# Analysis and Optimization of Output Low-Pass Filter
## 2 Filter Resonance and Damping
### Filter Damping  
In Figure 3, four potential connections for the damping resistor are illustrated. \(R_{D1}\) is placed in series with the capacitor, \(R_{D2}\) is connected in parallel with the capacitor, \(R_{D3}\) is  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Harmonic Order** & **Limit for Each Harmonic Based on \%** \\  & **of Fundamental** \\ \hline
2–9 & 4\% \\
10–15 & 2\% \\
16–21 & 1.5\% \\
22–33 & 0.6\% \\ Even harmonics & 25\% of equivalent odd harmonics \\ Total harmonic distortion (to the 50 th harmonic) & 5\% \\ \hline \hline \end{tabular}
\end{table}
Table 1: Current harmonic limits [11].  
positioned in parallel with the inductor, and \(R_{D4}\) is arranged in series with the inductor. The third configuration was previously shown in Figure 2.  
Equation (2) defines \(Z_{0}\), which is determined by the filter configuration, damping resistance, and characteristic impedance. These factors together determine the quality factor. If \(Q\) is less than, equal to, or larger than 0.71 (\(1/\sqrt{2}\)), then the filter is classed as over-damped, critically damped, or under-damped, respectively. A switched-mode power supply filter can be constructed with a quality factor ranging from 2 to 4, even though a quality factor below 1 effectively lowers harmonic amplification [1].  
\[Z_{0}=\sqrt{\frac{L}{C}} \tag{2}\]  
The quality factor \(Q\) is inversely proportional to damping and is associated with the filter's gain at the resonant frequency. It is important to mention that the filter gain in Figure 2 determines the output-to-input current ratio, defined as \(I_{G}/I_{UC}\). The quality factor is affected by the filter topology and the specific type of damping implemented (see Figure 3). The next section continues the investigation of a variety of practical filter topologies and the associated damping techniques. The quality factor of the filter shown in Figure 2 can be represented as  
\[Q=\sqrt{\left(\frac{R_{D}}{Z_{0}}\right)^{2}+1} \tag{3}\]  
where the characteristic impedance of the filter is \(Z_{0}=\sqrt{L/C}\). When the filter's \(Q\) is too high, THD increases at its resonant frequency. If \(Q\) is too low, however, there will be significant filter loss. Grid filters are recommended to have \(Q\) values between 2 and 4 [1]. As a result, \(Q\) was initially selected as 4 for the filter analysis.  
The CL filter has the capability to effectively attenuate high-frequency PWM harmonics to a significant extent (by an order of magnitude) provided that the resonant frequency is maintained at a lower value compared to the PWM switching frequency, \(f_{sw}\). Low resonant frequencies require large filter inductance, which increases loss in the low-pass filter. Conversely, switching losses increase if the switching frequency is set too high. In this application, a \(f_{sw}\) value of 4 kHz was selected for the analysis in Section 4. This value corresponds to that used experimentally in [38; 39] using the topology in Figure 2. In order to achieve a reasonable ratio of \(f_{c}/f_{sw}~{}=~{}0.125\), the value of \(f_{c}\) was selected as 500 Hz.