SOME THEORETICAL ASPECTS OP IMMITTANCE CONVERSION AND INVERSION IN THE CONTEXT OF ACTIVE RC NETWORKS by Manuel de Medeiros Silva A Thesis submitted for the Degree of Doctor of Philosophy of the University of London September 1976 Electrical Engineering Department Imperial College of Science and Technology London S.W.7 - 2 - ABSTRACT Two-port immittance (i.e, impedance or admittance) converters and inverters are important active units used, in signal processing applica- tions, for the realization of active RC networks suitable for micro,. electronic implementation. In this thesis a generalization to multiports of the concepts of 2-port converters and inverters is proposed, the special case of 3-port converters and inverters suitable for floating inductor simulation is studied in some detail, the realization of inverters with a minimum number of active components is discussed, and the concept of anti-reciprocity (which has close associations with converters and inverters) is investigated. According to the proposed generalization, conversion and inversion, which are scalar operations when performed by 2-ports, become matrix operations, when performed by multiports. One consequence is that a distinction, which does not exist for 2-ports, has to be made between multiport admittance and impedance converters or inverters. Most of the circuits used for the active RC simulation of floating inductors can be interpreted as 3-port admittance inverters or converters. A classification of these inverters and converters, based on their terminal description (as distinct from their port description), is developed, which clarifies the relationships between known realizations and shows the possibility of new ones. It is known that 2-port positive inverters containing resistors and only one operational amplifier can be realized if only one port is grounded. It is proved that this is impossible if both ports are grounded (a similar restriction concerning positive inverters using only one ideal - 3 - transistor is also proved). The consequences of this conclusion regarding the realization of 3-port inverters are examined. Some converters and inverters (e.g. the gyrator) have been regarded as possessing a kind of extreme non-reciprocity or "anti-reciprocity". It is pointed out that two existing definitions of anti-reciprocity are incor.patible. An investigation of this question leads to the introduction of a "complete set" of network properties related to the concept of reciprocity. Various rules concerning the interconnection of multiports possessing these• properties are derived. A generalization of the definitions of reciprocity and anti-reciprocity is presented, and the consequences of the proposed generalization are evaluated. 