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Sensitivity Analysis and Sensitivity Invariants of Similar Networks of Active Two-Ports in Cascade Connections
ELSEVIER
Copyright © IF AC Programmable Devices and Systems, Ostrava, Czech Republic, 2003
IFAC PUBLICATIONS www.elsevier.comllocatelifac
SENSITIVITY ANALYSIS AND SENSITIVITY INVARIANTS OF SIMILAR NETWORKS OF ACTIVE TWO-PORTS IN CASCADE CONNECTIONS
Andrzej Kukielka*
* Silesian
University o/Technology, Institute 0/ Electronics. Gliwice
Two-ports in cascade connections are considered. The transfer and sensitivity properties of similar two-ports are considered. The transfonn of connections for passive and active two-ports and changes of their internal structures have been noticed. Two types of similarity and their sensitivities are given. Sensitivities are considered with respect to corresponding elements of networks or with respect to parameters of two-ports. For these sensitivity analysis, sensitivity invariants of networks function of similar networks with two-ports are derived. Sensitivity invariants of networks with two-ports (in cascade connections) can be used to test and find fault in electronic networks with active twoports. Copyright © 2003 IFAC Keywords: sensitivity analysis, active elements, transfer functions, fault detection, active two-ports, sensitivity invariants, impedances inversion and conversion, dual networks, affined networks, passive two-ports.
1. INTRODUCTION
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, to each loop b' of first network corresponds a node B" of second network; to each element k' of the first network belonging to a node A' corresponds an element k" of the second network belonging to loop a". One can find a similarity between the node equations of the second network. Assuming the relations of the second immitances of elements k' and k" in papers (Zagajewski, 1972a,b) relations between currents, voltages and transmittances of two dual networks have been found.
For two cascaded two-ports (active and passive) matrix A is obtained by multiplying the matrices of the single two-ports. For cascade connection, the transfer functions are we given in the fonn (Chojca~, 2001):
1
Ku
K =_1_= I
1
= A11 = A(l). A(2) + A(l) . A(2) 11 11 12 21 A
22
A (1)
1
• .4 (2)
1 2 ' -'21
+ A22(1) • A22(2)
(1)
(2)
top index (1) indicates first two-port (passive), top index (2) indicates second two-port (active). and
According to above considerations, if the original network N' is cascade connections of two-ports, dual network N" is also a cascade connection of two-ports (table 1.1). The transfer functions of these networks are (4):
(3)
445
K~
and
K~
1 - A;'I
In consideration of the cascade connection of twoports, also dual network is cascade connection of two-ports (table 1.1). For these case cascade matrix can be used for evaluation of sensitivities.
(4)
Similarly as in (Zagajewski, 1972a,b) let resume two relations between immitances of elements k' and k", namely impedance inversion and conversion. For these relations, we shall derive the relations between sensitivities of similar networks.
Sensitivity invariants will be presented in table 1.2 and 1.3. Some sensitivities are: constant and - it's value is -1 or equal (tables 1.2 and 1.3).
2. DUAL INVERSION OF INSIDE INTENSIFY Inversion of inside intensify of corresponding dual active two-ports is (Chojcan, Karwan, Kukielka, 2000):
TABLE 1.1. CASCADE CONNECTION OF TWO TWO-PORTS (ACTIVE AND PASSIVE) RELATIONS BETWEEN SIMILAR NETWORK AND ORIGINAL NETWORK
(5) KIND OF SIMILARITY
where k~ is the inside intensify of the active element of the first network (original) and k~ is the inside intensify of the active element of the second network (dual). Impedances inversion between impedances of corresponding elements of branch k' and k" in the form are (Zagajewski, 1972a,b) - only for passive elements:
DUAL WITH DUAL WITH IMPEDANCES IMPEDANCES INVERSION CONVERSION (TYPE la) (TYPE Ib)
KIND OF TWO-PORTS CONNECTION
CASCADE
CASCADE
INSIDE STRUCTURES ANALYZED TWO-PORTS
DUAL
DUAL
(6) Where Z~ is the impedance inversion of passive elements, Z~ is the impedance of element of the
TABLE 1.2. CASCADE CONNECTION OF TWO TWO-PORTS - DUAL SIMILARITY
first network (original), Z~ is the impedance of k" element of the second network (dual).
