Novel comprehensive feedback theory—Part II: Unifying previous and new feedback models and further results

The objective of this Part II of the paper on a novel feedback approach is that of unifying all the previous general feedback models within a single framework and of obtaining new results. In order to pursue this goal, we exploit the multiple possibilities of implementation of the generic three‐terminal circuit (TTC) that has to be inserted in the cut of the feedback loop when the cut‐insertion theorem (CIT) is applied for the analysis of the network. We present models based on a direct opening of the feedback loop inside the TTC, TTC implementations (associated with circuital examples) that allow an arbitrary splitting of the signal flow between a part associated to the feedback loop and another due to a direct leakage path from input to output, and methods (with exemplifying networks) based on test signal injection that allow measuring the gain of the closed feedback loop. The asymptotic formula of the overall gain is generalized to any network. Finally, examples of application to a voltage‐feedback amplifier are reported to show, in particular, the values and the properties of the multiple possible feedback loop gains that can be defined for a given cut and also to compare our results with those available in the literature.


Summary
The objective of this Part II of the paper on a novel feedback approach is that of unifying all the previous general feedback models within a single framework and of obtaining new results. In order to pursue this goal, we exploit the multiple possibilities of implementation of the generic three-terminal circuit (TTC) that has to be inserted in the cut of the feedback loop when the cutinsertion theorem (CIT) is applied for the analysis of the network. We present models based on a direct opening of the feedback loop inside the TTC, TTC implementations (associated with circuital examples) that allow an arbitrary splitting of the signal flow between a part associated to the feedback loop and another due to a direct leakage path from input to output, and methods (with exemplifying networks) based on test signal injection that allow measuring the gain of the closed feedback loop. The asymptotic formula of the overall gain is generalized to any network. Finally, examples of application to a voltagefeedback amplifier are reported to show, in particular, the values and the properties of the multiple possible feedback loop gains that can be defined for a given cut and also to compare our results with those available in the literature.

K E Y W O R D S
circuit and system analysis, circuit theory, cut-insertion theorem, feedback theory

| INTRODUCTION
As well known, there are several general representations of a network as a single loop feedback system. The objective of the present Part II of our paper on a novel feedback theory, described in Pellegrini, 1 is to show that it unifies, under a single perspective, all the previous feedback models (and new ones proposed here), and that it allows reaching innovative results. This can be obtained by means of proper implementations of the generic three-terminal circuit (TTC) that, according to the cut-insertion theorem (CIT), 2 has to be inserted in the cut of the loop along which we define the feedback. In doing this, we exploit the possibility of choosing the number and type of components of the TTC, as well as its topology.
First, in Section 2, we recall some definitions and results of the Part I. 1,2 Then, Section 3 deals with the original CIT, which is based on two particular choices of TTCs which are "separable," that is, with direct opening of the feedback loop inside the TTC itself. In particular, the critiques advanced in Fianovsky 3 to the evaluation of the cut immittance X p (which was first introduced in Pellegrini 4 ) are refuted. Section 4 deals with other approaches that adopt "separable" TTCs. [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] In particular, as a new result, we show two choices of TTCs which allow to split the flow through a network in an arbitrary way between a part "associated with" the feedback loop and a "direct" input-output flow (and an example of this is shown for a shunt-voltage MOS amplifier).
Instead, Section 5 deals with approaches based on "non-separable" TTCs, in particular with the "test signal injection" (TSI) method, that is also suitable to measure the closed loop gain. [17][18][19][20][21][22][23] Moreover, an example of analysis of a shunt-voltage MOS amplifier by means of the TSI method is provided.
The bilinear expression of the overall gain in an element K of the circuit (in the cases in which feedback is defined in terms of feedback around such an element 2,5 ) and a generalization to any network of the asymptotic formula of the overall gain 24,25 are treated in Section 6.
In Section 7, several examples of analysis of a multistage voltage-feedback amplifier are reported, to highlight the previous results (and in particular to show the multiple possible definitions of feedback loop gain for the same cut, and their properties) and for comparison with other feedback approaches. 26 Finally, the conclusions are drawn in Section 8.

