ELEKTRYKA 0 Zeszyt (9) Rok LX Andrzej KUKIEŁKA Politechnika Śląska w Gliwicach DUAL SIMILARITY OF VOLTAGE TO CURRENT AND CURRENT TO VOLTAGE TRANSFER FUNCTION OF HYBRID ACTIVE TWO- PORTS WITH CONVERSION Summary. Dual active two-port networks in hybrid connection are discussed and their voltage to current (M) and current to voltage (N) transfer functions for parallel to serial (and serial to parallel) connections are determined. Relations between transformation of their internal structures and kinds of two-port network connections are analyzed. Appropriate use of: dual constant of conversion of internal gain, impedance inversion and overall transfer functions M and N allows to extend the theory of General Principles of Similarity of Electric Networks of Professor Tadeusz Zagajewski. Keywords: dual similarity, passive two-port networks, active two-port networks, impedance conversion, dual constant of conversion of internal again, overall transfer functions, voltage-to-current M transfer function, current-to-voltage N transfer function PODOBIEŃSTWO TRANSMITANCJI NAPIĘCIOWO-PRĄDOWYCH I PRĄDOWO-NAPIĘCIOWYCH DUALNYCH HYBRYDOWYCH CZWÓRNIKÓW AKTYWNYCH Z KONWERSJĄ Streszczenie. Analizie poddano połączone hybrydowo czwórniki (aktywne i pasywne) oraz określono ich transmitancje przejściowe (napięciowo-prądowe) M i N (prądowo-napięciowe). Obliczenia przeprowadzono dla odpowiednich połączeń tych czwórników: szeregowo-równoległych i równoległo-szeregowych. Dzięki wykorzystaniu transformacji dualnej z konwersją impedancji oraz konwersji wzmocnień wewnętrznych czwórników aktywnych wykazano, że transmitancje przejściowe zachowują się analogicznie do transmitancji napięciowych i prądowych dwójników pasywnych z konwersją impedancji dla obwodów zgodnych z definicją podobieństwa dualnego typu Ib. Wyprowadzone wzory rozszerzają teorię opracowaną przez Profesora Tadeusza Zagajewskiego. Słowa kluczowe: podobieństwo dualne, czwórniki pasywne, czwórniki aktywne, impedancja konwersji, stała dualna konwersji wzmocnień wewnętrznych, transmitancje przejściowe, transformacja napięciowo-prądowa M, transmitancja prądowo-prądowa N
76 A. Kukiełka. INTRODUCTION This article is a continuation of the works [], [] and [5]. The purpose of this article is to extend the general principles of similarity of electric networks [6] (derived by Professor Tadeusz Zagajewski) to appropriate overall transfer functions (voltage-to-current M and current-to-voltage N).. ORIGINAL CIRCUIT. PARALLEL-TO-SERIAL CONNECTION OF TWO- PORTS To simplify the calculation, it is assumed that the two-port networks have simple internal structures (Fig. ), analyzed in hybrid connection (Fig. ). two-port (passive) H A = ^ R A R A two-port (active) H B = ^ I R B R B k I UI Fig.. Original circuit. Two-port networks are used with simple internal structures: passive and active Rys.. Obwód oryginału. Dwa czwórniki o prostych strukturach wewnętrznych: pasywny i aktywny two-port (original) two-port (original) Fig.. Hybrid (parallel-to-serial) connection of two-port networks from Fig. Rys.. Hybrydowe (równoległo-szeregowe) połączenie czwórników z rys.
