Received: 12 May 2023
Accepted: 13 May 2023
DOI: 10.1002/cta.3674
ORIGINAL PAPER
Novel comprehensive feedback theory—Part I:
Generalization of the cut-insertion theorem and
demonstration of feedback universality
Bruno Pellegrini
| Massimo Macucci | Paolo Marconcini
Dipartimento di Ingegneria
dell'Informazione, Università di Pisa, Pisa,
Italy
Correspondence
Paolo Marconcini, Dipartimento di
Ingegneria dell'Informazione, Università
di Pisa, Via Caruso 16, 56122 Pisa, Italy.
Email: p.marconcini@iet.unipi.it
Funding information
Italian Ministry of University and
Research (MUR)
Summary
The previous cut-insertion theorem, defined for linear circuits and for particular three-terminal circuits (TTCs) to be inserted into the cut, is here extended
to any TTC and, in a simplified form, to non-linear networks. The TTC may
include dependent and independent sources and any other passive and/or
independent elements, and, together with the remaining part of the circuit,
determines a feedback loop for which the “transmission factors” T and T 0 can
be defined and computed. The theorem allows to obtain all the circuit properties: the overall gain, the driving-point immittances, a new properly defined
“cut immittance,” the sensitivity, and so on. For T 0 ¼ 0, T becomes the “classic”
loop gain. As will be shown in Part II of the paper, the new approach,
depending on the TTC implementation, includes and unifies all the previous
feedback models and enables the creation of new ones. Moreover, starting
from the paradigmatic definition of a system with a feedback as a system in
which the output quantity is in part an effect of itself, the universal presence
of feedback in any system is shown. Finally, the results are highlighted by the
application to unilateral non-linear systems, such as digital latches, and to
bilateral linear circuits, such as bridged-T networks.
KEYWORDS
circuit and system analysis, circuit theory, cut-insertion theorem, feedback theory
1 | INTRODUCTION
Feedback is one of the most ancient techniques exploited in the first complex systems in order to control their characteristics. Indeed, Ctesibius, a Greek inventor of Alexandria, Egypt, in the third century BC used feedback to keep the
water level constant in the clepsydras, in order to improve the precision of water watches. Later, feedback was
employed to control the air flux in furnaces, the velocity of windmills, and, above all, the steam flow in steam engines
(at the end of the 18th century AD, by means of the Watt regulator), thereby contributing to the first industrial revolution and to the birth of mechanical and electromechanical controls.
This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided
the original work is properly cited.
© 2023 The Authors. International Journal of Circuit Theory and Applications published by John Wiley & Sons Ltd.
Int J Circ Theor Appl. 2023;51:3979–3993.
wileyonlinelibrary.com/journal/cta
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PELLEGRINI ET AL.
Since the beginning of last century, with the invention of vacuum tubes and the beginning of the “electronics era,”
feedback has been extensively used in electronic systems to improve their characteristics, and research on this topic is
still very active today. Milestones of the research on feedback in electronics are the first patents of feedback amplifiers,1
the Nyquist stability criterion,2 the Blackman formula3 for evaluating driving-point impedances, and the fundamental
contributions by Bode to a general theory of feedback4 for computing, in particular, the stability and the sensitivity with
respect to the variation of a single element. A necessarily incomplete list of other contributions can be found in previous
studies.5–33 However, only a few of them are able to retrieve the intuitive concept of feedback (e.g., Pellegrini5,6).
Instead, in most of the approaches, the feedback analysis, performed in terms of mesh or nodal equations, leads to a
loss of the physically intuitive concept of feedback loop, that is central to understand the operation of the regulators
and of many electronic systems.
The main motivation of this paper is the proposal of a novel comprehensive feedback approach, with the objective
to unify all the pre-existing general feedback models and to get new results.
The method is based on the extension of the cut-insertion theorem (CIT),5,6 which is able to represent any linear
network as a single-loop feedback system. In particular, we will include the possibility to insert new three-terminal circuits (TTCs) in the cut and, although in a more basic version, we will extend its application to non-linear systems. The
flexibility resulting from the possibility to adopt a wide variety of TTCs will lead to new expressions of the overall gain
and of the driving-point immittances, as well as to a new purposely defined cut-insertion immittance, which is useful
also to implement some types of TTCs.
In Section 2, the CIT is applied to non-linear systems. Indeed, while the complete form of the CIT, in which a few
quantities characteristic of the cut are computed and then combined through the superposition theorem to obtain all
the circuit properties, can be used only for linear or linearized circuits, the core concept of the CIT (i.e., cutting the feedback loop and inserting the TTC, enforcing the proper constraints) can be adopted also for non-linear circuits, with one
caveat. For non-linear systems, unlike the linear ones, the equality of the currents and of the voltages at the input and
at the output of the TTC is not sufficient for the equivalence between the cut and the uncut network when more than
one state is possible (e.g., for the steady state), as the simple circuit example of a digital latch shows.
