CHAPTER 2 Power Theory in Electrical Circuits
1. Introduction
Defining the concept of power means opening a can of ‘scientific worms’.
Griffiths (2013:357) defines power as the rate of work:
where work or energy E · J is deployed in per unit (p.u.) time and volume (i.e., energy has both time and space attributes).
However, Lehrman wrote a paper entitled “Energy is Not the Ability to Do Work” (Lehrman, 1973).
Valkenburg (1964) defines power as only a time-derivative of energy:
He factorizes the expression for the time derivative of energy into a product of voltage and current:
Einstein’s definition of energy
has long been accepted, but according to Okun (1989, 2006), it is incorrect.
Moreover, the concept of energy is related to the concept of mass, and mass itself is a ‘messy’ concept, as discussed by Psarros (1996) in his article, entitled, ‘The Mess with the Mass Terms’.
No wonder that famous scientists like Feynman (1961) declare that physicists do not know what energy is.
The concept of energy, born in a period spanning the years 1800-1830 (Coopersmith, 2010), remains a subtle, elusive, unobservable, and evolving concept.
And what is true for the concept of energy is also true for the concept of electrical power. Power remains a subtle, elusive, unobservable, and evolving concept.
2. A Critical Assessment of the Existing Power Paradigm
The current paradigm expresses power in four mathematical forms: 1) power as a real-valued function in real algebra, 2) power as a complex-valued function in the algebra of complex numbers, 3) power as a trigonometric expression in trigonometric algebra, and 4) power as a vector-valued function in vector calculus.
In DC circuits, power is a real-valued function expressed as: p = vi.
In AC circuits, the current paradigm, based on Steinmetz’s symbolic method, is an entanglement of three different mathematical formalisms:1) trigonometric algebra, 2) complex algebra, and 3) vector calculus.
In addition, Janet’s expression for complex power: Ṡ = V̇I * is mistakenly considered to be equivalent to Steinmetz’s formulation.
2.1 Steinmetz's Assumptions Underpinning His Symbolic Method: A Critical Review
In 1890 and 1891, Steinmetz introduced the representations of electrical magnitudes as vectors in polar coordinates; he mentions Blakesley, Föppl, and Kapp as predecessors in using vectorial representations (Steinmetz, 1890, 1891).
In his communication at the International Electrical Congress, Steinmetz proposed the representation of electrical magnitudes as complex variables (Steinmetz, 1893). Therefore, he introduced a double mathematical representation: 1) electrical magnitudes as algebraic complex numbers (in rectangular coordinates), and 2) electrical magnitudes as geometric (polar) vectors.
In his book, Steinmetz (1897) brings new terms, concepts, and ideas such as:
- The terms reactive power and apparent power
- The idea that the power factor could differ from unity, even if the voltage and current are in phase
- The idea that electrical power is a wave of double frequency
In a paper published in 1899, Steinmetz expresses power as P = [Ėİ], in which the square brackets symbol [...] “denotes the transfer from the frequency Ė and İ to the double frequency of P.” His expression implies that the vector diagram for power should be separated from the vector diagram for voltage and current.
In the same paper, he gives a new interpretation of the symbol ‘j’ as follows:
Since j2 = –1, that is, 180° rotation for E and I, for the double frequency vector P, j2 = +1, or 360° rotation.
He further states that for the double frequency calculations,
(Steinmetz 1899:270).
He adds that “the introduction of the double frequency vector product P = [Ė İ] brings us outside of the limits of algebra...and the commutative principle of algebra: a × b = b × a does not apply any more... [EI] unlike [IE].”
On the basis of the above conjectures, he derived the expression of power in rectangular coordinates:
(Steinmetz 1899:272).
Steinmetz’s books published in German (Steinmetz, 1900 a) and French (Steinmetz, 1912) repeated the main ideas of the 1899 paper and ensured the wide diffusion of the symbolic method in Europe.
