Gheorghe Asachi Technical Universiy of Iasi Faculy of Elecronics, Telecommunicaions and Informaion Technologies Compuer-Aided Analysis of Elecronic Circuis Course Noes 3 Bachelor: Telecommunicaion Technologies and Sysems Year of Sudy: 2 Lecurer: Dan Burdia, PhD C www.ei.uiasi.rouiasi www.sudy.uiasi.ro www.learning.uiasi.ro
2 Conens I. Inroducion in Compuer-Aided Analysis. II. Compuer Circui Models of Elecronic Devices and Componens III. Nework Topology: The Key o Compuer Formulaion of he Kirchhoff Laws IV. Nodal Linear Nework Analysis: Algorihms and Compuaional Mehods V. Nodal Nonlinear Resisive Nework Analysis: Algorihms and Compuaional Mehods
3 Chaper III - Nework Topology: The Key o Compuer Formulaion of he Kirchhoff Laws 3.1 Basic conceps in nework opology 3.2 Inciden marix 3.3 Loops marix 3.4 Cuse marix 3.5 Basic relaions beween branch variables 3.6 Compuer-aided formulaion of A, B and D marices
4 3.1 Basic conceps in nework opology Graph - consiss of a se of nodes conneced by branches. Graph ypes: - Direced graph - Undireced graph - Conneced graph - Unconeced graph Circui schemaic Direced graph Undireced graph Conneced graph here is a pah beween any wo nodes from graph Unconneced graph a graph wih one or more isolaed (separaed) nodes. Unconneced graph
3.1 Basic conceps in nework opology (con) Loop A subgraph Gs of a graph Gn which fulfills he following wo condiions: Gs is conneced (connex) A each node in Gs here are exacly wo branhes of he Gs Example: 1,2,6; 1,3,4,5,6; 3,4,7,6,2; No loop: 3,4,5,2,7 Cu-se A se of branches which fulfills he following wo condiions: If he se is removed from graph hen he graph becomes unconneced Re-insering any branch from se he graph becomes again conneced. Example: 1,2,3; 3,5,7; 2,5,6; No cuse: 3,2,5,7 Tree A subgraph Gs of a graph Gn which fulfills he following hree condiions: Gs is conneced Gs conains all nodes from Gn Gn does no conain loops. Example: 3,2,5,6; 3,4,6,7; No ree: 1,2,6,5; 3,2,4,5,7
3.2 Inciden marix Complee inciden marix (Aa). For a direced graph Gd having n nodes and b branches, Aa is a n x b marix, Aa = [aij], where: - aij = 1, if branch j is inciden o node i, sense goes ou from node - aij = -1, if branch j is inciden o node i, sense goes in o node - aij = 0, if branch j is no inciden o node I Example: Reduced inciden marix (A) is obained from Aa removing a row (usually he row of ground node) Firs Kirchhoff Law for all nodes:
3.2 Inciden marix (con.) Theorem 3.1. For a direced graph G d he rows of reduced inciden marix A are linear independen. Corolar. The maximum se of independen Firs Kirchhoff Law equaions are wrien as: Ai Theorem 3.2. Given A he reduced inciden marix for a direced graph G d having n nodes. Then n-1 columns from A are linear independen if and only if he branches corresponding o hose columns form a ree in G d. Corolar. Le consider A pariioned as: A A MA where A T and A L have he columns corresponding o he ree branches and co-ree branches, respecively. Then de(a T ) 0. Example: 0 T L
3.3 Loops marix Definiion. Complee loops marix B a For a direced graph G d wih n nodes, b branches and n b direced loops, Ba = [b ij ] nb x b, where: bij= +1 if branch j belongs o he loop i, he same orienaion bij= -1 if branch j belongs o he loop i, he opposie orienaion bij= 0 if branch j does no belong o he loop i Kirchhoff II Law for all possible loops (no all are independen): Base loops marix B b : a submarix of B a having maximum independen rows from B a B b have b-n+1 lines. How we can selec a se of b-n+1 independen rows (loops)? Sysemaic approach: by deermining he fundamenal loops Fundamenal loop: conains only one branch from co-reeand he oher branches are from he ree. The loop orienaion is he same o branch from co-ree. Fundamenal loops marix B can be pariioned as B = [B T 1]
3.3 Loops marix (con.) Example: Fundamenal loops marix B for a graph. Tree = a,d,e,f, i. Theorem 3.3. If he columns of he marices A a and B a are arranged in he same order of branches, hen for any i and j: [row i from B a ] x [row j from A a ] = 0. Corolar: The following relaions are rue
3.3 Loops marix (con.) Theorem 3.4. For a direced graph G d having n nodes and b branches he maximum number of independen rows from B a is b-n+1. Demonsraion: consider he ree T. Then Ba can be pariioned as: B A 0 Saring from i is easy o show ha B 2 = B 2L B. This means only he rows of a B are he independen rows from B a. The maximum se of Kirchhoff II Law equaions can be wrien as B v=0
3.4 Cuses marix Definiion. Complee cuse marix D a For a direced graph G d wih n nodes, b branches and n s direced cuses, D a = [d ij ] ns x b, where: d ij = +1 if branch j belongs o he cuse i, he same orienaion d ij = -1 if branch j belongs o he cuse i, he opposie orienaion d ij = 0 if branch j does no belong o he cuse i Generalized Kirchhoff I Law for all possible cuses (no all are independen): Base cuses marix D b : a submarix of D a having maximum independen rows from D a D b have n-1 lines. How we can selec a se of n-1 independen rows (cuses)? Sysemaic approach: by deermining he fundamenal cuses Fundamenal cuse: conains only one branch from reeand he oher branches are from he co-ree. The cuse orienaion is he same o branch from he ree. Fundamenal cuses marix D can be pariioned as D= [1 D L ].
