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arXiv:2105.07762v1 [math.DG] 17 May 2021
1
A Geometrical Interpretation of Frequency
Federico Milano, Fellow, IEEE
Abstract—The letter provides a geometrical interpretation of frequency
in electric circuits. According to this interpretation, the frequency is
defined as a multivector with symmetric and antisymmetric components.
The conventional definition of frequency is shown to be a special case of
the proposed theoretical framework. Several examples serve to show the
features, generality as well as practical aspects of the proposed approach.
Index Terms—Frequency, differential geometry, curvature, inner prod-
uct, wedge product, geometrical product.
I. INT ROD UC TI ON
In power system applications, the frequency of an ac signal is
conventionally defined as the time derivative of the argument of the
cosine function of the signal itself [1]. This definition appears to have
some issues. First, it depends on the representation of the signal itself.
This has led to a tremendous number of publications, each of which
using as starting point a different representation [2]. Second, the
value of the frequency often depends on the transformation utilized to
represent the ac signal. A good criterion to decide if a transformation
is robust is to check whether a signal can be fully reconstructed
to its original state if the inverse transformation is applied to the
transformed signal [3]. This is a sensible criterion but does not
guarantee the correctness and consistency of the estimation of the
frequency itself. The value of the estimated frequency should be
always the same (invariant) independently from the transformation.
A third issue with the common definition of frequency and a large
number of existing techniques to estimate the frequency is that
they do not account for variations of the magnitude of the signal.
The theoretical framework and definition of generalized frequency
proposed in this paper address the issues above.
II. OU TL IN ES
We first provide some outlines of operations with vectors and space
curves. Let x= (x1, x2,...,xn)and y= (y1, y2,...,yn)be two
n-dimensional vectors in Rn. The inner product is defined as:
x·y=
n
X
i=1
xiyi.(1)
For example, in R3,x·y=x1y1+x2y2+x3y3. The inner product
is symmetric, associative, and commutative. In particular, the inner
product of a vector by itself gives:
x=|x|=√x·x.(2)
The outer product is defined as:
x⊗y=
x1y1. . . x1yn
.
.
.....
.
.
xny1. . . xnyn
.(3)
The wedge product is defined as:
x∧y=x⊗y−y⊗x.(4)
F. Milano is with School of Electrical and Electronic Engineering, Uni-
versity College Dublin, Belfield Campus, Dublin 4, Ireland. E-mail: fed-
erico.milano@ucd.ie
This work is supported by the European Commission by funding F. Milano
under project EdgeFLEX, Grant No. 883710.
For example, in R3, the wedge product gives:
x∧y=
0b12 −b31
−b12 0b23
b31 −b23 0
,(5)
where bij =xiyj−yixj. The result of the wedge product is a
bivector. In the reminder of this letter it will be indicated with an
uppercase bold symbol, e.g., B=x∧y, where Bis a skew-
symmetric matrix. The wedge product is antisymmetric, associative,
and anti-commutative. The latter means that x∧y=−y∧xand,
consequently x∧x=0. In R3, the wedge product is similar to the
cross product x×y, although the result of the cross product is a
vector not a tensor. For the developments of this letter, it is relevant
to note that in Euclidean metric, the magnitude of a bivector is given
by:
|x∧y|=|B|=v
u
u
t
n
X
i=1
n
X
j>i
b2
ij .(6)
For example, in R3:
|x∧y|=qb2
12 +b2
23 +b2
31 .(7)
The geometric product is defined as:
xy =x·y+x∧y.(8)
The result of the geometric product, which is called multivector,
consists of two components. The first component, x·y, is a scalar
that represents the projection of yonto the vector x. The second
component, x∧y, represents a bivector orthogonal to the space
defined by the vectors xand y. It may seem strange at first to sum a
scalar with a bivector but this is exactly the same kind of operation
that is intended when one writes a complex number as a+b. Section
IV shows that, in fact, the algebra of complex numbers is a special
case of the algebra of multivectors.
In this work, we are interested in time-dependent n-dimensional
curves (or trajectories), i.e., x(t) = (x1(t), x2(t), . . . , xn(t)), where
tis time. The time derivative of xis defined as:
x
′=dx
dt = (x′
1, x′
2,...,x′
n).(9)
From the geometrical point of view, x
′is the tangent vector of the
curve x. Let us define sas the arc length of the curve x, then the
following property holds:
s′=ds
dt =√x′·x′=x′.(10)
It is important to note that the arc length sand thus also its derivatives
are invariant with respect to the system of coordinates.
It is relevant to define the derivative of the curve xwith respect
to s, as follows:
˙
x=dx
ds =dx
dt
dt
ds =x
′
x′,(11)
where we have used (10) and the identity dt/ds = 1/s′. From (11),
it follows that ˙
x·˙
x= 1. The vector ˙
xis tangent to xand that the
tangent vector to a curve is the unit vector if the arc length is chosen
as a parameter.
