DE102019220103A1 - Method and device for operating an electrical machine - Google Patents

Abstract

The invention relates to a method for the controlled operation of a multiphase electrical machine (2) with several phase strands (23) connected in a delta circuit, with the following steps: - providing a control deviation, - regulating the operation of the electrical machine (2) depending on the computationally determined actual String currents to provide terminal potential information (Su, Sv, Sw) for controlling an inverter (3), which applies terminal potentials (Vu, Vv, Vw) to the electrical machine (2), whereby the actual string currents (Iaist, Ibist, Icist) can be determined with the help of a machine model depending on the terminal potentials (Vu, Vv, Vw), whereby the machine model depends on an inductance matrix that determines the magnetic properties of the electrical machine in such a way that these voltage differences between the terminal potentials (Vu, Vv, Vw) can be mapped onto the actual line currents (Iaist, Ibist, Icist).

Description

Technical area

The invention relates generally to electrical machines, in particular to methods for operating electrical machines that use a machine model for modeling an operating variable of the electrical machine.

Technical background

A machine model that depicts a relationship between phase voltages and phase currents in the electrical machine is used for both simulation and operation of electrical machines. The core of this machine model is an inductance matrix, the elements of which have phase inductances and coupling inductances that are dependent on the rotor position. To use this machine model, it is usually necessary to invert the inductance matrix, which is complex and not trivial due to the position dependency of the individual elements of the inductance matrix.

Previous approaches to the implementation of the machine model use a park transformation into a rotor-fixed coordinate system, which is usually lossy. Other approaches simplify the machine model by assuming that the coupling inductances are position-independent. However, this approach is not expedient in sequential pole machines, in which the inductances not only have 2nd order harmonies, but also 1st, 3rd, 4th and higher order harmonies, since the conventional inversion method can only take into account 2nd order harmonies the harmonies of other orders are not taken into account.

The determination of the phase currents requires an inversion of the inductance matrix in each time step. This is computationally and time consuming. In addition, a circulating current can occur with a delta connection, which negatively affects the performance of the electrical machine.

It is an object of the present invention to provide a method for operating an electrical machine, which has phase strings in a delta connection, with a machine model that requires little computing effort.

Disclosure of the invention

This object is achieved by the method for operating an electrical machine in a delta connection with a machine model according to claim 1 and by the device and the motor system according to the independent claims.

Further refinements are given in the dependent claims.

• - Bereitstellen einer Regelabweichung,
• - Regeln des Betriebs der elektrischen Maschine abhängig von rechnerisch ermittelten Ist- Strangströmen, um Klemmenpotenzialangaben für die Ansteuerung eines Wechselrichters bereitzustellen, der entsprechend Klemmenpotenziale an die elektrische Maschine anlegt,

wobei die Ist- Strangströme mithilfe eines Maschinenmodells abhängig von den Klemmenpotenzialen ermittelt werden, wobei das Maschinenmodell von einer Induktivitätsmatrix abhängt, die magnetische Eigenschaften der elektrischen Maschine so bestimmt, dass durch diese Spannungsdifferenzen zwischen den Klemmenpotenzialen auf die Ist- Strangströme (I_aist, I_bist, I_cist) abgebildet werden.According to a first aspect, there is a method for the controlled operation of a multiphase electrical machine with several phase strings connected in a delta circuit, with the following steps:

• - Provision of a control deviation,
• - Regulating the operation of the electrical machine depending on the computationally determined actual phase currents in order to provide terminal potential information for controlling an inverter, which applies the corresponding terminal potentials to the electrical machine,

The actual phase currents are determined using a machine model as a function of the terminal potentials, the machine model being dependent on an inductance matrix that determines the magnetic properties of the electrical machine in such a way that these voltage differences between the terminal potentials result in the actual _phase currents (I aist, I _bist , I _cist ).

mit $$\left[\begin{array}{c}Iu\\ Iv\\ Iw\end{array}\right]=\left[\begin{array}{c}{I}_a-I_c\\ I_b-I_a\\ I_c-I_b\end{array}\right]$$

$${\psi }_a=La\left(\phi \right)\cdot Ia+Mab\left(\phi \right)\cdot Ib+Mac\left(\phi \right)\cdot Ic+{\psi }_{ma}$$

$${\psi }_b=Mab\left(\phi \right)\cdot Ia+Lb\left(\phi \right)\cdot Ib+Mbc\left(\phi \right)\cdot Ic+{\psi }_{mb}$$

$${\psi }_c=Mac\left(\phi \right)\cdot Ia+Mbc\left(\phi \right)\cdot Ib+Lc\left(\phi \right)\cdot Ic+{\psi }_{mc}$$

wobei Vu, Vv, Vw den Klemmenpotenzialen der Phasenstränge, la,lb,lc den Strangströmen, R dem elektrischen Strangwiderstand, La, Lb, Lc den Stranginduktivitäten, Mab, Mac, Mbc den Kopplungsinduktivitäten, φ der Läuferlage, Ψa, Ψb, Ψc den Flüssen in den Phasensträngen und Ψma, Ψmb, Ψmc den durch die Rotation des Läufers jeweils induzierten Flüssen entsprechen. Die Indices a, b, c bezeichnen den jeweiligen Phasenstrang und die Indices u, v, w den jeweiligen Klemmenanschluss.Machine models that describe the relationship between phase voltages and phase currents and the dependence of the torque on the phase currents or the motor current are generally used to simulate and operate electrical machines. For a three-phase electrical machine in a delta connection (corresponds to a delta connection), for example, this results for the machine model $$\left[\begin{array}{c}Vu-Vv\\ Vv-Vw\\ Vw-Vu\end{array}\right]=\mathrm{R.}\cdot \left[\begin{array}{c}\mathrm{I.}a\\ \mathrm{I.}b\\ \mathrm{I.}c\end{array}\right]+\frac{d}{dt}\left[\begin{array}{c}{\psi }_a\left({\mathrm{I.}}_a,{\mathrm{I.}}_b,{\mathrm{I.}}_c,\phi \right)\\ {\psi }_b\left({\mathrm{I.}}_a,{\mathrm{I.}}_b,{\mathrm{I.}}_c,\phi \right)\\ {\psi }_c\left({\mathrm{I.}}_a,{\mathrm{I.}}_b,{\mathrm{I.}}_c,\phi \right)\end{array}\right]$$

With $$\left[\begin{array}{c}\mathrm{I.}u\\ \mathrm{I.}v\\ \mathrm{I.}w\end{array}\right]=\left[\begin{array}{c}{\mathrm{I.}}_a-{\mathrm{I.}}_c\\ {\mathrm{I.}}_b-{\mathrm{I.}}_a\\ {\mathrm{I.}}_c-{\mathrm{I.}}_b\end{array}\right]$$

$${\psi }_a=\mathrm{L.}a\left(\phi \right)\cdot \mathrm{I.}a+\mathrm{M.}ab\left(\phi \right)\cdot \mathrm{I.}b+\mathrm{M.}ac\left(\phi \right)\cdot \mathrm{I.}c+{\psi }_{ma}$$

$${\psi }_b=\mathrm{M.}ab\left(\phi \right)\cdot \mathrm{I.}a+\mathrm{L.}b\left(\phi \right)\cdot \mathrm{I.}b+\mathrm{M.}bc\left(\phi \right)\cdot \mathrm{I.}c+{\psi }_{mb}$$

$${\psi }_c=\mathrm{M.}ac\left(\phi \right)\cdot \mathrm{I.}a+\mathrm{M.}bc\left(\phi \right)\cdot \mathrm{I.}b+\mathrm{L.}c\left(\phi \right)\cdot \mathrm{I.}c+{\psi }_{mc}$$

where Vu, Vv, Vw the terminal potentials of the phase strands, la, lb, lc the phase currents, R the electrical phase resistance, La, Lb, Lc the phase inductances, Mab, Mac, Mbc the coupling inductances, φ the rotor position, Ψa, Ψb, Ψc den The flows in the phase strands and Ψma, Ψmb, Ψmc correspond to the flows induced by the rotation of the rotor. The indices a, b, c designate the respective phase line and the indices u, v, w the respective terminal connection.

