# Analysis and Optimization of Output Low-Pass Filter
## 3 Frequency and Filter Component Normalization
### Filter Configurations  
The placement of the damping resistor differs among the four CL filter configurations presented in Table 4. The table presents the transfer functions \(H(s)\), the quality factor \(Q\), and the rate of high-frequency attenuation or roll-off associated with each filter configuration. The first filter configuration utilizes a series resistor in conjunction with the capacitor of the CL filter. This arrangement leads to a first-order roll-off in the filter's high-frequency response. The filter's impedance at high frequencies is also increased. This configuration results in reasonable power dissipation \(P_{d}\) for the damping resistor, which primarily encounters high-frequency current components. In FC 2, a resistor placed in parallel with the capacitor functions as a resistive load to reduce the resonant frequency of the filter. Since the full fundamental output voltage is seen across this resistor in the circuit, the high power dissipation makes this solution unacceptable. Table 4 presents the FC 3 configuration, which incorporates a resistor functioning in parallel with the filter inductor, which causes a first-order high-frequency roll-off. The  
Figure 13: Normalized capacitance as a function of inductance, illustrating different cutoff frequencies and characteristic impedance values.The design region, indicated by the shaded area, limits inverter switching losses and meets specified power factor standards. power dissipation in \(R_{D}\) is expected to be considerably lower than in other types, due to the relatively small fundamental voltage drop across the inductor. The inductor and resistor are connected in series in filter configuration FC 4. This configuration faces two primary challenges: it results in substantial power losses and generates a resistive output impedance at lower frequencies. The impedance of the filter is similar to that of the damping resistor at low frequencies [1]. The entire grid current is passed through the resistor in series within FC 4, resulting in significant losses. It shows a second-order roll-off characteristic that is advantageous at higher frequencies. The performance of four different CL filter configurations was thoroughly examined. The analysis was carried out with a fixed switching frequency of 4 kHz and a filter capacitance of 0.12 pu, which was considered necessary to comply with the power-factor standards. Figure 14 presents \(Q\) as a function of the normalized \(f_{c}\) for each FC. The normalized cutoff frequency, which is the ratio of the cutoff frequency to the switching frequency, is shown in the picture. The contours of the graphs show the power loss and total harmonic distortion, with the solid lines indicating a constant value of 5%. The design space is divided into four distinct regions: (A) an area with total harmonic distortion below 5%, (B) a region where both THD and damping resistor power loss are below 5%, (C) a zone where power loss is under 5%, and (D) an area where both THD and power loss exceed 5%). The choice of a 5% limit for power loss was made without specific justification suggesting that a lower threshold might be more suitable for optimal design. It can be seen in Figures 14 and 15 that the only configurations that satisfied the recommendation of having \(Q\) between 2 and 4 are FC 1 and 3 [1]. Additionally, FC 3 could meet the requirement across a broader range of cutoff frequencies and at lower \(Q\) values compared to FC 1. Moreover, the power dissipation \(P_{d}\) was less affected by variations in \(Q\). On the other hand, FC 2 and FC 4 could not fulfill the required \(Q\) range; the minimum \(Q\) required for FC 4 to be in the design region is 6.5, while for FC 2 it is about 20 (for a cut-off frequency of 0.1 pu). Based on these results, FC 3 was selected as the low-pass filter for this study.