2
Finally, it is relevant to recall the definition of another invariant
quantity, namely the curvature, that plays an important role in
differential geometry. The curvature is defined as:
κ=|˙
x∧¨
x|=|x′∧x′′|
(x′)3,(12)
where
¨
x=d˙
x
ds =x
′′
(x′)2−x′′
x
′
(x′)3(13)
is the tangent vector to ˙
xand satisfies the condition ˙
x·¨
x= 0.
We are now ready to present the main contribution of this work.
III. FRE QU EN CY A S A MU LTI VE CT OR
Let us start with the vector of the magnetic flux, ϕ. According to
the Faraday’s law of induction, one has:
−ϕ
′=v,(14)
where the minus accounts for the Lenz’s law but is not crucial for
the discussion below. On the other hand, it is important to note that
ϕdoes not need to be known or to be measurable. In the context
of this work, ϕserves only to define the macroscopic effect of the
magnetic field. In this context, the most important property of ϕis
that its time derivative is the vector of the voltage. If one interprets
the components of the vector of the flux as the coordinates of a curve,
say x=−ϕ, then the voltage v=x
′is the tangent vector to this
curve.
According to the definitions given in Section II, one has:
s′=|ϕ
′·ϕ
′|=ϕ′=v , (15)
and
˙
x=−˙
ϕ=ϕ′
s′=v
v,(16)
and
¨
x=−¨
ϕ=v
′
v2−v′
v
v3.(17)
Since ˙
x·¨
x= 0, one obtains:
0 = ˙
ϕ·¨
ϕ=v·v
′
v3−v′
v·v
v4(18)
which leads to:
ρv=v′
v=v
v·v
′
v.(19)
Similarly, from the definition of curvature in (12), one obtains:
κv=|˙
ϕ∧¨
ϕ|=
v
v∧v′
v2−v′v
v3
,(20)
and remembering that v∧v= 0, one obtains:
κv=
v
v∧v′
v2
,(21)
We define the magnitude of the frequency of v, say ωv, as:
ωv=v κv.(22)
This definition, while admittedly a little obscure at this point, will
be apparent in the examples presented in Section IV. From (21) and
(22), one obtains:
ωv=
v
v∧v′
v
,(23)
and, hence, we can define the bivector Ωvas:
Ωv=v
v∧v′
v.(24)
Based on (19) and (22) and on the definition of geometric product
given in (8), we can finally provide the following novel and most
important expression of this work:
ρv+Ωv=vv
′
v2,(25)
where we define the term ρv+Ωvas the generalized frequency of
the voltage v. The left-hand side of (25) depends only on geometric
invariants, namely v=s′,v′=s′′ and the components of the bivec-
tor that define the magnitude of the curvature κv. The generalized
frequency, thus, does not depend on the system of coordinates with
which vis represented or measured, nor the number of “dimensions”
where vis defined. It is also interesting to observe that frequency is
defined in (25) as the sum of a symmetric (ρv) and an antisymmetric
term (Ωv). Finally, we note that (25) has been obtained without
any assumption on the dynamic behavior of the components of v.
Unbalanced and/or non-sinusoidal conditions, multi-phase systems
and even dc systems are consistent with this definition.
IV. EXAMP LE S
The examples presented below are aimed at illustrating the features
of the generalized frequency. The first two examples show that,
in stationary conditions, (25) leads to the well-known and widely
accepted definition of frequency in ac systems. Examples 3 and 4
illustrate the special cases of transient balanced three-phase systems
and dc systems, respectively. Example 5 extends the definition of
generalized frequency to the current. Finally, Example 6 illustrates
the link between the generalized frequency and the generalized
instantaneous reactive power proposed in [4]–[6] and shows a simple
way to estimate the generalized frequency in practice.
Example 1: Let us consider a stationary single-phase voltage with
constant angular frequency ωoand magnitude V. The voltage vector
can be defined as:
v=Vcos(θ)e1+Vsin(θ)e2=v1e1+v2e2,
where θ=ωot+φand (e1,e2)is the canonical basis of the system,
with e1and e2orthonormal vectors. Then, v=|v|=Vand:
v
′=v′
1e1+v′
2e2=−ωov2e1+ωov1e2,
from which one can deduce that, as expected:
ρv=1
V2(v1v′
1+v2v′
2) = ωo
V2(−v1v2+v2v1) = 0 ,
ωv=1
V2|v1v′
2−v2v′
1|=ωo
V2|v2
1+v2
2|=ωo.
It is relevant to note that, in R2, multivectors are isomorphic to
complex numbers. In fact, the bivector Ωvis:
Ωv=ωo(e1∧e2) = ωo.
where the imaginary unit is defined as ≡e1∧e2.
It is also worth observing that, in two dimensions, the curvature is
defined as [7]:
κv=dθ
ds =dθ
dt
dt
ds =θ′
s′=θ′
v,(26)
which is valid in any transient and non-sinusoidal conditions. Using
the chain rule and recalling (22), one has that θ′=ω, which is the
commonly accepted definition of frequency [1].