$$Lb\left(\phi \right)=l_0+l_1\cdot cos\left(\phi -\frac{2\pi }{3}\right)+l_2\cdot cos\left[2\cdot \left(\phi -\frac{2\pi }{3}\right)\right]+l_3\cdot cos\left[3\cdot \left(\phi -\frac{2\pi }{3}\right)\right]+\cdots$$

$$Lc\left(\phi \right)=l_0+l_1\cdot cos\left(\phi +\frac{2\pi }{3}\right)+l_2\cdot cos\left[2\cdot \left(\phi +\frac{2\pi }{3}\right)\right]+l_3\cdot cos\left[3\cdot \left(\phi +\frac{2\pi }{3}\right)\right]+\cdots$$

$$Mbc\left(\phi \right)=m_0+m_1\cdot cos\left(\phi \right)+m_2\cdot cos\left(2\cdot \phi \right)+m_3\cdot cos\left(3\cdot \phi \right)+\cdots$$

$$Mac\left(\phi \right)=m_0+m_1\cdot cos\left(\phi -\frac{2\pi }{3}\right)+m_2\cdot cos\left[2\cdot \left(\phi -\frac{2\pi }{3}\right)\right]+m_3\cdot cos\left[3\cdot \left(\phi -\frac{2\pi }{3}\right)\right]+\cdots$$

$$Mab\left(\phi \right)=m_0+m_1\cdot cos\left(\phi +\frac{2\pi }{3}\right)+m_2\cdot cos\left[2\cdot \left(\phi +\frac{2\pi }{3}\right)\right]+m_3\cdot cos\left[3\cdot \left(\phi +\frac{2\pi }{3}\right)\right]+\cdots$$

The phase voltages Vu - Vv, Vv - Vw, Vw - Vu are represented as a sum term made up of the phase resistance R, the phase inductances La, Lb, Lc and the fluxes Ψa, Ψb, Ψc in the phase phases. Due to the interactions of the inductances between the phase coils, the magnetization voltages are described with the aid of an inductance matrix, which contains both phase inductances La, Lb, Lc and coupling inductances Mab, Mac, Mbc. In reality, position-dependent functions result for the strand inductances La, Lb, Lc and the coupling inductances Mab, Mac, Mbc, which can be represented as series developments of cosine terms. $$\mathrm{L.}a\left(\phi \right)=l_0+{\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="DE102019220103A1\_0065.png"\; state="\{\{state\}\}">l</figure-callout>}}_1\cdot cOs\left(\phi \right)+{\mathrm{<figure-callout\; id="2"\; label="l"\; filenames="DE102019220103A1\_0064.png"\; state="\{\{state\}\}">l</figure-callout>}}_2\cdot cOs\left(2\cdot \phi \right)+{\mathrm{<figure-callout\; id="3"\; label="l"\; filenames="DE102019220103A1\_0064.png"\; state="\{\{state\}\}">l</figure-callout>}}_3\cdot cOs\left(3\cdot \phi \right)+\cdots$$

$$\mathrm{L.}b\left(\phi \right)=l_0+{\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="DE102019220103A1\_0065.png"\; state="\{\{state\}\}">l</figure-callout>}}_1\cdot cOs\left(\phi -\frac{2\pi }{3}\right)+{\mathrm{<figure-callout\; id="2"\; label="l"\; filenames="DE102019220103A1\_0064.png"\; state="\{\{state\}\}">l</figure-callout>}}_2\cdot cOs\left[2\cdot \left(\phi -\frac{2\pi }{3}\right)\right]+{\mathrm{<figure-callout\; id="3"\; label="l"\; filenames="DE102019220103A1\_0064.png"\; state="\{\{state\}\}">l</figure-callout>}}_3\cdot cOs\left[3\cdot \left(\phi -\frac{2\pi }{3}\right)\right]+\cdots$$

$$\mathrm{L.}c\left(\phi \right)=l_0+{\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="DE102019220103A1\_0065.png"\; state="\{\{state\}\}">l</figure-callout>}}_1\cdot cOs\left(\phi +\frac{2\pi }{3}\right)+{\mathrm{<figure-callout\; id="2"\; label="l"\; filenames="DE102019220103A1\_0064.png"\; state="\{\{state\}\}">l</figure-callout>}}_2\cdot cOs\left[2\cdot \left(\phi +\frac{2\pi }{3}\right)\right]+{\mathrm{<figure-callout\; id="3"\; label="l"\; filenames="DE102019220103A1\_0064.png"\; state="\{\{state\}\}">l</figure-callout>}}_3\cdot cOs\left[3\cdot \left(\phi +\frac{2\pi }{3}\right)\right]+\cdots$$

$$\mathrm{M.}bc\left(\phi \right)=m_0+{\mathrm{<figure-callout\; id="1"\; label="m"\; filenames="DE102019220103A1\_0065.png"\; state="\{\{state\}\}">m</figure-callout>}}_1\cdot cOs\left(\phi \right)+{\mathrm{<figure-callout\; id="2"\; label="m"\; filenames="DE102019220103A1\_0064.png"\; state="\{\{state\}\}">m</figure-callout>}}_2\cdot cOs\left(2\cdot \phi \right)+{\mathrm{<figure-callout\; id="3"\; label="m"\; filenames="DE102019220103A1\_0064.png"\; state="\{\{state\}\}">m</figure-callout>}}_3\cdot cOs\left(3\cdot \phi \right)+\cdots$$

$$\mathrm{M.}ac\left(\phi \right)=m_0+{\mathrm{<figure-callout\; id="1"\; label="m"\; filenames="DE102019220103A1\_0065.png"\; state="\{\{state\}\}">m</figure-callout>}}_1\cdot cOs\left(\phi -\frac{2\pi }{3}\right)+{\mathrm{<figure-callout\; id="2"\; label="m"\; filenames="DE102019220103A1\_0064.png"\; state="\{\{state\}\}">m</figure-callout>}}_2\cdot cOs\left[2\cdot \left(\phi -\frac{2\pi }{3}\right)\right]+{\mathrm{<figure-callout\; id="3"\; label="m"\; filenames="DE102019220103A1\_0064.png"\; state="\{\{state\}\}">m</figure-callout>}}_3\cdot cOs\left[3\cdot \left(\phi -\frac{2\pi }{3}\right)\right]+\cdots$$

