Wersja elektroniczna artykułu - Politechnika Śląska, Wydział

ELEKTRYKA
Zeszyt 1 (229)
2014
Rok LX
Andrzej KUKIEŁKA
Politechnika Śląska w Gliwicach
DUAL SIMILARITY OF VOLTAGE TO CURRENT AND CURRENT TO
VOLTAGE TRANSFER FUNCTION OF HYBRID ACTIVE TWOPORTS 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
A. Kukiełka
76
1. INTRODUCTION
This article is a continuation of the works [3], [4] 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).
2. ORIGINAL CIRCUIT. PARALLEL-TO-SERIAL CONNECTION OF TWOPORTS
To simplify the calculation, it is assumed that the two-port networks have simple internal
structures (Fig. 1), analyzed in hybrid connection (Fig. 2).
3
1
2
1
two-port 1'
(passive) H '
A
2
'
R A
^
=
'
R A
3
4
4
2
1
1
two-port 2'
(active) H B'
3
^
=
'
I 1
R'
4
2
R'
B
B
k
3
'
'
I
UI 1
4
Fig. 1. Original circuit. Two-port networks are used with simple internal structures: passive and active
Rys. 1. Obwód oryginału. Dwa czwórniki o prostych strukturach wewnętrznych: pasywny i aktywny
1
two-port 1’
(original)
3
two-port 2’
(original)
4
2
Fig. 2. Hybrid (parallel-to-serial) connection of two-port networks from Fig.1
Rys. 2. Hybrydowe (równoległo-szeregowe) połączenie czwórników z rys.1
Dual similarity of voltage…
77
The hybrid matrix of analyzed two-port network is given by the formula:
 R 'A +R 'B

H'A  H'B   k 'UI
  '
 R B


1
1 


R 'A R 'B 
0
Equations (1) and (2) present the voltage-to-current M and current-to-voltage N transfer
functions of analyzed two-port networks [2], [3].
M' =
U'2
I1'
N =
'
H'
= '21
H 22
I'2
U1'
=
 k 'UI 
H'21
'
H11

1
R 'B
1
1
 '
'
RA RB
 k 'UI 
R 'A
R 'A  R 'B
k 'UI
R 'B (R 'A  R 'B )
(1)
(2)
3. 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 ([2] 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. 3).
This also changes the internal structures of each of the dual two-port networks (Fig. 4)
[2], [5].
A. Kukiełka
78
2
two-port 1’’
(dual)
4
1 3
two-port 2”
(dual)
Fig. 3. Dual circuit. Hybrid (serial-to-parallel) connection of two-port networks
Rys. 3. Obwód dualny. Hybrydowe (szeregowo-równoległe) połączenie czwórników
1
3
2
2
1
two-port 1"
(passive) H ''
A
''
RA
^
=
4
3
4
2
1
two-port 2"
(active)
''
RA
1
2
U1
^
=
H B''
3
4
3
k ''IUU1
R''
B
R''
B
4
Fig. 4. Similar circuit [3]. Dual equivalents of two-port networks from Fig. 1
Rys. 4. Obwód podobny [3]. Dualne odpowiedniki czwórników z rys. 1
Common matrix of hybrid two-port networks in serial-to-parallel connection is specified by
formula:
 R ''A +R ''B

H ''A  H ''B   "
k  R"
 IU B



1
1 

R ''A R ''B 
0
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 [3],
indicated in the formulas (3) and (4) 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. (3) and Eq. (4).
M'' = 
H''21
H''22
HFOR


H''21 / ΔH''
''
H11
/ ΔH''

k"IU  R "A  (R "B ) 2
R ''A  R ''B
(3)
Dual similarity of voltage…
79
N'' = 
H''21
''
H11
HFOR


H''21 / ΔH''

H''22 / ΔH''
k"IU  R "B
R ''A  R ''B
(4)
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.
4. DUAL CIRCUIT WITH CONVERSION
If branches of circuit are associated with impedance conversion A [6], then
Z'k Z"k  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 [1] should be defined,
described by formula (5).
k 'UI  k"IU  A
( k"IU  A k 'UI )
(5)
After the appropriate substitutions to formulas (3) and (4), equations (6) and (7) are
obtained.
M'' =
conversion

M ''
 
i N
N '' =
conversion


'

V 
V 
 A 
 A
    1
  A   V 
i     
 V    A 
1 1
1


A k 'UI R 'A
N''
 
i M
'
 
(6)
 
(7)
1 k 'UI
1

 1 A  i N'
A 1 R 'B (1 R 'A  1 R 'B )
(
1
R 'A

1
A
1
R 'B
 A
 A
V 
V 
    1
 V    A 
i     
  A   V 
)
1
A
1
 i M'
A
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
1. Chojcan J.: Zasady podobieństwa obwodów z uwzględnieniem źródeł sterowanych.
IC-SPETO 2000, Ustroń, s. 241-245.
2. Bolkowski S.: Teoria obwodów elektrycznych. WNT, Warszawa 2012.
3. Kukiełka A.: Dual Hybrid Matrix Coefficients – Voltage and Current Transfer Functions,
IC-SPETO 2009, Gliwice-Ustroń, p. 63-64.
4. Kukiełka A.: Dual Similarity of Hybrid Connections of Active Two-Ports, IC-SPETO
2012, Gliwice-Ustroń 2012, p. 53-54.
5. Kukiełka A.: Dual Similarity with Impedance Inversion of Overall Transfer Functions
M and N of Active Two-Ports, IC-SPETO 2013, Gliwice-Ustroń, p. 59-60.
6. Zagajewski T.: Ogólne zasady podobieństwa obwodów elektrycznych. „Arch. Elektr.”
1973, Vol. 22, s. 427-438.
Dr inż. Andrzej Kukiełka
Politechnika Śląska, Wydział Automatyki, Elektroniki i Informatyki
Instytut Elektroniki
Akademicka 16
44-100 Gliwice
e-mail: [email protected]