u(t)=0for t≤t0implies the same
is true for y) and time invariant (if yis the output corresponding to some admissible uand if
τ∈R, then u(.−τ)is admissible and y(.−τ)is the corresponding output). If inputs from D(R)
are admissible and generate continuous outputs, then under mild continuity assumptions2there
is a distribution Φsupported on R+allowing us to describe the system through the convolution
equation:
y(t) = (Φ ∗u)(t), u ∈ D(R).(19)
Moreover inputs from D(R)turn out to generate smooth outputs [12, thm. 6.33]. The distri-
bution Φis called the impulse response of the system as it corresponds formally to the output
generated by a Dirac-δinput. What we defined really is a scalar system, that is, one whose
input and output are real or complex valued. The more general case of vector-valued (finite
dimensional) input and output can be represented as a matrix of scalar systems. For matters
under consideration here, results for vector-valued inputs and outputs are immediately deduced
by concatenation from their scalar-valued analogs.
Let now S(R)be the Schwartz space of smooth functions whose derivatives of any order
decrease faster than every polynomial at infinity. A distribution Ψis called tempered if it
extends to a form on Swhich is continuous in that there are integers n, m ≥0and a constant C
for which |hΨ, ϕi| ≤ Ck(1 + |.|m)ϕ(k)kL∞(R)for all 0≤k≤n, whenever ϕ∈ S(R). Now, if Φis
a distribution supported on R+such that the distribution e−A·Φis tempered for some A∈R,
then we say that Φis Laplace transformable (or A-Laplace transformable if some Aneeds to be
specified) and we can define its Laplace transform L(Φ) as a function of the complex variable
s=x+iy defined on the semi-infinite strip x>Aby the rule:
L(Φ)(s) = hΦ, e−s·i.(20)
Notation (20) deserves a word of explanation: although Φcannot be applied, strictly speaking, to
the function on R+given by t7→ e−st,<s>A, the latter can be smoothly extended for negative
tso as to vanish, say for t≤ −1. The tempered distribution e−A·Φmay then be applied to eA·
times this extension which lies in S(R), and the fact that Φis supported on R+entails that the
result is independent of the extension used. This is what is meant by (20). It can be shown
that the latter defines a holomorphic function of sfor <s>A[9, thm. 8.3.1]. If Φhappens
to be a locally integrable function, say f∈L1
loc(R), then it is easy to see that it is A-Laplace
transformable if and only if e−A·f∈L1(R)and that (20) reduces to the standard definition of
2If {un} ⊂ D(R)is supported on a compact set Kand u(k)
nconverges to u(k)uniformly on Kfor each k≥0
and some u∈ D(R), the corresponding outputs {yn}should converge pointwise to the output associated with u.
RR n°8659

32 L. Baratchart, S. Chevillard & F. Seyfert
Laplace transform:
L(f)(s) = Z∞
0
e−stf(t)dt. (21)
Note for later use that if Φis Laplace transformable, then its derivative is again Laplace trans-
formable and L(Φ(1))(s) = sL(Φ)(s). This follows easily from the definition.
When Φis A-Laplace transformable and u∈ D(R), then ygiven by (19) is in turn A0-Laplace
transformable for all A0> A. Indeed, it holds that
e−A0t|y(t)|=e−A0thΦ, u(t−.)i=e−(A0−A)the−A·Φ, e−A(t−.)u(t−.)i(22)
≤e−(A0−A)tC

(1 + |.|m)e−A·u(n)