The CIT extension for linear networks, made in Section 3, consists in the insertion into the cut of a generic TTC
including internal independent and/or dependent sources and other elements. After this cut-insertion procedure, the
TTC and the remaining portion of the network identify a feedback loop, characterized by a parameter T 0 , called “forward transmission factor,” and by a parameter T, called “backward transmission factor.” T and T 0 characterize all the
system quantities and, in the case of T 0 ¼ 0, T becomes the “classic” loop gain. In the same section, the superposition
constants and other cut functions necessary to develop the feedback model are defined.
Section 4 deals with a new cut-insertion immittance X q that is defined as the ratio of the voltage, present between
the split node q q0 and the reference one t t 0 , to the current in the connection q, q0 , generated by the external source
S, or vice versa (Figure 1); examples of the calculation of X q in bilateral bridged-T networks are also shown. We show
that X q is equal to the immittance X p , introduced for the first time in Pellegrini5,6 as closing loop element of the new
feedback model. Moreover, new general expressions of the overall gain function Af and of the driving-point immittances are shown, as well as the multiple alternative possibilities to define the parameters that characterize the same
cut network.
In Section 5, as a conclusion of the present general framework and on the basis of the paradigmatic definition of a
system with feedback as a system in which the output is partly a cause of itself, we show, through an equivalent “universal” scheme, that the feedback representation is ubiquitous, that is, it is possible in any system.
Then, in order to highlight the obtained results and to compare them to the “classic” ones, in Section 6, we perform
a feedback representation of a bridged-T circuit, by means both of the CIT and of the more general “universal” feedback
scheme, and we compare the results with those of nodal analysis.
Finally, after a brief discussion in Section 7 on the utility of feedback analysis, the conclusions are drawn in
Section 8.
2 | APPL ICATION O F THE C I T TO N O N - L I N E A R N E TW O R K S
The CIT5 is based on splitting an arbitrary node q of a given linear network N (Figure 1A) into two sub-nodes q and q0
and inserting between them and a third arbitrary “reference” node t t 0 a TTC, obtaining a linear network N 0
(Figure 1B, where the three terminals j, k, and m of the TTC are marked with solid dots). S is an input independent
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F I G U R E 1 (A) A general linear network N and (B) its equivalent one N 0 after splitting the q, q0 connection and inserting a threeterminal circuit (TTC) (delimited by the three marked nodes j, k, and m) between the new nodes q and q0 , thus obtained, and the “reference”
one t t0 . The arrow to the left (right) of the TTC indicates the current entering into (exiting from) it.
voltage or current source (placed between the nodes e and h), while U is the output voltage or current (measured on
the immittance X u , connected between the nodes u and w). The W quantities represent voltages or currents of the network; in particular, W r (W p ) is the voltage or current at the input (output) port of the TTC (in the figures we indicate
each generic quantity, which can be either a voltage or a current, both with a + and a sign, and with an arrow).
The CIT states5 that necessary and sufficient conditions for the equivalence of the networks N and N 0 are the
equalities
vr ¼ vp , ir ¼ ip ,
ð1Þ
between the voltages and the currents at the TTC input (index r) and output (index p) ports, for any value of the independent external signal source S. The CIT has been derived5 in the domain of the complex frequency s. In the same formal way it can be used, for either linear or non-linear networks, in the time domain, except that Equations (1) are not
sufficient to assure the equivalence between N and N 0 for non-linear networks when Equations (1) can be satisfied in
more than one condition.
As an example, let us consider the latch (and/or flip-flop) circuit of Figure 2A7 in which the TTC inserted among
the nodes q, q0 and t t 0 consists of the independent voltage source va and of the dependent current source ir ¼ ip , which
directly satisfies the second of Equations (1).