| A SUMMARY OF A FEW CONCEPTS FROM PART I
In order to present the second part of the paper, some concepts from Part I 1 have to be briefly recalled.
According to the generalized cut-insertion theorem introduced in Part I, starting from a general linear network N with input S and output U (see Figure 1A), one of its nodes (indicated with q) can be split into two sub-nodes q and q 0 , and an arbitrary TTC (containing an internal independent source W a ) can be introduced between q, q 0 , and a 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 between the new nodes q and q 0 , thus obtained, and the arbitrary "reference" node t t 0 . The arrow on the left (right) of the TTC indicates the current entering into (exiting from) it.
"reference" node t t 0 , in this way obtaining a split network N 0 (see Figures 1B and 2A). In this general formulation, S, U, and the quantities W can represent either voltages or currents, and thus in the figures are indicated both with +/À signs, and with an arrow. The TTC can be made up of two parts (a loop closing element CE and a testing circuit TE) connected only by the reference node t t 0 ("separable" TTC, shown in Figure 2B), or not ("non-separable" TTC). The original network N and the split network N 0 are equivalent if and only if the following two CIT constraints are satisfied: where the W 's and the W 's are dual quantities (i.e., if the W 's are voltages, the W 's are currents, and vice versa) and the indices r and p refer to the quantities at the input and output ports of the TTC, respectively. Considering the split network N 0 , it is possible to define the following quantities characteristic of the cut: From the first of the CIT constraints (1), we have that Figure 1B, in which we highlight the "forward" ("backward") chain FN (BN) that is 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 case of a "separable" TTC. In (B), CE and TE represent the loop closing element and the testing circuit of the TTC, respectively.
where we have defined the following loop factors ("gains"): (i.e., the "forward" and the "backward" transmission factors, respectively). The second of Equations (1) yields further constraints on the TTC, which depend on the TTC structure. We can define the cut immittance (due to the CIT constraints (1), the immittances X q and X p , defined according to Equation (6), have to coincide).
The immittance X p , which can be used as closing element CE of the loop in Figure 2B, is related to the other cut quantities by where 1 Defining the overall gain A f U=S is given by 1 Moreover, using the loop gain definition together with Equations (2), (3), (5), and (9), we obtain that Therefore, the overall gain A f can be expressed also as while the driving-point immittance X is given by where X 0 Xj W a ¼0 is the value of X when W a is switched off. This further extends the Blackman formula. 2,8 3 | ORIGINAL CIT METHOD The objective of the present part is to show that the new approach includes all the previous feedback models and yields novel results.
To this end, let us begin exploiting the TTCs of Figure 3, which are those used in the original CIT (which was proposed in previous works 2,4 and which represents the foundation of the proposed novel feedback theory). These are "separable" TTCs, in which the loop closing element CE is connected to the TE testing source W a only through the reference node (see Figure 2B). In particular, the W a nodes a and b coincide with the output nodes q 0 and t 0 of the TTC, respectively (i.e., a q 0 and b t 0 ), so that we have W a ¼ W p and, from the fourth of Equations (2), the fourth of Equations (3), the first of Equations (5), and Equation (4), we get These equations characterize the original CIT model 2,4 in the new feedback framework and, from (7) and (10), give Therefore, the original CIT can be seen as a particular case of the new feedback loop model, the "classical" results of previous studies 2,4 are retrieved, and further results can be achieved. In particular, the loop factor T represents the return ratio or gain of the whole open feedback loop.
The cut immittance X p (in the form of an impedance Z p ) is one of the bases of the CIT in its earlier formulation. 4 It is worth noticing that, in general, the first of Equations (16) gives the cut immittance X p through an implicit equation because each of the quantities MðsÞ in its right-hand side (i.e., X À1 i , ρ, T, and α) depends on X p ¼ Z p itself (considering the impedance case of previous works 2,4 and W p and W r as voltages) in a bilinear form 5-7 : TTCs with the testing source W a ¼ W p directly located at the output q 0 , t 0 of the TTC itself. 2,3 In (A), the loop closing element is represented by the immittance X p , while in (B), by the dependent generator W p where A M ðsÞ, B M ðsÞ, C M ðsÞ and D M ðsÞ are polynomials in s with real coefficients independent of Z p . In its turn, Δ L ¼ Δ L0 þ Δ rr Z p is the determinant of the open feedback loop, Δ L0 is its value for Z p ¼ 0, and Δ rr is the cofactor of its element in position ðr,rÞ, when only the rth mesh contains Z p (in position r). Now, if we directly substitute Equation (17) into the first of Equations (16) we get an apparently fifth-degree equation, as claimed in Fianovsky, 3 which is algebraically unsolvable. In reality, from the mesh analysis of the cut network exploiting the TTC of Figure 3A, we have that: Z À1 i , ρ, T, and α have the same denominator Δ L (the determinant of the open feedback loop); Z À1 i and ρ obey Equation (17); while, since V r ¼ I r Z p , α ¼ ðV r =SÞj V a ¼0 , and T ¼ ÀðV r =V p Þ S¼0 , α and T are directly proportional to Z p , that is, α ¼ B α Z p =Δ L and T ¼ B T Z p =Δ L . Therefore, the substitution of Equation (17) into the first of Equations (16) indeed yields a second degree equation for Z p . However, even though it is algebraically solvable, difficulties arise in computing its coefficients through the polynomials A M ðsÞ, B M ðsÞ, C M ðsÞ, D M ðsÞ, and the determinant Δ L ðsÞ.
An alternative is represented by the direct evaluation of X q ¼ X p (see Equation 6). Moreover, we can notice that Equations (15) and the first of Equations (16) hold true also when the loop closing element is a dependent source W p (as in the case of the TTC of Figure 3B, 2 which automatically satisfies the second constraint of Equation 1): with this choice, the first of Equations (16) gives explicitly Z p , because, for such a TTC, Z p does not appear in the quantities MðsÞ in the right-hand side. It is also important to underline that in the meaningful case of unilateral networks, for instance for the case of a cut on the gate of a field effect transistor or on the input terminal of an operational amplifier, for which we have ρ ðW p =SÞ W a ¼0 ¼ 0, from the first of Equations (16) we directly obtain X p ¼ X i , in which X i is the input immittance of the unidirectional element, which is independent from X p .
In the case of "separable" TTCs, beyond obtaining the cut immittance X p and the overall gain A f in the original forms of Equations (16), 3,4 from Equation (13), we obtain the new expression while from Equation (14) the driving-point immittance becomes again 2 where, in particular, the subscripts highlight the conditions for the evaluation of the cut functions. Equation (19) has been written considering U ¼ S; as shown in Bode, 5 when S is a voltage source Tj S¼0 T SC and T n j S¼0 T OC have to be computed when the input nodes are short-and open-circuited, respectively. Moreover, for the dual quantities (where the homogeneous quantities W r and W p are replaced by their dual ones, W r and W p , respectively, and vice versa), the Equations (26) and (27) of the Part I give the simplified relationships respectively. In the case in which we choose the TTC of Figure 3B as the three-terminal circuit to be inserted in the cut, it is (by definition) W r ¼ W p , so that we get T 0 ¼ ÀðW p =W r Þ W a ¼0 ¼ À1 and T ¼ ÀðW r =W p Þ S¼0 ¼ À1. In such a case, the expression (10) of the overall gain, if written in terms of the dual quantities, would be indefinite and thus not applicable (as we will discuss more in detail in Section 7.3.1).

