Editing Negative-feedback amplifier (section)

==Signal-flow analysis==

A principal idealization behind the formulation of the ''Introduction'' is the network's division into two ''autonomous'' blocks (that is, with their own individually determined transfer functions), a simple example of what often is called "circuit partitioning",<ref name=Sahu>
{{cite book |title=VLSI Design |author= Partha Pratim Sahu |chapter-url=https://books.google.com/books?id=C37vAwAAQBAJ&pg=PA253 |chapter=§8.2 Partitioning |page=253 |isbn= 9781259029844 |publisher=McGraw Hill Education |year=2013 |quote=dividing a circuit into smaller parts ...[so]...the number of connections between parts is minimized}}
</ref> which refers in this instance to the division into a forward amplification block and a feedback block. In practical amplifiers, the information flow is not unidirectional as shown here.<ref name=PalumboG>
{{cite book |author1=Gaetano Palumbo |author2=Salvatore Pennisi |title=Feedback Amplifiers: Theory and Design |quote=In real cases, unfortunately, blocks...cannot be assumed to be unidirectional. |url=https://books.google.com/books?id=VachCXS6BK8C&q=%22In+real+cases%2C+unfortunately%2C+blocks%22%2C%22cannot+be+assumed+to+be+unidirectional.%22&pg=PA66 |isbn= 9780792376439 |year=2002 |publisher=Springer Science & Business Media}}
</ref> Frequently these blocks are taken to be [[two-port network]]s to allow inclusion of bilateral information transfer.<ref name=ChenW>
{{cite book |title=Feedback, Nonlinear, and Distributed Circuits |author=Wai-Kai Chen |chapter-url=https://books.google.com/books?id=W0dPWAaRx6kC&q=%22A+second+approach+to+feedback+network+analysis+involves+modeling+the+%22&pg=SA1-PA3 |pages=1–3 |chapter=§1.2 Methods of analysis |isbn=9781420058826 |year=2009 |publisher=CRC Press}}
</ref><ref name=Pederson>
{{cite book |chapter=§5.2 Feedback for a general amplifier |pages=105 ''ff'' |title=Analog Integrated Circuits for Communication: Principles, Simulation and Design |author1=Donald O. Pederson |author2=Kartikeya Mayaram |chapter-url=https://books.google.com/books?id=MBZugbZ1UM0C&pg=PA105 |year=2007 |publisher=Springer Science & Business Media |isbn=9780387680309}}
</ref> Casting an amplifier into this form is a non-trivial task, however, especially when the feedback involved is not ''global'' (that is directly from the output to the input) but ''local'' (that is, feedback within the network, involving nodes that do not coincide with input and/or output terminals).<ref name=Burgess>
{{cite web |work=Generalized feedback circuit analysis |author1=Scott K. Burgess |author2=John Choma, Jr. |name-list-style=amp |title=§6.3 Circuit partitioning |url=http://www.te.kmutnb.ac.th/~msn/nitiphat.pdf |url-status=dead |archive-url=https://web.archive.org/web/20141230083914/http://www.te.kmutnb.ac.th/~msn/nitiphat.pdf |archive-date=2014-12-30 }}
</ref><ref name=Palumbo>
{{cite book  |author1=Gaetano Palumbo  |author2=Salvatore Pennisi  |name-list-style=amp |title=Feedback amplifiers: theory and design
|page=66 |year= 2002
|publisher=Springer Science & Business Media
|isbn=9780792376439
|url=https://books.google.com/books?id=VachCXS6BK8C&q=%22the+method+is+straightforwardly+applicable+to+only+those+circuits+that+implement+a%22+%22a+feedback+between+the+input+and+the+output%22+%22whereas+many+feedback+amplifiers+exploit+only+%22&pg=PA66}}
</ref> 
[[File:Signal flow graph for feedback amplifier.png|thumb|200px |A possible [[signal-flow graph]] for the negative-feedback amplifier based upon a control variable ''P'' relating two internal variables: ''x''<sub>''j''</sub> = ''Px''<sub>''i''</sub>. Patterned after D'Amico ''et al.''<ref name=Damico/>]]
In these more general cases, the amplifier is analyzed more directly without the partitioning into blocks like those in the diagram, using instead some analysis based upon [[Signal-flow graph|signal-flow analysis]], such as the [[Return ratio|return-ratio method]] or the [[asymptotic gain model]].<ref name=Sarpeshkar>For an introduction, see {{cite book |title=Ultra Low Power Bioelectronics: Fundamentals, Biomedical Applications, and Bio-Inspired Systems |pages=240 ''ff'' |chapter-url=https://books.google.com/books?id=eYPBAyDRjOUC&pg=PA240 |chapter=Chapter 10: Return ratio analysis |author=Rahul Sarpeshkar |isbn=9781139485234 |year=2010 |publisher=Cambridge University Press}}</ref><ref name=Chen>
{{cite book |title=Circuit Analysis and Feedback Amplifier Theory |author=Wai-Kai Chen |chapter-url=https://books.google.com/books?id=ZlJM1OLDQx0C&pg=SA11-PA2 |pages=11–2 ''ff'' |chapter=§11.2 Methods of analysis |publisher=CRC Press |year=2005 |isbn= 9781420037272}} 
</ref><ref name=Palumbo3>{{cite book |title=Feedback Amplifiers: Theory and Design  |author1=Gaetano Palumbo |author2=Salvatore Pennisi |chapter-url=https://books.google.com/books?id=VachCXS6BK8C&pg=PA69 |pages= 69 ''ff'' |chapter=§3.3 The Rosenstark Method and §3.4 The Choma Method |isbn=9780792376439 |year=2002 |publisher=Springer Science & Business Media }}
</ref> Commenting upon the signal-flow approach, Choma says:<ref name=ChomaJr>
{{cite journal |author=J. Choma, Jr |url=http://wenku.baidu.com/view/e046d9d528ea81c758f578c7.html |title=Signal flow analysis of feedback networks |journal=IEEE Transactions on Circuits and Systems |volume=37 |issue=4 |date=April 1990 |pages=455–463 |doi=10.1109/31.52748|bibcode=1990ITCS...37..455C }}
</ref>
:"In contrast to block diagram and two-port approaches to the feedback network analysis problem, signal flow methods mandate no ''a priori'' assumptions as to the unilateral or bilateral properties of the open loop and feedback subcircuits. Moreover, they are not predicated on mutually independent open loop and feedback subcircuit transfer functions, and they do not require that feedback be implemented only globally. Indeed signal flow techniques do not even require explicit identification of the open loop and feedback subcircuits. Signal flow thus removes the detriments pervasive of conventional feedback network analyses but additionally, it proves to be computationally efficient as well."