4 ACKNOWLEDGEMENTS I wish to express my deepest gratitude to Dr. W. Saraga for his advice and encouragement throughout every stage of the work reported in this thesis. Dr. Saraga has always been, not only an extremely kind and helpful supervisor, but also a very good friend. His contagious enthusiasm, patient guidance and friendly encouragement had a decisive influence in the work that led to the present thesis. The work reported here has benefited from many helpful discussions with various members of the Communication Section. Particular mention is due to Mr. D. Haigh, Dr. A. G. Constantinides, Dr. R. Spence and Mr. D. Perry. The encouragement provided by my colleagues and friends, especially Mr. R. W. Turpin, M. Ahmadi, M. Buckley, R. Galv5n, K. Karna, D. Horn and N. Esteves is gratefully acknowledged. I would like to express my gratitude to my colleague Mr. H. Abreu Santos who has been my representative in Lisbon during my absence, and to Dr. F. Borges da Silva who first stimulated my interest for Network Theory. The work reported in this thesis was done while the author was on leave from Instituto Superior Tecnico Lisbon, supported by a grant from INVOTAN. -5 TABLE OF CONTENTS TITLE PAGE ............................ OOOOOOOOOOO oo000000.0000000 p. 1 ABSTRACT OOOOO 0 OOOOOO 0000000000000000000000000 p. 2 ACKNOWLEDGEMENTS 0000 OOOOOO o0000000000000000000000000000.0.000000 P. 4 TABLE OF CONTENTS 0.0.0 0000000000000000000015000000000000000000000 p. 5 NOTATION 00000000000000000000000000000000000000000000000000000000 p. lo LIST OF FIGURES 0000000000000000000000000000000000000000000000000 p. 12 LIST OF TABLES .0 O OOOOO 0000000000000000000000000000000 OOOOOO 00000 p. 17 CHAPTER 1 - INTRODUCTION 1.1 - IMMITTANCE CONVERTERS AND INVERTERS - GENERAL CONSIDERATIONS p. 18 1.2 - OUTLINE OF THE THESIS 0000. O ......... p. 20 1.3 - STATEMENT OF ORIGINALITY p. 21 CHAPTER 2 - REVIEW OF SOME BASIC CONCEPTS 2.1 - MULTIPORTS Definition .............. 00000000000 .......... p. 23 General description of n-ports OOOOOO ................ p. 25 Hybrid descriptions . .......... p. 28 Well-defined multiports O 0000000000000000000 p. 29 Scattering description 0 OOOOOOO 000000000000000000000 p. 31 Separation of the ports into two sets: (n+k)-ports p. 35 2.2 - NULLORS Nullators, norators and nullors ..................... p. 39 Ideal transistors and ideal operational amplifiers 0.p. 45 Node analysis of networks containing nullors ........P• 47 -6 203 — RESISTIVE MULTIPORTS OOOOOOOOOO 0 OOOOOO 00000 OOOOOO 000000 P., 50 2.4 — ACTIVITY p. 52 2.5 — INTERRECIPROCITY Interreciprocal multiports OOO .......... P. 54 Interreciprocity of well—defined multiports OOOOO p. 57 The interreciprocity theorem OOOOO 00000000 opeoct000 p. 59 Reciprocal multiports ....... 0000000 0000000 ......... P. 60 CHAPTER 3 — REVIEW OF 2—PORT IMMITTANCE CONVERSION AND INVERSION IN THE CONTEXT OF ACTIVE RC NETWORKS 3.1 — ACTIVE RC NETWORKS 000000 o p. 63 3.2 — SIMULATION OF DOUBLY TERMINATED LC LADDER FILTERS 0000 P. 67 3.3 — TWO—PORT CONVERTERS AND INVERTERS P• 73 3.4 — ACTIVE REALIZATION OF CONVERTERS AND INVERTERS P• 80 3.5 — SIMULATION OF GROUNDED AND FLOATING INDUCTORS p. 88 3.6 — THE (ad, bc)—PLANE p. 96 CHAPTER 4 — MULTIPORT CONVERTERS AND INVERTERS 4.1 — INTRODUCTION ............... 0000000......... p. 99 4.2 — DEFINITIONS . .............. 00000000000000 ............. P. 101 4.3 — MATRIX DESCRIPTION 00000000 0 0 0000000000000 p• 104 4.4 — THE DIFFERENCE BETWEEN 'ADMITTANCE' AND 'IMPEDANCE* CONVERTERS AND INVERTERS 00000000000 0000000 00000000000 p. 108 4.5 — SOME PROPERTIES OF MULTIPORT CONVERTERS AND INVERTERS p. 111 4.6 — MULTIPORT TRANSFORMERS AND GYRATORS 0. 