IEEE Circuits and Systems Magazine - Q4 2022 - 52

Hence, we arrive at a very general law for sets of numbers
with the form of Kirchhoff's laws.
Along the same line, we can derive two invariances.
First, we consider a set of arbitrarily chosen real numbers
such as, for example, a, b, and c. Form pairs of
these real numbers where each real number occurs in
only two pairs and this as the first and as the second
real number. For each of these pairs, make the difference
between the first and second real numbers and
add up the differences. For the real numbers a, b and c
it is immediately clear that the following equation holds:
(a−b) + (b−c) + (c−a) = 0
(4)
This is a first invariant. Indeed, (4) holds for all possible
values of the variables. This also expresses a form
of conservation.
There is also a second invariant. In this case we add
to the numbers a, b and c in the left side of the formula
(4) a constant k. After this transformation, the righthand
side remains equal to zero and we get:
[(a+k)−(b+k)] + [(b+k)−(c+k)] + [(c+k)−(a+k)] = 0 (5)
The invariant that follows from this transformation
can be regarded as a symmetry of the variables but not
of the topology [29]. Indeed, (5) also remains valid for
all possible values of the constant k.
If we set the constant k equal to 0 in (5), then we find
the invariant (4). The first and second invariant can
therefore be derived from each other.
Note that equations (4) and (5) are identities using only
addition and subtraction. The left-hand side of (4) and of
(5) is consequently a function that is linear in each of
the variables.
After entering a constant k in (4), it holds that:
(k.a − k.b) + (k.b − k.c) + (k.c − k.a) =
k.[(a − b) + (b − c) + (c − a)] = 0
The left-hand side of (4) is consequently a function
that is scale invariant [13]. Equation (4) can be further
generalized for time and frequency dependent functions
and by applying Kirchhoff operators to it [1, pp.
11-14, 113-115].
x31
x12
x10
x20
x30
Figure 2. Graphical illustration of the invariant with lengths
along a straight line.
52
IEEE CIRCUITS AND SYSTEMS MAGAZINE
1
2
x23
3
Note also that the formulation of the invariances and
symmetries for sets of real numbers is also valid for natural
numbers and complex numbers. It is sufficient that
the numbers form a field [14]. A further generalization is
possible for vectors, matrices, and tensors.
IV. Invariants and Symmetries for Measurable
Quantities
It turns out to be possible to generalize Kirchhoff's laws
and analogous laws, starting from the measurement
concept. We base this generalization on the following
principles:
■ measuring is the assignment of a real number to
what is being measured.
■ since measurement results are real numbers, they
also have the properties of real numbers.
If
the currents in and the voltages across the
branches of a network can be quantified, then laws in
the form of Kirchhoff's laws can be obtained. We can
derive invariants based on the properties of sets of
real numbers.
To find a first invariant consider a set of, for example,
three measurement results: X10, X20 and X30. Then
make pairs of the measurement results where each
measurement result occurs in only two pairs. For
each of these pairs, calculate the difference between
the first and second real numbers and call it X12, X23,
and X31 such that:
X12 = X10 − X20
X23 = X20 − X30
X31 = X30 − X10
Then for the sum of the differences the following
identity holds:
(X10 − X20) + (X20 − X30) + (X30 − X10) = 0
This immediately leads to a second identity:
X12 + X23 + X31 = 0
(6)
(7)
These two identities are invariants. Moreover, note
that the measurement results X10, X20 and X30 may take
any real value.
The invariants (6) and (7) are clarified with the
lengths along a straight line in the graphical model
of Figure 2.
Note that the invariants (6) and (7) correspond to the
invariant (4). It can be shown easily that the following
invariant holds too:
X01 + X12 + X23 + X30 = 0
If the reference point 0 of the measurements X10, X20
and X30 is shifted, we find a third invariant which is
FOURTH QUARTER 2022
IEEE Circuits and Systems Magazine - Q4 2022

Table of Contents for the Digital Edition of IEEE Circuits and Systems Magazine - Q4 2022
