IEEE Circuits and Systems Magazine - Q4 2022 - 50

proposes a derivation of relations with a formulation of
Kirchhoff's laws, assuming invariances and symmetries
for a set of real numbers. It is strange that no appeal has
to be made to the laws of physics. This simple and direct
derivation opens the avenue to generalize Kirchhoff's laws.
Laws with a formula like that of Kirchhoff's laws can
also be found in other fields of science than electrical
network theory. Moreover, analogous laws appear to apply
to measurable quantities. This points to the possibility
of developing an overarching theory and unifying
various theories. Such a derivation forms the basis for
a generalized network, structure, and systems theory.
The fact that no use is made of physical explanations
raises questions about the deeper nature of reality.
II. Kirchhoff's Laws
Kirchhoff's laws play an important role in the theory of
electrical networks and also in the practice of electronic
circuits, electric systems, and electrical power grids [3],
[5], [8]. These are connection laws that, together with
the models of the components, describe the behaviour
of networks. The interconnection pattern of an electrical
network can also be modelled by a directed graph.
A network consists of branches connected to nodes.
In a simple network, components with two terminals are
considered as a branch. A voltage is applied across a
branch and a current flows through the branch.1 The
currents are considered positive when they arrive via
a terminal into a node or component. Kirchhoff's current
law describes the relationship between the currents
in a node of a network. Kirchhoff's voltage law determines
the relationship between the voltages across
the branches in a loop of the network. In this case, the
branches of a circuit are connected to each other in
such a way that they form a closed loop.
Kirchhoff's voltage law (KVL) can be derived very
easily from the concept of potential. An electrical voltage
between two nodes is defined as the difference in
potential between these nodes. Consider three nodes
1, 2 and 3 connected by branches that together form a
loop.2 If we call the potential in these nodes V1, V2 and
V3, then the voltages across the branches are:
U12 = V1 − V2
U23 = V2 − V3
U31 = V3 − V1
1We consider directed branches with an incoming and outgoing current
flowing from a positive to a negative potential through the terminals.
2To keep the formulas simple, a loop with only three branches is considered.
If
these equations are added together, we immediately
find:
U12 + U23 + U31 = (V1 − V2) + (V2 − V3) + (V3 − V1)
U12 + U23 + U31 = (V1 − V1) + (V2 − V2) + (V3 − V3)
U12 + U23 + U31 = 0
(1)
This equation expresses KVL. The sum of the voltages
across branches of a loop is equal to zero. An extension
for a loop with any number of branches is obvious.
Other elements of the network topology and electrical
circuit theory are not involved.
In the electrical network theory, the KVL is explained
with the concept of potential and the definition of voltage.
KVL can also be derived from a special case of one
of Maxwell's laws for electromagnetism [8].
A less known formulation of KVL is obtained by considering
the voltages U10, U20 and U30 for the nodes 1, 2
and 3, which are measured with respect to a common
reference point 0. It's clear that:
(U10−U20) + (U20−U30) + (U30−U10) = 0
(2)
Kirchhoff's current law (KCL) can also be easily derived
based on the definition of loop currents for planar
networks. Loop currents are also called mesh currents.
In the mesh current method for the analysis of planar
networks, one defines a current in each of the closed
loops always measured in the same way (e.g., clockwise)
[4, p. 437], [5], [6], [9]. Planar networks can be drawn on
a sheet without the branches crossing each other.3 The
loop currents are in fact fictitious currents.
In a branch that is part of two closed loops i and j,
the loop currents Ji and Jj flow in opposite directions
and the branch current is the difference Iij = Ji - Jj.
Let us now consider a slightly more complex network
(see Figure 1).
For a node n connected to three branches, we can
write the branch currents I12, I23 and I31 as follows:
I12 = J1 − J2
I23 = J2 − J3
I31 = J3 − J1
Adding up these equations leads to:
I12 + I23 + I31 = 0
(3)
This equation represents KCL for a node with three
branches. The sum of the currents in a node is zero. Conversely
it is possible to derive the loop currents J1, J2 and
J3 from the branch currents I12, I23 and I31. However, the
system of equations is underdetermined. This means
3For deriving the KCL, we consider only one node with the connected
branches. A network which is completely planar is not required.
Hubert Van Belle was with the Mechanical Engineering Department, KU Leuven, B-3001 Leuven, Belgium (e-mail: hubert.vanbelle@skynet.be) and
Joos P. Vandewalle is with the Electrical Engineering Department, KU Leuven, B-3001 Leuven, Belgium (e-mail: Joos.Vandewalle@esat.kuleuven.be).
50
IEEE CIRCUITS AND SYSTEMS MAGAZINE
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IEEE Circuits and Systems Magazine - Q4 2022

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