## 1 Introduction
### Thirty Six-Impulse Rectifiers  
The system with a parallel connection of two six-pulse rectifiers showing the properties of the 36-pulse system is shown in the figure below (Figure 3). This is achieved by switching on three  
Figure 1: Diagram of a classical 12-pulse rectifier in the auto-transformer rectifier unit (ATRU) system. Figure 2: Diagram of a classic 24-pulse rectifier powered by a three-phase transformer. thyristors in the order: T3-T1-T2-T2-T1-T3-T3-T1. The sequence of switching thyristors on in a specific order plays an important role due to the production of appropriate voltage and current components reaching the final equipment (devices, installations, systems) mounted on-board modern aircrafts. This condition must also be met due to the elimination of higher harmonics that negatively affect the AC voltage of the electrified aircraft's on-board network. Where: T\({}_{1}\), T\({}_{2}\) and T\({}_{3}\)- thyristors, V\({}_{1}\), V\({}_{2}\) and V\({}_{3}\)- three-phase voltage sources, LOAD- terminal devices. The proper sequence of switching on thyristors also results in better power efficiency in the field of its transfer to terminal equipment. In addition, the divided windings of the matching choke in the DC circuit provide natural commutation of the attached thyristors [14; 15]. Mathematical Model for Controlling the ATRU Unit in the Field of the PES System in Accordance with the Concept of MEA  
Currently, the most commonly used technique responsible for the process of controlling the work of circuits (used to convert AC three-phase voltages and currents to their DC equivalents) is realized by PWM (Pulse Width Modulation) and rectifier circuits. Its main advantages, which influenced the use of this method in more electric aircrafts were, among others: the possibility of stabilizing the input voltage, elimination of undesirable harmonic signals in the output voltage and a more effective PF. All the above-mentioned factors have a decisive influence on the overall functioning of the electrical network of the more electric aircraft, in particular in the field of improving the quality of the electro-energetic system. Subsequently, a mathematical model was created, describing the work of the PWM pulse width rectification circuit. During the process of creating a mathematical model of the rectifier, its key parameters (voltage, current) should first be considered. By analyzing these parameters in the scope of the numerical analysis process, it can be used to describe the mathematical method of converting AC voltage and current into their equivalents in the form of DC current. The whole process was analyzed on the basis of the more electric aircraft on-board network, based on the ATRU subunit. The input voltages supplied to the ATRU system were delivered from an alternating current AC generator. The key goal of the mathematical analysis was to determine the stabilized DC output voltage in a short response time. The entire analysis process refers to the change in  
Figure 3: Diagram of a classic 36-pulse rectifier powered by a three-phase transformer. the coordinate system, which determines the position of the PMSM (Permanent Magnet Synchronous Machine) poles _d-q_[16; 17]. The diagram on the basis of which a description of mathematical electrical phenomena taking place in the ATRU system is shown in the figure below (Figure 4). Where: AC1 and AC2- three-phase sources AC (generators), V\({}_{\text{p}}\) and V\({}_{\text{n}}\)- voltage value at the output of the system, LOAD- terminal devices. Turning to the mathematical description, it should take into account the three-phase voltage given to the input of the ATRU converter in the reference system _d-q_. The following expression will take the form [18; 19]:  
\[ \begin{array}{l}u_{SA}(t)=U_{m}\cos(\omega t)\\ u_{SB}(t)=U_{m}\cos(\omega t-\frac{2}{3}\pi)\\ u_{SC}(t)=U_{m}\cos(\omega t+\frac{2}{3}\pi)\end{array}