Instantaneous Power Theory Revisited with Classical Mechanics

TL;DR

This work proposes a physics-based reinterpretation of instantaneous power in electric circuits by treating voltages and currents as generalized velocities and forces within a differential-geometric Lagrangian framework. The central result is a compact multivector expression $\hat{W}$ that links instantaneous power to generalized kinetic energy and the geometric frequency operator, enabling a decomposition into active and reactive components via Frenet-frame invariants. The framework maps to electrical and magnetic domains through corresponding generalized quantities and accounts for losses, providing a robust set of formulas ($\hat{S}$ and $\hat{R}$) for a coordinate-free interpretation of power. Through stationary and non-stationary, sinusoidal and non-sinusoidal examples, the authors demonstrate how active power arises from velocity-force interactions while reactive power is tied to angular-momentum-related terms, such as centrifugal effects. The approach offers physical insight and a potential path toward improved control of unbalanced and non-sinusoidal power systems, with future work targeting nonlinear components and dynamic performance optimization.

Abstract

The paper revisits the concepts of instantaneous active and reactive powers and provides a novel definition for basic circuit elements based on quantities utilized in classical mechanics, such as absolute and relative velocity, momentum density, angular momentum and apparent forces. The discussion leverages from recent publications by the authors that interpret the voltage and current as velocities in generalized Lagrangian coordinates. The main result of the paper is a general and compact expression for the instantaneous active and reactive power of inductances, capacitances and resistances as a multivector proportional to the generalized kinetic energy and the geometric frequency multivector. Several numerical examples considering stationary and transient sinusoidal and non-sinusoidal conditions are discussed in the case study.

Figures (21)

• Figure 1: Representation of momentum $\boldsymbol{\rm p}$ and angular momentum $\boldsymbol{L}$ for a point particle of mass $m$ and position vector $\boldsymbol{r}$. The particle rotates with angular velocity vector $\boldsymbol{\omega}_r$ with respect to the origin $O$. By construction, $\boldsymbol{L} \, \| \, \boldsymbol{\boldsymbol{\omega}_r}$.
• Figure 2: Left: $\boldsymbol{v}$-controlled conductance; Right: $\boldsymbol{\imath}$-controlled resistance.
• Figure 3: A point spherical particle that rotates with constant radius and angular speed is analogous to a stationary balanced sinusoidal 3-phase system.
• Figure 4: A 3-phase capacitor with stationary balanced sinusoidal AC voltage is always subject to a current that is centrifugal: $\boldsymbol{\imath} = C\boldsymbol{\omega}_{\varphi} \times (\boldsymbol{\omega}_{\varphi} \times \boldsymbol{\varphi})$. The relative, Coriolis, and Euler components are zero.
• Figure 5: Balanced 3-phase capacitor with stationary unbalanced sinusoidal AC voltage: Relative, Coriolis, Euler and centrifugal components of current. The Coriolis and Euler components are opposite to each other.