## 1 Introduction

## 2 Continuous-time and discrete-time model equations of the electrical subsystem of inverter-fed three-phase drives

### 2.1 Continuous-time model equations of the PMSM

### 2.2 Continuous-time model equations of the IM

### 2.3 Summary of the continuous-time model equations of PMSM and IM

### 2.4 Discrete-time decoupling-relevant model equations of PMSM and IM

## 3 Discrete-time decoupling

### 3.1 Theoretical principles

_{1}element, is shown in Fig. 3b via the corresponding difference equation and marked as P–T

_{1}element.

### 3.2 Comparison of PI controller-based decoupling methods

Parameter | Symbol | Value | Unit |
---|---|---|---|

Direct axis stator inductance | \({L}_{\mathrm{S},\mathrm{d}}\) | 5.89 | mH |

Quadrature axis stator inductance | \({L}_{\mathrm{S},\mathrm{q}}\) | 5.89 | mH |

Stator resistance | \({R}_{\mathrm{S}}\) | 1.9 | Ω |

Absolute value of the permanent magnet flux space vector | \({\Psi }_{\mathrm{PM},\mathrm{d}}\) | 0.08 | Vs |

Number of pole pairs | \({n}_{\mathrm{P}}\) | 5 | |

Moment of inertia | \(J\) | 0.000113 | kg·m ^{2} |

Nominal speed | \({n}_{\mathrm{nom}}\) | 5500 | min ^{−1} |

Nominal stator frequency | \({f}_{\mathrm{S},\mathrm{nom}}\) | 458.3 | Hz |

Nominal current (rms) | \({I}_{\mathrm{S},\mathrm{nom}}\) | 2.4 | A |

Nominal phase-to-phase voltage (rms) | \({V}_{\mathrm{S},\mathrm{nom}}\) | 300 | V |

Nominal torque | \({T}_{\mathrm{nom}}\) | 2 | N·m |

DC-link voltage | \({v}_{\mathrm{dc}}\) | 565 | V |

^{−1}. Due to the number of pole pairs \({n}_{\mathrm{P}}=5\), the stator frequency is then 500 Hz. The nominal value 3.4 A of the quadrature axis current is applied as setpoint \({i}_{\mathrm{S},\mathrm{q},\mathrm{ref}}\). Because the nominal speed is 5500 min

^{−1}, no field weakening is required. I.e., \({i}_{\mathrm{S},\mathrm{d},\mathrm{ref}}=0\) is specified. When the stated maximum speed is reached, \({i}_{\mathrm{S},\mathrm{q},\mathrm{ref}}=-3.4 \; \mathrm{A}\) is applied until − 6000 min

^{−1}is reached, and so on (reversing process). A computational dead time of one sampling interval was taken into account, while Fig. 4a shows the ramp-up with a switching and sampling frequency of \({f}_{\mathrm{PWM}}=2 \; \mathrm{kHz}\), Fig. 4b shows the time characteristics for \({f}_{\mathrm{PWM}}=3 \; \mathrm{kHz}\) and Fig. 4c the time characteristics for \({f}_{\mathrm{PWM}}=4 \; \mathrm{kHz}\), each for the case of a time-continuously designed decoupling and disturbance rejection according to Eq. (38) (but see the note just before Fig. 4d). The controller parameters of the PI current controller were designed discrete-time. This was also based on a computational dead time of one sampling interval. In view of this, a double real pole at \(z=\frac{1}{2}\) was aimed at for the z-command transfer function of the closed current control loop, resulting in

_{1}elements with the settling time constant of the closed-loop current control as time constant. However, the use of setpoints does not represent decoupling, but is merely a precontrol of the voltages ideally required for decoupling. Accordingly, the complex poles of the controlled system model are at best approximately compensated, but in no case shifted into real areas where they do not cause coupling and oscillations. To demonstrate that the use of current setpoints for decoupling does not solve the decoupling task, Fig. 4d shows the time curves of the stator current direct and quadrature axis components as well as other quantities obtained during the described reversing process when, at a switching and sampling frequency of 4 kHz, this method was used for decoupling purposes and the current setpoint components are previously passed over the aforementioned P–T

_{1}elements. Furthermore, it should be noted that the decoupling law used in Fig. 4a through 4d still contains the rotation factor \({\mathrm{e}}^{ \mathrm{j}2{\omega }_{\mathrm{S}}{T}_{\mathrm{S}}}\) as in Eq. (59) (see Sect. 4). Without it, the course of \({i}_{\mathrm{S},\mathrm{d}}\) and \({i}_{\mathrm{S},\mathrm{q}}\) would have a less favorable appearance.

_{1}elements:

^{−1}, where a sign change of \({i}_{\mathrm{S},\mathrm{q},\mathrm{ref}}\) then takes place until the speed has dropped to 4200 min

^{−1}. As soon as this speed is fallen below, the sign of \({i}_{\mathrm{S},\mathrm{q},\mathrm{ref}}\) is changed again. As a result, the current control loop is excited within a few milliseconds with changing quadrature axis current setpoints, at a stator frequency in the range of 375 Hz. The maximum frequency was reduced in comparison to the profile in Fig. 4a–k in order to prevent the magnitude of the reference voltage space vector from being limited because the stator current setpoint steps now take place partly in motor operation and need therefore higher voltage.