Sensitivity of voltage transfer function (change of ,,k" parameter) Relations between appropriate two-ports parameters of dual and original circuit
3.DUAL CONVERSION BETWEEN INSIDE INTENSIFY
Ib
la Conversion between inside intensify of dual active two-ports of corresponding active elements of networks N' and Nil is (Chojcan, Karwan, Kukielka 2000):
S
K~ (Z ~ ,k ~)
=s
K~ A
;1 ·S
~
S
(7) Impedances conversion between passive elements of networks N' and Nil are (Zagajewski, 1972a,b):
K; (Z ~ ,k ~I)
=S
S AK~;1
(8)
SA;.I . =s A~ (Zkokru)
A;I (Z ~ ,k ;u)
~ K; A ~2
·S
A ~2 (Z ~ ,k ~d
S K; =-1
• (Zk,kud
A~2
~;! . =i(S A~ (Zk.kru)
. ) (Zk.kU)
Where A is the inversion of impedance passive elements, Z~ is the impedance of element of the fIrst network (original), Z~ is the impedance of k" element of the second network (dual). A is constant
TABLE 1.3. CASCADE CONNECTION OF TWO TWO-PORTS - DUAL SIMILARITY 446
Sensitivity of current transfer function (change of .,k" parameter) Relations between appropriate two-ports parameters of dual and original circuit
la
S
Z~
In consideration of the cascade connection of twoports, also affined network is cascade connection of two-ports (table 1.4). For these case cascade matrix can be used for evaluation of sensitivities.
Ib
K~ (Z ~ ,k ;u)
=s
K~
A;,
II
s K;
(Z ~ ,k ~,)
=s
K~ S A;,
SA ;} " = S A~2 (Z k ,k ru)
Z~=A
•
A;, (Z ~ ,k ~)
II
Sensitivity invariants will be presented in table 1.5 and 1.6.
A~2 A~2 ·S (Z ~ ,k ~,)
K;
S
(Z k .k UI )
-s
Some sensitivities are: constant and - it's value is -1 or equal (tables 1.2 and 1.3).
K; A
=-1 ~2
TABLE 1.4. CASCADE CONNECTION OF TWO TWO-PORTS (ACTIVE AND PASSIVE)RELATIONS BETWEEN SIMILAR NETWORK AND ORIGINAL NETWORK
~;! " =i(S A~ . ) (Zk,kru)
(Zk.k Ul )
KIND OF SIMILARITY
4. AFFINED INVERSION OF INSIDE INTENSIFY Inversion of inside intensify of corresponding affined active two-ports is (Chojcan, Karwan, Kukielka, 2000):
k 'UI . k"UI -- Zi2
AFFINED AFFINED WITH WITH IMPEDANCES IMPEDANCES INVERSION CONVERSION (TYPE la) (TYPE Ib)
KIND OF TWO-PORTS CONNECTION
CASCADE
CASCADE
AFFINED
AFFINED
(9) INSIDE STRUCTURES ANALYZED TWO-PORTS
where k~ is the inside intensify of the active element of the first network (original) and k~ is the inside intensify of the active element of the second network (affined).
TABLE 1.5. CASCADE CONNECTION OF TWO TWO-PORTS - AFFINED SIMILARITY
Impedances inversion between impedances of corresponding elements of branch k' and k" in the form are (Zagajewski, 1972a,b) - only for passive elements:
Sensitivity of voltage transfer function (change of ,,k" parameter) Relations between appropriate two-ports parameters of dual and original circuit
Ib
la 5.AFFINED CONVERSION BETWEEN INSIDE INTENSIFY
K~ S (ZK~~ ,k ~d =S A;, ·S (ZA;,~ ,k ~d
Conversion between inside intensify of affined active two-ports of corresponding active elements of networks N' and Nil is (Chojcan, Karwan, Kukielka 2000):
II
II
K~ S (ZK~~ ,k ~d =S A;, ·S (ZA;,~ ,k ~,)
(10)
K~ S A;\
K~ S A;\ =-1
Where A is constant.