$$\mathrm{M.}ab\left(\phi \right)=m_0+{\mathrm{<figure-callout\; id="1"\; label="m"\; filenames="DE102019220103A1\_0065.png"\; state="\{\{state\}\}">m</figure-callout>}}_1\cdot cOs\left(\phi +\frac{2\pi }{3}\right)+{\mathrm{<figure-callout\; id="2"\; label="m"\; filenames="DE102019220103A1\_0064.png"\; state="\{\{state\}\}">m</figure-callout>}}_2\cdot cOs\left[2\cdot \left(\phi +\frac{2\pi }{3}\right)\right]+{\mathrm{<figure-callout\; id="3"\; label="m"\; filenames="DE102019220103A1\_0064.png"\; state="\{\{state\}\}">m</figure-callout>}}_3\cdot cOs\left[3\cdot \left(\phi +\frac{2\pi }{3}\right)\right]+\cdots$$

$$\begin{array}{c}{\psi }_{mb}=a1\cdot cos\left(\phi -\frac{2\pi }{3}\right)+a3\cdot cos\left[3\cdot \left(\phi -\frac{2\pi }{3}\right)\right]+a5\cdot cos\left[5\cdot \left(\phi -\frac{2\pi }{3}\right)\right]+a7\\ \cdot cos\left[7\cdot \left(\phi -\frac{2\pi }{3}\right)\right]+\cdots \end{array}$$

$$\begin{array}{c}{\psi }_{mc}=a1\cdot cos\left(\phi +\frac{2\pi }{3}\right)+a3\cdot cos\left[3\cdot \left(\phi +\frac{2\pi }{3}\right)\right]+a5\cdot cos\left[5\cdot \left(\phi +\frac{2\pi }{3}\right)\right]+a7\\ \cdot cos\left[7\cdot \left(\phi +\frac{2\pi }{3}\right)\right]+\cdots \end{array}$$

Furthermore, there are position-dependent functions for the flows Ψma, Ψmb, Ψmc, which can be represented as series expansion of cosine terms. $${\psi }_{ma}=a1\cdot cOs\left(\phi \right)+a3\cdot cOs\left(3\cdot \phi \right)+a5\cdot cOs\left(5\cdot \phi \right)+a\mathrm{7th}\cdot cOs\left(\mathrm{7th}\cdot \phi \right)+\cdots$$

$$\begin{array}{c}{\psi }_{mb}=a1\cdot cOs\left(\phi -\frac{2\pi }{3}\right)+a3\cdot cOs\left[3\cdot \left(\phi -\frac{2\pi }{3}\right)\right]+a5\cdot cOs\left[5\cdot \left(\phi -\frac{2\pi }{3}\right)\right]+a\mathrm{7th}\\ \cdot cOs\left[\mathrm{7th}\cdot \left(\phi -\frac{2\pi }{3}\right)\right]+\cdots \end{array}$$

$$\begin{array}{c}{\psi }_{mc}=a1\cdot cOs\left(\phi +\frac{2\pi }{3}\right)+a3\cdot cOs\left[3\cdot \left(\phi +\frac{2\pi }{3}\right)\right]+a5\cdot cOs\left[5\cdot \left(\phi +\frac{2\pi }{3}\right)\right]+a\mathrm{7th}\\ \cdot cOs\left[\mathrm{7th}\cdot \left(\phi +\frac{2\pi }{3}\right)\right]+\cdots \end{array}$$

$$\begin{array}{l}\left[\begin{array}{c}La\left(\phi \right)Mab\left(\phi \right)Mac\left(\phi \right)\\ Mab\left(\phi \right)Lb\left(\phi \right)Mbc\left(\phi \right)\\ Mac\left(\phi \right)Mbc\left(\phi \right)Lc\left(\phi \right)\end{array}\right]\cdot \frac{d}{dt}\left[\begin{array}{c}Ia\\ Ib\\ Ic\end{array}\right]\\ =\left[\begin{array}{c}Vu-Vv\\ Vv-Vw\\ Vv-Vu\end{array}\right]-R\cdot \left[\begin{array}{c}Ia\\ Ib\\ Ic\end{array}\right]-\frac{d}{dt}\left[\begin{array}{c}{\psi }_{ma}\\ {\psi }_{mb}\\ {\psi }_{mc}\end{array}\right]-\frac{d}{dt}\left[\begin{array}{c}La\left(\phi \right)Mab\left(\phi \right)Mac\left(\phi \right)\\ Mab\left(\phi \right)Lb\left(\phi \right)Mbc\left(\phi \right)\\ Mac\left(\phi \right)Mbc\left(\phi \right)Lc\left(\phi \right)\end{array}\right]\cdot \left[\begin{array}{c}Ia\\ Ib\\ Ic\end{array}\right]\end{array}$$

The above equation is usually implemented and solved with the phase currents as state variables: $$\left[\begin{array}{c}Vu-Vv\\ Vv-Vw\\ Vv-Vu\end{array}\right]=\mathrm{R.}\cdot \left[\begin{array}{c}\mathrm{I.}a\\ \mathrm{I.}b\\ \mathrm{I.}c\end{array}\right]+\frac{d}{dt}\left[\left[\begin{array}{c}\mathrm{L.}a\left(\phi \right)\mathrm{M.}ab\left(\phi \right)\mathrm{M.}ac\left(\phi \right)\\ \mathrm{M.}ab\left(\phi \right)\mathrm{L.}b\left(\phi \right)\mathrm{M.}bc\left(\phi \right)\\ \mathrm{M.}ac\left(\phi \right)\mathrm{M.}bc\left(\phi \right)\mathrm{L.}c\left(\phi \right)\end{array}\right]\cdot \left[\begin{array}{c}\mathrm{I.}a\\ \mathrm{I.}b\\ \mathrm{I.}c\end{array}\right]\right]+\frac{d}{dt}\left[\begin{array}{c}{\psi }_{ma}\\ {\psi }_{mb}\\ {\psi }_{mc}\end{array}\right]$$

$$\begin{array}{l}\left[\begin{array}{c}\mathrm{L.}a\left(\phi \right)\mathrm{M.}ab\left(\phi \right)\mathrm{M.}ac\left(\phi \right)\\ \mathrm{M.}ab\left(\phi \right)\mathrm{L.}b\left(\phi \right)\mathrm{M.}bc\left(\phi \right)\\ \mathrm{M.}ac\left(\phi \right)\mathrm{M.}bc\left(\phi \right)\mathrm{L.}c\left(\phi \right)\end{array}\right]\cdot \frac{d}{dt}\left[\begin{array}{c}\mathrm{I.}a\\ \mathrm{I.}b\\ \mathrm{I.}c\end{array}\right]\\ =\left[\begin{array}{c}Vu-Vv\\ Vv-Vw\\ Vv-Vu\end{array}\right]-\mathrm{R.}\cdot \left[\begin{array}{c}\mathrm{I.}a\\ \mathrm{I.}b\\ \mathrm{I.}c\end{array}\right]-\frac{d}{dt}\left[\begin{array}{c}{\psi }_{ma}\\ {\psi }_{mb}\\ {\psi }_{mc}\end{array}\right]-\frac{d}{dt}\left[\begin{array}{c}\mathrm{L.}a\left(\phi \right)\mathrm{M.}ab\left(\phi \right)\mathrm{M.}ac\left(\phi \right)\\ \mathrm{M.}ab\left(\phi \right)\mathrm{L.}b\left(\phi \right)\mathrm{M.}bc\left(\phi \right)\\ \mathrm{M.}ac\left(\phi \right)\mathrm{M.}bc\left(\phi \right)\mathrm{L.}c\left(\phi \right)\end{array}\right]\cdot \left[\begin{array}{c}\mathrm{I.}a\\ \mathrm{I.}b\\ \mathrm{I.}c\end{array}\right]\end{array}$$

