{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T00:34:10Z","timestamp":1760229250279,"version":"build-2065373602"},"reference-count":15,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2022,6,6]],"date-time":"2022-06-06T00:00:00Z","timestamp":1654473600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Sciences and Engineering Research Council of Canada","award":["RGPIN-2017-03730"],"award-info":[{"award-number":["RGPIN-2017-03730"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>It is common practice to model the input\u2013output behavior of a single-loop feedback circuit using the two parameters, A and \u03b2. Such an approach was first proposed by Black to explain the advantages and disadvantages of negative feedback. Extensive theories of system behavior (e.g., stability, impedance control) have since been developed by mathematicians and\/or control engineers centered around these two parameters. Circuit engineers rely on these insights to optimize the dynamic behavior of their circuits. Unfortunately, no method exists for uniquely identifying A or \u03b2 in terms of the components of the circuit. Rather, indirect methods, such as the injection method of Middlebrook or the break-the-loop approach proposed by Rosenstark, compute the return ratio RR of the feedback loop and inferred the parameters A and \u03b2. While one often assumes that the zeros of (1 + RR) are equal to the zeros of (1 + A \u00d7 \u03b2), i.e., the closed-loop poles are equivalent, this is not true in general. It is the objective of this paper to present an exact method to uniquely identify each feedback parameter, A or \u03b2, in terms of the circuit components. Further, this paper will identify the circuit conditions for which the product of A \u00d7 \u03b2 leads to the correct closed-loop poles.<\/jats:p>","DOI":"10.3390\/s22114303","type":"journal-article","created":{"date-parts":[[2022,6,7]],"date-time":"2022-06-07T00:10:33Z","timestamp":1654560633000},"page":"4303","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Identifying A(s) and \u03b2(s) in Single-Loop Feedback Circuits Using the Intermediate Transfer Function Approach"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4880-0272","authenticated-orcid":false,"given":"Gordon Walter","family":"Roberts","sequence":"first","affiliation":[{"name":"Integrated Microsystems Laboratory, Department of Electrical and Computer Engineering, McGill University, Montreal, QC H3A 0G4, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,6,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1002\/j.1538-7305.1934.tb00652.x","article-title":"Stabilized Feedback Amplifiers","volume":"13","author":"Black","year":"1934","journal-title":"Bell Syst. Tech. J. Jan."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"485","DOI":"10.1080\/00207217508920421","article-title":"Loop Gain in Feedback Systems 1","volume":"38","author":"Middlebrook","year":"1975","journal-title":"Int. J. Electron."},{"key":"ref_3","unstructured":"Rosenstark, S. (1986). Feedback Amplifier Principles, Macmillan Publishing Company."},{"key":"ref_4","unstructured":"Sedra, A., and Smith, K. (2014). Microelectronic Circuits, Oxford University Press. [7th ed.]."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1991","DOI":"10.1109\/31.99170","article-title":"Exact Simulation of Feedback Circuit Parameters","volume":"38","author":"Hurst","year":"1991","journal-title":"IEEE Trans. Circuits Syst."},{"key":"ref_6","unstructured":"Tuinenga, P.W. (1995). SPICE: A Guide to Circuit Simulation and Analysis Using PSpice, Prentice-Hall. [3rd ed.]."},{"key":"ref_7","unstructured":"Roberts, G.W., and Sedra, A.S. (1997). SPICE, Oxford University Press. [2nd ed.]."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1045","DOI":"10.1109\/TCSI.2002.801256","article-title":"A Loop-Breaking Method for the Analysis and Simulation of Feedback Amplifiers","volume":"49","author":"Russel","year":"2002","journal-title":"IEEE Trans. Circuits Syst. I Fundam. Theory Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1109\/101.900125","article-title":"Striving for Small-Signal Stability: Loop-Based and Device-Based Algorithms for Stability Analysis of Linear Circuits in the Frequency Domain","volume":"17","author":"Tian","year":"2001","journal-title":"Circuit Devices Mag."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"625","DOI":"10.1109\/TCSI.2014.2370151","article-title":"Comparative analysis of simulation-based methods for deriving the phase-and gain-margins of feedback circuits with op-amps","volume":"62","author":"Neag","year":"2015","journal-title":"IEEE Trans. Circuits Syst."},{"key":"ref_11","unstructured":"Bode, H.W. (1945). Network Analysis and Feedback Amplifier Design, Van Nostrand."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Roberts, G.W. (2021, January 22\u201328). Single-Loop Feedback Parameter Extraction Method for Stability Analysis of Electronic Circuits. Proceedings of the 2021 IEEE International Symposium on Circuits and Systems (ISCAS), Daegu, Korea.","DOI":"10.1109\/ISCAS51556.2021.9401597"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1109\/TCS.1986.1085910","article-title":"Synthesis and analysis of state-space active filters using intermediate transfer functions","volume":"33","author":"Snelgrove","year":"1985","journal-title":"IEEE Trans. Circuits Syst."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"126","DOI":"10.1002\/j.1538-7305.1932.tb02344.x","article-title":"Regeneration Theory","volume":"11","author":"Nyquist","year":"1932","journal-title":"Bell Syst. Tech. J."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Roberts, G.W. (2021, January 13\u201316). A Modified Nyquist Stability Criteria that Takes into Account Input\/Output Circuit Loading Effects. Proceedings of the 2021 19th IEEE International New Circuits and Systems Conference (NEWCAS), Toulon, France.","DOI":"10.1109\/NEWCAS50681.2021.9462771"}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/11\/4303\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:25:01Z","timestamp":1760138701000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/11\/4303"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,6,6]]},"references-count":15,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2022,6]]}},"alternative-id":["s22114303"],"URL":"https:\/\/doi.org\/10.3390\/s22114303","relation":{},"ISSN":["1424-8220"],"issn-type":[{"type":"electronic","value":"1424-8220"}],"subject":[],"published":{"date-parts":[[2022,6,6]]}}}