# Instantaneous Reactive Power Theory in the Geometric Algebra Framework
## 3 The New Approach to Multi-Phase Systems
### Instantaneous Power Multivector in Multi-Phase Systems  
For a multi-phase power system of n+1 conductors, as illustrated in Figure 2, the voltage vector \(\mathbf{u}\) constituted by the instantaneous voltages \(u_{1}\),... \(u_{n}\) of each conductor relative to the neutral conductor is defined. Likewise, the current vector \(\mathbf{i}\) consisting of instantaneous line currents \(i_{1}\),... \(i_{n}\) flowing through the conductors, is defined:The voltage and current vectors can be considered as elements of a n-dimensional linear space and be expressed as a linear combination of the vectors of an orthonormal basis \(e_{j}\), \(j=1\),..., n, that is:  
\[ \begin{array}{l}\mathbf{u}=u_{1}e_{1}+\ldots+u_{n}e_{n}\\ \mathbf{i}=i_{1}e_{1}+\ldots+i_{n}e_{n}\end{array}  \tag{52}\]  
From an alternative perspective, the vectors \(\mathbf{u}\) and \(\mathbf{i}\) can be considered as tensors of order 1, in the same way that a scalar can be considered a zero-order tensor. **Definition 4**. : _In a multi-phase system, the instantaneous power \(p\) is defined as the inner product of voltage and current vectors:_  
\[p=\mathbf{u}\cdot\mathbf{i}=u_{1}i_{1}+\ldots+u_{n}i_{n} \tag{53}\]  
**Definition 5**. : _In a multi-phase system, the instantaneous reactive power tensor is defined as the second order hemi-symmetrical tensor determined by the relationship (54):_  
\[\mathbf{q}=\mathbf{u}\wedge\mathbf{i}=\mathbf{u}\otimes\mathbf{i}-\mathbf{i} \otimes\mathbf{u} \tag{54}\]  
_where the symbol '\(\wedge\)' means outer product and the symbol '\(\otimes\)' means tensor product._  
The definition of cross product only makes sense in a three-dimensional space; the tensor product, through the relationship (54), allows a definition of outer product in an n-dimensional space. The tensor product of the voltage vector to the current vector is:  
\[\mathbf{u}\otimes\mathbf{i}= \begin{bmatrix}u_{1}\\ u_{2}\\ \ldots\\ u_{n}\end{bmatrix}\begin{bmatrix}i_{1}&i_{2}&\ldots&i_{n}\end{bmatrix} = \begin{bmatrix}u_{1}i_{1}&u_{1}i_{2}&\ldots&u_{1}i_{n}\\ u_{2}i_{1}&u_{2}i_{2}&\ldots&u_{2}i_{n}\\ \ldots&\ldots&\ldots&\ldots\\ u_{n}i_{1}&u_{n}i_{2}&\ldots&u_{n}i_{n}\end{bmatrix}  \tag{55}\]  
and the tensor product of the current vector to the voltage vector is:  
\[\mathbf{i}\otimes\mathbf{u}= \begin{bmatrix}i_{1}\\ i_{2}\\ \ldots\\ i_{n}\end{bmatrix}\begin{bmatrix}u_{1}&u_{2}&\ldots&u_{n}\end{bmatrix} = \begin{bmatrix}i_{1}u_{1}&i_{1}u_{2}&\ldots&i_{1}u_{n}\\ i_{2}u_{1}&i_{2}u_{2}&\ldots&i_{2}u_{n}\\ \ldots&\ldots&\ldots&\ldots\\ i_{n}u_{1}&i_{n}u_{2}&\ldots&i_{n}u_{n}\end{bmatrix}  \tag{56}\]  
Thus, the instantaneous reactive power tensor takes the following form:  
Figure 2: Phase to neutral voltages and line currents in a multi-phase power system. \[\mathbf{q}=\mathbf{u}\wedge\mathbf{i}= \begin{bmatrix}0&i_{1}u_{2}-u_{1}i_{2}&\ldots& i_{1}u_{n}-u_{1}i_{n}\\ u_{1}i_{2}-i_{1}u_{2}&0&\ldots&i_{2}u_{n}-u_{2}i_{n}\\ \ldots&\ldots&\ldots&\ldots\\ u_{1}i_{n}-i_{1}u_{n}&u_{2}i_{n}-i_{2}u_{n}&\ldots&0\end{bmatrix}  \tag{57}\]  
The terms on each side of the main diagonal are opposite, so it is a hemisymmetric tensor. Thus, the tensor \(\mathbf{q}\) is composed of \(\mathrm{n}(\mathrm{n}-1)/2\) different terms [9]. Definition 6: For a multi-phase power system, the instantaneous power multivector \(\mathbf{s}\) is defined as the geometric product between the voltage vector \(\mathbf{u}\) and the current vector \(\mathbf{i}\) in the form (58):  
\[\mathbf{s}=\mathbf{u}\mathbf{i}=\mathbf{u}\cdot\mathbf{i}+\mathbf{u}\wedge \mathbf{i} \tag{58}\]  
The instantaneous power multivector \(\mathbf{s}\) is obtained from the inner product of voltage and current vectors (53), that is, the instantaneous real power \(p\), and the outer product of voltage and current vectors (54), which is the instantaneous reactive power tensor \(\mathbf{q}\).