For instance, it can be exploited to evaluate the quiescent point for is ¼ 0. To this end, one computes the characteristic vr ¼ vr ðva Þ vs. va and determines its points P1 , P0 , and P2 where, being vr ¼ va , also the first of Equations (1) is satisfied and, hence, the equivalence between N and N 0 could be verified according to the CIT. The characteristic is
qualitatively sketched in Figure 2B for the circuit of Figure 2A. A piecewise-linear approximation is also plotted,
obtained considering the average trans-characteristics iDi ¼ gM ðvGi V T ÞuðvGi V T Þ for the MOS transistors, where iDi
and vGi are the drain current and gate voltage, respectively, V T is an average threshold voltage, gM is an average transconductance, and uðxÞ ¼ 1 (0) for x > 0 (<0). With reference to the symbols of Figure 2A, using the piecewise-linear
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F I G U R E 2 Analysis of a shunt voltage amplifier or of a flip-flop (depending on the slope of the vr ðva Þ characteristic). (A) Cut network
N 0 after splitting the q, q0 connection and inserting a TTC between q, q0 and the “reference” node t t0 . (B) Characteristic curve vr ¼ vr ðva Þ
versus va for is ¼ 0 that determines three possible steady states: P0 , P1 , and P2 . In these states, being vr ¼ va ¼ vp and, by definition, ir ¼ ip , we
can have, according to the CIT equations (1), three possible equivalences between the original network N and the cut one N 0 , or none if, for
instance, N and N 0 are in different states.
approximation and, for the sake of simplification, the assumption that RD ¼ R1 þ R2 , the shape of the characteristics
vr ¼ vr ðva Þ in Figure 2B requires V DD > 2RD V T =R1 and gM > 2=R1 , in order to have P0 to the right of V T and to the left of
V~ (see Figure 2B), and ðdvr =dva Þ > 1, respectively. In the three points P1 , P0 , and P2 , we have vr ¼ va ¼ vp . Therefore,
according to the CIT conditions (1), we can have three possible equivalences (same currents and voltages in all the branches and nodes of the networks) between N and N 0 if N and N 0 are in the same state Pi (with i ¼ 1,2, 3), or no one if,
for instance, N and N 0 are in different states. Therefore, the CIT conditions (1) are not sufficient to warrant the equivalence between cut and uncut non-linear networks. Moreover, according to the Nyquist2 criterion, the circuit in the state
P0 is unstable and it will end up into one of the stable points P1 or P2 , that is, the circuit becomes a binary storage
element.
Otherwise, for different values of the parameters such that ðdvr =dva Þ < 1 (and thus, P0 becomes stable, while the
states P1 and P2 disappear), the circuit becomes a shunt voltage amplifier. This could in turn be analyzed, for instance,
by means of the procedure (similar to that of Russel9) reported in Section 4.2 of Part II, that is, by first computing the
bias point P0 (according to the method described above) and the corresponding dynamic parameters, and then, for the
AC analysis of the linearized system, by evaluating the overall network functions by means of the original5 or the present feedback model.
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3 | EXTENSION OF THE C IT F O R TH E L I N E A R N E T W O R K S
3.1 | Feedback loop with generic TTC
The original version of the CIT5 was applied in Pellegrini5 to linear networks N (Figure 1A) and exploits a TTC made
up of an independent source at the TTC output, and of an immittance or a dependent source at the TTC input (used as
a closing element of the feedback loop). The CIT is here extended to a generic TTC containing internal independent
and dependent sources, together with other elements. Such a generic TTC (Figure 3A) can be made up of two parts
(a loop closing element CE and a testing circuit TE) connected only by the reference node, as in the original formulation (case of “separable” TTCs, shown in Figure 3B), or not (case of “non-separable” TTCs). As shown in Figure 1B and
detailed in Figure 3, the TTC and the remaining part of N identify a loop, usable as foundation of any feedback model.
Adopting this CIT extension, both a “backward” transmission factor T and a “forward” transmission one T 0 can be
defined, that characterize the overall network functions.
Since the proof of the CIT in the s domain, reported in Pellegrini,5 does not depend on the TTC structure, the equivalence between the uncut N and cut N 0 networks is still guaranteed if and only if (using the symbols of Figures 1B and
3), for any S, it is
W r ¼ W p, W r ¼ W p,
ð2Þ
because, owing to the linearity of the networks N, TTC, and N 0 , the solution of their equation systems is unique (unlike
in the non-linear circuits discussed in the preceding section). The bar over a symbol indicates the dual quantity (if W is
a voltage, W is a current, and vice versa).
F I G U R E 3 Sketch of the loop of the split network N 0 of Figure 1B, in which we highlight the “forward” (“backward”) chain FN (BN)
that are made up of the TTC (of the remaining part of the uncut network N). In (A), we represent a generic TTC, while in (B), we show the
particular case of a “separable” TTC. In (B), CE and TE represent the loop closing element and the testing circuit of the TTC, respectively.
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3.2 | Superposition constants and functions of the cut
In order to develop the model, we need to define several quantities concerning the cut network N 0 . We use the symbols
of Figures 1 and 3: the input source S is placed between nodes e and h; U is the output quantity, measured on a branch
of immittance X u connecting two nodes u and w; W r and W p are homogeneous quantities, measured at the input and
output ports of the inserted TTC; the TTC contains an internal independent source W a , which is necessary in order to
evaluate the above mentioned quantities of the cut network N 0 . The various nodes are arbitrarily chosen.
