# Comparison of Hysteresis Based PWM Schemes \(\Delta\Sigma\)-PWM and Direct Torque Control
## 3 Hysteresis Based Pulse Width Modulation Schemes
### \(\Delta\Sigma\)-Pwm  
\(\Delta\Sigma\) signal processing operates internally on bitstreams with a datarate of 10 MBit/s. This reduces the wiring effort between calculation units on chip, respectively, in an FPGA. Since modern inverters can handle input signals at switching frequencies between4 and 40 kHz [6], a decimation has to be implemented. For this purpose, the authors of [6] introduce a two-dimensional hysteresis element, which derives the switching decisions from the accumulated voltage time error vector. As shown in Figure 1, the two components of this vector (\(\text{Int}_{1},\text{Int}_{2}\)) are calculated by two integrators. The input to each integrator is the difference of two high frequent bit streams [9]. The advantages and disadvantages of several schemes in bit stream signal processing are compared in, e.g., [31]. As a result, the _bipolar interpretation_ is shown to be suitable to apply linear algebra on a high frequent bit stream. In this approach, the values of the high frequent bit stream are mapped to \"1\" and \"\(-1\)\", followed by an optional bipolar arithmetic operation. Afterwards, a \(\Delta\Sigma\)-modulator generates a bit stream from the multi bit signal [31]. Furthermore, all input signals of \(\Delta\Sigma\)-PWM are converted to high frequent bit streams by using \(\Delta\Sigma\)-modulators [19]. Those transformation blocks can be found in Figure 1, which shows the inner control loop of \(\Delta\Sigma\)-PWM. Unless otherwise stated, second order \(\Delta\Sigma\)-modulators, as in [6; 32] and [33], are utilized. Those modulators are known to push the quantization noise to higher frequencies [6]. However, the control loop presented in Figure 1 combines several calculation tasks within a single step operation. Additionally to the error calculation, a coordinate transformation is applied. The voltage error is calculated in a \(120^{\circ}\) two phase coordinate system according to the following equation [6]  
\[\underline{u}_{s}=(u_{s1}-u_{s3})\cdot e^{j\cdot 0^{\circ}}+(u_{s2}-u_{s3}) \cdot e^{j\cdot 120^{\circ}} \tag{15}\]  
The integrator output, i.e., the voltage time error trajectory, moves with each bit stream clock in a plane spanned by the \(120^{\circ}\) coordinate system. The associated magnitude and angle boundaries are shown in Figure 2[9]. Figure 1: First control loop of \(\Delta\Sigma\)-PWM, adapted with permission from [6], [PERSON], 2016. Figure 2: (a) Amplitude boundaries as well as (b) angle boundaries for V2 and for V1 voltage space vectors in DS-PWM, adapted with permission from [6], [PERSON], 2016. The boundaries of \(\Delta\Sigma\)-PWM can be divided into two groups. The constants \(N_{1}\) and \(N_{2}\) are magnitude boundaries. While \(N_{1}\) mainly controls the average switching frequency of \(\Delta\Sigma\)-PWM, \(N_{2}\) defines the functionality in the case of overmodulation [9]. Figure 2b explains the angle boundaries which are used in \(\Delta\Sigma\)-PWM. The angle boundaries are activated depending on the applied vector. As long as the error trajectory stays within active boundaries, the switching state is kept. Only when the error trajectory crosses an angle or magnitude boundary, the voltage space vector is changed [10]. The new vector is selected according to the sequence [6]  
\[(000)\to V_{1}\to V_{2}\rightarrow(111)\to V_{2}\to V_{1} \rightarrow(000). \tag{16}\]  
This sequence of voltage space vectors is well known from the concept of space vector modulation [26]. At the beginning of the sequence, i.e., the voltage vector (000) is applied to the inverter, the voltage time error trajectory is usually close to the origin. As (000) results in zero voltage at the inverter output, the trajectory is driven only by the reference bit stream [19]. Consequently, the error trajectory moves apart from the origin and the boundary \(N_{1}\) is exceeded in one of the three sectors connected to the vectors of type \(V_{1}\) (001, 010, 100), see Figure 2b. Then the respective vector is switched on and the error trajectory is directed to the origin.