# 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  
The term reactive power has been used since the dawn of alternating current to take into account the phenomena of phase shift between voltage and current. The IRPT took the name of instantaneous reactive power to designate new time-dependent power variables that, in principle, had nothing to do with classical reactive power. The coincidence in the denomination could lead to confusion. However, instantaneous reactive power variables are, in fact, an extension of the sinusoidal steady-state reactive power concept. To show this idea, and in order not to be exhaustive, we will consider the case where the voltages and currents of a three-phase system do not include zero-sequence components, that is, \(u_{1}\)\(+u_{2}\)\(+u_{3}\)\(=0\); \(i_{1}\)\(+i_{2}\)\(+i_{3}\)\(=0\). Under these conditions it can be easily verified that the three components of the instantaneous reactive power bivector, \(\mathbf{q}\), (7), have the same value:  
\[\mathbf{q}=\frac{q}{\sqrt{3}}(e_{12}+e_{31}+e_{23}) \tag{37}\]  
where q is the bivector norm \(\mathbf{q}\). The instantaneous reactive current vector is, therefore, from (30):  
\[\mathbf{i_{q}}=-\frac{q}{u^{2}}(\frac{u_{23}}{\sqrt{3}}e_{1}+\frac{u_{31}}{ \sqrt{3}}e_{2}+\frac{u_{12}}{\sqrt{3}}e_{3}) \tag{38}\]in which line to line voltages appear \(u_{ij}=u_{i}-u_{j}\). A comparison with reactive current, and from there with classic reactive power, requires a transition to the sinusoidal steady state. A sinusoidal direct sequence voltage and an asymmetrical sinusoidal current of frequency \(h\omega\) is considered here first. The extra subscript '\(h\)' is used to characterize voltages, currents, and reactive power for this situation. Let be the phase voltages be:  
\[\mathbf{u}_{h}= \begin{bmatrix}u_{1h}\\ u_{2h}\\ u_{3h}\end{bmatrix} =\sqrt{2}\Re e\left\{ \begin{bmatrix}V_{h}\\ V_{h}e^{-j\pi/3}\\ V_{h}e^{j\pi/3}\end{bmatrix} e^{jh\omega t}\right\} \tag{39}\]  
where \(Re\) means 'take real part'. Line currents are considered in the same way:  
\[\mathbf{i}_{h}= \begin{bmatrix}i_{1h}\\ i_{2h}\\ i_{3h}\end{bmatrix} =\sqrt{2}\Re e\left\{ \begin{bmatrix}I_{h}^{+}e^{i\varphi_{h }^{+}}\\ I_{h}^{+}e^{i(\varphi_{h}^{+}-2\pi/3)}\\ I_{h}^{+}e^{i(\varphi_{h}^{+}+2\pi/3)}\end{bmatrix} e^{jh\omega t}\right\}+ \sqrt{2}\Re e\left\{ \begin{bmatrix}I_{h}^{-}e^{i(\varphi_{h}^{+}+2\pi/3)}\\ I_{h}^{-}e^{i(\varphi_{h}^{+}+2\pi/3)}\\ I_{h}^{-}e^{i(\varphi_{h}^{-}-2\pi/3)}\end{bmatrix} e^{jh\omega t}\right\} \tag{40}\]  
where a current component of the same phase sequence as the voltage and a current component of opposite phase sequence to the voltage are distinguished. For these waveforms the instantaneous reactive power (\(q_{h}\)) is:  
\[q_{h}=3V_{h}I_{h}^{+}\sin\varphi_{h}^{+}-3V_{h}I_{h}^{-}\sin(2h\omega t+\varphi _{h}^{-})=Q_{h}+\widetilde{q}_{h} \tag{41}\]  
The average value of \(q_{h}\) represents the classical reactive power corresponding to the phase shift between voltages and currents of the same phase sequence. When the currents (40) include only a positive sequence component, \(q_{h}\) is a constant of value equal to the reactive power \(Q_{h}\). The oscillatory part in (41) is related to the phase-sequence current component opposite to the voltage. Instantaneous reactive power extends the concept of classical reactive power. From (29) we can find the instantaneous reactive current \(\mathbf{i}_{qh}\):  
\[\mathbf{i}_{qh}=-\frac{Q_{h}}{u_{h}^{2}}\mathbf{u}_{qh}-\frac{\widetilde{q}_{ h}}{u_{h}^{2}}\mathbf{u}_{qh} \tag{42}\]  
where for convenience the orthogonal voltage vector \(\mathbf{u}_{qh}\) has been introduced as:  
\[\mathbf{u}_{qh}=\frac{1}{\sqrt{3}} \begin{bmatrix}u_{23}\\ u_{31}\\ u_{12}\end{bmatrix} =\sqrt{2}\Re e\left\{-j \begin{bmatrix}V_{h}\\ V_{h}e^{-j\pi/3}\\ V_{h}e^{j\pi/3}\end{bmatrix}