All the above quantities (S, U, W ) can be either voltages or currents in the s domain. Any possible initial condition
can be considered, by using an appropriate independent source S and exploiting the superposition theorem.
By using the independent sources S and W a , let us define the following superposition constants:
W p U W r 1 W p ε
, θ
, , δ
,
W a S¼0 x i W a S¼0
W a S¼0
W a S¼0
ð3Þ
W p W p U W r 0
, α
, ρ
, θ ,
γ S W a ¼0
S W a ¼0
S W a ¼0
S W a ¼0
ð4Þ
from which, from the superposition theorem and from the first of Equations (2) of the CIT, we obtain, for any S, the
relationship
α þ θ0
S,
δþθ
ð5Þ
α ð1 þ T 0 Þ
S,
δ ð1 þ TÞ
ð6Þ
Wa ¼
that can be written in the form
Wa ¼
where we define the cut functions
W p θ0
θ
W r , T ¼
,
T ¼
α
W r W a ¼0
W p S¼0
δ
ð7Þ
W r :
Tn W p U¼0
ð8Þ
0
The feedback loop (Figure 3) is formed by a “forward” path FN, that is, the TTC, and by a “backward” path BN, that
is, the part of the N network external to the TTC.
In these definitions, T 0 represents the “forward” ratio of the output to the input quantities of the “forward” path FN
(i.e., the ratio of the input to the output quantities of the “backward” path BN), caused by S, for W a ¼ 0.
Instead, T is the “backward” ratio of the input to the output quantities of the “forward” path FN (i.e., the ratio of
the output to the input quantities of the “backward” path BN), determined by W a , for S ¼ 0.
Finally, T n , which will be used in the following, is the value of W r =W p for U ¼ 0.
Therefore, the first of the CIT conditions (2) enforces the constraint (6) on the TTC; the second of Equations (2)
gives a further constraint on it that depends on the structure of the TTC.
As we will see in Section 4.4, if, in the place of the homogeneous quantities W r and W p of the TTC, we exploit their
dual ones W r and W p , we obtain different (dual) results.
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4 | CUT I MMITTANCE, OVERALL GAIN, AND DRIVING-POINT
IMMITTANCES
4.1 | Cut immittance
Equation (6) enforces the first of the CIT conditions (2), that, however, as above stated, is not sufficient for the equivalence between N and N 0 . To this end, it is useful to define a new quantity, the cut immittance
1 Wq
,
Xq Wq
ð9Þ
where W q and W q (Figure 1A) are the current in the connection q-q0 (flowing from q to q0 ) and the voltage between
q q0 and t t 0 , generated by S, that hence depend on the position of S, of the cut, and of its nodes in the network.
There are several ways to compute it.
For instance, the cut impedance in the unilateral circuit of Figure 2A (linearized for small signals) is Rq ! ∞. More
interesting cases are shown for the cuts of the bilateral bridged-T of Figure 4, for which we have
Rq1 ¼ R5 ðR2 þ R3 þ R4 Þ
:
R3 þ R4
ð10Þ
F I G U R E 4 Bridged-T circuit (A) with examples for the calculation of the cut impedance and for the feedback representation according
to its paradigmatic definition (see Section 5) and (B) with the CIT applied to Cut 1.
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Rq2 ¼ R5 ðR2 þ R3 þ R4 Þ=R2 and Rq3 ¼ R5 for Cuts 1, 2, and 3, respectively. More in general, according to (9), X q is
equal to an element K if K directly connects q q0 and t t 0 and the current flowing through K coincides with the current in the connection q–q0 ; moreover, its sign depends on the reciprocal position of q and q0 .
Other approaches can be the direct evaluation of W q and W q or the method of Cherry8 that inserts an impedance
X ! 0 into the connection q–q0 and can express X q through the network admittance matrix, with the limitation that
one of the source S terminals coincides with the reference node t t0 .
