# Analysis and Optimization of Output Low-Pass Filter
## 3 Frequency and Filter Component Normalization  
Normalization is a widely recognized technique that facilitates the generalization of analyses. The cutoff frequency (or resonant frequency) and filter components are normalized. This makes the analysis much easier and makes it easy for the designer to choose component values for any system size. The representation of the normalized cutoff frequency, denoted as \(\omega_{cn}\), is associated with the base frequency, \(\omega_{B}\), as demonstrated in Equation (4).  
\[\omega_{cn}\equiv\frac{\omega}{\omega_{B}}\quad\Big{\}}\qquad\text{where} \qquad\omega_{B}=2\pi f_{1} \tag{4}\]  
where the fundamental frequency of the grid (or inverter) is denoted by \(f_{1}\). For a single-phase inverter that delivers rated power, \(P_{B}\), to a grid with rated voltage, \(V_{B}\), the base impedance, \(Z_{B}\), can be calculated using Equation (5). This base impedance is then used to normalize the filter components. Essentially, the base impedance is the ratio of the base voltage, \(V_{B}\), to the base current, \(I_{B}\).  
\[Z_{B}\equiv\frac{V_{B}}{I_{B}}=\frac{V_{B}^{2}}{P_{B}} \tag{5}\]  
Equation (6) provides the value of the normalized filter capacitance, denoted as \(C_{n}\), in terms of the base capacitance \(C_{B}\). The definition of the normalized filter inductance, \(L_{n}\), is defined with respect to the base inductance, \(L_{B}\).  
\[\left.\begin{aligned} C_{n}&=\frac{1}{w_{cn}^{2}\,L_{n }}=\frac{C}{C_{B}}\\ L_{n}&=\frac{1}{w_{cn}^{2}\,C_{n}}=\frac{L}{L_{B}} \end{aligned}\right\}\qquad\text{where}\qquad\qquad\qquad C_{B}=\frac{1}{ \omega_{B}\,Z_{B}} \tag{6}\]  
The normalized and simplified resonant frequency of the filter is represented by Equation (1). The resulting product shows an inverse relationship with the square of the filter's resonant frequency when \(C_{n}\) and \(L_{n}\) are included in this expression. The relationship is illustrated by Equation (7) and Figure 6a, showing the contours of \(\omega_{cn}\) at 1, 5, 10, and 20 pu. Additionally, Figure 6b presents the contours of the normalized characteristic impedance, \(Z_{0n}\), at 0.5, 1, 1.5, 2, and 2.5 pu, as defined by Equation (8).  
\[L_{n}C_{n}=\frac{1}{\omega_{cn}^{2}} \tag{7}\]  
\[Z_{0n}=\sqrt{\frac{L_{n}}{C_{n}}}=\frac{Z_{0}}{Z_{B}} \tag{8}\]  
\begin{table}
\begin{tabular}{l c c c c} \hline \hline \begin{tabular}{c} **Variable** \\ **Parameter** \\ \end{tabular} & \begin{tabular}{c} **Resonant** \\ **Frequency, \(f_{R}\)** \\ \end{tabular} & \begin{tabular}{c} **Quality** \\ **Factor, \(Q\)** \\ \end{tabular} & \begin{tabular}{c} **Harmonic** \\ **Attenuation** \\ **Loss, \(P_{d}\)** \\ \end{tabular} \\ \hline \begin{tabular}{l} Inductance, \(L\) \\ Capacitance, \(C\) \\ \end{tabular} & \begin{tabular}{c} **✓** \\ **✓** \\ **✓** \\ **✓** \\ **✓** \\ **✓** \\ **✓** \\ **✓** \\ **✓** \\ **✓** \\ \end{tabular} & \
\begin{tabular}{c} **✓** \\ **✓** \\ **✓** \\ **✓** \\ **✓** \\ **✓** \\ **✓** \\ **✓** \\ **✓** \\ \end{tabular} &
\begin{tabular}{c} **Damping** \\ **Resistor** \\ **Power** \\ **Loss, \(P_{d}\) \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table}
Table 2: Variables of a low-pass filter and the essential parameters they influence.