IEEE Circuits and Systems Magazine - Q4 2022 - 51

that one of the loop currents must be chosen in order to
calculate the other two loop currents.
The KCL is explained in the electrical network theory
with the law of conservation of charge. Since a node
does not store a charge, the charges that flow into a
node also leave the node, the currents are in balance.
The KCL thus expresses a continuity principle.4 KCL can
be considered as special case of another of Maxwell's
laws for electromagnetism [8].
Along the same lines but with more equations and variables
it can be shown that Kirchhoff's KCL also holds for a
node with more than three branches. More generally, the
KCL also applies to a part of a circuit comprising several
nodes. If we add the incoming and leaving currents respectively,
the sum of the totals becomes zero. The totals of the
incoming currents and leaving currents of the separately
examined part of the circuit are in balance. This is also the
case for a single component with more than two terminals
such as a multiport or a multiterminal [3], [10].
For nonplanar graphs the derivation can easily be
generalized by constructing several loops in the graph.
One can start with any tree in the graph and can construct
a set of loops by adding to the tree one by one a
branch that does not belong to the tree, called a link.
The link has a current that is now the loop current in the
same way as it was for the planar graph in Figure 1. If such
loops are constructed for all the links, we have a collection
of loop currents for the graph. Any branch current is
then the algebraic sum of the loop currents of the loops
where that branch is involved. For any node one can
now make the algebraic sum of all the branch currents
of the branches connected to that node and arrive at an
expression like (3).
Note that KCL and KVL are each other's dual [3]. If we
interchange the voltages and currents, these laws are
transformed into one another. This curious property
can be regarded as a form of analogy.
We can now show another remarkable property,
called Tellegen's theorem. It tells that the dual vectors
of currents in and voltages across the branches of an
electrical network are orthogonal [1, pp. 19-21], [4]. In
other words, this theorem says that the sum S =∑Ib.Ub
of the products of the current Ib and voltage Ub in all
branches of a network is zero.
The proof follows easily in two steps. First the voltage
in branch b, that is connected to nodes i and j, is
expressed as the difference Ub = Vi - Vj of the node potentials
Vi and Vj. When branch b is part of loops k and
m, that branch current Ib can be expressed in terms of
the loop currents as follows Ib = Jk - Jm. These two steps
4The continuity principle for electrical networks holds too for the components.
This principle can be considered as a special case of Kirchhoff's
KCL.
FOURTH QUARTER 2022
Figure 1. Network example for the derivation of KCL.
quadruple the number of terms in S. Next, consider any
node n and a loop l that crosses that node. There are
only two terms in S that carry the product of the variables
Vn and Jl and these are +Vn..Jl and -Vn.Jl since the
loop l enters node n with one branch and leaves with
another. Hence these terms cancel, and the theorem is
proven since this argument is valid for all pairs of nodes
and loops. This theorem can also be proven with the
help of the node-to-branch incidence matrix [1].
As we have already noted, KVL and KCL are usually
based on the definitions and laws of physics. However, it
turns out to be possible to derive Kirchhoff's laws without
consideration of the laws of physics, purely using
notions of numbers, graphs, or networks.
III. Invariants and Symmetries in Number Sets
It is surprising that the derivation of Kirchhoff's laws in
the previous section remains valid without reference to
the physics of currents and voltages, but purely using
properties of numbers. We can consider sets of arbitrary
numbers Ni, Nj and Nk and write algebraic identities
for these sets:
(Ni − Nj) + (Nj − Nk) + (Nk − Ni) = 0
Next, we define new numbers:
Nij = (Ni − Nj)
Njk = (Nj − Nk)
Nki = (Nk − Ni)
that immediately lead to an equation:
Nij + Njk + Nki = 0
This expression has the same form as (1) or (3) and
hence reminds us of the KVL or KCL. That general approach
was already mentioned in the book [1, pp. 114-
115] for numbers assigned to the nodes of a network.5
Remarkable is too that a law corresponding to Kirchhoff's
laws applies to the node numbers of a network.
5In [1, pp. 114-115] it is shown that the formulation of Kirchhoff's laws is
preserved for linear or nonlinear functions of the numbers assigned to
the nodes of a network.
IEEE CIRCUITS AND SYSTEMS MAGAZINE
51
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