Another way to compute X q exploits the CIT theorem,5 starting from the definition of the immittance
1 Wr
:
Xp W r
ð11Þ
In order to have the equivalence of the cut and uncut networks, it has to be W q ¼ W r and W q ¼ W r (Figures 1 and
3), so that, from (9) and (11), we get
X q ¼ X p:
ð12Þ
We can compute X p from the cut network (Figure 1B) by defining the following function of the cut:
1
1 W p ¼
:
X i δx i W p S¼0
ð13Þ
Indeed, from the superposition theorem and from the second of Equations (3), the second of Equations (4),
Equation (6), and the second of Equations (7), we get
Wr ¼ α
ð1 TT 0 Þ
S:
ð1 þ TÞ
ð14Þ
Again from the superposition theorem, from the second of Equations (2) of the CIT, and from the third of Equations (3), the third of Equation (4), Equation (6), and Equation (13), we get
α ð1 þ T 0 Þ
S ¼ W r,
Wp ¼ ρþ
X i ð1 þ TÞ
ð15Þ
1
1
ð1 þ T 0 Þ ρð1 þ TÞ
:
¼
þ
X p ð1 TT 0 Þ
Xi
α
ð16Þ
and then from (11),
According to (12), this gives another way to compute X q .
A useful application of X q ¼ X p is its usage as closing element CE of the “separable” TTC shown in Figure 3B; since
it has been computed according to the CIT constraints (2), the equivalence of the cut and uncut networks is
guaranteed.
It is to be noticed that, in general, such an equivalence holds true only for the computations that do not require
changes of the position of S and/or of the choice of the cut. For instance, no change in the position of S is needed for
the calculation of the input driving-point immittance, while changes are required, for instance, for the evaluation of the
output immittance5,6 and of any other driving-point immittance different from the input one (see Section 4.3).
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4.2 | Overall gain
An important characteristic quantity of a linear network is the overall gain Af U=S; in particular, let us examine it in
the present framework, when it is computed by means of the new feedback approach. From the superposition theorem,
the first of Equations (3), and the first of Equations (4),
U ¼ γS þ εW a ,
ð17Þ
and from Equation (6) and from the definition of the cut function,
ε U ,
A ¼
δ W p S¼0
ð18Þ
we obtain the overall gain Af in the form
Af ¼ αA
ð1 þ T 0 Þ
þ γ,
ð1 þ TÞ
ð19Þ
in which both the “backward” transmission factor T and the “forward” one T 0 appear.
Moreover,
in Equation (19), we can express αA=γ by means of the gain T n constrained by U ¼ 0, that is, T n ¼
ðW r =W p ÞU¼0 (Equation (8)), that, together with Equations (3), (4), (7), and (18), leads to
αA T n T
¼
,
γ
1 T 0Tn
ð20Þ
ð1 þ T n Þð1 TT 0 Þ
,
ð1 þ TÞð1 T n T 0 Þ
ð21Þ
so that Equation (19) gives also
Af ¼ γ
expression that reduces the number of the cut functions appearing in Af .
4.3 | Driving-point immittances
Another meaningful network function is the driving-point immittance between two arbitrary nodes e and h
(Figure 1B), defined as4
X S=S, X 0 ðS=SÞW a ¼0 :
ð22Þ
X 0 being the X value when the source W a is switched off (W a ¼ 0).
X is normally computed by means of mesh or nodal analysis in the form of a determinant ratio or of a bilinear
expression.4
Let us compute it also by means the new feedback approach.
Due to the arbitrariness in the choice of the nodes, we can select u e and w h, that is, U ¼ S as the output quantity, whereas S remains the input quantity, so that from the first of Equations (4), from Equation (22), and from the definition Af ¼ U=S, we get
X ¼ 1=ðAf U¼S Þ, X 0 ¼ 1=ðγjU¼S Þ,
ð23Þ
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from which and from (21), we finally obtain
X ¼ X0
ð1 þ TÞð1 T n T 0 Þ
ð1 þ T n Þð1 TT 0 ÞU¼S
ð24Þ
that further extends the Blackman formula.3,5
Actually, to obtain the driving-point immittance between e and h, we can exploit the equivalence between the cut
and uncut networks, as previously discussed in Section 4.1, because in this case, we do not change the position of S, of
the cut, and of its nodes.
4.4 | Dual and alternative definitions of the cut functions
For each choice of cut and of TTC, there are several alternative ways to define the various cut functions L of the cut
network.
Let us begin to compute the various dual functions of the circuit that we obtain when, in the place of the homogeneous quantities W r and W p , we use their dual ones W r and W p , respectively, and vice versa. From the first of Equations (4) and from the Af definition, we obtain that γ ¼ γ and Af ¼ Af , because they do not depend on such quantities.
From the second of Equations (4) and from Equation (11), we get α ¼ α=X p , and from Equation (18) and from
¼ X i A. From the second of Equations (7), from Equation (11), and from Equation (13), we
Equation (13), we have A
obtain T, while from the first of Equations (7), from Equation (11), from the third of Equations (4), and from the second
of Equations (4), we have T 0 : they are given by
T¼
ρX p
Xi
,
T, T 0 ¼ Xp
α
ð25Þ
respectively.
