# Analysis and Optimization of Output Low-Pass Filter
## 3 Frequency and Filter Component Normalization
### Harmonic Distortion and Attenuation  
This section analyzes the filter's ability to reduce the overall harmonic distortion of a unipolar PWM current. Figure 7 depicts the harmonic spectrum of a 4 kHz unipolar PWM signal. The spectrum was provided by the filter gains of four different filter designs (FCs 1-4). The quality factor \(Q\) and the cutoff frequency \(f_{c}\) were arbitrarily selected as 1 kHz and 2, respectively. In particular, the filter gains of FC 1 and FC 3 were identical, which led to comparable THD values. Similarly, FC 2 and FC 4 showed identical gain, which resulted in identical THD. It is clearly seen from Figure 7 that filter configurations 2 and 4 have more attenuation than those 1 and 3. Based on observations indicating a gain roll-off of \(-\)20 dB/decade, configurations 1 and 3 function as first-order filters. Configurations 2 and 4 operate as second-order low-pass filters and show a roll-off of \(-\)40 dB per decade. Factors like \(Q\) and \(f_{c}\) also influence filter gain, which influences the THD of the output current. Figure 8 depicts the impact of adjusting the normalized cutoff frequency, \(f_{cn}\), on filter gain with a quality factor of 2. Figure 9 shows the effect of varying the quality factor while maintaining the cutoff frequency at 800 Hz. The PWM switching frequency of 4 kHz was used. Figure 6: Plots of normalized capacitance, \(C_{n}\), vs. inductance, \(L_{n}\), illustrating the contours for (**a**) cutoff frequency, \(\omega_{cn}\), and (**b**) characteristic impedance, \(Z_{0n}\), contours. The contours for \(\omega_{cn}\) correspond to values of 2, 5, 10, and 20 pu, while those for \(Z_{0n}\) correspond to values of 0.5, 1, 1.5, and 2 pu. Please note that in Figure 5(a), the normalized cutoff frequency, denoted as \(\omega_{cn}\), is defined relative to the fundamental frequency, \(f_{1}\), as shown in Equation (4). Conversely, Figures 8 and 9, utilize a normalization based on the inverter switching frequency, \(f_{sw}\), denoted as \(f_{cn}=f_{c}/f_{sw}\). Figure 8 shows that when \(f_{c}\) approaches \(f_{sw}\), low-pass filters become less effective, allowing higher magnitude high-frequency components to pass through and increase output current THD. Figure 9 demonstrates that the quality factor for FC 1 and FC 3 affects harmonic mitigation and consequently, THD. A low \(Q\) (around 1) leads to minimal attenuation at higher frequencies, whereas a higher \(Q\) (such as 5) considerably improves attenuation in the high-frequency range. For FC 2 and FC 4, \(Q\) has no impact on the output current's THD at low cutoff frequencies, yet its effect becomes notable as the \(f_{c}\) nears the switching frequency, \(f_{sw}\). Figure 10 provides the previously discussed points by illustrating the calculated THD of the output current for each filter, considering both \(f_{c}\) and \(Q\). The dashed line marks the grid THD limit at 5%. The results indicate that FC 1 and FC 3 perform effectively when \(Q\) is greater than 2 and \(f_{c}\) is roughly between 0.2 and 0.3 pu; this finding supports Nave's recommendation of a quality factor between 2 and 4 [1]. Furthermore, the data show that the total harmonic distortion for configurations 2 and 4 remains unaffected by the quality factor when the normalized cutoff frequency, \(f_{cn}\), is below approximately 0.3 pu. Figure 8: The analysis of the filter gain for a 4 kHz unipolar PWM harmonic spectrum shows how varying the filter cutoff frequency affects (**a**) FC 1 and FC 3 and (**b**) FC 2 and FC 4. The cutoff frequencies are normalized with respect to the switching frequency, with normalized values of 0.1, 0.2, 0.5, and 1 pu. Additionally, the PWM harmonics are now presented up to only two times the switching frequency. Figure 7: The unipolar PWM waveform’s harmonic spectrum and the gain properties of two different filter configuration sets. FC 1 and FC 3 correspond to first-order filter responses, while configurations 2 and 4 correspond to second-order responses.