Dual similarity of voltage 77 The hybrid matrix of analyzed two-port network is given by the formula: R A+RB 0 UI RB RA RB HA HB k Equations () and () present the voltage-to-current M and current-to-voltage N transfer functions of analyzed two-port networks [], []. U H RB RA UI UI I H RA RB RA RB M = = k k H kui B A B I N = = U H R (R R ) () (). DUAL CIRCUIT. SERIAL TO PARALLEL CONNECTION OF TWO PORT NETWORKS Two planar networks N and N" are structurally dual if each node A consisting of n branches of first network corresponds to a loop a" of the second network constituted of n branches. Similarly, each loop b of first network corresponds to a node B" of second network; to each element k of the first network belonging to a node A corresponds to an element k" of the second network belonging to loop a". To each element k of the first network belonging to a loop a corresponds to an element k" of the second network belonging to node A" [6]. In addition to the similarity of the structures of the first and second circuit also determines the relationship between the impedances [6] (in the presented case - the constant of conversion). In dual circuit ([] and [5]) the way of connection is changing: parallel-to-serial connection of the original two-port circuit is transformed into a serial-to-parallel connection (Fig. ). [], [5]. This also changes the internal structures of each of the dual two-port networks (Fig. )
78 A. Kukiełka two-port (dual) two-port (dual) Fig.. Dual circuit. Hybrid (serial-to-parallel) connection of two-port networks Rys.. Obwód dualny. Hybrydowe (szeregowo-równoległe) połączenie czwórników two-port " (passive) H A = ^ R A R A two-port " (active) H B = ^ U R B k IU U R B Fig.. Similar circuit []. Dual equivalents of two-port networks from Fig. Rys.. Obwód podobny []. Dualne odpowiedniki czwórników z rys. Common matrix of hybrid two-port networks in serial-to-parallel connection is specified by formula: R A+RB 0 HA HB " " kiu RB RA R B Due to the choice of a simple two-port network (passive two-port network has zero values in the matrix outside the main diagonal), a hybrid form of the inverted matrix [], indicated in the formulas () and () as "transformation HFOR (hybrid form of reverse) must be used. In this case, transfer functions of two-port networks: voltage-to-current M and current-to-voltage N in dual circuit assume the form shown in Eq. () and Eq. (). " " " H HFOR H / ΔH kiu R A (R B) A B M= H H / ΔH R R ()
Dual similarity of voltage 79 " " H HFOR H / ΔH kiu RB A B N= H H / ΔH R R () In addition to the similarity of the structures of the first and second circuit, the relationship between the impedances of corresponding elements should be defined.. DUAL CIRCUIT WITH CONVERSION If branches of circuit are associated with impedance conversion A [6], then " k k Z Z A [6] ( A is constant, and is impedance conversion of passive elements, Z k and Z k are impedances of k -element of the original and dual networks. Because one of two-port networks is active, dual constant of conversion of internal gain [] should be defined, described by formula (5). " UI IU k k A ( k " A k ) (5) After the appropriate substitutions to formulas () and (), equations (6) and (7) are obtained. IU conversion kui i (6) A R B ( RA R B) M = A N V V M A A in A V A i V A conversion i (7) A k A UI RA RA RB N = ( ) M A A N V V im V A A i A V UI
80 A. Kukiełka 5. CONCLUSIONS The article proved that the general principles of similarity of electric networks also take effects for overall transfer functions M and N of dual active two-port networks with constant of conversion A (similarity type Ib) [6]. Use of: dual constant of conversion of internal gain, impedance inversion and overall transfer functions M and N allows to extend the theory of General Principles of Similarity of Electric Networks of Professor Zagajewski. BIBLIOGRAPHY. Chojcan J.: Zasady podobieństwa obwodów z uwzględnieniem źródeł sterowanych. IC-SPETO 000, Ustroń, s. -5.. Bolkowski S.: Teoria obwodów elektrycznych. WNT, Warszawa 0.. Kukiełka A.: Dual Hybrid Matrix Coefficients Voltage and Current Transfer Functions, IC-SPETO 009, Gliwice-Ustroń, p. 6-6.. Kukiełka A.: Dual Similarity of Hybrid Connections of Active Two-Ports, IC-SPETO 0, Gliwice-Ustroń 0, p. 5-5. 5. Kukiełka A.: Dual Similarity with Impedance Inversion of Overall Transfer Functions M and N of Active Two-Ports, IC-SPETO 0, Gliwice-Ustroń, p. 59-60. 6. Zagajewski T.: Ogólne zasady podobieństwa obwodów elektrycznych. Arch. Elektr. 97, Vol., s. 7-8. Dr inż. Andrzej Kukiełka Politechnika Śląska, Wydział Automatyki, Elektroniki i Informatyki Instytut Elektroniki Akademicka 6-00 Gliwice e-mail: andrzej.kukielka@polsl.pl