und durch Umformung wie folgt bekannt. $$\begin{array}{l}\frac{d}{dt}\left[\begin{array}{c}Ia\\ Ib\\ Ic\end{array}\right]=\left[\left[\begin{array}{c}Vu-Vv\\ Vv-Vw\\ Vv-Vu\end{array}\right]-R\cdot \left[\begin{array}{c}Ia\\ Ib\\ Ic\end{array}\right]-\left[\begin{array}{c}\mathrm{\Psi }ma\\ \mathrm{\Psi }mb\\ \mathrm{\Psi }mc\end{array}\right]-\frac{d}{dt}\left[\begin{array}{c}La\left(\phi \right)Mab\left(\phi \right)Mac\left(\phi \right)\\ Mab\left(\phi \right)Lb\left(\phi \right)Mbc\left(\phi \right)\\ Mac\left(\phi \right)Mbc\left(\phi \right)Lc\left(\phi \right)\end{array}\right]\cdot \left[\begin{array}{c}Ia\\ Ib\\ Ic\end{array}\right]\right]\ast \\ \frac{1}{\text{det}\left(L\right)}{\left[\begin{array}{c}La\left(\phi \right)Mab\left(\phi \right)Mac\left(\phi \right)\\ Mab\left(\phi \right)Lb\left(\phi \right)Mbc\left(\phi \right)\\ Mac\left(\phi \right)Mbc\left(\phi \right)Lc\left(\phi \right)\end{array}\right]}^{-1}\end{array}$$

So far, the implementation of the above machine model for an operation of an electrical machine, in which the phase currents are not measured but rather modeled and are intended to serve as a control variable for current regulation, has only been done with inversion of the inductance matrix $$\left[\begin{array}{c}\mathrm{L.}a\left(\phi \right)\mathrm{M.}ab\left(\phi \right)\mathrm{M.}ac\left(\phi \right)\\ \mathrm{M.}ab\left(\phi \right)\mathrm{L.}b\left(\phi \right)\mathrm{M.}bc\left(\phi \right)\\ \mathrm{M.}ac\left(\phi \right)\mathrm{M.}bc\left(\phi \right)\mathrm{L.}c\left(\phi \right)\end{array}\right]$$

and known by forming as follows. $$\begin{array}{l}\frac{d}{dt}\left[\begin{array}{c}\mathrm{I.}a\\ \mathrm{I.}b\\ \mathrm{I.}c\end{array}\right]=\left[\left[\begin{array}{c}Vu-Vv\\ Vv-Vw\\ Vv-Vu\end{array}\right]-\mathrm{R.}\cdot \left[\begin{array}{c}\mathrm{I.}a\\ \mathrm{I.}b\\ \mathrm{I.}c\end{array}\right]-\left[\begin{array}{c}\mathrm{\Psi }ma\\ \mathrm{\Psi }mb\\ \mathrm{\Psi }mc\end{array}\right]-\frac{d}{dt}\left[\begin{array}{c}\mathrm{L.}a\left(\phi \right)\mathrm{M.}ab\left(\phi \right)\mathrm{M.}ac\left(\phi \right)\\ \mathrm{M.}ab\left(\phi \right)\mathrm{L.}b\left(\phi \right)\mathrm{M.}bc\left(\phi \right)\\ \mathrm{M.}ac\left(\phi \right)\mathrm{M.}bc\left(\phi \right)\mathrm{L.}c\left(\phi \right)\end{array}\right]\cdot \left[\begin{array}{c}\mathrm{I.}a\\ \mathrm{I.}b\\ \mathrm{I.}c\end{array}\right]\right]\ast \\ \frac{1}{\text{det}\left(\mathrm{L.}\right)}{\left[\begin{array}{c}\mathrm{L.}a\left(\phi \right)\mathrm{M.}ab\left(\phi \right)\mathrm{M.}ac\left(\phi \right)\\ \mathrm{M.}ab\left(\phi \right)\mathrm{L.}b\left(\phi \right)\mathrm{M.}bc\left(\phi \right)\\ \mathrm{M.}ac\left(\phi \right)\mathrm{M.}bc\left(\phi \right)\mathrm{L.}c\left(\phi \right)\end{array}\right]}^{-1}\end{array}$$

This type of implementation requires an inversion in every time step due to the dependence on the rotor position. Furthermore, due to the delta connection, depending on the values of the parameters l ₁ , l ₂ , l ₃ , m ₁ , m ₂ , m ₃ , Ψma, Ψmb, Ψmc, a circulating current 'II' can occur which can negatively affect the performance of the electrical machine. With this implementation it is difficult to see at which values of these parameters the circulating current can arise.

By suitable transformation of the above machine model, the conditions for the generation of circular currents should be determinable _{for the general case l 1} ≠ 0, l ₂ ≠ 0, l ₃ ≠ 0, m ₁ ≠ 0, m ₂ ≠ 0, m _{3 ≠ 0, so that these} can be taken into account for the operation of the electrical machine.

The above procedure suggests to formulate the machine model using two differential equations that use a simple invertible 2x2 inductance matrix and two state variables that do not contain a circulating current component.

For this purpose, a linked flow is defined for two of the phase strands.

mit $$dLMa\left(\phi \right)=La\left(\phi \right)-Mac\left(\phi \right)$$

$$dLMb\left(\phi \right)=Mab\left(\phi \right)-Mac\left(\phi \right)$$

$$dLMc\left(\phi \right)=Mab\left(\phi \right)-Mbc\left(\phi \right)$$

$$dLMd\left(\phi \right)=Lb\left(\phi \right)-Mbc\left(\phi \right)$$

The machine model results from: $$\frac{d}{dt}\left(\left[\begin{array}{c}d\mathrm{L.}\mathrm{M.}a\left(\phi \right)d\mathrm{L.}\mathrm{M.}b\left(\phi \right)\\ d\mathrm{L.}\mathrm{M.}c\left(\phi \right)d\mathrm{L.}\mathrm{M.}d\left(\phi \right)\end{array}\right]\cdot \left[\begin{array}{c}\mathrm{I.}a\\ \mathrm{I.}b\end{array}\right]\right)=\left[\begin{array}{c}Vu-Vv\\ Vv-Vw\end{array}\right]-\mathrm{R.}\cdot \left[\begin{array}{c}\mathrm{I.}a\\ \mathrm{I.}b\end{array}\right]-\frac{d}{dt}\left[\begin{array}{c}{\psi }_{ma}\\ {\psi }_{mb}\end{array}\right]$$

With $$d\mathrm{L.}\mathrm{M.}a\left(\phi \right)=\mathrm{L.}a\left(\phi \right)-\mathrm{M.}ac\left(\phi \right)$$

$$d\mathrm{L.}\mathrm{M.}b\left(\phi \right)=\mathrm{M.}ab\left(\phi \right)-\mathrm{M.}ac\left(\phi \right)$$

$$d\mathrm{L.}\mathrm{M.}c\left(\phi \right)=\mathrm{M.}ab\left(\phi \right)-\mathrm{M.}bc\left(\phi \right)$$

$$d\mathrm{L.}\mathrm{M.}d\left(\phi \right)=\mathrm{L.}b\left(\phi \right)-\mathrm{M.}bc\left(\phi \right)$$