3.4 Cuses marix (con.) Example 1. Complee cuse marix D a Example 2. Fundamenal cuse marix D for a graph. Tree T = a,d,e,f,i.
3.4 Cuses marix (con.) Theorem 3.5. If he columns of he marices B a and D a are arranged in he same order of branches, hen for any i and j: [row i from B a ] x [row j from D a ] = 0. Corolar: The following relaions are rue Theorem 3.6. For a direced graph G d having n nodes and b branches he maximum number of independen rows from D a is n-1. Demonsraion: consider he ree T. Then Da can be pariioned as: D 1 M DL Da D 2 D2T M D 2L Saring from DaB 0 i is easy o show ha D 2 = D 2T D. This means only he rows of D are he independen rows from D a. The maximum se of generalized Kirchhoff I Law equaions can be wrien as D i = 0
3.5 Basic relaions beween branch variables Given G d wih n nodes and b branches, le pariion he marices A, B and D as: 1 1 A AT MAL B BT M D MDL b n 1 Branch volages andbranch currens: vt it v i v L i L 1 BD 0 => BT 1 M L 0 BT DL 0 => D L v T Bv0 BT M1 0 vl => B v v 0 => n1 B D => T T L L T T Di 0 => 1 it 0 0 DL it DLiL i L T L or v B v i D i M => T L L D B L T
3.5 Basic relaions beween branch variables (con.) Branch currens as a funcion of co-ree branch currens it BT i B L T i i B 1 i B i i L i 1 M L L T L L Branch volages as a funcion of ree branch volages vt vt 1 v v 1 D v D v v L DL v T D M L T L T T Nodal volages vecor: v n v v n1 n2 vnn, 1 Relaion beween branch volages and nodal volages: M va v n
3.6 Compuer-aided generaion of he marices A, B and D Generaion of inciden marix A - Assign consecuive numbers o nodes. 0 is for reference node (ground) - Assign consecuive numbers o nework branches - If branch k is beween nodes i and j, his can be sored as a riple (k, i, j). Thus: a a 1 1 ik jk Generaion of fundamenal marices B and D - he nework ree mus be deermined firsly There are wo sages: 1 - Deerminaion of he nework ree T wih a preferenial order of branches o be included in he ree. Such ree is named normal ree. 2 - Generaion of he marices D and B
3.6.1 Nework ree deerminaion The nework ree can be deermined saring from A marix The columns of A marix are rearranged in he preferenial order of branches o be included in he ree: Example: Independen volage sources V9 C1 C6 R2 R5 R8 L10 I3 J4 J7 Conrolled volage sources 1 Capaciors 2 Resisors A Inducors M Conrolled curren sources n1 Independen curren sources In he marix A firs independen n-1 columns are deermined. This is possible by ransforming A marix ino an echelon form marix using elemenary row operaions: row inerchanging muliply a row wih a non-zero value muliply a row wih a non-zero value and add o anoher row Echelon form marix: c1 c2 c3 c k 11 x x x x L x x L x 2 0 0 1 x x x x x L L Q30 0 0 1 x L x x L x M M M M M M M M k0 0 0 0 0 L 0 1 L x 0 0 0 0 0 L 0 0 L 0
Nework ree deerminaion con. Algorihm o ransform he marix A ino echelon form marix Le As a submarix of A. Iniial As=A Sep 1: he firs nonzero column is searched in As. Name c his column. Sep 2: In column c, firs non-zero value is searched from up o down, for example in he row r. If r=1 go o sep 4. Sep 3: Wihin As, rows 1 and r are inerchanged. Sep 4: If elemen (1,c) is -1 he row 1 from As is muliplied by -1. Sep 5: In column c all non-zero elemens below firs row from As are reduced o zero. Sep 6: New As marix is defined. I consis of rows below firs row from old As and columns siuaed a he righ side of column c. If new As doesn exis, algorihm is finished, else go o sep 1. Independen columns in A are he columns of firs elemen of 1 from each row wihin echelon form marix Branches corresponding o hese columns form henework ree.