The author of this monograph stumbled over Steinmetz’s assertion that for double frequency (2w), the square of the imaginary j is equal to plus one: j2 = +1. It was obvious that Steinmetz had committed a mathematical blunder, and nothing is so exciting as to discover a failure in the thinking of an important scientist. Steinmetz was an accomplished mathematician, although with an unfinished mathematical dissertation at the University of Breslau. He obtained an additional engineering degree from ETH Zürich. It was surprising for somebody of this stature to state that
For this writer, this discovery represented a stumbling stone, the end of the symbolic method orthodoxy, and the beginning of inquiry into the validity of the symbolic method and the existing power theory paradigm. The path of his subsequent research, and of this monograph, was similar to opening a set of Russian dolls – a riddle, wrapped in a mystery, wrapped in an enigma, wrapped in a puzzle, wrapped in a conundrum, wrapped...The riddle: if Steinmetz is wrong, why is his method heuristically right? The mystery: if Steinmetz is right, how could it be that Janet is also right? The puzzle: could it be that both are wrong, or both are right? The conundrum: could we have a plurality of mathematical expressions for the same physical entity?
Steinmetz’s symbolic method has long been the subject of various debates and criticisms.
Kennelly (1893) supported the use of complex variables to represent electrical magnitudes, whereas Franklin (1903) considered that complex algebra is insufficient for representing the concept of power.
Kafka (1925) stated that a complex representation ‘hides’ the essence of the physical phenomena and considered that the electrical magnitudes should be represented as planar vectors.
Natalis (1924) supported a purely geometric representation, whereas Creedy (1910) proposed a representation of electrical magnitudes based on Möbius algebra.
Punga (1901, 1938) proposed to use, instead of the algebraic product, Grassmann’s geometric product; in fact, he was the first scientist to promote Grassmann algebra in electrical engineering, before Bolinder (1959).
Perrine (1897) stated that complex numbers are not vectors. Besso (1900), a close collaborator of Einstein, criticized Steinmetz for not distinguishing between vector multiplication and complex multiplication. Emde (1901) criticized Steinmetz for confusing vectors with complex numbers.
Patterson (1911) stated that treating harmonic quantities as vector quantities gives wrong results in multiplication. Pomey (1918) stressed the difference between the commutative rule of multiplication in complex algebra and the non-commutative rule of vector multiplication in Gibbs’ vector calculus.
Kennelly and Valander (1919) and Sah Pen-Tung (1936) also criticized Steinmetz for confusing complex numbers with vectors.
Oberdorfer (1929) considered that the use of planar vectors confines the analysis to a two-dimensional space.
However, even an influential textbook authored by M.I.T. professors as late as 1948 stated that there is “a common engineering use of the term vector interchangeably with complex quantity” (E.E. Staff, M.I.T., 1948: 266).
Despite its theoretical inconsistencies, Steinmetz’s symbolic method entered the arsenal of analytical tools as an efficient use of complex numbers for power calculations in AC circuits. It is known under two expressions:
Steinmetz’s expression for power:
and Janet’s expression for complex power:
The symbolic method’s resilience is based on Steinmetz’s huge scientific authority and the weight of authoritative support he received from his peers, such as A.E. Kennelly, P. Janet, C.-F. Guilbert, J.L. La Cour, and O.S. Bragstad. This resilience could be an interesting topic for the sociology or psychology of science.
The author considers that:
- The symbolic method is mathematically inconsistent because it relies on two irreconcilable structures: 1) complex algebra (a commutative algebra) and 2) vector calculus (which is not an algebra).
- The symbolic method considers that electrical magnitudes, from an algebraic point of view, are complex numbers. Complex algebra imposes a limit on the dimensionality of its entities; complex numbers exist only in a two-dimensional (2-D) space.
- The symbolic method considers that electrical magnitudes, from a geometric point of view, are vectors. Vector calculus is axiomatically endowed with two types of multiplication: the inner product and the outer product. The inner or dot product of two vectors results in a scalar. The outer or vector product of two (polar) vectors results in an axial vector. Vector calculus is not a closed set; each type of multiplication results in an entity different from the entities that were multiplied. Vector calculus is not an algebra.
