## 1 Introduction
### Thirty Six-Impulse Rectifiers
 \tag{1}\]  
where: \(U_{m}\)--is the amplitude of the maximum three-phase voltage of the generator AC, \(u_{SA}(t)\), \(u_{SB}(t)\) and \(u_{SC}(t)\)--are the control signal voltages in individual phases given to the converter input ATRU, and \(\theta=\omega t\), \(\alpha\)- is the phase angle of the initial variable signal. The mathematical record of the equations showing the induced voltages in each phase and the capacitive current can be written as follows:  
\[ \begin{array}{l}L\frac{di_{SA}}{dt}=u_{SA}-f_{A}U_{DC}\\ L\frac{di_{SB}}{dt}=u_{SB}-f_{B}U_{DC}\\ L\frac{di_{SC}}{dt}=u_{SC}-f_{C}U_{DC}\\ C\frac{du_{DC}}{dt}=f_{A}i_{SA}+f_{B}i_{SB}+f_{C}i_{SC}-I_{LOAD}\end{array}  \tag{2}\]  
where: L and C- are the inductance and capacity, occurring in the rectifier system,\(f_{A}\),\(f_{B}\),\(f_{C}\)- represent the switching function of the AC/DC converter, and \(i_{SA}(t)\), \(i_{SB}(t)\) and \(i_{SC}(t)\)- define the control signal  
Figure 4: Diagram of a classic 36-pulse rectifier powered by a three-phase transformer. currents in a particular phase, respectively, given to the input of the ATRU converter, while ILOAD and UDC- indicate the values of the load current and the DC voltage at the output of the rectifier. The mathematical dependencies describing the operation of the function are as follows:  
\[ \begin{array}{l}f_{A}=\frac{(2S_{A}-S_{B}-S_{C})}{\left.\begin{array}{l}f_{B} =\frac{(2S_{B}-S_{A}-S_{C})}{3}\\ f_{C}=\frac{(2S_{C}-S_{A}-S_{B})}{3}\end{array}\right.\end{array}  \tag{3}\]  
Parameters \(S_{A}\), \(S_{B}\), \(S_{C}\)- represent a control signal that is used to control the ATRU system in any phase of the input voltage. Each of the control signals emits a \"0\" or \"1\" state, which means if in the control system marked by \(S_{A}\) there is a value of \"1\", the upper switch operates in phase A and is \"switched on\" (bottom switch in voltage phase A is \"off\"). Otherwise, when \(S_{A}\) is set to state \"0\", the upper switch in phase A is \"off\" and the lower switch is \"on\". The process of switching modes of operation in the ATRU system of the aircraft electrical network in accordance with the MEA concept is mainly based on the value of three-phase voltage; in particular, its phase. In turn, the vector defining the operating state of the generator is transformed in the two axes of the reference system \(d\)-\(q\). The notation is as follows:  
\[\left[ \begin{array}{c}X_{d}\\ X_{q}\end{array} \right]=\frac{2}{3}\left[ \begin{array}{cc}\cos\omega t&\cos \!\left(\omega t-\frac{2}{3}\pi\right)&\cos\!\left(\omega t+\frac{2}{3}\pi \right)\\ -\sin\omega t&\sin\!\left(\omega t-\frac{2}{3}\pi\right)&-\sin\!\left(\omega t +\frac{2}{3}\pi\right)\end{array} \right]\right]\left[ \begin{array}{c}X_{A} \\ X_{B}\\ X_{C}\end{array} \right] \tag{4}\]  
Based on the above mathematical relations, the ATRU converter output voltage can be controlled by a combination of eight switching states. These states are described by using vectors \(V\) (\(0\)) to \(V\) (\(7\)). Vector positioning states in the \(d\)-\(q\) reference system depend on the items in the control state in the \(d\)-\(q\) system, as illustrated in the figure below (Figure 5). It should be noted that for practical applications, six states are taken into account, defining the ATRU system operating voltage and ensuring appropriate values of the output voltage. Figure 5: Voltage vector relations in the \(d\)-\(q\) reference system.