IEEE Circuits and Systems Magazine - Q4 2022 - 54

By considering duality as a starting principle, it is possible
to derive the KCL from the KVL without introducing
loop currents. Duality can be regarded as a symmetry.
The dual laws of Kirchhoff are very important in the
proof of Tellegen's theorem. This interesting energy theorem
is still not so widely known [1, pp. 19-21], [4]. Tellegen's
theorem expresses the orthogonality of currents
in and voltages across the branches of an electrical network.
As shown Kirchhoff's laws can be derived from
symmetries which make it possible to acquire objective
knowledge. Pairs of dual measurement results offer the
possibility to generalize Tellegen's theorem and to define
generalized energy concepts.
As already noticed, different observers performing measurements
with a different reference point must be able to
derive the same laws from their measurement results. Invariance
for transformation of reference points and for scale
invariance are conditions that enable objective knowledge
and impose requirements for the laws of physical systems.7
VI. Analogous Laws and Unification of Sciences
Kirchhoff's laws have a remarkably simple formulation.
Moreover, in various scientific domains we find laws
with the linear form of Kirchhoff's laws. For example,
the KCL and KVL are analogous to the equilibrium and
compatibility condition of the mechanics and strength
theory. An interesting example based on these conditions
is the thought8 experiment to prove the friction
law of Amontons.9 Analogous laws can also be discovered
in hydraulics (fluid mechanics) and for transport
phenomena.10 In these domains, however, different
physical laws apply. Kirchhoff's laws and the analogous
laws appear to have a formulation that is independent of
the scientific domain.
The analogy with electrical networks can be used to
describe analogous systems. Equivalent electrical networks
were used to model these systems. A striking example
is the modelling of Schrödinger's equation for the
wave function [20], [21], [22]. Kirchhoff's laws could therefore
be regarded as a basic axiom of quantum mechanics.
The analogies point also to the possibility of developing
an overarching theory that unifies laws with
the form of Kirchhoff's laws. The purpose of unification
is to 'explain more with less'. This can be done, for
7The requirement of scale invariance restricts the formulation of the
laws. This is also the case for dimensional analysis which can be derived
from scale invariance. Scale invariance leads to power laws [5],
[13], [16].
8Tellegen's theorem is analog to the theorem of virtual work [17], [18].
9The equilibrium and compatibility condition are used in the thought
experiment of Appendix A of [19]
10The use of the word 'current' in 'electrical current' is an example of a
hydraulic analogy.
54
IEEE CIRCUITS AND SYSTEMS MAGAZINE
example, by means of abstraction and generalization.11
It appears possible to generalize Kirchhoff's laws and
to unify important branches of engineering sciences.
The modelling of multi-physics systems by graph and
network theories is becoming increasingly important in
engineering. The unification with an overarching theory
has a large reach and is also one of the ambitions of the
bond graph theory [24].
Unification through abstraction and generalization
also offers the possibility to structure a collection of
separate theories that are studied independently one of
another. We can probably apply the generalized network
theories in more sciences than is currently the case. Unification
also allows for a 'reparcelling' of the scientific
landscape. Separately studied theories can be reduced
to an overarching whole. Moreover, we can axiomatically
restructure domains of science by considering the generalized
form of Kirchhoff's laws as the basic axiom.
Starting from Kirchhoff's laws for the connections
and a continuity principle for the components, we can
prove Tellegen's theorem [1], [3]. It is a very powerful
yet easy to demonstrate energy theorem. Tellegen's
theorem expresses the conservation of energy across
the components of a network. This very general theorem
applies to linear and non-linear, time-invariant, and
time-variant networks. Many theorems of electrical network
theory can be derived from it.
Energy theorems such as the 'adjoint network theory'
[10], [11] are examples. At first sight it is strange
that Tellegen's theorem remains valid if we consider a
given and an adjoint network. Both networks have the
same topology, but their components may be different
and even correspond to different physical objects. This
leads to abstract energy concepts. Important is the application
of the adjoint network theory for sensitivity
analysis and tolerance analysis of networks. Energy
theorems lead also to variational approaches which are
applied for the optimization of networks and structures
[11]. Sensitivity analysis can be used in the 'hill climbing
technique' for mathematical optimization [25], [26].
A generalization of Tellegen's theorem for systems is
Lee's theorem [12]. Tellegen's generalized theorem and Lee's
theorem often offer interesting possibilities for approaching
and solving problems in the study of networks, structures,
and systems. They make it possible to apply the 'tools' from
the electrical network theory outside this domain.
Laws similar to Kirchhoff's laws can be found not
only in electrical network theory but also in other fields
of science. This means that the theorems that can be
11The energy concept and energetic methods also make it possible to link
different theories from physics and engineering sciences. The first law of
thermodynamics, for example, is a conservation law which establishes a
connection between thermal energy and mechanical energy [23].
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