## 2 Assessment of the SPWM Angles
### Basic Definitions  
The SPWM technique requires several important parameters for its correct application. First, we need a sinusoidal signal with an amplitude designated as \(A_{s}\) (see Figure 1) and a frequency of usually 60 Hz.  
Another parameter is the amplitude of the triangular signal, represented by \(A_{t}\), which is the maximum value reached by this signal. From these definitions, we can calculate the modulation index (\(M\)), which is the ratio between the amplitude of the sine signal and the amplitude of the triangular signal, which is expressed by the following equation:  
\[M=\frac{A_{s}}{A_{t}} \tag{1}\]  
One of the key parameters in the SPWM technique is the modulation index (\(M\)). It is crucial to keep this value less than 1 to avoid overmodulation, which can negatively affect system performance. Overmodulation can lead to distortion in the output waveform, reducing the efficiency and accuracy of the modulation process.  
The ratio between the frequency of the sine signal \(f_{o}\) and the frequency of the triangular signal \(f_{c}\), also known as the carrier frequency, is the frequency modulation ratio (\(m\)), shows in Equation (2). Proper tuning of these frequencies is essential for efficient and accurate modulation.  
\[m_{f}=\frac{f_{c}}{f_{o}} \tag{2}\]  
In addition, it is important to know the number of triangular pulses per half-cycle \(p_{t}\), which the following expression can calculate:  
\[p_{t}=\frac{f_{c}}{2f_{o}} \tag{3}\]Hence, \(p_{t}\) represents the number of triangular pulses per half-cycle. Since \(p_{t}\) is based on frequency modulation, we can substitute \(f_{c}/f_{o}\) and express it as  
\[p_{t}=\frac{m_{f}}{2} \tag{4}\]  
To maintain 1/4-wave symmetry, \(p_{t}\) must be even  
Having obtained the number of triangular pulses per half-cycle, it is relevant to calculate the total number of output pulses per half-cycle \(p\), which is obtained by the following expression:  
\[p=p_{t}-1 \tag{5}\]  
This expression indicates that the number of pulses equals the number of triangular pulses minus one. By considering \(p_{t}\) to be half of the frequency modulation, we can rewrite the expression as  
\[p=\frac{m_{f}}{2}-1 \tag{6}\]  
Also, to maintain 1/4 wave symmetry, \(p\) must be odd.  
These equations are fundamental to understanding and controlling the comparison process between the triangular signal and the sinusoidal signal in the SPWM technique.  
Another condition is to have a unipolar signal for each half-cycle with quarter-wave symmetry with the largest pulse centered at \(\frac{1}{4}\). If these conditions are satisfied, it is possible to reduce the harmonics.  
Meeting these conditions not only reduces unwanted components but also simplifies the process. With this specific quarter-wave symmetry, it is not necessary to calculate all the pulses but only the angles present in a quarter-wave, since these will be replicated throughout the period.  
The number of angles required for a quarter-wave is calculated using the following expression:  
\[N_{\alpha}=\frac{p_{t}}{2} \tag{7}\]  
\(N_{\alpha}\) is the number of angles for a quarter-wave, and \(p_{t}\) is the number of triangular pulses.