More in general, with reference to the schematics of Figures 1B and 3, let us indicate with j, k, and m the three terminals of the TTC, and with q, q0 and t t0 the nodes of the external network connecting it to the TTC, while we label
its external quantities (W r , W r , W p , and W p ) with r and p. We can use the symbol Lrd
qj to indicate the cut function L
evaluated when the nodes q, j, and r are on the left side of the TTC, specifying with d or d if the (homogeneous) quantities W r and W p , or their dual ones W r and W p are considered, respectively.
rd
For instance, for the quantity T rd
qj , we have the dual one T qj T ¼ ðW r =W p Þ S¼0 that, from Equation (16) and from
the first of Equations (25), becomes
rd
T qj
¼
h
i
1
ρ
0
0 ð1 þ T Þ þ X i ð1 þ TÞ T:
ð1 TT Þ
α
ð26Þ
rd
In an analogous way, we have T 0 qj
T 0 ¼ ðW p =W r ÞW a ¼0 that, from Equation (16) and from the second of Equations (25), gives
1
T 0 qj
rd
¼
1
α
0
ð1
þ
TÞ
þ
ð1
þ
T
Þ
:
ð1 TT 0 Þ
ρX i
ð27Þ
0
Since in Lrd
qj we can have two independent possibilities (q or q , j or k, r or p, d or d) for each super-(sub-)script, the
total number of possible alternative definitions for each choice of cut function, cut, and TTC is 16.
5 | UNIVERSAL FEEDBACK
The above network feedback theory based on the CIT theorem5,6 relies on finding a loop around an arbitrary node pair
(q q0 and t t 0 ).
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From another point of view, it can be based on the paradigmatic definition of a system with feedback as a system in
which an output effect U is partly a cause of itself, through an intermediate quantity W , besides partially depending
directly on the external action S. To this end, the expression (19) of the overall gain can be represented by means of the
block diagram of Figure 5 that, according to the superposition theorem, allows one to express the output quantity
U ¼ AUS S þ AUW W ,
ð28Þ
through the external signal (cause) S and the intermediate variable
W ¼ AWS S þ AW U U,
ð29Þ
AUS ¼ γ, AUW ¼ A,
ð30Þ
AWS ¼ αð1 þ T 0 Þ þ γT=A, AW U ¼ T=A:
ð31Þ
where
Indeed, by substituting the expression (29) of W into Equation (28), we retrieve Equation (19).
Actually, Equation (28) shows how the effect U, besides depending directly on the external source S through the
“leakage path” γ, is partly a cause of itself through the intermediate quantity W , according to the paradigmatic definition that we can give of feedback.
Most importantly, on the basis of the previous definition, let us show that the feedback theory can be exploited to
represent any linear system, the n quantities x k s of which, in the complex frequency space, depend on each other
according to the set of the n linear equations
n
X
aik x k ¼ bi S ðwith i, k ¼ 1,2, …,nÞ,
ð32Þ
k¼1
where S is the only quantity in the system associated to an external action. Let x j U and x l W (with j ≠ l) be an output and an intermediate quantity, respectively. By eliminating the mth equation from Equation (32) (at the condition
that bm is not the only nonzero coefficient bi , in order to preserve the dependence of x j on S) and by moving the lth column to the right-hand side, the solution x j U of the remaining system becomes of the type of Equation (28), where
AUS and AUW result from the described procedure. Instead, by applying the same method for x l W , that is, by eliminating the oth equation (with o ≠ m, i.e., an equation different from the mth one that we have eliminated in the previous case, in order to obtain a relationship that is linearly independent from Equation (28)) and by moving the jth
column to the right-hand side, the solution x l W of the remaining system becomes of the type (29), where AWS and
AWU derive from the illustrated procedure.
Hence, in any linear system, the output U can be expressed as a partial effect of itself through an intermediate cause
W , according to the paradigmatic definition of feedback.
FIGURE 5
Block diagram of a feedback system according to Equation (19) and to Equations (28)–(31).
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PELLEGRINI ET AL.
PELLEGRINI ET AL.
Therefore, on the basis of its paradigmatic definition, feedback exists in any linear and (at least for small signals that
allow a linearization around the quiescent point) non-linear system in which more than one observable quantity exists
that is interconnected by equations to the others. This is true whatever the nature of the system may be (physical, economic, biological, and so on) and even in the “startling” “minimum” case of a single mesh of two impedances stimulated by a voltage source (see Pellegrini5).
In this sense, we can claim that feedback is a ubiquitous feature of the observable systems, that is, it can be
exploited as a universal representation of any system, or, from another point of view, feedback is a universal feature of
nature.
