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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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