L∞(R)=C0e−(A0−A)t,(23)
where we used continuity properties of tempered distributions. Thus, since Laplace transform
converts convolutions into ordinary products much like Fourier transform does [9, eqn. 8.5.8],
we get that
L(y)(s) = L(Φ)(s)L(u)(s),<s > A. (24)
Equation (24) is the so-called frequency description of (19), and L(Φ) is called the transfer
function of the system. Note that the restriction of L(y)to the vertical line {<s=x > A}is
the Fourier transform of e−x·y∈L1(R), hence L(y)cannot vanish identically unless ydoes [11,
thm. 9.11]. Consequently, by (24), a system is uniquely defined by its transfer-function when it
exists.
Although (19) is initially valid for u∈ D(R)only, the class of admissible inputs is much
bigger in cases of interest and (24) will usually extend to more general functions. For instance
if continuous inputs with compact support are admissible and generate continuous outputs, and
if moreover bounded pointwise convergence of inputs entails pointwise convergence of outputs,
then by the Riesz representation theorem [11, thm. 6.19] the impulse response Φis locally a
finite measure, say ν, and (19) becomes
y(t) = Z[0,t]
u(t−τ)dν(τ)(25)
(compare [1, Ch. 1]). Letting |ν|be the total variation of ν, which is a (not necessarily finite)
positive measure on R+[11, ch. 6, sec.1], we find that Φis A-Laplace transformable if and only
if t7→ e−At is summable against |ν|. In this case (24) will hold in some strip <s > A0as soon
as u∈L1
loc(R)is of exponential type, meaning there is B∈Rsuch that e−B·u∈L∞(R). This
level of generality is about right for most applications to circuit theory, for solutions to linear
differential equations with constant coefficients initially at rest can be expressed via convolution
against exponential kernels while delays introduced by transmission lines introduce Dirac delta
measures in the impulse response [2, 3].
It has become customary to denote signals (i.e. functions of t) by lower case letters and
their Laplace transforms with corresponding upper case letters. Real-valued signals t7→ u(t)
correspond under Laplace transform to conjugate-symmetric functions: U(¯s) = U(s). System
(19) maps real signals to real signals if and only if Φis real-valued on real functions, which
results in the transfer function being conjugate symmetric. System theory is mostly concerned
with real-valued signals although such a restriction is rather immaterial as far as theory goes.
What precedes is enough to bring formal meaning to most computations, as long as pointwise
evaluation in (24) is restricted to some strip where every Laplace transform involved does exist.
However, when dealing with issues of passivity and stability, square summable inputs and outputs
Inria

Transfer functions realizable with active electronic components 33
are of special significance (these are the signals of finite energy) and a more refined interpretation
of (24), valid on the imaginary axis, becomes necessary.
Clearly if f∈L2(R+), then it is A-Laplace transformable for each A > 0because f(t)e−at is
summable by the Schwarz inequality. Its Laplace transform Fis thus defined and holomorphic
for <s > 0and, as can be surmised from (21) and rigorously proved [7, ch. 8], the limit
F(iy) = limx→0F(x+iy)exists for almost every y∈Rand coincides with the Fourier transform
ˆ
f(y)defined for almost every y∈Rby the formula
ˆ
f(y) = lim
B→+∞ZB
0
e−iytf(t)dt. (26)
The limit in (26) holds in the L2-sense by Plancherel’s theorem [11, thm. 9.13]. The fact that
it also holds pointwise almost everywhere is a deep result after the work of Carleson [6, thm.
11.1.1]. Furthermore, putting Fx(y) = F(x+iy), we have that limx→0kFx−ˆ
fkL2(R)= 0 and
also that
sup
x>0Z+∞
−∞ |F(x+iy)|2dy < ∞.(27)
In fact the supremum in (27) is kˆ
fk2
L2(R), which is in turn equal to 2πkfk2
L2(R+)by the essentially
isometric character of the Fourier transform. It is a theorem of Paley and Wiener that the space
of all functions Fholomorphic in the right half-plane and satisfying (27) coincides with the set
of Laplace transforms of functions f∈L2(R+). When normed with the square root of the left
hand side of (27), the latter becomes a Hilbert space known as the Hardy space of exponent 2
of the right half plane [7, Ch. 8]3, hereafter denoted by by H2. The norm of a Hardy function
F∈ H2, indicated with kFk2, is thus √2πtimes the energy of the signal of which it is the
Laplace transform. Conjugate symmetric H2-functions. form a real Hilbert subspace of H2.
We need also introduce the Hardy space H∞of bounded analytic functions in {<s > 0}
endowed with the sup norm. As is the case for H2functions, the limit F(iy) = limx→0F(x+iy)
exists for almost every y∈Rwhen F∈ H∞[7, ch. 8]. Moreover, this limit function lies in
L∞(iR), and it holds that
kFkL∞(iR)= sup
<s>0|F(s)|.(28)
The quantity in (28) is the H∞-norm of Fthat we abbreviate as kFk∞.
Consider now a linear dynamical system mapping L2(R+)into itself, that is, the system maps
inputs of finite energy to outputs of finite energy Such a system we call stable. It is remarkable
that it is automatically continuous as a map, that is, there is a constant C≥0such that
kykL2(R+)≤CkukL2(R+)[10, thm. 4.1.1]. Taking Laplace transforms, we find that the system
induces a continuous linear map from H2into itself that commutes with multiplication by e−sτ
for all τ∈R+. As it turns out [10, Cor. 3.2.4], such a linear map is just multiplication by some
H∈ H∞, and its norm is kHk∞. Next, pick a < 0and consider the function G(s) = H(s)/(s−a).
Because H∈ H∞, it is easy to check that G∈ H2hence by the Paley Wiener theorem there is
g∈L2(R+)such that G=L(g). If we regard gas a distribution, we find that H=L(g(1) −ag).
Thus, since a system is uniquely defined by its transfer function, we conclude that our initial
system has Laplace transformable impulse response g(1) −ag and transfer function H. To recap,
a system is stable if and only if it has a transfer function lying in H∞, and the maximum gain
is the H∞-norm of the transfer function. We also gather from what precedes that the impulse
response is of the form g(1) −ag with g∈L2(R+),a < 0, and that every function in H∞arises
3More generally, one defines the Hardy space Hp,1≤p < ∞, to consist of holomorphic functions in the right
half plane such that supx>0kF(x+.)kLp(iR)<∞., see [7, ch.8]
RR n°8659

