# Geometric Objects: A Quality Index to Electromagnetic Energy Transfer Performance in Sustainable Smart Buildings

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Motivation

#### 1.2. Literature Review

#### 1.3. Contribution

## 2. Geometric Objects

^{3}and are shown in Figure 1.

## 3. The System Model

- -
- Electric power (energy) is dissipated.
- -
- Electric power (energy) in the system derives from an electric field and magnetic field interactions.

## 4. Generalized Poynting Multivector

## 5. Formulation of Electromagnetic Energy Quality Index

#### 5.1. Energy Flow on Electrical Systems

#### 5.2. Electromagnetic Quality Index (EQI)

#### 5.3. Illustrative Comparison between Different Non-Active Electromagnetic Geometric Objects

- Sinusoidal case: $p=q=1\Rightarrow N=\left\{1\right\}$$${\tilde{\Pi}}^{\mathrm{sin}}={\tilde{\mathcal{P}}}_{1}=\underset{{\tilde{\mathcal{P}}}_{{1}_{\mathit{a}\mathit{c}\mathit{t}}}}{\underbrace{\mathrm{Re}\left\{{\tilde{\mathcal{P}}}_{1}\right\}}}+\mathit{j}\text{\hspace{0.17em}}\underset{{\tilde{\mathcal{P}}}_{{1}_{\mathit{non}-\mathit{act}}}}{\underbrace{\mathrm{Im}\left\{{\tilde{\mathcal{P}}}_{1}\right\}}}$$$${\tilde{\xi}}^{\mathrm{sin}}=1+\mathit{j}\frac{\mathrm{Im}\left\{{\displaystyle \underset{\varsigma}{\iint}{\displaystyle \sum _{\mathit{p}\in 1}{\overrightarrow{\mathbf{1}}}_{\mathit{Z}}\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}}{\tilde{\mathcal{P}}}_{1}\text{\hspace{0.17em}}\mathit{d}\varsigma}\right\}}{\mathrm{Re}\left\{{\displaystyle \underset{\varsigma}{\iint}{\displaystyle \sum _{\mathit{p}\in 1}{\overrightarrow{\mathbf{1}}}_{\mathit{Z}}\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}}{\tilde{\mathcal{P}}}_{1}\text{\hspace{0.17em}}\mathit{d}\varsigma}\right\}}$$

_{1}is the fundamental power factor, also known as the displacement power factor

- Multi-sinusoidal linear case: $p\in N,q\in N$

- Multi-sinusoidal non-linear case: $p\in N,\text{\hspace{0.17em}}q\in N\cup M$

## 6. Numerical Example

#### 6.1. Linear Load Supplied by A Sinusoidal Voltage Source

#### 6.2. Linear Load Supplied by A Non-Sinusoidal Sinusoidal Voltage Source

#### 6.3. Non-Linear Load Supplied by A Non-Sinusoidal Sinusoidal Voltage Source

## 7. Conclusions

## Author Contributions

## Conflicts of Interest

## Glossary of Symbols

$\mathbb{R}$ | real numbers |

$\mathcal{C}$ | complex vector space |

${\mathcal{G}}_{n}$ | Geometric Algebra in n-dimensional real space |

$\mathcal{C}{\mathcal{G}}_{\mathit{n}}$ | Complex Geometric Algebra |

$\mathcal{C}{\mathcal{G}}_{\mathit{n}}^{\mathit{t}}-{\mathbb{R}}^{3}$ | time generalized geometric euclidean space |

$\mathcal{C}{\mathcal{G}}_{\mathit{n}}-{\mathbb{R}}^{3}$ | frequency generalized geometric euclidean space |

${\overrightarrow{\mathbf{1}}}_{X},{\overrightarrow{\mathbf{1}}}_{Y},{\overrightarrow{\mathbf{1}}}_{Z}$ | Euclidean canonical basis |

${\sigma}_{1\dots k}$ | canonical basis of ${\mathcal{G}}_{\mathit{n}}$ |

${\sigma}_{p}$ | basis vector of ${\mathcal{G}}_{\mathit{n}}$ |

${\sigma}_{p}{\sigma}_{q}={\sigma}_{pq}$ | basis bivectors of ${\mathcal{G}}_{\mathit{n}}$ |

${\sigma}_{1}{\sigma}_{2}{\sigma}_{3}$ | trivector or pseudoscalar of ${\mathcal{G}}_{\mathit{n}}$ |

λ, μ | scalars or 0-grade geometric object |

a, b | vectors or 1-grade geometric object |

B | bivector or 2-grade geometric object |

J | pseudoscalar or n-grade geometric object |

$\tilde{M}$ | generic multivector |

$\tilde{\mathit{\Pi}}$ | generalized Poynting multivector (GPM) |

$\tilde{\mathcal{P}}$ | Poynting multivector (PM) |

$\tilde{\mathcal{D}}$ | Complementary Poynting multivector (CPM) |

$\tilde{\mathit{E}}$ | electric field geometric phasor |

$\tilde{\mathit{H}}$ | magnetic field geometric phasor |

$\odot $ | generalized geometric product |

$\xb7$ | inner product |

$\wedge $ | outer product |

j | imaginary unit |

* | conjugated operation |

† | reverse operation |

~ | multivector characterization |

${U}_{p}$ | p-th harmonic voltage rms value |

${I}_{q}$ | q-th harmonic current rms value |

P | active power or real part of 0-grade power multivector |

Q | reactive power or imaginary part of 0-grade power multivector |

D | distortion power or 2-grade power |

$\tilde{S}$ | power multivector |

$\Vert \tilde{S}\Vert ,S$ | apparent power multivector |

ω_{p} | p-th harmonic frequency |

α_{p} | phase angle of p-th voltage geometric phasor |

β_{q} | phase angle of q-th current geometric phasor |

φ_{q} | q-th impedance phase angle |

$\tilde{\xi}$ | electromagnetic quality index multivector (EQI) |

PF | power factor |

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**MDPI and ACS Style**

Bravo, J.C.; Castilla, M.V.
Geometric Objects: A Quality Index to Electromagnetic Energy Transfer Performance in Sustainable Smart Buildings. *Symmetry* **2018**, *10*, 676.
https://doi.org/10.3390/sym10120676

**AMA Style**

Bravo JC, Castilla MV.
Geometric Objects: A Quality Index to Electromagnetic Energy Transfer Performance in Sustainable Smart Buildings. *Symmetry*. 2018; 10(12):676.
https://doi.org/10.3390/sym10120676

**Chicago/Turabian Style**

Bravo, Juan C., and Manuel V. Castilla.
2018. "Geometric Objects: A Quality Index to Electromagnetic Energy Transfer Performance in Sustainable Smart Buildings" *Symmetry* 10, no. 12: 676.
https://doi.org/10.3390/sym10120676