- Vector calculus imposes a limit on the dimensionality of its entities. If voltage and current are considered planar vectors, the result of a vector or outer multiplication is a vector perpendicular to the plane in a third dimension (i.e. it contradicts the assumptions of the two-dimensional representation). In addition, vector calculus permits vector multiplication only in three-dimensional and seven-dimensional spaces (Eckmann, 1943, 2006; Gogberasvili, 2005; Silagadze, 2002). The vector product cannot be extended to spaces of other dimensions.
- The failure of Steinmetz’s symbolic method is rooted in his adherence to Peacock’s principle of permanence. This principle states that algebra should follow the rules of arithmetic, and thus, the multiplication obeys the rule of commutativity, i.e.,
With regard to the expression Ṡ = V̇I* proposed by Janet, there is no mathematical proof why the canonical and axiomatic definition of power p = vi should be rewritten in complex as V̇I*.
The author agrees with the dictum of Steinmetz (1911:540): “In some fields of electrical engineering or of electrical science, we might almost say that we know less now than we knew, or rather believed we knew, a quarter of a century ago. There are things which had been investigated a quarter of a century ago and which were explained in a satisfactory manner to our limited knowledge in the early days, but this explanation does not seem satisfactory now, with our greater knowledge.”
In this monograph, the author re-assesses and rigorizes Steinmetz’s symbolic method and gives a mathematical proof for Janet’s mnemonic formula.
2.2 Steinmetz's Symbolic Method: A Disguised Geometric Algebra
Table 2.1 reformulates Steinmetz’s symbolic method in the mathematical formalism of geometric algebra.
| Steinmetz’s Original Expressions as in His Symbolic Method | Steinmetz’s Expressions Reformulated in Geometric Algebra | Comments |
|---|---|---|
Ė = e1 + je11 Voltage as a complex number. Imaginary unit squares to minus one.j2 = –1 Voltage is represented in the Argand plane of complex numbers C(1, j) | Ē = e1ē1 +e11ē2 Voltage as a grade-1 multivector Unit vectors square to plus one.(|ē1|)2 = (|ē2|)2 = + 1 | Voltage as a complex number is represented in an anisotropic complex plane (1, j). Voltage as a grade-1 multivector is represented in an isotropic plane of real numbers ℜ2 (ē1, ē2) |
İ = i1 + ji11 Current as a complex number. Current is represented in the Argand plane of complex numbers C(1, j) | Ī = i1ē1 + i11ē2 Current as a grade-1 multivector. Unit vectors square to plus one.(|ē1|)2 = (|ē2|)2 = + 1 | Current as a complex number is represented in an anisotropic complex plane (1, j). Current as a grade-1 multivector is represented in an isotropic plane of real numbers ℜ2 (ē1, ē2) |
| [P] = [Ėİ] ≠ [İĖ] [P] = [P]1 + j[P]j = (e1i1 + e11i11) + j(e1i11 – e11i1) Steinmetz defines power [P] as a double-frequency vector product of voltage and current vectors. Steinmetz also represents it as a complex number. Steinmetz’s double representation is mathematically ambiguous. | S = ĒĪ ≠ ĪĒ
S = ĒĪ = Ė · Ī +Ē ∧ Ī
S = P + QJ
P = Ē · Ī as the inner product of voltage and current vectors is a scalar.
P = e1i1 + e11i11 QJ = Ē ∧ Ī as the outer (wedge) product of voltage and vectors is a bivector.QJ = (e1i11 – e11i1)e12 J = e12 is a pseudo scalar and squares to minus one. In Geometric Algebra we define apparent power as a geometric product of voltage and current as grade-1 multivectors. Apparent power is therefore a linear combination of a scalar (active power) and a bivector (reactive power) and it could be (geometrically) interpreted as a spinor. | Steinmetz’s expression for electrical power contains contradicting elements:
|
The following calculations contain the mathematical proof that Steinmetz’s symbolic method is based on geometric algebra:
Calculation of the Inner Product: Active Power
The expression
in geometric algebra is identical to Steinmetz’s expression for active power [P]1.
The expression for active power (inner product of grade-1 multivector voltage and grade-1 multivector current) represents a geometric invariant.
Calculation of the Outer (Wedge) Product: Reactive Power
The expression
in geometric algebra is identical to Steinmetz’s expression for reactive power [P]j.