Under the condition that l _k = m _k , k ∉ {0, 3, 6, ...} and a _j = 0, each {3, 6, ...}

This can then be converted to determine the phase currents: $$\begin{array}{c}\frac{d}{dt}\left(\left[\begin{array}{c}\mathrm{I.}a\\ \mathrm{I.}b\end{array}\right]\right)=\frac{1}{d\mathrm{L.}\mathrm{M.}a\left(\phi \right)\cdot d\mathrm{L.}\mathrm{M.}d\left(\phi \right)-d\mathrm{L.}\mathrm{M.}b\left(\phi \right)\cdot d\mathrm{L.}\mathrm{M.}c\left(\phi \right)}\cdot \left[\begin{array}{c}d\mathrm{L.}\mathrm{M.}d\left(\phi \right)-d\mathrm{L.}\mathrm{M.}b\left(\phi \right)\\ -d\mathrm{L.}\mathrm{M.}c\left(\phi \right)d\mathrm{L.}\mathrm{M.}a\left(\phi \right)\end{array}\right]\\ \cdot \left\{\left[\begin{array}{c}Vu-Vv\\ Vv-Vw\end{array}\right]-\mathrm{R.}\cdot \left[\begin{array}{c}\mathrm{I.}a\\ \mathrm{I.}b\end{array}\right]-\frac{d}{dt}\left[\begin{array}{c}{\psi }_{ma}\\ {\psi }_{mb}\end{array}\right]-\frac{d}{dt}\left(\left[\begin{array}{c}d\mathrm{L.}\mathrm{M.}a\left(\phi \right)d\mathrm{L.}\mathrm{M.}b\left(\phi \right)\\ d\mathrm{L.}\mathrm{M.}c\left(\phi \right)d\mathrm{L.}\mathrm{M.}d\left(\phi \right)\end{array}\right]\right)\cdot \left[\begin{array}{c}\mathrm{I.}a\\ \mathrm{I.}b\end{array}\right]\right\}\end{array}$$

The above original machine model can be reshaped as follows $$\left[\begin{array}{c}Vu-Vv\\ Vv-Vw\\ Vu-Vv+Vv-Vw+Vw-Vu=0\end{array}\right]=\mathrm{R.}\cdot \left[\begin{array}{c}\mathrm{I.}a\\ \mathrm{I.}b\\ \mathrm{I.}a+\mathrm{I.}b+\mathrm{I.}c\end{array}\right]+\frac{d}{dt}\left(\left[\begin{array}{c}\psi a\\ \psi b\\ {\psi }_a+{\psi }_b+{\psi }_c\end{array}\right]\right)$$

umschreiben zu $$\left[\begin{array}{c}Vu-Vv\\ Vv-Vw\\ 0\end{array}\right]=R\cdot \left[\begin{array}{c}Ia\\ Ib\\ 3\cdot Iloop\end{array}\right]+\frac{d}{dt}\left(\left[\begin{array}{c}\psi a\\ \psi b\\ {\psi }_a+{\psi }_b+{\psi }_c\end{array}\right]\right),$$

wobei I_loop einem Kreisstrom entspricht.This can be done with $$\mathrm{I.}a+\mathrm{I.}b+\mathrm{I.}c=3\cdot {\text{I.}}_{\text{loop}}$$

rewrite to $$\left[\begin{array}{c}Vu-Vv\\ Vv-Vw\\ 0\end{array}\right]=\mathrm{R.}\cdot \left[\begin{array}{c}\mathrm{I.}a\\ \mathrm{I.}b\\ 3\cdot \mathrm{I.}lOOp\end{array}\right]+\frac{d}{dt}\left(\left[\begin{array}{c}\psi a\\ \psi b\\ {\psi }_a+{\psi }_b+{\psi }_c\end{array}\right]\right),$$

where I _loop corresponds to a circular current.

The sum of the above equations leads to: $$\begin{array}{c}{\psi }_a+{\psi }_b+{\psi }_c=\left(\mathrm{L.}a\left(\phi \right)+\mathrm{M.}ab\left(\phi \right)+\mathrm{M.}ac\left(\phi \right)\right)\cdot \mathrm{I.}a+\\ \left(\mathrm{L.}b\left(\phi \right)+\mathrm{M.}ab\left(\phi \right)+\mathrm{M.}bc\left(\phi \right)\right)\cdot \mathrm{I.}b+\\ \left(\mathrm{L.}c\left(\phi \right)+\mathrm{M.}ac\left(\phi \right)+\mathrm{M.}cc\left(\phi \right)\right)\cdot \mathrm{I.}c+\\ {\psi }_{ma}+{\psi }_{mb}+{\psi }_{mc}\end{array}$$

$$\begin{array}{l}Lb\left(\phi \right)+Mab\left(\phi \right)+Mbc\left(\phi \right)=l_0+l_1\cdot cos\left(\phi -\frac{2\pi }{3}\right)+l_2\cdot cos\left[2\cdot \left(\phi +\frac{2\pi }{3}\right)\right]+l_3\cdot \\ cos\left[3\cdot \left(\phi -\frac{2\pi }{3}\right)\right]+2\cdot m_0+m_1\cdot \left[cos\left(\phi +\frac{2\pi }{3}\right)+cos\left(\phi \right)\right]+m_2\cdot \left[cos\left[2\cdot \left(\phi +\frac{2\pi }{3}\right)\right]+cos\left(2\cdot \phi \right)\right]\\ +m_3\cdot \left[cos\left[3\cdot \left(\phi +\frac{2\pi }{3}\right)\right]+cos\left(3\cdot \phi \right)\right]+\cdots \end{array}$$

$$\begin{array}{l}Lc\left(\phi \right)+Mab\left(\phi \right)+Mbc\left(\phi \right)=l_0+l_1\cdot cos\left(\phi -\frac{2\pi }{3}\right)+l_2\cdot cos\left[2\cdot \left(\phi +\frac{2\pi }{3}\right)\right]+l_3\cdot \\ cos\left[3\cdot \left(\phi +\frac{2\pi }{3}\right)\right]+2\cdot m_0+m_1\cdot \left[cos\left(\phi -\frac{2\pi }{3}\right)+cos\left(\phi \right)\right]+m_2\cdot \left[cos\left[2\cdot \left(\phi -\frac{2\pi }{3}\right)\right]+cos\left(2\cdot \phi \right)\right]\\ +m_3\cdot \left[cos\left[3\cdot \left(\phi -\frac{2\pi }{3}\right)\right]+cos\left(3\cdot \phi \right)\right]+\cdots \end{array}$$