| Models with arbitrary signal flow splitting
In order to provide further new results of the CIT extension, let us show, for example, how particular TTCs ( Figure 4) lead to an arbitrary splitting of the input-output signal flow between the leakage path γ and the feedback loop core (A, À T=A) ( Figure 5). To this end, let us again exploit a "separable" TTC of the type of Figure 2B.
For the implementation of the testing element TE of Figure 2B, we use the dependent sources μV a and μ 0 V r (as shown in Figure 4A), where μ ¼ δð1 À TT 0 Þ and μ 0 ¼ ÀT 0 (here, for the sake of simplicity, we specify the quantities W as voltages V ). In Figure 4A, we have that V p ¼ μ 0 V r þ μV a . This agrees with the definition of the parameters because if, in μ 0 V r þ μV a , we substitute to V r the expression V r ¼ αS À δTV a (obtained, using the superposition theorem, from the second of Equations 3, the second of Equations 2, and the second of Equations 5), we obtain (using the first of Equations 5) Àθ 0 S þ δV a , that is (according to the fourth of Equations 3 and to the fourth of Equations 2), V p .
As in the original CIT, the first of the constraints (1) is satisfied if V a is given by Equation (4): in this case, The loop closing element CE consists of the cut impedance Z p ; if Z p is given by Equation (7), also the second of the CIT constraints (1) is satisfied.
Alternatively, one can adopt the dual TTC shown in Figure 4B.
Making again reference to Figure 4A, let us now analyze the network properties deriving from the arbitrariness in the choice of the TTC and in particular of the quantity μ 0 ¼ ÀT 0 , which describes how, inside the TTC, V p is split between the contributions of V r and V a . Defining α 0 αj T 0 ¼0 and γ 0 γj T 0 ¼0 , from the definitions of α and γ (Equations 3), we obtain that where A and T are independent of T 0 (note that also Z p does not depend on T 0 , since Z p Z q , which depends only on the cut). The relations (22) can be derived expressing both V r j V a ¼0 and Uj V a ¼0 as sums of two contributions of S, a direct one (not passing through V p ) and an indirect one (passing through V p ): TTCs with the loop testing element made up of a combination of the quantity at the TTC input and of the independent source present inside the TTC: Panel (A) is the configuration described in the text, while Panel (B) is the dual one.
F I G U R E 5 Block diagram of a feedback system according to Equation (25).
with μ 0 ¼ ÀT 0 . Alternatively, Equations (22) can be achieved observing that the quantities V p ¼ V r ¼ ½αð1 À TT 0 Þ=ð1 þ TÞS and U ¼ ½αAð1 þ T 0 Þ=ð1 þ TÞ þ γS obtained in the network N 0 equivalent to original circuit N do not have to change if we use in our analysis the TTC of Figure 4A with a generic T 0 or with T 0 ¼ 0]. From Equations (10) and (22), we derive that the overall voltage gain A f of the network (i.e., its most meaningful quantity) can be expressed as The schematic representation of this relationship is given in Figure 5.
The arbitrariness of μ 0 , the relationship (22) and the corresponding block diagram in Figure 5 highlight how the signal flow throughout a network can be split in an arbitrary way into a part which (after passing through α ∘ ðT 0 Þ) is "handled" by the feedback loop core (A, À T=A) (which amplifies it by its "intrinsic" amplification A i A=ð1 þ TÞ, independent of T 0 ) and into a "direct" input-output flow through the leakage path γ ∘ ðT 0 Þ (which, as well as α ∘ ðT 0 Þ, depends on T 0 ).
In order to illustrate the obtained results, let us analyze the shunt-voltage amplifier in Figure 6, to which the TTC of Figure 4A is applied, according to the generalized CIT. From the second of Equations (5) we get the loop gain T ¼ g m R S R L =ðR S þ R F þ R L Þ, g m being the transistor trans-conductance, while from the first of Equations (5) we obtain the F I G U R E 6 Analysis of a shunt-voltage FET amplifier, with the TTC of Figure 4A applied according to the generalized CIT.
F I G U R E 7 Qualitative plots of the cut functions α ∘ ðT 0 Þ and γ ∘ ðT 0 Þ of Equation (25). arbitrary value of the forward gain T 0 ¼ Àμ 0 . The plots of α ∘ ðT 0 Þ and γ ∘ ðT 0 Þ sketched in Figure 7 show that α ∘ ðT 0 Þ and γ ∘ ðT 0 Þ increases and decreases, respectively, for increasing T 0 , that is, they show the arbitrary splitting of the inputoutput signal flow between the leakage path γ and the feedback loop core (A, À T=A) ( Figure 5).
However, this arbitrary splitting does not mean that A f is undetermined. Indeed, the further development of Equation (25) gives again Equation (10) with T 0 ¼ 0, α ¼ α 0 , and γ ¼ γ 0 , without any arbitrariness deriving from the use of the TTC in Figure 4A; therefore, this does not limit the applicability of this TTC. In particular, the complete analysis leads to the overall amplification (which can be obtained also by means of the direct nodal calculation). One obtains the same results using the dual TTC shown in Figure 4B.