6 | F E E D B A C K R E P R E S E N T A T I O N O F A BR I D G E D -T CIRCUIT
As an example, let us analyze, by means of the presented feedback tools, and, for a proper comparison, of the “classic”
nodal analysis, the bridged-T circuit of Figure 4, split with Cut 1, in which we insert a TTC made up of the testing current source I a and of the cut resistance Rq1 , given by Equation (10), as closing element of the loop (Figure 4B). A first
objective is to compute the various feedback functions of the circuit, in particular the loop factors T and T 0 and the
overall gain Af , exploiting for this the currents W r ¼ I r and W p ¼ I p as cut quantities. From Equations (7) and (10), we
obtain
T¼
R1 ðR3 þ R4 Þ
, T 0 ¼ 0,
R2 ðR1 þ R3 þ R4 Þ þ R5 ðR2 þ R3 þ R4 Þ
ð33Þ
while from the second of Equations (4) and from the second of Equations (7), we have α ¼ T=R1 , and from the first of
Equations (4) and from Equation (18), we get γ ¼ A=R1 , being
A¼
R2 R3 þ R5 ðR2 þ R3 þ R4 Þ
,
R3 R4 R5 ðR2 þ R3 þ R4 ÞΔ
ð34Þ
where Δ is the system determinant given by
1
R2 ðR1 þ R3 þ R4 Þ
Δ¼
:
1þ
R1 R3 R4
R5 ðR2 þ R3 þ R4 Þ
ð35Þ
According to the second of Equations (7), the backward factor T becomes the loop gain, and the overall gain Af is
given in the “usual” form (19) with T 0 ¼ 05 and, from the preceding results, we get
Af Vu
R2 R3 þ R5 ðR2 þ R3 þ R4 Þ
:
¼
V s R2 ðR1 þ R5 Þ þ ðR3 þ R4 ÞðR1 þ R2 þ R5 Þ
ð36Þ
Moreover, let us continue the example applying the feedback “ubiquitous” representation of Section 5 again to the
bridged-T circuit of Figure 4A by computing its nodal equation system (32) written in the form
2
6
4
G1 þ G2 þ G4
G4
G4
G3 þ G4
G2
G3
G2
32
V1
3
2
G1 V s
3
76 7 6
7
54 V 2 5 ¼ 4 0 5,
G2 þ G3 þ G5
V3
0
G3
ð37Þ
where Gi 1=Ri . According to the previous procedure, we choose m ¼ 3, that is, we eliminate the third equation; then,
we choose j ¼ 2 and l ¼ 3, that is, we move the third column to the right-hand side and we define U V 2 and W V 3 ,
thereby obtaining the equation (28), where now
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3991
AUS ¼
G1 G4
G2 ðG3 þ G4 Þ þ G3 ðG1 þ G4 Þ
, AUW ¼
,
ΔU
ΔU
ð38Þ
ΔU ¼ G3 G4 þ ðG3 þ G4 ÞðG1 þ G2 Þ:
ð39Þ
in which
In an analogous way, we choose o ¼ 2, that is, we eliminate the second equation from the equation system (37);
then, having chosen j ¼ 2 and l ¼ 3, we move the second column to the right-hand side, thus obtaining Equation (29),
where we have
G1 G2
G3 ðG1 þ G2 Þ þ G4 ðG2 þ G3 Þ
, AW U ¼
,
ΔW
ΔW
ð40Þ
ΔW ¼ ðG1 þ G4 ÞðG2 þ G3 þ G5 Þ þ G2 ðG3 þ G5 Þ:
ð41Þ
AW S ¼
in which now
Finally, with regard to the network voltage gain Af V u =V s , let us conclude the example by performing its direct
computation by means of “classic” nodal analysis, that is, by solving the equation system (37): of course, we obtain that
Af is given again by (36).
In conclusion, the bridged-T circuit has been represented by means of the feedback theory, both through the “mere”
CIT method and through the “universal” approach, obtaining the same results as with the nodal analysis.
7 | C O N S I D E R A T I O N S O N F E E D B A C K A N A LY S I S
While in cases like this particular passive network the calculations are actually more complex in the feedback analysis
than by means of the mesh current or node voltage equations, in general feedback analysis presents several advantages.
First of all, in a wide range of practical situations (such as the sample network, containing active elements, shown
in Pellegrini5), the CIT method, with a proper cut choice, can significantly simplify the calculations with respect to classical methods and is able to yield more accurate results with respect to other feedback approaches.
Moreover, feedback analysis is often able to reveal the fundamental operating principle of a system. From this point
of view, approaches like ours have the advantage to maintain the elementary and intuitive meaning of feedback (which
is instead lost in the matrix approaches4).