The coefficients La (φ) + Mab (φ) + Mac (φ), Lb (φ) + Mab (φ) + Mbc (φ) and Lc (φ) + Mac (φ) + Mbc (φ) can be rewritten as follows : $$\begin{array}{l}\mathrm{L.}a\left(\phi \right)+\mathrm{M.}ab\left(\phi \right)+\mathrm{M.}ac\left(\phi \right)=l_0+{\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="DE102019220103A1\_0065.png"\; state="\{\{state\}\}">l</figure-callout>}}_1\cdot cOs\left(\phi \right)+{\mathrm{<figure-callout\; id="2"\; label="l"\; filenames="DE102019220103A1\_0064.png"\; state="\{\{state\}\}">l</figure-callout>}}_2\cdot cOs\left(2\cdot \phi \right)+{\mathrm{<figure-callout\; id="3"\; label="l"\; filenames="DE102019220103A1\_0064.png"\; state="\{\{state\}\}">l</figure-callout>}}_3\cdot cOs\left(3\cdot \phi \right)+2\cdot \\ m_0+{\mathrm{<figure-callout\; id="1"\; label="m"\; filenames="DE102019220103A1\_0065.png"\; state="\{\{state\}\}">m</figure-callout>}}_1\cdot \left[cOs\left(\phi +\frac{2\pi }{3}\right)+cOs\left(\phi -\frac{2\pi }{3}\right)\right]+{\mathrm{<figure-callout\; id="2"\; label="m"\; filenames="DE102019220103A1\_0064.png"\; state="\{\{state\}\}">m</figure-callout>}}_2\cdot \left[cOs\left[2\cdot \left(\phi +\frac{2\pi }{3}\right)\right]+cOs\left[2\cdot \left(\phi -\frac{2\pi }{3}\right)\right]\right]\\ +{\mathrm{<figure-callout\; id="3"\; label="m"\; filenames="DE102019220103A1\_0064.png"\; state="\{\{state\}\}">m</figure-callout>}}_3\cdot \left[cOs\left[3\cdot \left(\phi +\frac{2\pi }{3}\right)\right]+cOs\left[3\cdot \left(\phi -\frac{2\pi }{3}\right)\right]\right]+\cdots \end{array}$$

$$\begin{array}{l}\mathrm{L.}b\left(\phi \right)+\mathrm{M.}ab\left(\phi \right)+\mathrm{M.}bc\left(\phi \right)=l_0+{\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="DE102019220103A1\_0065.png"\; state="\{\{state\}\}">l</figure-callout>}}_1\cdot cOs\left(\phi -\frac{2\pi }{3}\right)+{\mathrm{<figure-callout\; id="2"\; label="l"\; filenames="DE102019220103A1\_0064.png"\; state="\{\{state\}\}">l</figure-callout>}}_2\cdot cOs\left[2\cdot \left(\phi +\frac{2\pi }{3}\right)\right]+{\mathrm{<figure-callout\; id="3"\; label="l"\; filenames="DE102019220103A1\_0064.png"\; state="\{\{state\}\}">l</figure-callout>}}_3\cdot \\ cOs\left[3\cdot \left(\phi -\frac{2\pi }{3}\right)\right]+2\cdot m_0+{\mathrm{<figure-callout\; id="1"\; label="m"\; filenames="DE102019220103A1\_0065.png"\; state="\{\{state\}\}">m</figure-callout>}}_1\cdot \left[cOs\left(\phi +\frac{2\pi }{3}\right)+cOs\left(\phi \right)\right]+{\mathrm{<figure-callout\; id="2"\; label="m"\; filenames="DE102019220103A1\_0064.png"\; state="\{\{state\}\}">m</figure-callout>}}_2\cdot \left[cOs\left[2\cdot \left(\phi +\frac{2\pi }{3}\right)\right]+cOs\left(2\cdot \phi \right)\right]\\ +{\mathrm{<figure-callout\; id="3"\; label="m"\; filenames="DE102019220103A1\_0064.png"\; state="\{\{state\}\}">m</figure-callout>}}_3\cdot \left[cOs\left[3\cdot \left(\phi +\frac{2\pi }{3}\right)\right]+cOs\left(3\cdot \phi \right)\right]+\cdots \end{array}$$

$$\begin{array}{l}\mathrm{L.}c\left(\phi \right)+\mathrm{M.}ab\left(\phi \right)+\mathrm{M.}bc\left(\phi \right)=l_0+{\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="DE102019220103A1\_0065.png"\; state="\{\{state\}\}">l</figure-callout>}}_1\cdot cOs\left(\phi -\frac{2\pi }{3}\right)+{\mathrm{<figure-callout\; id="2"\; label="l"\; filenames="DE102019220103A1\_0064.png"\; state="\{\{state\}\}">l</figure-callout>}}_2\cdot cOs\left[2\cdot \left(\phi +\frac{2\pi }{3}\right)\right]+{\mathrm{<figure-callout\; id="3"\; label="l"\; filenames="DE102019220103A1\_0064.png"\; state="\{\{state\}\}">l</figure-callout>}}_3\cdot \\ cOs\left[3\cdot \left(\phi +\frac{2\pi }{3}\right)\right]+2\cdot m_0+{\mathrm{<figure-callout\; id="1"\; label="m"\; filenames="DE102019220103A1\_0065.png"\; state="\{\{state\}\}">m</figure-callout>}}_1\cdot \left[cOs\left(\phi -\frac{2\pi }{3}\right)+cOs\left(\phi \right)\right]+{\mathrm{<figure-callout\; id="2"\; label="m"\; filenames="DE102019220103A1\_0064.png"\; state="\{\{state\}\}">m</figure-callout>}}_2\cdot \left[cOs\left[2\cdot \left(\phi -\frac{2\pi }{3}\right)\right]+cOs\left(2\cdot \phi \right)\right]\\ +{\mathrm{<figure-callout\; id="3"\; label="m"\; filenames="DE102019220103A1\_0064.png"\; state="\{\{state\}\}">m</figure-callout>}}_3\cdot \left[cOs\left[3\cdot \left(\phi -\frac{2\pi }{3}\right)\right]+cOs\left(3\cdot \phi \right)\right]+\cdots \end{array}$$

$$\begin{array}{l}Lb\left(\phi \right)+Mab\left(\phi \right)+Mbc\left(\phi \right)=l_0+2\cdot m_0+\left(l_1-m_1\right)\cdot cos\left(\phi -\frac{2\pi }{3}\right)+\left(l_2-m_2\right)\cdot \\ cos\left[2\cdot \left(\phi -\frac{2\pi }{3}\right)\right]+\left(l_3+2\cdot m_3\right)\cdot cos\left[3\cdot \phi \right]+\cdots \end{array}$$

$$\begin{array}{l}Lc\left(\phi \right)+Mac\left(\phi \right)+Mbc\left(\phi \right)=l_0+2\cdot m_0+\left(l_1-m_1\right)\cdot cos\left(\phi +\frac{2\pi }{3}\right)+\left(l_2-m_2\right)\cdot cos[2\cdot \\ \left(\phi +\frac{2\pi }{3}\right)]+\left(l_3+2\cdot m_3\right)\cdot cos\left[3\cdot \phi \right]+\cdots \end{array}$$