| Models based on the insertion of a transformer
According to the premise of Section 3, let us include other previous models, based on breaking a feedback loop, in the present framework, and let us add new models and results.
With reference to the schematics of Figures 1B and 2B, let us begin by exploiting the TTC of Figure 8A, in which a 1: n transformer is employed and voltages are again used in the place of the quantities W for the sake of simplicity.
(The presence of the dependent current generator ð1 À nÞI p is necessary in order to satisfy the second of Equations (1), since the current in the primary winding of a 1: n transformer is n times greater than that in the secondary winding.) From the first of Equations (5) we have T 0 ¼ ÀðV p =V r Þ V a ¼0 ¼ Àn and from Equation (10) The case of n ¼ 1 requires V a ¼ 0 in order to satisfy the condition (4) deriving from the CIT; in that case, the TTC becomes the mere transformer of Figure 8B, which is at the basis of the model of Codecasa 9 in which no independent source is necessary and that, by definition, obeys the CIT.
An analogous case is represented by the TTCs of Figure 8C,D, that exploit dependent sources. These TTCs give the same results as those in Figures 8A and 8B, respectively, since they are simply an implementation of the transformers of Figure 8A,B (in the case of Figure 8C, they include also the effect of the generator ð1 À nÞI p of Figure 8A). In particular, in the case of n ¼ 1 one only has to insert into the network a unitary ratio transformer or its equivalent of Figure 8D (which, in agreement with the CIT, leaves the currents and the voltages unchanged), without inserting any testing source W a . This makes it impossible to define the quantities and to develop the feedback approach of Part I. 1 However, we can use such a TTC adopting an approach similar to the model introduced in Russel. 10 Representing the feedback loop by means of the two-port sub-circuits BN and FN ( Figure 9A), in the model of Russel, 10 the FN is then duplicated ( Figure 9B) and connected through VCVSs (voltage-controlled voltage sources), in such a way as to leave the voltages and currents unchanged, and to enable DC and AC analysis. With respect to that described in Russel, 10 in our loop-breaking model ( Figure 9C) the sub-circuit FN on the left side and the two connected dependent sources are replaced with the TTC of Figure 8D. This is useful to perform DC computations, also for non-linear devices and using computer-aided simulations. Following the procedure described in Part I of this paper, 1 the bias point of the circuit can be found studying how the voltage between the nodes q and t varies when we change the voltage applied between the nodes q 0 and t 0 : when the two voltages coincide, the network is equivalent to the original one and we have identified the bias point of the circuit.
Instead, for the analysis of small signals (which is what is mainly considered in the present Part II), the network can be linearized around the bias point, and the AC loop-breaking can be performed employing one dependent and one F I G U R E 9 Network (A) analyzed by opening up a loop as in Russel 10 (B) and according to the CIT (C). independent source (i.e., replacing one of the dependent sources with an AC independent source), in order to perform the AC analysis with the CIT-based approach proposed here.

