# Random PWM Technique Based Two-State Markov Chain for Permanent Magnet Synchronous Motor Control
## 2 Conventional SVPWM Analysis  
### Operating Principle of the SVPWM  
SVPWM regards the inverter system and the motor as a whole. By combining the on and off states of the switching power devices of each bridge arm of the three-phase inverter, the actual flux linkage trajectory generated by the permanent magnet synchronous motor approaches the ideal circular flux linkage trajectory. By comparing the actual flux linkage with the target flux linkage, the combination of the switch states of the inverter is determined, thereby generating the PWM signal waveform and achieving efficient and stable motor control [19].  
The three-phase inverter has six switching power devices, divided into three groups of upper and lower bridge arms. The same bridge arm's upper and lower switching devices cannot conduct at the same time. The structure of the diagram is shown in Figure 1. In the figure, \(\mathrm{U_{dc}}\) represents the DC bus voltage, and \(i_{a}\), \(i_{b}\), and \(i_{c}\) represent the three-phase stator currentsThere are eight switching combinations in the three-phase inverter, corresponding to 8 different voltage space vectors, including six non-zero vectors (100, 110, 010, 011, 001, 101) and two zero vectors (000, 111), of which \"1\" represents the upper arm bridge conduction, and \"0\" represents the lower arm bridge conduction. Six non-zero vectors have amplitudes of \(2/3\mathrm{U}_{dc}\), dividing the plane into six sectors, with a 60\({}^{\circ}\) difference in angle between the two vectors, and the two zero vectors are located in the center of the plane. At a certain moment, the rotation of the voltage space vector to a certain sector is obtained from the different combinations of the two neighboring nonzero and zero vectors in that sector in time [20]. Taking the first sector as an example, the synthesized space vector is \(V_{m}\), and the voltage space vector diagram is shown in Figure 2. In the figure, I, II, III, IV, V, and VI respectively represent the six sectors of the spatial voltage plane.  
\[TV_{m}=T_{1}V_{1}+T_{2}V_{2}+T_{0}V_{0} \tag{1}\]  
where \(T\) represents the switching period, \(T_{0}\) is the time of action of the zero vector \(V_{0}\), and \(T_{1}\) and \(T_{2}\) indicate the times of action of the non-zero vectors \(V_{1}\) and \(V_{2}\), respectively.  
\[T_{1}=\sqrt{3}\frac{|V_{m}|}{V_{dc}}T\sin(\frac{\pi}{3}-\theta) \tag{2}\]  
\[T_{2}=\sqrt{3}\frac{|V_{m}|}{V_{dc}}T\sin(\theta) \tag{3}\]  
\[T_{0}=T-T_{1}-T_{2} \tag{4}\]  
where \(V_{dc}\) represents the DC bus voltage value, while \(\theta\) denotes the angle between the synthesized vector and the principal vector.  
The modulation index of the SVPWM is defined as follows:  
\[M=\sqrt{3}\frac{|V_{m}|}{V_{dc}} \tag{5}\]  
When the amplitude of the target synthetic vector precisely equals the radius of the circumcircle of the SVPWM hexagon, the following conditions are fulfilled:  
\[|V_{m}|=\frac{2}{3}V_{dc} \tag{6}\]  
By substituting Equation (6) into Equation (5), the maximum modulation index of the SVPWM is derived as 1.15. At this point, SVPWM enters the overmodulation state (i.e., PWM saturation). In comparison, the maximum modulation index of sinusoidal PWM  
Figure 1: Structure diagram of the three-phase voltage source inverter.  
is limited to 1. Consequently, SVPWM demonstrates significant advantages with a notably higher voltage utilization rate [21].