Using cos (p) + cos (q) = 2 · cos [(p + q) / 2] · cos [(pq / 2)], equations can be simplified: $$\begin{array}{l}\mathrm{L.}a\left(\phi \right)+\mathrm{M.}ab\left(\phi \right)+\mathrm{M.}ac\left(\phi \right)=l_0+2\cdot m_0+\left(l_1-m_1\right)\cdot cOs\left(\phi \right)+\left(l_2-m_2\right)\cdot \\ cOs\left(2\cdot \phi \right)+\left({\mathrm{<figure-callout\; id="3"\; label="l"\; filenames="DE102019220103A1\_0064.png"\; state="\{\{state\}\}">l</figure-callout>}}_3+2\cdot m_3\right)\cdot cOs\left(3\cdot \phi \right)+\cdots \end{array}$$

$$\begin{array}{l}\mathrm{L.}b\left(\phi \right)+\mathrm{M.}ab\left(\phi \right)+\mathrm{M.}bc\left(\phi \right)=l_0+2\cdot m_0+\left(l_1-m_1\right)\cdot cOs\left(\phi -\frac{2\pi }{3}\right)+\left(l_2-m_2\right)\cdot \\ cOs\left[2\cdot \left(\phi -\frac{2\pi }{3}\right)\right]+\left({\mathrm{<figure-callout\; id="3"\; label="l"\; filenames="DE102019220103A1\_0064.png"\; state="\{\{state\}\}">l</figure-callout>}}_3+2\cdot m_3\right)\cdot cOs\left[3\cdot \phi \right]+\cdots \end{array}$$

$$\begin{array}{l}\mathrm{L.}c\left(\phi \right)+\mathrm{M.}ac\left(\phi \right)+\mathrm{M.}bc\left(\phi \right)=l_0+2\cdot m_0+\left(l_1-m_1\right)\cdot cOs\left(\phi +\frac{2\pi }{3}\right)+\left(l_2-m_2\right)\cdot cOs[2\cdot \\ \left(\phi +\frac{2\pi }{3}\right)]+\left({\mathrm{<figure-callout\; id="3"\; label="l"\; filenames="DE102019220103A1\_0064.png"\; state="\{\{state\}\}">l</figure-callout>}}_3+2\cdot m_3\right)\cdot cOs\left[3\cdot \phi \right]+\cdots \end{array}$$

nämlich $$\begin{array}{l}\left(La\left(\phi \right)+Mab\left(\phi \right)+Mac\left(\phi \right)\right)\cdot Ia+\\ \left(Lb\left(\phi \right)+Mab\left(\phi \right)+Mbc\left(\phi \right)\right)\cdot Ib+\\ \left(Lc\left(\phi \right)+Mac\left(\phi \right)+Mbc\left(\phi \right)\right)\cdot Ic\end{array}$$

der obigen Gleichung vereinfachen mit: $$\begin{array}{c}\left(La\left(\phi \right)+Mab\left(\phi \right)+Mac\left(\phi \right)\right)\cdot Ia+\left(Lb\left(\phi \right)+Mab\left(\phi \right)+Mbc\left(\phi \right)\right)\cdot Ib+\\ \left(Lc\left(\phi \right)+Mac\left(\phi \right)+Mbc\left(\phi \right)\right)\cdot Ic=\left[l_0+2\cdot m_0+\left(l_3+2\cdot m_3\right)\cdot cos\left(3\cdot \phi \right)\right]\cdot \left(Ia+Ib+Ic\right)\\ \\ {\psi }_{ma}+{\psi }_{mb}+{\psi }_{mc}=a1\cdot \left[cos\left(\phi \right)+cos\left(\phi -\frac{2\pi }{3}\right)+cos\left(\phi +\frac{2\pi }{3}\right)\right]+\\ a3\cdot \left[cos\left(3\cdot \phi \right)+cos\left[3\cdot \left(\phi -\frac{2\pi }{3}\right)\right]+cos\left[3\cdot \left(\phi +\frac{2\pi }{3}\right)\right]\right]+\\ a5\cdot \left[cos\left(5\cdot \phi \right)+cos\left[5\cdot \left(\phi -\frac{2\pi }{3}\right)\right]+cos\left[5\cdot \left(\phi +\frac{2\pi }{3}\right)\right]\right]+\\ a7\cdot \left[cos\left(7\cdot \phi \right)+cos\left[7\cdot \left(\phi -\frac{2\pi }{3}\right)\right]+cos\left[7\cdot \left(\phi +\frac{2\pi }{3}\right)\right]\right]+\end{array}$$

If l _k = m _k , k ∉ {0, 3, 6, ...}, the first term of $${\psi }_a+{\psi }_b+{\psi }_c,$$

namely $$\begin{array}{l}\left(\mathrm{L.}a\left(\phi \right)+\mathrm{M.}ab\left(\phi \right)+\mathrm{M.}ac\left(\phi \right)\right)\cdot \mathrm{I.}a+\\ \left(\mathrm{L.}b\left(\phi \right)+\mathrm{M.}ab\left(\phi \right)+\mathrm{M.}bc\left(\phi \right)\right)\cdot \mathrm{I.}b+\\ \left(\mathrm{L.}c\left(\phi \right)+\mathrm{M.}ac\left(\phi \right)+\mathrm{M.}bc\left(\phi \right)\right)\cdot \mathrm{I.}c\end{array}$$

the above equation simplify with: $$\begin{array}{c}\left(\mathrm{L.}a\left(\phi \right)+\mathrm{M.}ab\left(\phi \right)+\mathrm{M.}ac\left(\phi \right)\right)\cdot \mathrm{I.}a+\left(\mathrm{L.}b\left(\phi \right)+\mathrm{M.}ab\left(\phi \right)+\mathrm{M.}bc\left(\phi \right)\right)\cdot \mathrm{I.}b+\\ \left(\mathrm{L.}c\left(\phi \right)+\mathrm{M.}ac\left(\phi \right)+\mathrm{M.}bc\left(\phi \right)\right)\cdot \mathrm{I.}c=\left[l_0+2\cdot m_0+\left(l_3+2\cdot m_3\right)\cdot cOs\left(3\cdot \phi \right)\right]\cdot \left(\mathrm{I.}a+\mathrm{I.}b+\mathrm{I.}c\right)\\ \\ {\psi }_{ma}+{\psi }_{mb}+{\psi }_{mc}=a1\cdot \left[cOs\left(\phi \right)+cOs\left(\phi -\frac{2\pi }{3}\right)+cOs\left(\phi +\frac{2\pi }{3}\right)\right]+\\ a3\cdot \left[cOs\left(3\cdot \phi \right)+cOs\left[3\cdot \left(\phi -\frac{2\pi }{3}\right)\right]+cOs\left[3\cdot \left(\phi +\frac{2\pi }{3}\right)\right]\right]+\\ a5\cdot \left[cOs\left(5\cdot \phi \right)+cOs\left[5\cdot \left(\phi -\frac{2\pi }{3}\right)\right]+cOs\left[5\cdot \left(\phi +\frac{2\pi }{3}\right)\right]\right]+\\ a\mathrm{7th}\cdot \left[cOs\left(\mathrm{7th}\cdot \phi \right)+cOs\left[\mathrm{7th}\cdot \left(\phi -\frac{2\pi }{3}\right)\right]+cOs\left[\mathrm{7th}\cdot \left(\phi +\frac{2\pi }{3}\right)\right]\right]+\end{array}$$

ungleich Null ist, j ∈ {3, 6, ...} , nämlich $$3\cdot cos\left(j\cdot \phi \right)$$

entspricht, was für $$a_j\text{ mit }j\in \left\{\mathrm{3,6,}\dots \right\}$$

der Fall ist.Whereby only the term with $$\left[cOs\left(j\cdot \phi \right)+cOs\left[j\cdot \left(\phi -\frac{2\pi }{3}\right)\right]+cOs\left[j\cdot \left(\phi +\frac{2\pi }{3}\right)\right]\right]$$

is not equal to zero, j ∈ {3, 6, ...}, namely $$3\cdot cOs\left(j\cdot \phi \right)$$

corresponds to what for $$a_j\text{With}j\in \left\{\mathrm{3.6,}...\right\}$$

the case is.