| Models defining feedback around an element
Now let us take in consideration other models in which, following Bode's approach, 5 feedback is defined around a (unilateral or bilateral) element K of the circuit, rather than around a pair of nodes as in the CIT. 2 Let us first consider approaches that cut a well identified loop around a unilateral element (represented by a generator KQ S k between the nodes 2 and 2', that depends, through the trans-immittance K, on the quantity Q measured on the immittance X s between the nodes 1 and 1', as shown in Figure 10A) and insert an independent source that controls it. [11][12][13][14][15] They can be dealt with by means of the CIT, 2 using the TTC shown in Figure 3A. A possible splitting (reminiscent of previous works 11,12 ) can be performed at the terminals of the immittance X s (Figure 10A), using as TTC nodes q c, q 0 c 0 and t t 0 c 00 . Another possible splitting (reminiscent of previous studies [12][13][14][15] can be performed at the terminals of the controlled source KQ S k (Figure 10A), considering as TTC nodes q d, q 0 d 0 and t t 0 d 00 [in this case, the presence of X p is irrelevant, since it is in parallel with (series to) the ideal voltage (current) source KQ]. In both cases, we can use the source W a of the TTC of Figure 3A, 2 inserted between q 0 and t 0 , as independent controlling source.
The CIT allows us also to deal with (in a new way with respect to Pellegrini 2 ) the case of a bilateral element K, for instance of an admittance Y ¼ K connecting the two nodes 1 and 2. As shown in Figure 10B, this case can be equivalently represented by the two-port circuit 1, 1 0 3, 2, 2 0 3 (where 3 is an arbitrary node), which preserves the relationships V 1 ¼ V 2 þ I 1 =Y and I 2 ¼ ÀI 1 between the quantities at the left and right side. In this two-port circuit, the TTC must be inserted between the cut nodes q q 0 and t t 0 ( Figure 10B). The above approaches [11][12][13][14][15] are the circuit counterpart of the analytic procedure, based on the matrix approach, in which first the superposition theorem is applied to the input source S and to the dependent source S k , considered as independent sources, and then the value KQ is attributed to S k in order to obtain the feedback results. 16 Therefore, the cut-insertion method contains and extends the preceding approaches of feedback around an element K, both unilateral and bilateral, facilitating the computation of the sensitivity of A f with respect to K. 2,5,6 5 | MODELS BASED ON "NON-SEPARABLE" TTCS

