Independent Device Currents

Transcription

1 Independent Dece Currents j Snce KCL = j k k Only one ndependent current can be defned for each termnal dece.

2 Snce KCL = Only ndependent currents can be defned for a termnal dece. Snce KVL = Only ndependent oltages can be defned for a termnal dece.

3 nport dece There are electrcal deces hang an een number of termnals and constructed n such a way that the termnals can be grouped nto pars, called ports, such that the current enterng one termnal of each par s always equal to the current leang the other termnal. A dece wth n pars of such currentfollowng termnals s called an nport.

4 N Example of a port A termnal dece In ths example, the separate physcal wndngs guarantee that : = and In ths case, only current and oltage needs to be defned for each par of termnals, henceforth called ports. =

5 e e e KCL Equatons: = = = A= node no. Branch no. =

6 e e e KCL Equatons: = = = A= node no. Branch no. A = A s called the reduced Incdence Matrx of the dagraph G relate to datum node.

7 e e e Matrx Formulaton: node Branch no. no. INCIDENCE MATRIX a jk = f branch f branch f branch k KCL Node Equatons: = = = = These equatons are lnearlydependent. = Aa = k k node No. leaes node enters node s not connected to node j j j

8 e e e = A= T A KCL Equatons: = = = node Branch no. no. e e e e KVL Equatons: = e e = e e = e e = e = e = e = KVL: = T A e Snce j s present only n the jth equaton, these equatons are lnearly ndependent. k

9 Theorem A= ges the maxmum possble number of lnearlyndependent KCL equatons for a connected crcut.

10 Reduced Incdence Matrx Let G be a connected dgraph wth n nodes and b branches. Let A a be the Incdence Matrx of G. The (n) x b matrx A obtaned by deletng any one row of A a s called a Reduced Incdence Matrx of G.

11 Obseraton : The KCL node equatons are not lnearly ndependent. Addng the left sde of the KCL node equatons, we obtan: ( ) ( ) ( ) ( ) Ths means we can dere any one of these equatons from the other. Example: Dere KCL equatons at node : Addng the frst node equatons ges: ( ) ( ) ( ) =

12 Reduced Incdence Matrx A Let G be a connected dgraph wth n nodes and b branches, the reduced ncdence matrx A relate to datum node n s an (n) x b matrx whose coeffcents a jk are obtaned from the (n) KCL equatons wrtten at the n nondatum nodes: a jk = f branch k f branch k f branch k leaes node enters node s not connected to node j j j

13 By applyng the arous ersons of KCL, we can wrte many dfferent KCL equatons for each crcut. Howeer, these equatons are usually not lnearly ndependent n the sense that each equaton can be dered by a lnear combnaton of the others. How can we wrte a maxmum set of lnearlyndependent KCL equatons?

14 Smplest Method to wrte lnearlyindependent KCL Equatons. Gen a connected crcut wth n nodes, choose an arbtrary node as datum. Wrte a KCL equaton at each of the remanng (n) nodes.

15 Relatonshp between A and A a Let A a be the n x b Incdence matrx of a connected dgraph G wth n nodes and b branches. By deletng any row correspondng to node m from A a, we obtan the reduced ncdence matrx A of G relate to the datum node m.

16 D port dece dsconnected dagraph KCL at : KCL at : = KVL around : KVL around : = = =

17 D port dece HINGED DIAGRAPH Snce nodes and are now the same node, they can be combned nto one node, and the redrawn dagraph s called a hnged graph.

18 D port dece Addng a wre connectng one node from each separate component does not change KVL or KCL equatons.

19 D port dece 7 =? Addng a wre connectng one node from each separate component does not change KVL or KCL equatons.

20 D port dece 7 = Addng a wre connectng one node from each separate component does not change KVL or KCL equatons. 7 s a cut set = { } 7

21 Assocated Reference Conenton : D D port Dece Dece Graph n n nport Dece n

22 e D D D e D D D e KCL at : = KVL around : =

23 Assocated Reference Conenton : A current drecton s chosen enterng each postelyreferenced termnal. D D D j j j j Dece Graph : DIGRAPH (Drected Graph) j j

24 e datum node 9 e 8 e 7 e e KVL = e e = e, = e e = e e, = e e = e e = e, = e e e

25 KCL Gaussan Surface D 8 D D 7 Gaussan Surface : 8 = Ths KCL equaton can be decomposed nto the sum of two KCL cut set equatons: { } = { } 8 cut set, cut set,,8 =