~;l
Impedance conversion between passive elements of networks N' and Nil is (Zagajewski, 1972a,b):
-i(S A;l
(Z~,k~) -
447
(Z~,k~)
)
~;! . =S A~ (Zk.k ru )
. (Zk,ku)
TABLE 1.6 CASCADE CONNECTION OF TWO TWO-PORTS - AFFINED SIMILARITY Sensitivity of current transfer function (change of .,k" parameter) Relations between appropriate two-ports parameters of dual and original circuit la
In paper (Li & Woo 1999) we can find some relations with this theory and it's potential application to fault detection in electronic circuits.
REFERENCES Chojcan J., Karwan L., Kukielka A: Zasady podobieilstwa obwod6w z uwzgll(dnieniem :h6del sterowanych. XXIII SPETO, Ustron 2000, ss. 241-241. Chojcan J. przy wsp6lpracy Karwana L., Drygajly A i Kolmera A: Zbi6r zadan z teorii obwod6w I, wydanie VII, Wydawnictwo Politechniki Slllskiej, nr 2264, Gliwice 2001. Li F., Woo P-G.: The Invariance of Node-Voltage Sensitivity Sequence and Its Application in a Unified Fault Detection Dictionary Method, IEEE Transactions on Circuits & Systems - I, vol.46, no.lO, October 1999, pp. 1222-1226. Zagajewski T.: General principles of similarity of electric networks. Bull. Acad. Po Ion. Sci., Ser. Sci. Techn. 20 (1972). Zagajewski T.: Some applications of the general principle of similarity of electric networks. Bull. Acad. Polon. Sci., Ser. Sci. Techn. 20 (1972).
Ib
s (ZK;~ ,k ~I =s )
K~ A ;2
·S
A ;2 (Z ~ ,k ~I)
n n
s (ZK;~,k ~I) =S K; ·S A ~2
A ~2 (Z ~,k ~d
=-1
6. CONCLUSIONS For dual and affined networks with cascade connections of two-ports (active and passive) one can obtain the sensitivity invariants. These results may be used to check the accuracy of various methods of sensitivity analysis. It is very interesting to use general principles of the network similarity to extend for other cases of sensitivity invariants.
448
Copyright © IF AC Programmable Devices and Systems, Ostrava, Czech Republic, 2003
IFAC PUBLICATIONS www.elsevier.comllocatelifac
SENSITIVITY ANALYSIS AND SENSITIVITY INVARIANTS OF SIMILAR NETWORKS OF ACTIVE TWO-PORTS IN CASCADE CONNECTIONS
Andrzej Kukielka*
* Silesian
University o/Technology, Institute 0/ Electronics. Gliwice
Two-ports in cascade connections are considered. The transfer and sensitivity properties of similar two-ports are considered. The transfonn of connections for passive and active two-ports and changes of their internal structures have been noticed. Two types of similarity and their sensitivities are given. Sensitivities are considered with respect to corresponding elements of networks or with respect to parameters of two-ports. For these sensitivity analysis, sensitivity invariants of networks function of similar networks with two-ports are derived. Sensitivity invariants of networks with two-ports (in cascade connections) can be used to test and find fault in electronic networks with active twoports. Copyright © 2003 IFAC Keywords: sensitivity analysis, active elements, transfer functions, fault detection, active two-ports, sensitivity invariants, impedances inversion and conversion, dual networks, affined networks, passive two-ports.
1. INTRODUCTION
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, to each loop b' of first network corresponds a node B" of second network; to each element k' of the first network belonging to a node A' corresponds an element k" of the second network belonging to loop a". One can find a similarity between the node equations of the second network. Assuming the relations of the second immitances of elements k' and k" in papers (Zagajewski, 1972a,b) relations between currents, voltages and transmittances of two dual networks have been found.
