Novel comprehensive feedback theory—Part I: Generalization of the cut‐insertion theorem and demonstration of feedback universality

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′ 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,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.


| 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.
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 formula 3 for evaluating driving-point impedances, and the fundamental contributions by Bode to a general theory of feedback 4 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.  However, only a few of them are able to retrieve the intuitive concept of feedback (e.g., Pellegrini 5,6 ). 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 q 0 and the reference one t t 0 , to the current in the connection q, q 0 , 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 Pellegrini 5,6 as closing loop element of the new feedback model. Moreover, new general expressions of the overall gain function A f 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.

| APPLICATION OF THE CIT TO NON-LINEAR NETWORKS
The CIT 5 is based on splitting an arbitrary node q of a given linear network N ( Figure 1A) into two sub-nodes q and q 0 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 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 À sign, and with an arrow).
The CIT states 5 that necessary and sufficient conditions for the equivalence of the networks N and N 0 are the equalities 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 derived 5 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 2A 7 in which the TTC inserted among the nodes q, q 0 and t t 0 consists of the independent voltage source v a and of the dependent current source i r ¼ i p , which directly satisfies the second of Equations (1).
For instance, it can be exploited to evaluate the quiescent point for i s ¼ 0. To this end, one computes the characteristic v r ¼ v r ðv a Þ vs. v a and determines its points P 1 , P 0 , and P 2 where, being v r ¼ v a , 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 i Di ¼ g M ðv Gi À V T Þuðv Gi À V T Þ for the MOS transistors, where i Di and v Gi are the drain current and gate voltage, respectively, V T is an average threshold voltage, g M 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 F I G U R E 1 (A) A general linear network N and (B) its equivalent one N 0 after splitting the q, q 0 connection and inserting a threeterminal circuit (TTC) (delimited by the three marked nodes j, k, and m) between the new nodes q and q 0 , thus obtained, and the "reference" one t t 0 . The arrow to the left (right) of the TTC indicates the current entering into (exiting from) it. approximation and, for the sake of simplification, the assumption that R D ¼ R 1 þ R 2 , the shape of the characteristics v r ¼ v r ðv a Þ in Figure 2B requires V DD > 2R D V T =R 1 and g M > 2=R 1 , in order to have P 0 to the right of V T and to the left of V (see Figure 2B), and ðdv r =dv a Þ > 1, respectively. In the three points P 1 , P 0 , and P 2 , we have v r ¼ v a ¼ v p . 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 P i (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 Nyquist 2 criterion, the circuit in the state P 0 is unstable and it will end up into one of the stable points P 1 or P 2 , that is, the circuit becomes a binary storage element.
Otherwise, for different values of the parameters such that ðdv r =dv a Þ < 1 (and thus, P 0 becomes stable, while the states P 1 and P 2 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 Russel 9 ) reported in Section 4.2 of Part II, that is, by first computing the bias point P 0 (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 original 5 or the present feedback model. F I G U R E 2 Analysis of a shunt voltage amplifier or of a flip-flop (depending on the slope of the v r ðv a Þ characteristic). (A) Cut network N 0 after splitting the q, q 0 connection and inserting a TTC between q, q 0 and the "reference" node t t 0 . (B) Characteristic curve v r ¼ v r ðv a Þ versus v a for i s ¼ 0 that determines three possible steady states: P 0 , P 1 , and P 2 . In these states, being v r ¼ v a ¼ v p and, by definition, i r ¼ i p , 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.

| Feedback loop with generic TTC
The original version of the CIT 5 was applied in Pellegrini 5 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 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).

| 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: from which, from the superposition theorem and from the first of Equations (2) of the CIT, we obtain, for any S, the relationship that can be written in the form where we define the cut functions 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 ÀW 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.  (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 where W q and W q ( Figure 1A) are the current in the connection q-q 0 (flowing from q to q 0 ) and the voltage between q q 0 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 R q ! ∞. More interesting cases are shown for the cuts of the bilateral bridged-T of Figure 4, for which we have R q2 ¼ R 5 ðR 2 þ R 3 þ R 4 Þ=R 2 and R q3 ¼ R 5 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 q 0 and t t 0 and the current flowing through K coincides with the current in the connection q-q 0 ; moreover, its sign depends on the reciprocal position of q and q 0 .
Other approaches can be the direct evaluation of W q and W q or the method of Cherry 8 that inserts an impedance X ! 0 into the connection q-q 0 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 t 0 .
Another way to compute X q exploits the CIT theorem, 5 starting from the definition of the immittance 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 We can compute X p from the cut network ( Figure 1B) by defining the following function of the cut: 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 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 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 immittance 5,6 and of any other driving-point immittance different from the input one (see Section 4.3).

| Overall gain
An important characteristic quantity of a linear network is the overall gain A f 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), and from Equation (6) and from the definition of the cut function, we obtain the overall gain A f in the form 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 ÀðW r =W p Þ U¼0 (Equation (8)), that, together with Equations (3), (4), (7), and (18), leads to so that Equation (19) gives also expression that reduces the number of the cut functions appearing in A f .

| Driving-point immittances
Another meaningful network function is the driving-point immittance between two arbitrary nodes e and h ( Figure 1B), defined as 4 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 A f ¼ U=S, we get from which and from (21), we finally obtain 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.

| 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 A f definition, we obtain that γ ¼ γ and A f ¼ A f , 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 Equation (13), we have A ¼ X i A. From the second of Equations (7), from Equation (11), and from Equation (13), we 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 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, q 0 and t t 0 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 L rd 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.
For instance, for the quantity T rd qj , we have the dual one T rd qj T ¼ ÀðW r =W p Þ S¼0 that, from Equation (16) and from the first of Equations (25), becomes In an analogous way, we have T 0 rd qj T 0 ¼ ÀðW p =W r Þ W a ¼0 that, from Equation (16) and from the second of Equations (25), gives Since in L rd qj we can have two independent possibilities (q or q 0 , 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.

| UNIVERSAL FEEDBACK
The above network feedback theory based on the CIT theorem 5,6 relies on finding a loop around an arbitrary node pair (q q 0 and t t 0 ). 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 through the external signal (cause) S and the intermediate variable 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 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 b m is not the only nonzero coefficient b i , 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 A US and A UW 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 A WS and A WU 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.
F I G U R E 5 Block diagram of a feedback system according to Equation (19) and to Equations (28)-(31). 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 Pellegrini 5 ).
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.

| FEEDBACK REPRESENTATION OF A BRIDGED-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 R q1 , 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 A f , 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 while from the second of Equations (4) and from the second of Equations (7), we have α ¼ ÀT=R 1 , and from the first of Equations (4) and from Equation (18), we get γ ¼ A=R 1 , being where Δ is the system determinant given by According to the second of Equations (7), the backward factor T becomes the loop gain, and the overall gain A f is given in the "usual" form (19) with T 0 ¼ 0 5 and, from the preceding results, we get 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 where G i 1=R i . 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 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 in which now Finally, with regard to the network voltage gain A f 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 A f 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.

| CONSIDERATIONS ON FEEDBACK ANALYSIS
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 Pellegrini 5 ), 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 approaches 4 ).
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 features 4 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 A f 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 way 5,6 ) 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 output 6 ). 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).
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.

| CONCLUSIONS
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.

AUTHOR CONTRIBUTIONS
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.