0000000 P• 113 4.7 — FLOATING INDUCTOR SIMULATION p. 115 4.8 — HYBRID CONVERTERS 0000000000000 00, 0000000000 P. 120 7 409 - CONCLUSIONS ........ ...........0.00000.0.0.0.000 P- 122 APPENDIX 0000000000.00000000....000000000000000000000 P. 124 CHAPTER 5 - REALIZATION OF MULTIPORT CONVERTERS AND INVERTERS USING 2-PORT CONVERTERS AND INVERTERS 5 01 - INTRODUCTION 0000000000000000000000000000000000000000 P. 125 5.2 - A SIMPLIFIED REPRESENTATION OF THE INTERCONNECTION OF MULTIPORTS 0000000.000000000000000 P- 126 5.3 - REALIZATION OF WELL-DEFINED MULTIPORTS USING • 2-PORTS p. 130 5.4 - REALIZATION:OF MULTIPORT CONVERTERS AND INVERTERS p. 136 5.5 - MULTIPORT CONVERTERS AND INVERTERS USING GROUNDED 2-PORT CONVERTERS AND INVERTERS p. 141 5.6 - CONCLUSIONS .0.00.0. OOOOOOOOOOOOOO 0000 OOOOO 0.000 OOOOO 144 CHAPTER 6 - A CLASSIFICATION OF 3-PORT INVERTERS AND CONVERTERS USED IN THE SIMULATION OF FLOATING INDUCTORS 6.1 - INTRODUCTION 000000000000 OOOOOOO .000000000 OOOOOO 00000 p. 149 6.2 - TERMINAL DESCRIPTION OF 3 -PORT INVERTERS FOR FLOATING INDUCTOR SIMULATION O P. 152 6.3 - FOUR-TERMINAL INVERTERS .0.0 OOOOOOO .............0.00. p. 156 6.4 - FLOATING INVERTERS p. 158 6.5 - FLOATING 3-TERMINAL INVERTER OOOOOOOOOO 0.0000.0000 p. 162 p. 161+ 6.6 - THE '.GENERAL' INVERTER .00000000000000000000000000000 6.7 - THREE-PORT CONVERTERS FOR FLOATING INDUCTOR SIMULATION OOOOOO ..0.00.000.000.000.00.00.00.00.00000 p. 169 6.8 - A FAMILY OF 3 -PORT INVERTERS USING 3 OPERATIONAL AMPLIFIERS 00900000000000 0000000000000000000000000000 p. 174 6.9 - CONCLUSIONS 000000000000000 00000000000000000000000000 p. 181 8 CHAPTER 7 - INVERTERS WITH A MINIMUM NUMBER OF ACTIVE COMPONENTS 7.1 - INTRODUCTION P. 184 7.2 - A REVIEW OF POSITIVE INVERTERS WITH A MINIMUM NUMBER OF ACTIVE COMPONENTS .00...0... P• 186 7.3 - THE ADMITTANCE PARAMETERS OF A GROUNDED 2-PORT CONTAINING RESISTORS AND ONE OPERATIONAL AMPLIFIER 000000 OOOOOOO 0000000000000 OOOOO P. 195 7.4 - ON THE REALIZATION OF VARIOUS TYPES OF INVERTERS USING OPERATIONAL AMPLIFIERS ID. 204 705 - ON THE REALIZATION OF POSITIVE INVERTERS USING TRANSISTORS Proof of the non-realizability of grounded positive inverters with one transistor p. 207 An unsolved question p. 213 7.6 - CONCLUSIONS 0 00 0. 0000 .. ... 0 0 0 0 0 0 0 0 0 .. 0...0000 0 000000 P. 216 CHAPTER 8 - A SET OF NETWORK PROPERTIES RELATED TO THE CONCEPT OF RECIPROCITY 8.1 - INTRODUCTION 00 OOOOOOOO 0000 OOOOOOOOOOO 000000000000000 P. 218 8.2 - RECIPROCITY p. 220 8.3 - TWO TYPES OF ANTI-RECIPROCITY ..... p. 227 8.4 - BIRECIPROCITY p. 234 8.5 - ANTI-BIRECIPROCITY p. 236 8.6 - THE MEANING OF BIRECIPROCITY AND ANTI-BIRECIPROCITY OOOOOOOOOO 000000 OOOOO 000 OOOOO P. 238 8.7 - CASCADE CONNECTION OF VARIOUS TYPES OF MULTIPORTS P. 241 8.8 - MULTIPORT TRANSFORMERS AND GYRATORS p. 21+1+ 8.9 - THE DETERMINANT OF THE TRANSMISSION MATRIX ........... p. 247 9 8.10 - RELATIONSHIPS WITH PASSIVITY AND ACTIVITY .......... P• 248 8.11 - CONCLUSIONS ........... 00000000 000000000000060000000 P. 2+9 CHAPTER 9 - A GENERALIZATION OF RECIPROCITY AND ANTI-RBCIPROCITY 9.1 - INTRODUCTION ....................................... p. 251 9.2 - GENERALIZED RECIPROCITY