| Models based on test signal injection
In general, and also for the proposed models, problems can arise in the practical measurement of the feedback loop gain. Indeed, these models require the implementation (in the closing element CE, see Figures 3A and 4) of the immittance X p , which can even be negative, or (in the closing element CE, see Figure 3B, or in the testing element TE, see Figures 4 and 9) of dependent sources, that are hard to obtain over wide frequency bands.
The problem can be solved by means of a direct measurement that leaves the feedback loop closed, without any cut. [17][18][19] Indeed, a test signal can be injected into the circuit, as shown in Figure 11A, by means of an independent source that injects the current I z ¼ I x þ I y into the connection between two sub-nodes 1 2 from a node 3 (the nodes are arbitrary) and that generates the currents I x and I y in the other two branches departing from the two sub-nodes 1 2.
The dual case of voltages, with V z ¼ V x þ V y , is shown in Figure 11B. The unifying CIT model proposed here includes also the test signal injection (TSI) method, 17-19 by using a "non-separable" TTC, in which we choose the nodes in this way: q 1, q 0 2, and t t 0 3 (see Figures 11A and 12), and, moreover, for the sake of simplicity, we define the quantities W of the TTC as currents I. Our objective is to relate the cut functions of the CIT model with those of the TSI method, and, in particular, to compute through them the F I G U R E 1 1 TTC for the test signal injection method: (A) with a current injected into the connection between two sub-nodes 1 2 from a reference node 3, and (B) its dual.
F I G U R E 1 2 Network analyzed by means of the TSI method. overall gain of the TSI method from the relation (10) of the CIT. The relationships between the quantities used, in the cut, in the two approaches are the following ones (see Figures 11A and 12): The definitions of the functions of the TSI method are 19 First, let us find the relationships between the loop gains in the two approaches. From the first of Equations (5), from the second and the third of Equations (28), and from the fact that when I z ¼ 0 we have that I z ¼ I x þ I y ¼ 0, we get the "forward gain" of the closed loop while from the second of Equations (5), the second and the third of Equations (28), and the first of Equations (29) we obtain its "backward gain" and from Equation (11), the second and the third of Equations (28), and the second of Equations (29) we have that It has to be noticed that the second of the CIT conditions (1) is directly satisfied by the fourth of Equations (28), while the first of the CIT conditions (1) is imposed by the constraint (4), that, due to Equation (30), requires W a ¼ 0 (i.e., I z ¼ 0) and, therefore, requires to leave the network unmodified to obtain its equivalence with the original one. The fact that it has to be W a ¼ 0 for the equivalence to exist, of course, does not prevent the arbitrariness of W a necessary to define the various functions of the models.
Next, let us compute the relationship between the overall gains for the two approaches that, from Equations (10) and (30), the first of Equations (3), and the fourth of Equations (29), becomes which outlines the fact that for W a ¼ I z ¼ 0 the cut and uncut networks are identical. Finally, let us compute the expression of the overall gain only in terms of the network functions (29) of the TSI framework. [17][18][19] From the second of Equations (2), the second of Equations (3), the second of Equations (5), and the superposition theorem, we have that W r ¼ αS À δTW a , while, in a similar way, from the first of (2), the first of Equations (3), Equation (9), and the superposition theorem, we get U ¼ γS þ δAW a . These relationships allow to obtain ðW a =SÞj W r ¼0 ¼ α=ðδTÞ and ðU=SÞj W r ¼0 ¼ γ þ αA=T, so that, from the third of Equations (29), we get From Equations (12) and (34), we obtain Finally, exploiting Equation (33) and the second of Equations (35), the CIT model leads to the expression of the overall gain of the TSI method, 19 reached as a result of the proposed unifying model. Analogous results hold true for the dual TTC of Figure 11B. The injection method is suitable for the measurement of the gain of the closed feedback loop T d I y =I x S¼0 ¼ T, by means of specific instrumentation that implements the signal injection and the signal detection by means of current probes. [20][21][22][23] Notice also that the ratio I z =I x , which can be measured at the same time as T, directly gives the return difference 5 : is useful, in particular, for the system stability analysis.
Let us finish this section with the example of the calculation of the loop gain T in the shunt-voltage amplifier dealt with in Figure 6, by means of both the CIT method of Figure 3A and the TSI method of Figure 11B (as shown in Figure 13A and 13B for the first and second case, respectively). Actually, by cutting the MOS gate and inserting the TTC of Figure 3A (or the TTC of Figure 4A and Figure 6 with μ 0 ¼ 0), we obtain the same gain T ¼ g m R S R L =ðR S þ R F þ R L Þ that we get by means the TSI method of Figure 11B, inserting the voltage testing source on the gate terminal (thus avoiding the measurement difficulties claimed at the beginning of this section).

| BILINEAR EXPRESSION AND ASYMPTOTIC FORMULA OF THE OVERALL GAIN
Let us focus again on cases in which the feedback can be defined around an element K of the circuit (see Section 4.3).
As previously stated, the CIT 2 and the Bode 5 feedback models are based on the definition of feedback around a generic pair of nodes, and around a generic element K, respectively. In particular, they lead to the same expression of the return difference F, that is the basis for the analysis of the stability and of the sensitivity of a single loop network 2,5,27 : in the case of feedback around a bilateral element K in which Δ and Δ 0 are the system determinant and its value for K ¼ 0, respectively. Let us find how this is affected by the CIT extension and by the TSI method in the case of feedback around a bilateral K element.