34 L. Baratchart, S. Chevillard & F. Seyfert
as the transfer function of a stable system. However, not every g∈L2(R+)gives rise to the
impulse response of a stable system via Φ = g(1) −ag for some a < 0. In fact, no non-tautological
characterization is known of those Laplace transformable distributions whose Laplace transform
belongs to H∞, compare [9, thm. 8.7.1].
A stable system can be fed with an input u∈L∞
loc(R+)(extended by 0for negative t) because,
by causality, the output at time t≤t0is the same as if we used the input χ[0,t0]uwhich lies in
L2(R+). In order to gain physical understanding of the transfer function, it is worth estimating
the asymptotic behavior, as t→+∞, of the output generated by the input u(t) = χR+(t)eiωt
for fixed ω∈R. Dwelling on what precedes, let the system have transfer function H∈ H∞
and impulse response Φ = g(1) −ag with g∈L2(R),a < 0. Due to the distributional nature
of the derivative g(1) appearing in Φ, we cannot bluntly plug in (19) the input uwhich is non-
smooth at zero and has infinite support. To circumvent this, let ϕ∈ D(R)be non-negative,
supported on [−1,1], even, and such that Rϕ(t)dt = 1. For ε∈(0,1], set ϕε(t) = ϕ(t/ε)/ε and
φε(t) = Rt
−1ϕε(τ)dτ. Pick T > 2, define φε,T (t) = φε(t)φε(T−t), and put uε,T (t) = φε,T (t)eiωt.
By construction uε,T ∈ D(R), hence using (19) we obtain the corresponding output yε,T by the
formula:
yε,T (t) = eiωt ZR
g(τ)φ(1)
ε,T (t−τ)+(iω −a)φε,T (t−τ)e−iωτ dτ (29)
=eiωtge−iω·∗ϕε(t)−ge−iω ·∗ϕε(T−.)(t)+(iω −a)ge−iω·∗φε,T (t).
When εtends to 0, then uε,T converges in L2(R)to uT=χ[0,T]eiω ·so that yε,T must converge to
the corresponding output yT, as the system is stable. Now, since ge−iω·∈L2(R)(when extended
by zero for negative arguments), it is standard that ge−iω·∗ϕε→ge−iω·in L2(R)[13, thm. 1.6.1]
and by the same token we get that ge−iω·∗ϕε(.−T)→g(.−T)e−iω (.−T). Moreover, because
φε,T converges to χ[0,T]in L1(R), it follows from Minkowski’s inequality for integrals [11, ch. 7,
ex. 4] that ge−iω·∗φε,T →ge−iω·∗χ[0,T ]in L2(R). Altogether, we find that
yT(t) = g(t)−g(t−T)eiωT +eiωt (iω −a)Zt
max(0,t−T)
g(τ)e−iωτ dτ.
Next, let ybe the output associated with u(t) = χR+(t)eiωt. By causality it must coincide with
yTon [0, T ], therefore using that gis supported on R+we obtain:
y(t) = g(t) + eiωt(iω −a)Zt
0
g(τ)e−iωτ dτ. (30)
If we let now t→+∞and take into account that (26) converges pointwise almost everywhere
while using the relation (iω −a)L(g)(iω) = H(iω), we find that for almost every ω
y(t) = g(t) + H(iω)eiωt +o(1), t →+∞.(31)
which is the formula we are aiming for. It says that, for almost every ω, the output y(t)
asymptotically winds at constant angular speed ωon the boundary of a tubular neighborhood
of t7→ g(t)having radius |H(iω)|. In general, the behavior at exceptional frequencies for which
(31) does not hold can be quite chaotic. Note that gdepends on abut not on ω, hence (31)
shows that the limiting behavior of g(t)for large tdepends only on the system. Whereas gneeds
Inria