Nework ree deerminaion Example Marix A wih columns arranged in he preferenial order of branches: V C C C R R R L L I 3 2 4 8 1 9 10 5 6 7 1 0 1 1 0 1 0 0 1 0 0 2 1 1 0 0 1 0 0 0 0 0 L2L1 A L1L5L5 3 0 0 0 0 0 0 0 0 1 1 4 0 0 0 1 0 1 0 1 1 0 5 1 0 1 0 0 1 1 0 0 0
Nework ree deerminaion Example (con.) Transform marix A ino echelon form using elemenary row operaions: Normal ree: V3, C2, C8, R9, L6
3.6.2 Generaing marices B and D Assume he nework ree T is deermined and marices A, B and D are pariioned as A A MA B B D 1 T L T M1 MD L Marices B and D can be obained from marix A. bn1 n number of nodes n1 b number of branches B T AB 0 AT MAL 0 AT BT AL 0 1 1 T T L B A A B D T L 1 L T L D A A 1 1 1 1 D D 1 L AT AT AT A M M L AT AT MAL AT A BBT 1 DL 1 M M Disadvanage: calculus of 1 A T Anoher mehod: based on elemenary operaions
3.6.2 Generaing marices B and D (con.) Alernaive mehod can be deduced using elemenary marices. Definiion: An elemenary marix is a marix obained from uni marix afer an elemenary operaion. Theorem: performing an elemenary operaion on a marix Q is equivalen wih he produc εq, where ε is he elemenary marix corresponding o ha operaion. Example: 3 2 1 4 1 6 5 2 ( 2) L2L1L1 Q 2 4 2 1 Q 1 2 4 2 1 0 5 3 2 0 5 3 2 1 2 0 1 6 5 2 0 1 0 Q 2 4 2 1 Q 0 0 1 0 5 3 2 If Q is a square and non-singular marix, hen here is an sequence of elemenary operaion o reduce Q o a uniy marix. A T is a square and non-singular marix! m 1... Q1 Q... 3 2 1 m 3 2 1 1 A... 1 T m 3 2 1
3.6.2 Generaing marices B and D (con.) A... 1 T m 3 2 1 1 T m... 321AT AL... A... A 1 D D A A m 3 2 1 T m 3 2 1 L L Thus, D can be obained from A by elemenary operaions unil sub-marix A T is reduced o a uniy marix. Algorihm o reduce a square and non-singular marix Q o a uniy marix: 1) A firs Q is reduced o echelon form marix Qes 2) Non-zero elemens above diagonal are reduced o zero using he ones from diagonal
3.6.2 Generaing marices B and D (con.) Example 1: Deerminaion of marices D and B saring from following inciden marix a b c d e f g 1 1 1 0 1 0 0 1 AAT MAL2 1 0 0 1 1 0 0 3 0 0 0 1 0 1 0 4 0 0 1 0 1 1 1 Sep 1: Reduce A o an echelon marix 1 1 0 0 0 0 1 1 1 0 0 0 0 1 L1L2L2 0 1 0 1 1 0 1 L3L4 0 1 0 1 1 0 1 0 0 0 1 0 1 0 L3( 1) 0 0 1 0 1 1 1 L4( 1) 0 0 1 0 1 1 1 0 0 0 1 0 1 0 Sep 2: Reduce o zeros all non-zero elemens above diagonal a b c d e f g 1 0 0 0 1 1 0 ( 1) L4L2L2 0 1 0 0 1 1 1 1MD ( 1) L2 L1 L1 L D 0 0 1 0 1 1 1 0 0 0 1 0 1 0
3.6.2 Generaing marices B and D (con.) Loops marix B a b c d e f g 1 1 1 0 1 0 0 BBT M1 DL 1 1 1 1 1 0 1 0 M 0 1 1 0 0 0 1 Example 2 V C C C R R R L L I 3 2 4 8 1 9 10 5 6 7 1 0 1 1 0 1 0 0 1 0 0 2 1 1 0 0 1 0 0 0 0 0 A 3 0 0 0 0 0 0 0 0 1 1 4 0 0 0 1 0 1 0 1 1 0 5 1 0 1 0 0 1 1 0 0 0
3.6.2 Generaing marices B and D (con.) Columns of A are rearranged in form A=[A T, A L ] V C C R L C R R L I 3 2 8 9 6 4 1 10 5 7 1 0 1 0 0 0 1 1 0 1 0 2 1 1 0 0 0 0 1 0 0 0 AAT MAL 3 0 0 0 0 1 0 0 0 0 1 4 0 0 1 1 1 0 0 0 1 0 5 1 0 0 1 0 1 0 1 0 0 The algorihm o reduce A T o a uniy marix is applied: Echelon form marix Marix D A es V C C R L C R R L I 3 2 8 9 6 4 1 10 5 7 1 1 0 0 0 0 1 0 0 0 0 1 0 0 0 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 1 V3 C2 C8 R9 L6 C4 R1 R10 L5 I7 L5L3L3 1 0 0 0 0 1 0 0 1 0 ( 1) L4L3L3 ( 1) L2L1L1 0 1 0 0 0 1 1 0 1 0 1MDL D 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 1
3.6.2 Generaing marices B and D (con.) Loops marix: can be wrien direcly from D, aking ino accoun B D T L V3 C2 C8 R9 L6 C4 R1 R10 L5 I7 1 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 BBT M1 DLM1 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 0 0 0 1 0 0 0 1 0 1 0 0 0 0 1