# Instantaneous Reactive Power Theory in the Geometric Algebra Framework
## 2 Instantaneous Reactive Power Theory in Three-Phase Systems: A New Approach
### Instantaneous Reactive Power versus Reactive Power
e^{jh\omega t}\right\} \tag{43}\]  
Thus, the first term of (42) corresponds to the reactive current component \(\mathbf{i}_{\text{rh}}\) of the current \(\mathbf{i}_{\text{h}}\), which has the same sequence as the voltage:  
\[\mathbf{i}_{th}=\sqrt{2}\Re e\left\{j\frac{Q_{h}}{u_{h}^{2}} \begin{bmatrix}V_ {h}\\ V_{h}e^{-j\pi/3}\\ V_{h}e^{j\pi/3}\end{bmatrix} e^{jh\omega t}\right\} \tag{44}\]  
In the above expression the instantaneous norm of the \(\mathbf{u}_{\text{h}}\) vector for a system of balanced sinusoidal voltages is constant and corresponds to its rms value:  
\[u_{h}^{2}=U_{h}^{2}=3V_{h}^{2} \tag{45}\]  
Thus, the norm squared (rms squared value) of the reactive current of the harmonic \(h\omega t\) is:\[\left\|\mathbf{i}_{rh}\right\|^{2}=I_{rh}^{2}=\frac{Q_{h}^{2}}{U_{h}^{2}} \tag{46}\]  
In the most general conditions of asymmetry and distortion there are difficulties in obtaining a definition of steady-state reactive power, and, therefore, there is no unanimous agreement about its computation. However, two definitions remain noteworthy either because of their historical use or their practical usefulness. In fact, the Budeanu reactive power, which computes the sum of the reactive power terms of each common harmonic between voltage and current, was traditionally used for a long time. In contrast, the definition used by Std 1459 takes only the reactive power of the fundamental harmonic [19]. The expression for reactive power in non-sinusoidal asymmetrical regimes in the frequency domain that is most commonly used today, is that suggested by [PERSON] and adopted by [PERSON] in the current's physical components (CPC) theory [19]. An identification with IRPT can be established by considering a system of non-sinusoidal voltages and non-sinusoidal unbalanced currents. That is, let the voltage and current vectors be:  
\[\mathbf{u}=\sum_{h}\mathbf{u}_{h};\quad\mathbf{i}=\sum_{h}\mathbf{i}_{h} \tag{47}\]  
where the sum extends to all harmonics of interest. In accordance with the results obtained for a generic harmonic of order \(h\), the current vector \(\mathbf{i}\) will include a reactive component resulting from the computation of the reactive currents of each individual harmonic, (41):  
\[\mathbf{i}_{r}=\sum_{h}\mathbf{i}_{rh}=\sum_{h}-\frac{Q_{h}}{U_{h}^{2}} \mathbf{u}_{qh}=\sum_{h}\sqrt{2}\Re e\left\{j\frac{Q_{h}}{U_{h}^{2}}\left[V_{ h}e^{-j\pi/3}\right]e^{ihuct}\right\} \tag{48}\]  
and its rms value is:  
\[\left\|\mathbf{i}_{r}\right\|^{2}=I_{r}^{2}=\sum_{h}\frac{Q_{h}^{2}}{U_{h}^{2}} \tag{49}\]  
The definition of reactive power in non-sinusoidal regimes in the classical sense, i.e., that which admits compensation with only the use of reactive passive elements, and which gives rise to a maximum power factor is, therefore:  
\[Q_{r}=UI_{r}=\sum_{h}U_{h}^{2}\sum_{h}\frac{Q_{h}^{2}}{U_{h}^{2}} \tag{50}\]  
In which the average value of the instantaneous reactive power corresponding to each of the harmonics appears. The following analysis allows a comparison of the current terms in the IRPT and the frequency domain theories.