IEEE Circuits and Systems Magazine - Q4 2022 - 53

analogous to (5). For a displacement of the reference
point 0 with a real number Xk it follows immediately
from (6):
[(X10 − Xk) − (X20 − Xk)] + [(X20 − Xk) − (X30 − Xk)]
+ [(X30 − Xk) − (X10 − Xk)] = 0
(8)
The change of the reference point can be considered
as a transformation. The equation (8) describes the invariance
for a transformation. We may also call this invariance
a symmetry.6
If the shift of the reference point Xk in (8) is set equal
to zero, we obtain after simplification (6) and also (7).
The invariants and the symmetry can be derived from
one another.
The action to measure is the allocation of a real number
to the quantity being measured. In addition to a voltage
and a current, we can among others also measure a
pressure and a flow, a force and a speed, a temperature
difference, and a heat flow. Again, this is not restricted
to scalar variables but also valid for vectors. If something
is measurable, it becomes possible to derive laws
with a formula similar to Kirchhoff's laws.
A measurement is made relative to a reference
point. Observers using different reference points to
measure must be able to find the same measure for
a quantity irrespective of their reference point. The
measurement results must therefore be invariant for
the choice of reference point. The symmetry (8) is a
property of reality that makes it possible to acquire
objective knowledge.
It has already been pointed out that (8) is invariant for
a shift Xk. The graphic model shows that we can write that:
X32 = X30 − X20
X32 = X31 − X21
X31 = X30 − X10
X21 = X20 − X10
From these equations it follows that:
X32 = X31 − X21 = (X30 − X10) − (X20 − X10)
Starting from the last term, we also find:
(X30 − X10) − (X20 − X10) = X30 − X20 = X32
This proves that X32 is invariant for a displacement
of the reference point from 1 to 0. This reminds us of
Noether's theorem [34]. With this theorem it is possible,
for example, to show the invariance of a momentum for a
displacement in space. According to Noether's theorem,
every differentiable symmetry corresponds to a conservation
law. Conservation laws consider quantities that
6The concepts of invariance and symmetry are often used interchangeably.
FOURTH
QUARTER 2022
are invariant. Invariants for transformations are also
called symmetries.
Also note that in the derivation of the invariants and
symmetry for measurement results it was only assumed
that the considered quantities are measurable and can
be represented by real numbers. The proofs are based
on the properties of sets of real numbers. They lead to a
'physics-free' approach.
For the application of the invariants and symmetry in
various disciplines of science, no additional information
is needed about domain-specific definitions and laws.
This insight provides the basis for the development of an
overarching and axiomatically structured network and
system theory. Symmetry (8) seems to be the best axiom.
The invariants (6) and (7) can be derived directly from it.
Hence there is no need to rely on the mathematical
models of physics in order to derive Kirchhoff's laws. It
is possible to derive these laws directly from the properties
of sets of real numbers. After all, we can recognize
invariances and symmetries in these number sets.
V. Derivation of Kirchhoff's Laws From Invariances
and Symmetries
As already noted, the simple and linear form of Kirchhoff's
laws is striking. Moreover, the KCL and KVL are
each other's dual. It is possible to derive Kirchhoff's
laws without involving concepts from physics.
To derive Kirchhoff's laws in a general way, we assume
that the voltages and currents in an electrical network
are measurable and can therefore be represented
by real numbers. If the voltage differences between the
nodes of a network are entered in the invariance (7), we
obtain KVL in the formulation (1).
Entering the differences between the voltages of the
nodes with respect to the reference point in the invariance
(6) leads to KVL in the form (2). In the latter case, it
is also possible to replace the voltages in the nodes with
potentials. Indeed, a voltage is defined as a difference
between two potentials:
(V1−V2) + (V2−V3) + (V3−V1) = U12 + U23 + U31 = 0
Starting from the invariance (6), we can also find KCL
in the formulation (3). In this case we are dealing with
branch currents which are defined as a difference between
two fictitious loop currents.
Instead of the invariants (6) and (7), we could also
have taken the symmetry (8) as a starting point to derive
Kirchhoff's laws. It has been shown that this symmetry
and invariants can be derived from each other.
Note that both the KVL (1) and the KCL (3) follow
from the invariance of measurement results. This
explains the duality between the two Kirchhoff's laws.
IEEE CIRCUITS AND SYSTEMS MAGAZINE
53
IEEE Circuits and Systems Magazine - Q4 2022

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