| Feedback around a bilateral element according to the extended CIT
To this end, referring to the schematics of Figures 2B and 14A, let us apply the procedure of Figure 3A of Pellegrini, 2 that differs from Figure 14A for the fact that here, in general, W a ≠ W p . However, the second of Equations (5) gives again 2 Moreover, from the second of Equations (3), the first of Equations (5), and Equation (10) we have, respectively, α ¼ ÀρK, , and the explicit expression of A f as function of K in the form where A, X i , X t , γ, and ρ are independent of K. Therefore, A f has a bilinear expression in K. 5,6 Moreover, applying the method of Pellegrini, 2 in the case of unilateral K we obtain the same results for the loop gain and the return difference (and hence for the stability analysis 27 ), while the formula for the sensitivity 2,5 will be changed according to the expression of A f and to the definition (28) of Pellegrini. 2 F I G U R E 1 4 Feedback around an element K in a network analyzed by means of (A) the extended CIT and (B) the TSI method.
6.2 | Feedback around a bilateral element according to the TSI method Now let us study the feedback around the bilateral element K using the TSI method, that is, the schematic of Figure 14B, where K is the only element inserted between the node t and the node q, in series with the branch carrying the current I y , so that, from Equations (28) and (29), we obtain again Equations (30), (31), and (38). On the other hand, from the nodal analysis of the network of Figure 14B, beyond obtaining again Equation (38), we achieve that the return difference, which is suitable for the system stability analysis, 5,27 is given by Equation (37): F ¼ Δ=Δ 0 . Moreover, from the same nodal analysis and from Equation (33), we obtain again for the overall system gain A f a bilinear expression in K, as in Equation (39), and the same consequence about the sensitivity with respect to K variations.

| Asymptotic formula of the overall gain
Previous studies 24,25 proposed an asymptotic formula for the overall gain of a network in the form where This formula coincides with Equation (39), obtained with the extended CIT model, with T given by Equation (38), G ∞ ¼ ÀρAX i þ γ and G 0 ¼ ÀρAX t þ γ. More in general, since Equation (10) coincides with Equation (40)-(41), with G ∞ ¼ γ and G 0 ¼ αAð1 þ T 0 Þ þ γ, we can conclude that such a formula is always valid. Therefore, the CIT model includes and generalizes also the asymptotic formula method, that was originally obtained for the case of feedback around a unilateral element K. 24,25 This confirms that the new approach represents a unifying analysis tool, in the sense that it includes and extends all the previous general models and the new ones that we have proposed here.

| EXAMPLE OF CIRCUIT ANALYSIS
The objective of the following example is applying the preceding model to evaluate some significant results concerning the network of Figure 15A, chosen also in order to compare the results with those obtained on the same network in Cherry. 26 In Figure 15B, we represent its small signal equivalent circuit, for the AC analysis (the g m 's and r d 's are the trans-conductances and output differential resistances of the MOS transistors, evaluated in their bias points). We will consider the cut impedance, the overall gain, and the multiple loop gains that can be defined for the same cut when different TTCs are used. In order to do this, we cut the source terminal of the left MOS and we define the left side of the cut as the node q 0 , while the ground is the reference node t t 0 .

| Cut resistance and overall voltage gain
Let us evaluate the cut resistance and the overall gain of the network of Figure 15B, in which we use the TTC of Figure 3B First, before cutting the network, let us evaluate the cut resistance R q ¼ X q by means of (6) where W q ¼ I c is the current entering into the source of the left MOS and W q is its source voltage. We get where R B ¼ r d2 ==R L , obtained also in Cherry 26 by means of an intuitive approach (instead of the appropriate method based on the y-parameter matrix, that cannot be applied in this case. 26 ) Moreover, before cutting the network and exploiting the CIT, let us compute the overall voltage gain by evaluating the source currents of the MOS transistors by means of the corresponding mesh equations. In accordance with Cherry, 26 we get the relationship where Δ is the determinant of the circuit given by

| Analysis by means of the cut superposition constants
Now, let us perform the network analysis by means of the proposed feedback model, by evaluating and exploiting the cut functions (i.e., the superposition constants) defined by Equations (2)-(5), (8), and(9), corresponding to the cut and to the choice of TTC (of the type of Figure 3B) shown in Figure 15B. We have the relationships (15) and T À I r I a Although unnecessary and not used in the present formulation, we can also compute the cut function introduced in which, together with Equation (50)