The expression for reactive power (outer product of grade-1 multivector voltage and grade-1 multivector current) represents a geometric invariant.
In conclusion, the author demonstrates that underlying Steinmetz’s method is a geometric algebra mathematical formalism. Steinmetz’s symbolic method is neither complex algebra nor vector calculus. Steinmetz’s method is based on the old Grassmann-Clifford geometric algebra; it is a geometric algebra in disguise.
It is painful to admit that, for such a long time, power engineers have been calculating power flows, stability, and state estimation without understanding the mathematical foundations of our algorithms. However, to quote Bertrand Russell (1929:58), “The fact that an opinion has been widely held is no evidence whatever that it is not utterly absurd; indeed in view of the silliness of the majority of mankind, a widespread belief is more likely to be foolish than sensible.”
2.3 Rigorization of Janet's Expression
2.3.1 Hilbert Algebra
In a Hilbert space of complex vectors, the multiplication of two complex vectors – x, y follows the rule:
This means that Janet’s mnemonic rule conforms to the rule of multiplication introduced much later by Hilbert’s algebra of complex vector space.
2.3.2 Geometric Reasoning
Represented in an Argand diagram, the voltage and the current coordinates are: v(v′, v″) and i(i′, i″) Active power as vi cos φ is also represented by the surface v′i′ + v″i″; similarly, reactive power vi sin φ is also represented by the difference of surfaces v′i″ – v″i′. These surfaces correspond to geometric invariants representing active and reactive powers. The multiplication of complex voltage × complex current destroys the geometric invariance, whereas the multiplication of complex voltage × the conjugate of complex current preserves the geometric invariance. Janet’s heuristic rule S = V̇I * preserves the geometric invariance of the surfaces corresponding to active and reactive power.
The author’s demonstration is based on a geometric reasoning method developed by Hongbo Li (2008).
2.3.3 A New Definition of Complex Power Based on an Extension of Complex Algebra
Andreescu and Andrica (2006) extend complex algebra and define two types of multiplication of complex numbers:
- A real product of complex numbers
- A complex product of complex numbers
Let us take two complex numbers, using Steinmetz’s notation (Steinmetz, 1900a:179):
The real product ȧ · ḃ is defined as:
Performing all the calculations, we obtain the result:
The complex product a × b is defined as:
Performing all the calculations, we obtain the result:
Replacing the symbols in these equations with Steinmetz’s symbols, we obtain, apart from the sign before j in the imaginary part, Steinmetz’s expressions for power:
and
Thus, we preserve the complex representation of the electrical magnitudes in AC circuits; at the same time, we maintain consistency between the expressions for instantaneous power and complex power:
In addition, the introduction of complex multiplication allows us to keep the expressions of power (active and reactive) consistent with the expression based on Grassmann’s geometric product.
3. Conclusion
The actual paradigm used in power engineering for more than a century is based on a confusion: it mixes Janet’s expression:
with Steinmetz’s expression:
that is re-written, in all textbooks, as:
Underlying Steinmetz’s algorithm or method is Grassmann-Clifford geometric algebra: the electrical entities (voltage and current) are grade-1 multivectors, and apparent power is interpreted as a spinor; active power is interpreted as a scalar, whereas reactive power is interpreted as momentum and represented mathematically as a bivector.
Underlying Janet’s expression for complex power is a Hilbert algebra of complex vector spaces.
The author’s contribution consists in mathematically rigorizing these two different approaches; however, this does not mean that he endorses them. On the contrary, from a physical point of view, they are incorrect. In a short form, the author’s position is a mathematical rigorization but a physical refutation.
In fact, the author dissents from all the existing mathematical methods of representing electrical magnitudes in that he considers it false to represent fundamentally different physical observables (entities) as having the same mathematical identity.
Back to Steinmetz’s symbolic method or Janet’s ubiquitous expression, the question is not which of these two methods will prevail; the question is whether the two have to be buried as twin brothers in the same grave or in two different graves.
The conclusions of this chapter are based on a large body of literature on electrical circuits, power engineering, electromagnetism, electrical machines, and signal processing. The most important contributions and critical opinions on the power paradigm published between 1890 and 2020 were examined, as well as more than 100 textbooks.