《电路》（英文版）16-2 Incidence(关联) matrix and KCL

§16-2 Incidence(关联) matrix and KcL 1. Augmented(#r) incidence matrix of a directed graph nodes(N+1), branches(B) The(N+1)B matrix Aa=lkj +1when bi is incidance to nk and directed away from it; aki=3-lwhen bi is incidance to nk and directed away toward it 0 when b; is not incidance to nk

§16-2 Incidence(关联) matrix and KCL 1. Augmented(增广) incidence matrix of a directed graph nodes(N+1),branches(B) The (N+1)B matrix   Aa = akj • . j nk when b is not incidance to      − + = 0 1 1 kj a when b i s incidance t o n and directed away from it; j k when b i s incidance t o n and directed away toward it; j k

3 n 5 6 29 100 1000 b00 0 0 1 0000 000 00 001 00100 (Each column has exactly two nonzero elements

(Each column has exactly two nonzero elements.)                     − − − − − − − − − = • 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 1 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 1 1 1 1 1 0 0 0 1 0 0 0 1 0 0 1 1 0 0 0 0 6 5 4 3 2 1 n n n n n n Aa b1 b2 b3 b4 b5 b6 b7 b8 b9 n4 1 2 3 4 5 6 9 8 7 n1 n2 n3 n6 n5

KCL: Let i(+)--away from a node; i(-)--toward a node. +i4+is=0 5 i -io +i=0 29 0 15+ 6+l7 0 10011000 11000-100 1-1000-1-1 0000 01-10000 00100 000-1110 .4...1.8 000000

KCL:(Let i(+)-- away from a node; i(-)--toward a node.) n1 : −i 1 + i 4 + i 5 = 0 n2 :i 1 + i 2 − i 6 = 0 n3 : −i 2 − i 3 − i 7 − i 8 + i 9 = 0 n4 :i 3 − i 4 = 0 n5 : −i 5 + i 6 + i 7 = 0 n6 :i 8 − i 9 = 0                     =                                                 − − − − − − − − − 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 1 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 1 1 1 1 1 0 0 0 1 0 0 0 1 0 0 1 1 0 0 0 0 9 8 7 6 5 4 3 2 1 i i i i i i i i i          = = = • • • • • T b B a kj a b i i i i A a A i 1 2  0 n4 1 2 3 4 5 6 9 8 7 n1 n2 n3 n6 n5

3 100 110 100 0-00 00 00 0 000 1-11 000 0 01 000 0…0…0…0…000…1…1;/0 000 11100 0 Re n equations are linearly independent: b A--ncidence matrix

                    =                                                 − − − − − − − − − 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 1 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 1 1 1 1 1 0 0 0 1 0 0 0 1 0 0 1 1 0 0 0 0 9 8 7 6 5 4 3 2 1 i i i i i i i i i N equations are linearly independent: • • • Ai b = 0 A− −Incidence matrix • Re f . n4 1 2 3 4 5 6 9 8 7 n1 n2 n3 n6 n5

2. The augmented cutset matrix of a directed graph. To assign an orientation to each .h3 3 c cutset, assign(+) to the currents in a branch whose direction is same as the orientation of the cutset C1:i-i2=0 1000 C2:2+i-i4=0 C3:i4+i=0 00011‖ 1-i3-is=0 5:i2+i3+i=0 000000 C:i+i3-i4=0 2aib=0 2a--augmented cutset matrix

2. The augmented cutset matrix of a directed graph. n1 n2 n3 n4 1 2 4 3 5 C1 C3 C4 C2 C5 C6 To assign an orientation to each cutset,assign (+) to the currents in a branch whose direction is same as the orientation of the cutset. C1 :i 1 − i 2 = 0 C2 :i 2 + i 3 − i 4 = 0 C3 :i 4 + i 5 = 0 C4 : −i 1 − i 3 − i 5 = 0 C5 :i 2 + i 3 + i 5 = 0 C6 :i 1 + i 3 − i 4 = 0                     =                                     − − − − − − 0 0 0 0 0 0 1 0 1 1 0 0 1 1 0 1 1 0 1 0 1 0 0 0 1 1 0 1 1 1 0 1 1 0 0 0 5 4 3 2 1 i i i i i or Qa i b = Qa − −augmented cutset matrix • • • • 0

Fundamental cutset matrix Let the orientation of Ck be the same as the tree branch b j(ork). +1 when b i is in Ck and has the same orientation; Aki=-1 when b; is in Ck and has the opposite orientation; 0 when bi is not in Ck

Fundamental cutset matrix Let the orientation of Ck be the same as the tree branch bj(ork).   Qf = qkj • . j Ck when b is not in      − + = 0 1 1 qkj when b i s inC and has the same orientation; j k when b i s i nC and has the opposite orientation; j k

11000 2 nI 1010 5 +i,=0 KCL: C2 i,+i3+is=0 i, ti=0 「-110001乙,「0 0 0 00

n1 n2 n3 n4 1 2 4 3 5 C1 C3 C2          − = • 0 0 0 1 1 1 0 1 0 1 1 1 0 0 0 Qf      + = + + = − + = 0 0 0 : 3 4 5 2 1 3 5 1 1 2 C i i C i i i C i i KCL           =                          − 0 0 0 0 0 0 1 1 1 0 1 0 1 1 1 0 0 0 5 4 3 2 1 i i i i i • • • or Qf i b = 0

Tree branches: 1.2.3 nI 100 10‖i, 5 01011‖3=0 0010 corresponding to the tree branches F--corresponding to the chords of t

n1 n2 n3 n4 1 2 4 3 5 C1 C2 C3 Tree branches:1,2,3           =                           − 0 0 0 0 0 1 0 1 0 1 0 1 1 1 0 0 1 0 5 4 3 2 1 i i i i i • • • or Qf i b = 0   • • • Qf = 1 F . 1 . F corresponding t o the chords of T corresponding t o the tree branches − − − − • •