## 4 Distortion Power Bivector
### Distortion Power bivector: Admittance Matrix Representation  
When the load impedance is linear, the only current harmonics that will flow must correspond to the supply voltage harmonics. Thus, in linear operation, the harmonic indexes of voltage and current, \(p,q\in N\) can define the quantities:  
\[\hat{Y}=\left(Y_{pp}\right)_{p=1}^{n}=Y_{p}\ \Rightarrow\ Y_{pp}=G_{p}-jB_{p} \tag{29}\]  
\[G_{p}=\frac{I_{p}}{U_{p}}cos\phi_{p},B_{p}=\frac{I_{p}}{U_{p}}sin\phi_{p} \tag{30}\]  
where \(\hat{Y}\) is called the \(n\)-frequency admittance matrix, \(G_{p}\)is the \(p\)-_th_ conductance and \(B_{p}\)is the \(p\)-_th_ susceptance. The multivectorial Ohm's law is now given by:  
\[\widetilde{I}=\widetilde{U}\ \hat{Y}=\sum_{q}\widetilde{I}_{q}\Rightarrow\ \widetilde{I}^{*}=\widetilde{U}^{*}\widetilde{Y}^{*}=\sum_{q} \widetilde{I}_{q}^{*} \tag{31}\]  
where:  
\[\widetilde{I}_{q}=\widetilde{U}_{q}\left(G_{q}-jB_{q}\right)\sigma_{q}= \widetilde{I}_{||}-j\widetilde{I}_{\bot} \tag{32}\]  
\[\widetilde{I}_{q}^{*}=\widetilde{U}_{q}^{*}\left(G_{q}+jB_{q}\right)\sigma_{q} =\widetilde{I}_{||}^{*}+j\widetilde{I}_{\bot}^{*} \tag{33}\]  
that coincide with Equations (15) and (16), respectively. In this case, \(\widetilde{S}\) multivector may be written in the form:  
\[\widetilde{S}=\widetilde{U}\odot\ \widetilde{I}^{*}=\widetilde{U}\odot\ \widetilde{I}_{||}^{*}+j \widetilde{U}\odot\ \widetilde{I}_{\bot}^{*} \tag{34}\]Now it is seen from Equations (A2) and (34) that:  
\[\widetilde{U}\odot\ \widetilde{I}_{\parallel}^{*}=\widetilde{U}\cdot\widetilde{I}_{ \parallel}^{*}+\widetilde{U}\wedge\widetilde{I}_{\parallel}^{*} \tag{35}\]  
and the following magnitude expressions of the _active_ and _active rotated distortion_ power components are obtained:  
\[\widetilde{P}=\widetilde{U}\cdot\widetilde{I}_{\parallel}^{*}=\sum_{P} \widetilde{U}_{p}\widetilde{U}_{p}^{*}G_{p}\Rightarrow P=\sum_{P}U_{p}^{2}G_{p} \tag{36}\]  
\[\widetilde{\Delta}_{Act}=\widetilde{U}\wedge\widetilde{I}_{\parallel}^{*}= \sum_{ \begin{subarray}{c}p<q\\ p,q\in N\end{subarray} }\widetilde{U}_{p}\wedge\widetilde{I}_{q\parallel}^{*} \
orm{\widetilde{\Delta}_{Act}}^{2}=\sum_{p<q}U_{p}^{2}I_{q}^{2}(G_{p}-G_{q})^{2} \tag{37}\]  
where from Appendix A, Equation (A4):  
\[\widetilde{U}_{p}\wedge\widetilde{I}_{q\parallel}^{*}=\overline{U}_{p} \sigma_{p}\wedge\overline{U}_{q}^{*}G_{q}\sigma_{q}+\overline{U}_{q}\sigma_{ q}\wedge\overline{U}_{p}^{*}G_{p}\sigma_{p}=e^{i(\alpha_{p}-\alpha_{q})} \widetilde{D}_{pq,Act}=\widetilde{\Delta}_{pq,Act} \tag{38}\]  
On the other hand, from Equations (A5) and (34) can be obtained that:  
\[\widetilde{U}\odot\ \widetilde{I}_{\perp}^{*}=\widetilde{U}\cdot\widetilde{I}_{ \perp}^{*}+\widetilde{U}\wedge\widetilde{I}_{\perp}^{*} \tag{39}\]  
Similarly, another two magnitude expressions for _reactive_ and reactive _rotated distortion_ powers are obtained:  
\[\widetilde{Q}=\widetilde{U}\cdot\widetilde{I}_{\perp}^{*}=\sum_{p}\overline{U} _{p}\overline{U}_{p}^{*}B_{p}\Rightarrow Q=\sum_{p}U_{p}^{2}B_{p} \tag{40}\]  
\[\widetilde{\Delta}_{Re\ act}=\widetilde{U}\wedge\widetilde{I}_{\perp}^{*}= \sum_{ \begin{subarray}{c}p<q\\ p,q\in N\end{subarray} }\widetilde{U}_{p}\wedge\widetilde{I}_{q\perp}^{*}\
orm {\widetilde{\Delta}_{Re\ act}}^{2}=\sum_{p<q}U_{p}^{2}I_{q}^{2}(B_{p}-B_{q})^{2} \tag{41}\]  
where from Appendix A, Equation (A7):  
\[\widetilde{U}_{p}\wedge\widetilde{I}_{q\perp}^{*}=\overline{U}_{p}\sigma_{p} \wedge U_{q}^{*}B_{q}\sigma_{q}+\overline{U}_{q}\sigma_{q}\wedge\overline{U}_ {p}^{*}B_{p}\sigma_{p}=e^{i(\alpha_{p}-\alpha_{q})}\widetilde{D}_{pq,Re\ act}= \widetilde{\Delta}_{pq,Re\ act} \tag{42}\]  
Combination of Equations (36)-(40) gives the apparent power multivector:  
\[\widetilde{S}=\widetilde{U}\odot\ \widetilde{I}^{*}=\widetilde{U}\odot\ \widetilde{I}_{ \parallel}^{*}+j\widetilde{U}\odot\ \widetilde{I}_{\perp}^{*}=(P+jQ)\sigma_{0}+\sum_{ \begin{subarray}{c}p<q\\ P,q\in N\end{subarray}