| Multiple loop gains for the same cut
In the feedback representation of a network, when the feedback loop is split and a TTC is inserted in the cut according to the CIT, the preeminent entities are the feedback loop, the closing element (dependent source or cut immittance) and the independent testing source. As a result of the above point of view, the quantity with a paramount role in any analysis by means of the feedback approach is the loop gain T, that, according to section 4.4 of Part I 1 can have different expressions for the same cut. 7.3.1 | Open loop gains with a test current source and a VCVS closing element (TTC of Figure 3B) With reference to the circuit of Figure 15B, in which we have inserted a TTC of the type of Figure 3B, with a test current source and a VCVS (voltage-controlled voltage source) closing element (in which the quantities W are currents, while the quantities W are voltages), let us further develop the above example. First, let us notice that (following the nomenclature introduced in the section 4.4 of Part I 1 ) the loop gain T given by Equation (49) is nothing else but T pd q 0 k , while the other quantities L pd q 0 k are given by the remaining expressions from Equation (45) to Equation (50). Instead, the dual case of L pd q 0 k is significantly different. Indeed, by definition and from the TTC structure, it is W r V r ¼ V p W p , which automatically satisfies the second of the CIT conditions (1). From this, we get T pd This turns Equation (4), written in terms of the dual quantities L pd q 0 k , into the indefinite relationship I a ¼ ðα pd q 0 k =δ pd q 0 k Þð0=0ÞV s . Indeed, such a relation, written in terms of the dual quantities L pd q 0 k , derives from the condition W r ¼ W p , which in this case is directly enforced by the structure of the TTC and thus does not yield further analytical constraints. As a consequence, also Equation (10) (which derives from Equation 4), if written in terms of the dual quantities L pd q 0 k , becomes indefinite and thus is not applicable. Therefore, in the cases in which the structure of the inserted TTC automatically enforces W r ¼ W p (with W r and W p currents or voltages, depending on the TTC), in order to calculate the overall gain A f through Equation (10) one has to adopt the normal set of parameters (in which T and T 0 are defined as ratios between quantities W ), rather than the dual one (in which T and T 0 are defined as ratios between quantities W ). 7.3.2 | Open loop gains with a test voltage source and the cut resistance as closing element (TTC of Figure 3A) Among the many other possible choices for the same cut, another interesting example is the use in Figure 15B of a TTC of the type of Figure 3A, with the cut impedance between q and t and a testing voltage source between q 0 and t 0 . With this choice, for example we obtain that, of course, is quite different from the one given by (49) and replaces it in this situation. Finally, in order to show the variety of representations allowed by the new analysis method, we can interchange the position of the TTC elements, obtaining again a new loop gain, whereas the overall gain (10) continues to hold true using the L pd qj 's corresponding to the new choice.

| Measurement of open loop gain by means of test signal injection into the closed loop
Finally, let us compute the gain by means of the signal injection method. By inserting the TTC of Figure 11A in the place of that of Figure 15B, that is, by injecting the current I z ¼ I x þ I y into the connection q 0 -q, we obtain the current gain T ðI y =I x Þ V s ¼0 ¼ T pd q 0 k given by Equation (49) that, therefore, can be measured by means of the methods [17][18][19]22 and of the dedicated instrumentation 20,21,23 for the injection methods, which, in particular, allow to directly measure the loop difference F ¼ 1 þ T from the measurements of I x , I y , and I z .
Instead, if we use the dual TTC of Figure 11B, we get (and we can measure) the loop gain T ðV y =V x Þ V s ¼0 ¼ T pd q 0 k given by Equation (52).

| CONCLUSIONS
Here we have shown some of the potentialities of a new method, described in Part I of this paper, for the feedback analysis of an electrical network. Following this approach, a circuit is studied splitting a node into two sub-nodes and inserting among them and a third node a three-terminal circuit (TTC), containing an independent testing source and with a generic structure, with the only constraints that the voltages and the currents at the input and output port of the TTC have to be identical, in such a way as not to alter the rest of the network. In the case of linear circuits, all the quantities of the system, such as the overall gain and the driving-point immittances, can be obtained properly combining a number of superposition functions evaluated for the cut.
The novelty and the possibilities of the proposed approach as an analysis and synthesis tool result from the arbitrariness of the cut-insertion, which can generate a feedback loop around any node pair, and on the flexibility in choosing the elements of the three-terminal circuit inserted into the cut.
Leveraging this versatility, we have shown that this new formulation is able to include in a unitary description the previous feedback models (both those based on breaking a pre-existing loop inside the TTC, and those which leave the feedback loop closed, such as the test signal injection approach), and to yield innovative results and analysis methods, exploiting proper TTC choices.
Since, how we have formally demonstrated in Part I of this paper, starting from its paradigmatic definition, feedback exists in any system, this method represents a universal mathematical representation tool, which can be applied to any electrical circuit (and which could potentially be generalized to systems of any nature).
Finally, we expect that the models, results and examples which we have presented will give rise to new questions that will stimulate further investigations.

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.