## 2 Analysis of the Apparent Power of Buchholz
### Three-Phase Sinusoidal System with Unbalanced Loads and Balanced Voltages  
Under these conditions, the negative-sequence and zero-sequence voltages are zero. Therefore, (5) and (6) have null values, and the total apparent power of the system is given by (7).  
\[{S_{1}}^{2}=S_{p+}^{2}=9\ V_{+}^{2}\Big{(}I_{+}^{2}+I_{-}^{2}+I_{0}^{2}\Big{)} \tag{7}\]  
Figure 4: Equivalent system with zero-sequence voltage.  
Using the Fortescue transformation matrix, the line currents moduli in terms of symmetric components are expressed by (8)-(10).  
\[\begin{array}{l}|I_{+}|^{2}=\frac{1}{9}[I_{a}\cos\beta_{a}+I_{b}\cos(\beta_{b} +120)+I_{c}\cos(\beta_{c}+240)]^{2}\\ \quad\quad+\frac{1}{9}[I_{a}\sin\beta_{a}+I_{b}\sin(\beta_{b}+120)+I_{c}\sin( \beta_{c}+240)]^{2}\end{array} \tag{8}\]  
\[\begin{array}{l}|I_{-}|^{2}=\frac{1}{9}[I_{a}\cos\beta_{a}+I_{b}\cos(\beta_{b }+240)+I_{c}\cos(\beta_{c}+120)]^{2}\\ \quad\quad+\frac{1}{9}[I_{a}\sin\beta_{a}+I_{b}\sin(\beta_{b}+240)+I_{c}\sin( \beta_{c}+120)]^{2}\end{array} \tag{9}\]  
\[\begin{array}{l}|I_{0}|^{2}=\frac{1}{9}[I_{a}\cos\beta_{a}+I_{b}\cos\beta_{b }+I_{c}\cos\beta_{c}]^{2}+\frac{1}{9}[I_{a}\sin\beta_{a}+I_{b}\sin\beta_{b}+I_{ c}\sin\beta_{c}]^{2}\end{array} \tag{10}\]  
Substituting the sequence values from (8), (9), and (10) into (7), the following expression is obtained  
\[S_{p+}^{2}=3\;V_{+}^{2}\big{(}I_{a}^{2}+I_{b}^{2}+I_{c}^{2}\big{)} \tag{11}\]  
Considering (11) and taking into account \(V_{+}=V_{a+}=V_{b+}=V_{c+}\), we get  
\[S_{p+}^{2}=3\big{(}V_{a+}^{2}I_{a}^{2}+V_{b+}^{2}I_{b}^{2}+V_{c+}^{2}I_{c}^{2} \big{)} \tag{12}\]  
The three addends inside the parenthesis of (12) represent the apparent powers in each phase of the system. Their values can be determined from the active and reactive powers, considering the following expressions [10]:  
\[\big{(}S_{a+}^{a}\big{)}^{2}=\big{(}P_{a+}^{a}\big{)}^{2}+\big{(}Q_{a+}^{a} \big{)}^{2} \tag{13}\]  
\[\big{(}S_{b+}^{b}\big{)}^{2}=\big{(}P_{b+}^{b}\big{)}^{2}+\big{(}Q_{b+}^{b} \big{)}^{2} \tag{14}\]  
\[\big{(}S_{c+}^{c}\big{)}^{2}=\big{(}P_{c+}^{c}\big{)}^{2}+\big{(}Q_{c+}^{c} \big{)}^{2} \tag{15}\]  
where  
* \(P_{a+}^{a}\), \(P_{b+}^{b}\), and \(P_{c+}^{c}\) are the active powers due to the positive-sequence voltage in each phase; see Figure 2. Their values according to Steinmetz are calculated from (16).
* \(Q_{a+}^{a}\), \(Q_{b+}^{b}\), and \(Q_{c+}^{c}\) are the reactive powers due to the positive-sequence voltage in each phase; see Figure 2. Their values according to Steinmetz are calculated from (17). \[P_{z+}^{z}=V_{z+}I_{z}\cos(\alpha_{z+}-\beta_{z})\quad z=\{a,b,c\}\] (16) \[Q_{z+}^{z}=V_{z+}I_{z}\sin(\alpha_{z+}-\beta_{z})\quad z=\{a,b,c\}\] (17)  
Here, \(\alpha_{z+}\) is the angle of positive-sequence voltages in each phase and \(\beta_{z}\) is the angle of line currents in each phase.  
Substituting (13) and (14) and (15) into (12), we get  
\[S_{p+}^{2}=3\Big{[}\sum\big{(}P_{z+}^{z}\big{)}^{2}+\sum\big{(}Q_{z+}^{z} \big{)}^{2}\Big{]}\quad z=\{a,b,c\} \tag{18}\]  
Therefore, when a system is powered by a balanced voltage source, the apparent power modulus \(S_{1}\) is equal to \(S_{p+}\) and is given by (19).  
\[S_{p+}=\sqrt{3}\sqrt{\sum\big{(}P_{z+}^{z}\big{)}^{2}+\sum\big{(}Q_{z+}^{z} \big{)}^{2}}\quad z=\{a,b,c\} \tag{19}\]