3
Example 2: We consider a stationary balanced three-phase system.
The voltage vector is:
v=Vsin(θa)ea+Vsin(θb)eb+Vsin(θc)ec
=vaea+vceb+vcec,
where Vis constant; θa=ωot,θb=θa−αand θc=θa+αwith
ωoconstant and α=2π
3; and (ea,eb,ec)is the canonical basis of
the system, with ea,eband ecorthonormal vectors. Then:
v2=|v|2=V2(sin2θa+ sin2θb+ sin2θc) = 3
2V2,
and
v
′=ωoVcos(θa)ea+ωoVcos(θb)eb+ωoVcos(θc)ec
=v′
aea+v′
ceb+v′
cec.
Then, one has:
ρv=vav′
a+vbv′
b+vcv′
c
v2
=ωoV21
2(sin 2θa+ sin 2θb+ sin 2θc)
v2= 0 ,
Ωv=1
v2
0vav′
b−vbv′
avav′
c−vcv′
a
vbv′
a−vav′
b0vbv′
c−vcv′
b
vcv′
a−vav′
cvcv′
b−vbv′
c0
,
ωv=1
v2q(vav′
b−vbv′
a)2+ (vbv′
c−vcv′
b)2+ (vcv′
a−vav′
c)2
=pω2
oV4(2 sin2(α) + sin2(2α))
3
2V2=ωo.
Example 3: We consider a balanced tree-phase system in transient
conditions. For illustration, we use the dqo reference frame. The
voltage vector is v=vded+vqeq+voeo, where (ed,eq,eo)is
the canonical basis of the system, with ed,eqand eoorthonormal
vectors, and the vectors edand eqare rotating at angular speed ωo.
Since the system is balanced, vo= 0. Then, v2=v2
d+v2
q, and:
v
′= (v′
d−ωovq)ed+ (v′
q+ωovd)eq= ˜v′
ded+ ˜v′
qeq,
where, assuming that the q-axis leads the d-axis, one has:
e′
d=ωoeq,e′
q=−ωoed.
The components of the generalized frequency are:
ρv=vdv′
d+ωovdvq+vqv′
q−ωovdvq
v2=vdv′
d+vqv′
q
v2,
Ωv=0vd˜v′
q−vq˜v′
d
vq˜v′
d−vd˜v′
q0,
ωv=v′
qvd+ωov2
d−v′
dvq+ωov2
q
v2=ωo+v′
qvd−v′
dvq
v2,
The equations above show that the definition of the Park vector as
v=vd+vqand the time derivative in the Park reference frame,
namely d
dt +ωo, are an equivalent formulation for balanced three-
phase systems in transient conditions [8].
Example 4: We show that (25) is valid also for dc voltages. In dc
circuits, the voltage has only one component along the unique basis
of the system, say edc, hence, v=vdc edc and v
′=v′
dc edc. From
the definitions of inner and wedge product one has:
v·v
′=vdc v′
dc ,v∧v
′=0.
In dc, then, the generalized frequency is equal to ρv=v′
dc/vdc and,
as expected, ωv=|Ωv|= 0.
Example 5: Similarly to the voltage, one can define the generalized
frequency of the current. Consider the vector of the electric charge q
as an abstract curve in Rn. This vector does not have to be intended
as a charge moving in space, but rather as the macroscopic effect of
the electric field in a given part of a circuit. Then:
q
′=ı,(27)
and, analogously to the discussion on the voltage, the generalized
frequency associated with the current is given by:
ρı+Ωı=ıı
′
ı2.(28)
In general, for any given element of a circuit, the generalized
frequency of the voltage is not equal to that of the current. Rel-
evant exceptions are resistances. For a balanced resistive branch,
vv
′=R2ıı
′, which indicates that, from a geometrical point of view,
resistances are scaling factors.
Example 6: We further elaborate on the link between voltage and
current vectors. For balanced capacitive elements, one has:
ı=Cv
′,(29)
Merging (25) and (29), one obtains:
ρv+Ωv=vı
Cv2=p−Q
Cv2=−Q
Cv2,(30)
where p=v·ıis the instantaneous active power, which is null for
capacitances as the current is orthogonal to the voltage, and Q=ı∧v
is the generalized instantaneous reactive power as defined in [6]. It
is interesting to note that (30) provides an expression to calculate the
frequency of an electric circuit in any transient condition through
instantaneous voltage and current measurements. Interestingly, no
discrete Fourier transforms with mobile windows or other standard
numerical techniques are required.
V. CONC LUSI ON S
The proposed formal framework generalizes and solves known
issues of the conventional definition of frequency. A strength of
the proposed approach is that it is based on invariant quantities,
hence it is compatible with any reference frame, e.g., abc,dqo and
even dc circuits. It is interesting to note that the proposed approach
defines the frequency as a geometrical object with symmetric and
antisymmetric parts. The last example shows the link between the
generalized frequency and the power of a circuit. This link between
geometry and energy appears worth further research.
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