# Instantaneous Reactive Power Theory in the Geometric Algebra Framework
## 2 Instantaneous Reactive Power Theory in Three-Phase Systems: A New Approach
### Instantaneous Power Multivector
Thus, (5) would be expressed as:  
\[\mathbf{s}=\mathbf{u}\mathbf{i}=p+\mathbf{q}=\left\langle\mathbf{s}\right\rangle _{0}+\left\langle\mathbf{s}\right\rangle_{2} \tag{8}\]  
The reverse of any multivector \(\mathbf{M}\), denoted as \(\mathbf{M}^{\dagger}\), is defined from expression (9):  
\[\left\langle\mathbf{M}^{\dagger}\right\rangle_{k}=(-1)^{k(k-1)/2}\left\langle \mathbf{M}\right\rangle_{k} \tag{9}\]  
From there, the reverse of the instantaneous power multivector \(\mathbf{s}^{\dagger}\) is given by (10):  
\[\mathbf{s}^{\dagger}=p-\mathbf{q} \tag{10}\]  
Since (11) is verified:  
\[\left\langle\mathbf{s}^{\dagger}\right\rangle_{0}=\left\langle\mathbf{s} \right\rangle_{0} \tag{11}\]The introduction of the reverse of a k-vector allows establishment of the instantaneous magnitude or norm of a multivector; definition 2 defines the instantaneous norm of the instantaneous power multivector:  
**Definition 2**. : _The instantaneous norm \(s\) (\(\|\mathbf{s}\|\)) of the instantaneous power multivector \(\mathbf{s}\) is defined according to the expression given by (12):_  
\[\mathbf{s}=\|\mathbf{s}\|=\left(\mathbf{s}^{\dagger}\mathbf{s}\right)^{1/2}= \left(\sum_{k=0}^{3}\left\langle\mathbf{s}^{\dagger}\right\rangle_{k}\langle \mathbf{s}\rangle_{k}\right)^{1/2} \tag{12}\]  
The summation in relation (12) extends to all the blades included in the multivector. Note that both the multivector \(\mathbf{s}\) and its norm are time-dependent entities. **Theorem 1**. : _The instantaneous norm of the instantaneous power multivector \(\mathbf{s}\) satisfies the relationship between the squares of the instantaneous power and the instantaneous reactive power given by (13):_  
\[s^{2}=p^{2}+q^{2} \tag{13}\]  
In fact, from (8), (10), and (12) it follows that the square value of the instantaneous power multivector norm can be obtained by:  
\[ \begin{array}{ll}\|\mathbf{s}\|^{2}&=s(t)^{2}=\left\langle\mathbf{s}^{ \dagger}\right\rangle_{0}\langle\mathbf{s}\rangle_{0}+\left\langle\mathbf{s}^ {\dagger}\right\rangle_{2}\langle\mathbf{s}\rangle_{2}\\ &=p^{2}+\left(q_{12}^{2}+q_{31}^{2}+q_{23}^{2}\right)\end{array}  \tag{14}\]  
Consider that, from (4), (9), and (12), for the instantaneous reactive power bivector, we have the relationship:  
\[ \begin{array}{ll}\|\mathbf{q}\|^{2}&=\left\langle\mathbf{s}^{\dagger} \right\rangle_{2}\langle\mathbf{s}\rangle_{2}=-\mathbf{q}\mathbf{q}\\ &=(q_{12}e_{12}+q_{31}e_{31}+q_{23}e_{23})(-q_{12}e_{12}-q_{31}e_{31}-q_{23}e_ {23})\\ &=\left(q_{12}^{2}+q_{31}^{2}+q_{23}^{2}\right)=q^{2}\end{array}  \tag{15}\]  
Relation (14) determines the instantaneous apparent power partition, \(\mathbf{s}\). The first term represents the square of the instantaneous power and the second term, obtained from the norm of the instantaneous reactive power bivector, (15), represents the square of the instantaneous reactive power \(q\)[1]. The model established by (3) and (14) describes three-phases systems in the same way as followed in single-phase systems. Thus, the squared norm of the instantaneous power multivector has an orthogonal decomposition in terms of the instantaneous power and the instantaneous reactive power, of the same type as the decomposition of the apparent power as a function of the active power and the reactive power, typical of the analysis of the power in the sinusoidal steady state. In Section 3, this is generalized for the analysis of multi-phase systems.