Following up on this suggestion, a signal-flow graph for a negative-feedback amplifier is shown in the figure, which is patterned after one by D'Amico ''et al.''.<ref name=Damico>
{{cite journal |title=Resistance of Feedback Amplifiers: A novel representation |author=Arnaldo D’Amico, Christian Falconi, Gianluca Giustolisi, Gaetano Palumbo |journal=IEEE Transactions on Circuits and Systems – II Express Briefs |url=http://piezonanodevices.uniroma2.it/wp-content/uploads/2013/04/Rosenstark.pdf |date=April 2007 |volume=54 |issue=4 |pages=298–302|doi=10.1109/TCSII.2006.889713 |citeseerx=10.1.1.694.8450 |s2cid=10154732 }}
</ref> Following these authors, the notation is as follows: 
:"Variables ''x''<sub>S</sub>, ''x''<sub>O</sub> represent the input and output signals, moreover, two other generic variables, ''x<sub>i</sub>, x<sub>j</sub>'' linked together through the control (or critical) parameter ''P'' are explicitly shown. Parameters ''a<sub>ij</sub>'' are the weight branches. Variables ''x<sub>i</sub>'', ''x<sub>j</sub>'' and the control parameter, ''P'', model a controlled generator, or the relation between voltage and current across two nodes of the circuit.

:The term ''a''<sub>11</sub> is the transfer function between the input and the output [after]  setting the control parameter, ''P'', to zero; term ''a''<sub>12</sub> is the transfer function between the output and the controlled variable ''x<sub>j</sub>''  [after]  setting the input source, ''x''<sub>S</sub>, to zero; term ''a''<sub>21</sub> represents the transfer function between the source variable and the inner variable, ''x<sub>i</sub>'' when the controlled variable  ''x<sub>j</sub>'' is set to zero (i.e., when the control parameter, ''P'' is set to zero); term ''a''<sub>22</sub> gives the relation between the independent and the controlled inner variables setting control parameter, ''P'' and input variable, ''x''<sub>S</sub>, to zero."

Using this graph, these authors derive the generalized gain expression in terms of the control parameter ''P'' that defines the controlled source relationship ''x<sub>j</sub>'' = ''Px<sub>i</sub>'':
:<math>x_\text{O} = a_{11} x_\text{S} + a_{12} x_j,</math>
:<math>x_i = a_{21} x_\text{S} + a_{22} x_j,</math>
:<math>x_j = P x_i.</math>

Combining these results, the gain is given by
:<math>\frac{x_\text{O}}{x_\text{S}} = a_{11} + \frac{a_{12} a_{21} P}{1 - P a_{22}}.</math>

To employ this formula, one has to identify a critical controlled source for the particular amplifier circuit in hand. For example, ''P'' could be the control parameter of one of the controlled sources in a [[two-port network]], as shown for a particular case in D'Amico ''et al.''<ref name=Damico/> As a different example, if we take ''a''<sub>12</sub> = ''a''<sub>21</sub> = 1, ''P'' = ''A'', ''a''<sub>22</sub> = –&beta; (negative feedback) and ''a''<sub>11</sub> = 0 (no feedforward), we regain the simple result with two unidirectional blocks.