106
3 RLC Networks Equations and Analysis Methods
a jk =
⎧
⎪ ⎨
⎪ ⎩
+1 if branch k leaves node j
−1 if branch k enters node j
0 if branch k does not touch node j
(3.11)
A set of complete linearly independent KVL equations can be derived in terms
of the node-to-datum voltage vector v e = (v e,1 , v e,2 , . . . , v e,n−1 ) T as follows:
v = A
T v e .
(3.12)
Comparing (3.8) with (3.12), it is apparent that v e coincides with v t and A = Q.
Example 3.3 (Fundamental Cut-Set Matrix Associated with the Tree T 3 in Fig. 3.3)
With regard to the cut-sets C 1 , C 2 , and C 3 in Fig. 3.3, the fundamental cut-set matrix
associated with tree T 3 = {β 2 , β 4 , β 5 } reads
Q =
⎛
⎝
+1 0 0 −1 −1 0
0 +1 0 −1 −1 +1
0 0 +1 0 −1 +1
⎞
⎠
(3.13)
where twig currents and voltages are i t = (i 2 , i 4 , i 5 ) T and v t = (v 2 , v 4 , v 5 ) T ,
respectively, whereas link currents and voltages are i l = (i 1 , i 3 , i 6 ) T and v l =
(v 1 , v 3 , v 6 ) T , respectively. It follows that the KCL equations can be written as
i 2 − i 1 − i 3 = 0
(3.14)
i 4 − i 1 − i 3 + i 6 = 0
(3.15)
i 5 − i 3 + i 6 = 0
(3.16)
whereas the KVL equations are
v 1 = −v 2 − v 4
(3.17)
v 3 = −v 2 − v 4 − v 5
(3.18)
v 6 = v 4 + v 5 .
(3.19)
3.1.4 The Fundamental Loop Matrix B Associated with a Tree
A dual approach with respect to that based on the fundamental cut-sets permits
to express Kirchhoff equations by means of link currents instead of twig voltages.
Theorem 3.1 guarantees that there exist l = b − (n − 1) links and each link identifies
a unique loop. Hence, KVL equations can be derived from the l fundamental
loops. Such KVL equations are linearly independent because each loop equation
3 RLC Networks Equations and Analysis Methods
a jk =
⎧
⎪ ⎨
⎪ ⎩
+1 if branch k leaves node j
−1 if branch k enters node j
0 if branch k does not touch node j
(3.11)
A set of complete linearly independent KVL equations can be derived in terms
of the node-to-datum voltage vector v e = (v e,1 , v e,2 , . . . , v e,n−1 ) T as follows:
v = A
T v e .
(3.12)
Comparing (3.8) with (3.12), it is apparent that v e coincides with v t and A = Q.
Example 3.3 (Fundamental Cut-Set Matrix Associated with the Tree T 3 in Fig. 3.3)
With regard to the cut-sets C 1 , C 2 , and C 3 in Fig. 3.3, the fundamental cut-set matrix
associated with tree T 3 = {β 2 , β 4 , β 5 } reads
Q =
⎛
⎝
+1 0 0 −1 −1 0
0 +1 0 −1 −1 +1
0 0 +1 0 −1 +1
⎞
⎠
(3.13)
where twig currents and voltages are i t = (i 2 , i 4 , i 5 ) T and v t = (v 2 , v 4 , v 5 ) T ,
respectively, whereas link currents and voltages are i l = (i 1 , i 3 , i 6 ) T and v l =
(v 1 , v 3 , v 6 ) T , respectively. It follows that the KCL equations can be written as
i 2 − i 1 − i 3 = 0
(3.14)
i 4 − i 1 − i 3 + i 6 = 0
(3.15)
i 5 − i 3 + i 6 = 0
(3.16)
whereas the KVL equations are
v 1 = −v 2 − v 4
(3.17)
v 3 = −v 2 − v 4 − v 5
(3.18)
v 6 = v 4 + v 5 .
(3.19)
3.1.4 The Fundamental Loop Matrix B Associated with a Tree
A dual approach with respect to that based on the fundamental cut-sets permits
to express Kirchhoff equations by means of link currents instead of twig voltages.
Theorem 3.1 guarantees that there exist l = b − (n − 1) links and each link identifies
a unique loop. Hence, KVL equations can be derived from the l fundamental
loops. Such KVL equations are linearly independent because each loop equation