3.1 Introduction to Kirchhoff Laws
107
has one and only one link voltage. In matrix form the KVL equations based on the
fundamental loops of a tree can be written as
Bv = 0
(3.20)
where B is an l × b matrix called the fundamental loop matrix associated with a tree
T and v = (i 1 , . . . , i b ) T is the b × 1 column vector of branch voltages. The jk-th
entry of B is defined as:
b jk =
⎧
⎨
⎩
+1 if branch k is in loop j and their reference directions are the same
−1 if branch k is in loop j and their reference directions are opposite
0 if branch k is not in loop j .
(3.21)
Here, the reference direction of each loop j is defined by the direction of its
associated link. Using the same labeling scheme as before, i.e., the twigs are marked
from 1 to n − 1, and the links from n to b, the fundamental loop matrix B can be
decomposed as follows:
B = [B n−1 , I l ]
(3.22)
and KCL equations using link currents are of the form
i = B
T i l
(3.23)
from which it is derived that
i t = B
T
n−1 i l .
(3.24)
3.1.5 Link Between Q and B: Tellegen’s Theorem
The following fundamental result expresses the relationship between the two
matrices Q and B [1].
Theorem 3.2 Let Q and B be the fundamental cut-set matrix and the fundamental
loop matrix, respectively, of a connected digraph G for a specified tree T . Then, we
have
BQ
T
= 0.
(3.25)
The transpose of (3.25) gives QB T = 0. Premultiplying such equation by v T
t and
then postmultiplying by i l we obtain
v
T
t QB
T i l = 0
(3.26)