Furthermore, the feedback representation can highlight meaningful intrinsic properties, useful for specific purposes,
that instead may be hidden in the application of “classical” compact approaches.
In particular, a feedback representation provides direct insight into essential system features4 through just the loop
gain or return ratio T, or, more in general, through the return difference F 1 þ T. Indeed, beyond affecting the overall
gain according to Equation (19), it reduces by a quantity F the effects on the output of spurious signals, as well as the
sensitivity of Af with respect to the variations of an element K when the feedback is performed around it.4,5 Since, in
such a case, F is proportional to the determinant of the system,4,5 it can be used to analyze its stability by means of the
Nyquist criterion.2 According to the Blackman formula,3 which has been generalized here in the form (24), F determines also the driving-point immittance. In particular, it affects (even in a more direct way5,6) the input and the output
impedances, on the basis of which one can define various types of feedback (series or shunt at the input and current or
voltage at the output6). Moreover, the condition jFj jαj guarantees the validity of the virtual short-circuit method,
that largely simplifies circuit analysis.5,7 Therefore, feedback analysis is particularly useful when specific meaningful
properties of the system, such as sensitivity, stability, and driving-point immittances, are investigated.
Finally, the feedback approach represents a very valuable design tool, which can help the designer to impose the
required circuit characteristics, and to validate or change the chosen solutions, orienting the designer in the synthesis
phase (also with the help of computer simulations, which are made simpler as a result of the feedback formulation).
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PELLEGRINI ET AL.
PELLEGRINI ET AL.
More in general, feedback methods can be seen as a mathematical representation applicable to a wide range of natural phenomena, which go beyond the mere field of electrical networks.
8 | C ON C L U S I ON S
We have presented a new method for the feedback analysis of electrical systems.
Feedback models are based on the definition of a loop in the network to be analyzed. In the present formulation,
such a loop is defined by splitting a node into two sub-nodes and by inserting among them and a third node a threeterminal circuit (TTC) that leaves the currents and the voltages (resulting from an arbitrary external source) unchanged
in the two-port circuit represented by the rest of the network. The loop consists of this two-port circuit, which contains
both the external signal sources and the output port, and of the added two-port circuit made up of the TTC, which contains in its turn an independent testing source, possibly a dependent one, and/or other circuit elements.
First, we have shown that, in its more simplified form, this approach can be adopted also for non-linear circuits.
Then, we have adopted this method (which represents an extension of the CIT to the usage of generic TTCs) to linear networks. We have defined a set of superposition functions, characteristic of the chosen cut and TTC, and we have
shown how it is possible to combine them to calculate all the quantities of the system, such as the cut immittance,
introduced here, the overall gain, and the driving-point immittances.
We have also shown that, in any system of interconnected entities, each of them can be considered as a partial cause
of itself through another intermediate one, thus finding (apparent or hidden) closed paths that can be named as “feedback loops.” In this sense, “feedback” can be considered as a ubiquitous (universal) property of any system (physical or
not) and can be exploited as a “universal” mathematical representation tool.
From another point of view, the feedback representation of a system can disclose and/or highlight its general meaning and foundations and can be useful both in the cases, like the Watt regulator, where the forward and feedback
chains and the feedback loop are clearly identifiable, and in situations in which they are physically less apparent.
Further developments of the above general approach consist in using the extended CIT theorem, exploiting its flexibility in the choice of the cut and of the TTC that has to be inserted, as a tool capable of unifying the previous feedback
models (and newly introduced ones) and of yielding new interesting results. This is the purpose of Part II of the paper.
A U T H O R C ON T R I B U T I O NS
This represents one of latest efforts of Prof. Bruno Pellegrini. After his death in 2018, starting from his latest manuscript, Paolo Marconcini and Massimo Macucci have re-elaborated and modified its content in order to obtain the present version.
A C K N O WL E D G M E N T S
We wish to thank Dr. Fabrizio Iacopetti for his help in the preparation of the figures. This work was partially supported
by the Italian Ministry of University and Research (MUR) in the framework of the CrossLab project (Departments of
Excellence) and of the FoReLab project (Departments of Excellence).
CONFLICT OF INTEREST STATEMENT
The authors declare no potential conflict of interests.
DATA AVAILABILITY STATEMENT
The data that support the findings of this study are available from the corresponding author upon reasonable request.
ORCID
Paolo Marconcini
https://orcid.org/0000-0003-3714-0026
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How to cite this article: Pellegrini B, Macucci M, Marconcini P. Novel comprehensive feedback theory—Part I:
Generalization of the cut-insertion theorem and demonstration of feedback universality. Int J Circ Theor Appl.
2023;51(9):3979‐3993. doi:10.1002/cta.3674
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