For two cascaded two-ports (active and passive) matrix A is obtained by multiplying the matrices of the single two-ports. For cascade connection, the transfer functions are we given in the fonn (Chojca~, 2001):
1
Ku
K =_1_= I
1
= A11 = A(l). A(2) + A(l) . A(2) 11 11 12 21 A
22
A (1)
1
• .4 (2)
1 2 ' -'21
+ A22(1) • A22(2)
(1)
(2)
top index (1) indicates first two-port (passive), top index (2) indicates second two-port (active). and
According to above considerations, if the original network N' is cascade connections of two-ports, dual network N" is also a cascade connection of two-ports (table 1.1). The transfer functions of these networks are (4):
(3)
445
K~
and
K~
1 - A;'I
In consideration of the cascade connection of twoports, also dual network is cascade connection of two-ports (table 1.1). For these case cascade matrix can be used for evaluation of sensitivities.
(4)
Similarly as in (Zagajewski, 1972a,b) let resume two relations between immitances of elements k' and k", namely impedance inversion and conversion. For these relations, we shall derive the relations between sensitivities of similar networks.
Sensitivity invariants will be presented in table 1.2 and 1.3. Some sensitivities are: constant and - it's value is -1 or equal (tables 1.2 and 1.3).
2. DUAL INVERSION OF INSIDE INTENSIFY Inversion of inside intensify of corresponding dual active two-ports is (Chojcan, Karwan, Kukielka, 2000):
TABLE 1.1. CASCADE CONNECTION OF TWO TWO-PORTS (ACTIVE AND PASSIVE) RELATIONS BETWEEN SIMILAR NETWORK AND ORIGINAL NETWORK
(5) KIND OF SIMILARITY
where k~ is the inside intensify of the active element of the first network (original) and k~ is the inside intensify of the active element of the second network (dual). Impedances inversion between impedances of corresponding elements of branch k' and k" in the form are (Zagajewski, 1972a,b) - only for passive elements:
DUAL WITH DUAL WITH IMPEDANCES IMPEDANCES INVERSION CONVERSION (TYPE la) (TYPE Ib)
KIND OF TWO-PORTS CONNECTION
CASCADE
CASCADE
INSIDE STRUCTURES ANALYZED TWO-PORTS
DUAL
DUAL
(6) Where Z~ is the impedance inversion of passive elements, Z~ is the impedance of element of the
TABLE 1.2. CASCADE CONNECTION OF TWO TWO-PORTS - DUAL SIMILARITY
first network (original), Z~ is the impedance of k" element of the second network (dual).
Sensitivity of voltage transfer function (change of ,,k" parameter) Relations between appropriate two-ports parameters of dual and original circuit
3.DUAL CONVERSION BETWEEN INSIDE INTENSIFY
Ib
la Conversion between inside intensify of dual active two-ports of corresponding active elements of networks N' and Nil is (Chojcan, Karwan, Kukielka 2000):
S
K~ (Z ~ ,k ~)
=s
K~ A
;1 ·S
~
S
(7) Impedances conversion between passive elements of networks N' and Nil are (Zagajewski, 1972a,b):
K; (Z ~ ,k ~I)
=S
S AK~;1
(8)
SA;.I . =s A~ (Zkokru)
A;I (Z ~ ,k ;u)
~ K; A ~2
·S
A ~2 (Z ~ ,k ~d
S K; =-1
• (Zk,kud
A~2
~;! . =i(S A~ (Zk.kru)
. ) (Zk.kU)
Where A is the inversion of impedance passive elements, Z~ is the impedance of element of the fIrst network (original), Z~ is the impedance of k" element of the second network (dual). A is constant
TABLE 1.3. CASCADE CONNECTION OF TWO TWO-PORTS - DUAL SIMILARITY 446
Sensitivity of current transfer function (change of .,k" parameter) Relations between appropriate two-ports parameters of dual and original circuit
la
S
Z~
In consideration of the cascade connection of twoports, also affined network is cascade connection of two-ports (table 1.4). For these case cascade matrix can be used for evaluation of sensitivities.
Ib
K~ (Z ~ ,k ;u)
=s
K~
A;,
II
s K;
(Z ~ ,k ~,)
=s
K~ S A;,
SA ;} " = S A~2 (Z k ,k ru)
Z~=A
•
A;, (Z ~ ,k ~)
II
Sensitivity invariants will be presented in table 1.5 and 1.6.
A~2 A~2 ·S (Z ~ ,k ~,)
K;
S
(Z k .k UI )
-s
Some sensitivities are: constant and - it's value is -1 or equal (tables 1.2 and 1.3).
