# Analysis and Optimization of Output Low-Pass Filter
## 3 Frequency and Filter Component Normalization
### Power Factor Requirements  
This section demonstrates how the selection of filter components affects the inverter power factor and provides guidance on determining the suitable filter inductance and capacitance to achieve the required power factor. Australian Standards define the inverter as a grid load that has to run within a power factor range of 0.8 leading to 0.95 lagging for all output apparent powers ranging from 20% to 100% of its rated capacity (0.2 to 1 pu) [11]. Table 3 and Figure 11 summarize the normalized apparent, real, and reactive powers (S, P, and Q) under these circumstances. The fundamental component of the inverter's output current is assumed to be synchronized with the grid voltage. Figure 10: The relationship between the quality factor, filter cutoff frequency, and output current’s THD is illustrated for configurations (**a**) FC 1 and FC 3 and (**b**) FC 2 and FC 4. The quality factors, \(Q\), of 1, 2, and 5 are depicted in the contours for all filter configurations. The critically damped case is also present in FC 2 and FC 4, which corresponds to a quality factor of \(1/\sqrt{2}\). Figure 9: Filter gain for configurations (**a**) FC 1 and FC 3 and (**b**) FC 2 and FC 4, on a 4 kHz unipolar PWM harmonic spectrum, demonstrating the impact of changing the filter quality factor. For every filter design, the quality factors of 1, 2, and 5 are displayed; additionally, the critically damped case, \(Q=0.71\)\((1/\sqrt{2})\), is displayed in (**b**). Note that 800 Hz is the cutoff frequency (0.2 pu, in relation to the \(f_{sw}\)). The reactive power in the figure above is a critical indicator of the overall reactive power exchange between the system and the grid within the specified circuit configuration. The difference between the reactive power produced by the capacitor, \(Q_{C}\), and the reactive power absorbed by the inductor, \(Q_{L}\), is a quantitative measurement of this reactive power. The formula for this is \(Q=Q_{L}-Q_{C}\). Figure 12 illustrates the reactive power flow within the framework of an undamped CL low-pass filter. The illustration further clarifies the dynamics of real power transfer from the inverter to the grid. It is crucial to note that the CSI does not contribute to reactive power generation. The fundamental output current is designed to operate at a unity power factor. The capacitor exclusively supplies reactive power, compensating for the demand of the inductor, while the grid absorbs that power. The analytical determination of the reactive power absorbed by the inductor and generated by the capacitor is presented here. Both \(Q_{L}\) and \(Q_{C}\) are represented as normalized  
Figure 11: According to the grid power factor limits, the real, \(P\), reactive, \(Q\), and apparent power, \(S\), triangles display (**a**) 0.8 lead and (**b**) 0.95 lag. \begin{table}
\begin{tabular}{l c c c c} \hline
**Grid ExtremesPower** & **Apparent (S)** & **Real (\(P\))** & **Reactive (\(Q\))** & \(\phi\) \\ \hline Power factor = 0.80 lead & 0.20 pu & 0.16 pu & \(-\)0.12 pu & 36.9\({}^{\circ}\) \\ Power factor = 0.95 lag & 1.00 pu & 0.95 pu & 0.31 pu & \(-\)18.2\({}^{\circ}\) \\ \hline \end{tabular}
\end{table} 
Table 3: Overview of normalized \(S\), \(P\), and \(Q\), along with the power factor angle (\(\phi\)), in relation to the extremes of grid power factor requirements. Figure 12: The real power, \(P\), and reactive power, \(Q\), of an undamped CL low-pass filter are displayed in the power flow diagram. The expression \(Q_{L}-Q_{C}\) indicates that reactive power is either delivered or received by the grid. Additionally, the filter does not absorb real power. values. The normalized inverter output current is considered to be 1 pu for rated operation and 0.2 pu for operation at 20% of rated volt-amperes. The inductor absorbs reactive power, \(Q_{L}\), which is proportional to the square of the normalized current (\(I_{n}\)) and reactance, \(X_{L}\). This is summarized in Equation (9), which is simplified to Equation (10).