# Comparison of Hysteresis Based PWM Schemes \(\Delta\Sigma\)-PWM and Direct Torque Control
## 1 Introduction  
Electric drive systems are used in automotive industry more often. Because of climate change and the increasing global awareness of reducing greenhouse gas emission the demand on zero emission drive systems rises as well [1, 2]. However, widespread requirements has to be considered in automotive industry. Weight, performance and constructed size of the drive system are just a few of the numerous requirements for automotive applications. Furthermore, the electric consumption including the efficiency of the electric drive is an important design criterion for manufacturers [1, 3, 4]. Several papers compare different control or modulation schemes with respect to efficiency [4]. Some of them use hysteresis elements, others fixed frequency modulation schemes. In this paper, a simulative comparison of the novel, \(\Delta\Sigma\) signal processing based, \(\Delta\Sigma\) pulse width modulation (\(\Delta\Sigma\)-PWM) with direct torque control (DTC) is performed. \(\Delta\Sigma\) signal processing has become increasingly important in recent years. It is characterized by a high small-signal bandwidth [5]. The high frequent binary bit streams, which are the basis of \(\Delta\Sigma\) signal processing, cannot be transferred directly to power electronics [6]. To ensure a widespread application of \(\Delta\Sigma\) signal processing in electrical drive systems, a decimation is required, to reduce the sampling rate. In [7, 8], the highly dynamic coupling of bit streams to the power electronics was demonstrated by the application of a three-phase hysteresis modulator. While the average switching frequencies of the modulator in [8] are asymmetrical with respect to the three phases, equal switching frequencies for all phases were achieved in [6] by adapting the vector sequence of the space vector modulation. This helps to keep the equal distribution of switching power losses among the phases. In contrast to other pulse width modulation (PWM) methods, such as classical carrier-based techniques or the space vector modulation, \(\Delta\Sigma\)-PWM has a variable switching frequency [9; 10]. In this paper, the differences and similarities of the novel \(\Delta\Sigma\)-PWM with an established PWM method are presented. The comparison is made using an example from the field of electric drives. In order to analyze the differences and similarities in detail, the comparison is limited to one method. Since the switching behavior has a great influence on the control quality of the method, a fair comparison can only be taken to other variable frequency modulation schemes [11]. DTC is one of the most famous control schemes working with a variable switching frequency [12]. Although DTC was developed in 1986, it is still frequently used in various applications [13; 14; 15; 16; 17; 18]. Due to its history and familiarity with electrical drive systems, DTC was chosen to be compared with \(\Delta\Sigma\)-PWM. In addition to the variable switching frequency, there are other similarities between DTC and \(\Delta\Sigma\)-PWM, which make a comparison of the two methods reasonable. In this context, the hysteresis controller as the base component of both methods is the first thing to mention. While DTC controls the flux and torque with two independent hysteresis elements, \(\Delta\Sigma\)-PWM consists of a two-dimensional hysteresis element [6; 12]. This element is fed with the two components of the integrated voltage error vector [6]. Since the flux is calculated by integrating the voltage, thus \(\Delta\Sigma\)-PWM features a flux controller as well. Furthermore, due to the proportionality between \(q\)-current and torque, torque control can be achieved by embedding \(\Delta\Sigma\)-PWM in a field-oriented current control loop. Thus, equivalent control structures can be set up with both methods. Furthermore, both methods are characterized by a low voltage reserve, since both use almost the entire inverter hexagon [6; 12].