K; A
=-1 ~2
TABLE 1.4. CASCADE CONNECTION OF TWO TWO-PORTS (ACTIVE AND PASSIVE)RELATIONS BETWEEN SIMILAR NETWORK AND ORIGINAL NETWORK
~;! " =i(S A~ . ) (Zk,kru)
(Zk.k Ul )
KIND OF SIMILARITY
4. AFFINED INVERSION OF INSIDE INTENSIFY Inversion of inside intensify of corresponding affined active two-ports is (Chojcan, Karwan, Kukielka, 2000):
k 'UI . k"UI -- Zi2
AFFINED AFFINED WITH WITH IMPEDANCES IMPEDANCES INVERSION CONVERSION (TYPE la) (TYPE Ib)
KIND OF TWO-PORTS CONNECTION
CASCADE
CASCADE
AFFINED
AFFINED
(9) INSIDE STRUCTURES ANALYZED TWO-PORTS
where k~ is the inside intensify of the active element of the first network (original) and k~ is the inside intensify of the active element of the second network (affined).
TABLE 1.5. CASCADE CONNECTION OF TWO TWO-PORTS - AFFINED SIMILARITY
Impedances inversion between impedances of corresponding elements of branch k' and k" in the form are (Zagajewski, 1972a,b) - only for passive elements:
Sensitivity of voltage transfer function (change of ,,k" parameter) Relations between appropriate two-ports parameters of dual and original circuit
Ib
la 5.AFFINED CONVERSION BETWEEN INSIDE INTENSIFY
K~ S (ZK~~ ,k ~d =S A;, ·S (ZA;,~ ,k ~d
Conversion between inside intensify of affined active two-ports of corresponding active elements of networks N' and Nil is (Chojcan, Karwan, Kukielka 2000):
II
II
K~ S (ZK~~ ,k ~d =S A;, ·S (ZA;,~ ,k ~,)
(10)
K~ S A;\
K~ S A;\ =-1
Where A is constant.
~;l
Impedance conversion between passive elements of networks N' and Nil is (Zagajewski, 1972a,b):
-i(S A;l
(Z~,k~) -
447
(Z~,k~)
)
~;! . =S A~ (Zk.k ru )
. (Zk,ku)
TABLE 1.6 CASCADE CONNECTION OF TWO TWO-PORTS - AFFINED SIMILARITY Sensitivity of current transfer function (change of .,k" parameter) Relations between appropriate two-ports parameters of dual and original circuit la
In paper (Li & Woo 1999) we can find some relations with this theory and it's potential application to fault detection in electronic circuits.
REFERENCES Chojcan J., Karwan L., Kukielka A: Zasady podobieilstwa obwod6w z uwzgll(dnieniem :h6del sterowanych. XXIII SPETO, Ustron 2000, ss. 241-241. Chojcan J. przy wsp6lpracy Karwana L., Drygajly A i Kolmera A: Zbi6r zadan z teorii obwod6w I, wydanie VII, Wydawnictwo Politechniki Slllskiej, nr 2264, Gliwice 2001. Li F., Woo P-G.: The Invariance of Node-Voltage Sensitivity Sequence and Its Application in a Unified Fault Detection Dictionary Method, IEEE Transactions on Circuits & Systems - I, vol.46, no.lO, October 1999, pp. 1222-1226. Zagajewski T.: General principles of similarity of electric networks. Bull. Acad. Po Ion. Sci., Ser. Sci. Techn. 20 (1972). Zagajewski T.: Some applications of the general principle of similarity of electric networks. Bull. Acad. Polon. Sci., Ser. Sci. Techn. 20 (1972).
Ib
s (ZK;~ ,k ~I =s )
K~ A ;2
·S
A ;2 (Z ~ ,k ~I)
n n
s (ZK;~,k ~I) =S K; ·S A ~2
A ~2 (Z ~,k ~d
=-1
6. CONCLUSIONS For dual and affined networks with cascade connections of two-ports (active and passive) one can obtain the sensitivity invariants. These results may be used to check the accuracy of various methods of sensitivity analysis. It is very interesting to use general principles of the network similarity to extend for other cases of sensitivity invariants.
448