Transfer functions realizable with active electronic components 35
not tend to 0at infinity4it is nevertheless small “most of the time” since obviously
m{t:t≥t0,|g(t)|> α}≤kgk2
L2([t0,+∞))
α2
which goes to 0as t0→+∞. Thus, asymptotically in time, the output resulting from a periodic
input with pulsation ωis “most of the time and for almost every ω” close to a signal having
same period, gain |H(iω)|, and phase shift arg H(iω). When gtends to zero at infinity, the
result is neater as the words “most of the time” can be omitted from the previous sentence. This
occurs for instance if Φ∈L1(R+), as follows easily by dominated convergence from the formula
g= Φ ∗(χR+(.)ea·); in this case (31) holds in fact for every ω.
Difficulties connected with the asymptotic behavior of gdisappear if instead of χR+(t)eiωt
we use an input achieving a smooth transition between the zero function and t7→ eiωt, like
u(t) = φ1(t)eiωt where φ1(t) = Rt
0ϕ(τ)dτ was defined earlier. Then, taking into account that
supp ϕ⊂[−1,1], a computation similar to (29) with Φε,T (t)replaced by φ(t)φε(T−t)leads to
y(t)=(g∗ϕ)(t)++eiωt(iω −a)Zt−1
0
g(τ)e−iωτ dτ +Zt+1
t−1
g(τ)e−iωτ ϕ(t−τ)dτ (32)
instead of (30). Because g∈L2(R+), both (g∗ϕ)(t)and the last integral in (32) tend to zero
as t→+∞by the Schwarz inequality, so the term g(t)is no longer present in (31). It is worth
observing that a similar conclusion is reached when Φ∈L2(R+), a condition that does not
subsume stability but allows for some unstable systems.
This interpretation of the transfer function as an asymptotic multiplier frequency-wise in
response to harmonic inputs is of great importance in design.
References
[1] Laurent Baratchart. Sur l’approximation rationnelle L2pour les systèmes dynamiques
linéaires. thèse de doctorat ès sciences, Université de Nice, Septembre 1987.
[2] Belevitch. Classical network theory. series in information systems. Holden Days, 1968.
[3] Herbert Carlin and Pier Paolo Civalleri. Wideband circuit design. Electronic engineering
systems series. CRC Press, 1998.
[4] H.J. Carlin and D.C. Youla. Network synthesis with negative resistors. Proceedings of the
IRE, 49(5):907–920, 1961.
[5] Pierre Faure, Michel Clerget, and cois Germain, Fran˙ Opérateurs rationnels positifs. Méth-
odes Mathématiques de l’Informatique. Dunod, 1979.
[6] Loukas Grafakos. Modern Fourier analysis. Number 250 in Grad. Texts in Maths. Springer,
2009.
[7] Kenneth Hoffman. Banach spaces of analytic functions. series in modern analysis. Prentice-
Hall, 1962.
4If we let Φ = Pk∈NχR+(.−tk)eαk(.−tk)with αk=b+iβkwhere b < 0and βk∈R, then H(s) =
Pke−tks/(s−αk)will lie in H∞if |βk|tends to infinity sufficiently fast but g= Φ ∗(χR+(.)ea·)does tend to
zero at infinity. A real-valued example is obtained upon taking the αkin conjugate pairs.
RR n°8659

36 L. Baratchart, S. Chevillard & F. Seyfert
[8] K. Kurokawa. Power waves and the scattering matrix. Microwave Theory and Techniques,
IEEE Transactions on, 13(2):194–202, Mar 1965.
[9] O. P. Misra and J.L. Lavoine. Transform analysis of generalized functions, volume 119 of
Mathematics Studies. North-Holland, 1986.
[10] Jonathan Partington. Linear operators and linear systems. Number 60 in Student texts.
London Math. Soc., 2004.
[11] Walter Rudin. Real and complex analysis. Mc Graw-Hill, 1982.
[12] Walter Rudin. Functional Analysis. International Series in Pure and Applied Mathematics.
Mc Graw-Hill, 2 edition, 1991.
[13] P. Ziemer, William. Weakly Differentiable Functions. Number 120 in Grad. Texts in Maths.
Springer, 1989.
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