Associated Reference Convention :1i+−1vDIGRAPH (Directed Graph : )Device GraphA current direction is chosenentering each positively-referenced terminal.DD++--1v2v1i2i1v2v2ijiDjv1jv−1ji−1i11212j1j−Associated Reference Convention :D1i2i+−2v+−1v2-port Device1i+−1v2i+−2vni+−nvDn-port DeviceDevice Graph1212n1e124D3D1D6D5D2D41i2i3i4i5i+−1v+−6v+−3v+−4v+−5v+−2v2e3e213456123421540vvvv=−+−=3KVL around closed node sequence:13 2 1231:0vvv+−=23 4 2354:0vvv−+−=13 4 2 12541:0vvvv+−−=These 3 KVL equations are not linearly-independent because the 3rd equation can be obtained by adding the first 2 equations:231 354()( )vvv vvv+− +−+−• Circuits containing n-terminal devices can have many distinct digraphs, due to different (arbitrary) choices of the datum terminal for each n-terminal device.• Although the KCL and KVL equations associated with 2 different digraphs of a given circuit are different, they contain the same information because each set of equations can be derived from the other.A Circuit with 3 different digraphs1. Choose as datum for DD1v+-1i2i+-2v+-3v4v+-3i4i31 21231234⇒32. Choose as datum for DD1v+-1i2i+-4v+-3i4i31 21231234⇒23. Choose as datum for DD1v+-1i+-4v+-3i4i31 21231234⇒1+-2v3v3v-+2v2iD1i+−1v+-2v2i+-3v3i2-portdevice+-4v+-5v6i+−6v123451234512341 236545disconnected digraphKCL at :KCL at :KVL around :KVL around :340ii+=560ii+=430vv−=650vv−=242 324 544i5iD1i+−1v+-2v2i+-3v3i2-portdevice+-4v+-5v6i+−6v123451234512341 2365454i5iAdding a wire connecting one node from each separate component does not changeKVL or KCL equations.70i={}77 is a 0i⇒=cut setD1i+−1v+-2v2i+-3v3i2-portdevice+-4v+-5v6i+−6v123451234512341 2365454i5iAdding a wire connecting one node from each separate component does not changeKVL or KCL equations.D1i+−1v+-2v2i+-3v3i2-portdevice+-4v+-5v6i+−6v123451234512341 2365454i5iHINGED DIGRAPH12341 2654Since nodes andare now the same node, they can be combined into one node, and the redrawn digraph is called a hinged graph.35322e3e1e134561234=⇒Ai 0KCL Equations:1260iii+−=1340iii−−+=2350iii−++=12311000110 1100011010−−−−000=123456Branch no.nodeno.123AA is called the reduced Incidence Matrixof the diagraph G relative to datum node .4123456iiiiiii022e3e1e134561234=⇒Ai 0KCL Equations:1260iii+−=1340iii−−+=2350iii−++=12311000110 1100011010−−−−000=123456Branch no.nodeno.123A123456iiiiiii0Choose as datum node for digraph G412131232233111223344556611010 1011010001100vvvvvvvvvvvveeeeeeeeeeee=−−=−−=−+−=⇒===−−vTAeIndependentKCL EquationsIndependentKVL Equations22ˆe1ˆe134561234ˆ=⇒Ai 0KCL Equations:1260iii+−=1340iii−−+=4560iii−−+=12411000110110000 0 1 1 1−−−−−000=123456Branch no.nodeno.124ˆA123456iiiiiii03121122244411223344556 46 1ˆˆˆˆˆˆˆˆˆ110100010011001101ˆˆˆeeeeeeeevvvvvvvvvveeveev=−−==−−=⇒=−−=−−=−+−vˆTAˆeIndependentKCL EquationsIndependentKVL Equationschoose as datum and let124node-to-ˆˆdaˆ, , be new volttu ae.mgseee4ˆeReduced Incidence Matrix ALet G be a connected digraph with “n” nodes and “b” branches. Pick any node as the datum node and label the remaining nodes arbitrarily from 1 to n-1 . Label the branches arbitrarily from 1 to b.The reduced incidence matrix of Gis an (n-1) x b matrix where each row jcorresponds to node j , and each column k, corresponds to branch k, and where the jkthelement ajkof is constructed as follow:AA1,1,0,jka=−if branch k leaves node jif branch k enters node jif branch k in not connected to node jHow to write An Independent System of KCL and KVL EquationsLet N be any connected circuit and let the digraphG associated with N contain “n” nodes and “b” branches. Choose an arbitrary datum node and define the associated node-to-datum voltagevector , the branch voltage vector , and the branch current vector . Then we have the following system of independent KCL and KVL equations.(n-1) Independent KCL Equations : =Ai 0T=vAeb Independent KVL Equations :
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