wenn $$l_k=m_k,k\notin \left\{\mathrm{0,3,6,}\dots \right\}und{a}_{\text{j}}=\mathrm{0,}j\in \left\{\mathrm{3,6,}\dots \right\}$$

gilt: $${\psi }_a+{\psi }_b+{\psi }_c=3\cdot \left[l_0+2\cdot m_0+\left(l_3+2\cdot m_3\right)\cdot cos\left(3\cdot \phi \right)\right]\cdot Iloop$$

If a _j = 0, j ∈ {3, 6, ...} then the above formula is simplified to $${\psi }_{ma}+{\psi }_{mb}+{\psi }_{mc}=0$$

if $$l_k=m_k,k\notin \left\{\mathrm{0.3.6,}...\right\}und{a}_{\text{j}}=\mathrm{0,}j\in \left\{\mathrm{3.6,}...\right\}$$

applies: $${\psi }_a+{\psi }_b+{\psi }_c=3\cdot \left[l_0+2\cdot m_0+\left({\mathrm{<figure-callout\; id="3"\; label="l"\; filenames="DE102019220103A1\_0064.png"\; state="\{\{state\}\}">l</figure-callout>}}_3+2\cdot m_3\right)\cdot cOs\left(3\cdot \phi \right)\right]\cdot \mathrm{I.}lOOp$$

mit I_loop(t=0)=0 lautet die Lösung der Gleichung I_loop(t ≥ 0) = Ia + Ib + Ic = 0The third line of the reshaped machine model above then corresponds to $$0=3\cdot \mathrm{R.}\cdot \mathrm{I.}lOOp+\frac{d}{dt}\left\{3\cdot \left[l_0+2\cdot m_0+\left({\mathrm{<figure-callout\; id="3"\; label="l"\; filenames="DE102019220103A1\_0064.png"\; state="\{\{state\}\}">l</figure-callout>}}_3+2\cdot m_3\right)\cdot cOs\left(3\cdot \phi \right)\right]\cdot \mathrm{I.}lOOp\right\}$$

with I _loop (t = 0) = 0 the solution to the equation is I _loop (t ≥ 0) = Ia + Ib + Ic = 0

mit $${\psi }_a=\left[La\left(\phi \right)-Mac\left(\phi \right)\right]\cdot Ia+\left[Mab\left(\phi \right)-Mac\left(\phi \right)\right]\cdot Ib+{\psi }_{ma}$$

$${\psi }_b=\left[Mab\left(\phi \right)-Mbc\left(\phi \right)\right]\cdot Ia+\left[Lb\left(\phi \right)-Mbc\left(\phi \right)\right]\left(\phi \right)\cdot Ib+{\psi }_{mb}$$

This means that the third line of the system of equations is no longer relevant and it is therefore sufficient to implement and solve the first and second lines of the system of equations: $$\left[\begin{array}{c}Vu-Vv\\ Vv-Vw\end{array}\right]=\mathrm{R.}\cdot \left[\begin{array}{c}\mathrm{I.}a\\ \mathrm{I.}b\end{array}\right]+\frac{d}{dt}\left(\left[\begin{array}{c}\psi a\\ \psi b\end{array}\right]\right)$$

With $${\psi }_a=\left[\mathrm{L.}a\left(\phi \right)-\mathrm{M.}ac\left(\phi \right)\right]\cdot \mathrm{I.}a+\left[\mathrm{M.}ab\left(\phi \right)-\mathrm{M.}ac\left(\phi \right)\right]\cdot \mathrm{I.}b+{\psi }_{ma}$$

$${\psi }_b=\left[\mathrm{M.}ab\left(\phi \right)-\mathrm{M.}bc\left(\phi \right)\right]\cdot \mathrm{I.}a+\left[\mathrm{L.}b\left(\phi \right)-\mathrm{M.}bc\left(\phi \right)\right]\left(\phi \right)\cdot \mathrm{I.}b+{\psi }_{mb}$$

Furthermore, the inductance matrix can have elements made of inductances, which are formed from phase inductances and coupling inductances, which are each position-dependent, and the inductances are defined by coefficients I, m of harmonics of a rotation of the electrical position angle.

In particular, the elements of the inductance matrix can take into account all or at least the first five harmonics of the phase inductances and coupling inductances between the phase phases, with the third harmonics and multiples thereof being neglected in particular.

According to further embodiments, the regulation can comprise a speed regulation, a torque regulation or a flux regulation.

It can be provided that the target terminal potentials are specified as pulse-width-modulated signals for controlling an inverter, the terminal potentials being measured in order to determine the actual phase currents with the aid of the machine model.

Furthermore, the actual phase currents can be used to determine an actual motor torque and / or actual motor flows.

In particular, the actual engine torque and / or actual engine flows can be used for torque regulation or flux regulation.

Figure list

• 1 eine schematische Darstellung eines Schaltbilds einer dreiphasigen elektrischen Maschine in Deltaverschaltung mit einer Drehzahlregelung; und
• 2 ein Flussdiagramm zur Beschreibung des Verfahrens zum Betreiben des Motorsystems.

Embodiments are explained in more detail below with reference to the accompanying drawings. Show it:

• 1 a schematic representation of a circuit diagram of a three-phase electrical machine in delta connection with a speed control; and
• 2 a flow chart describing the method of operating the engine system.

Description of embodiments

The method for operating an electrical machine is described below using an example of a motor system 1 described in more detail with a three-phase synchronous motor in delta connection and speed control.

The engine system 1 the 1 has a synchronous machine as an electrical machine 2 with a stator 21 and a runner 22nd on. The stator 21 has three phase lines connected in delta connection 23 with coil arrangements. The phase strands are for this purpose 23 interconnected as a ring and a corresponding terminal connection for applying a terminal potential is provided between each two coil arrangements

Each of the phase strands 23 is via its own inverter circuit 31 of an inverter 3 (Power driver) controlled. The inverter 3 can for example be designed as a B6 inverter and any inverter circuit 31 a pull-up transistor 33 and a pull-down transistor 32. The inverter 3 is used to specifically target terminal potentials Vu, Vv, Vw for application to the phase strands 23 according to a commutation process, such as block commutation or sine commutation.

The commutation process can control the phase strands 23 provide by means of a pulse width modulation with a respective duty cycle. The control signals S _u , S _v and S _w correspond to the last applied terminal potential information with which the inverter 31 was controlled.

The control signals Su, Sv, Sw are from a control unit 4th provided that the inverter 3 drives.

Furthermore, the runner can 22nd of the synchronous motor 2 with a position sensor 5 be coupled in order to provide a rotor position φ (electrical rotor position) and an actual speed n _ist through time derivation.

The control unit 4th can, for example, perform speed control. The speed control unit 4th a setpoint speed n set _is specified as the manipulated variable.

The speed control in the control unit 4th uses estimated actual phase currents laist, Ibist, Icist and the control deviation Δn between n _soll and n _ist according to a control function.

The control unit 4th includes a differential term 41 to determine the system deviation Δn. There is also a target torque determination unit 42 , which can be designed as a PI controller, for example, for determining a setpoint torque Msoll as a control variable and a setpoint flow determination unit 43 provided for determining a setpoint motor flux in a manner known per se from the setpoint torque Msoll. For this, reference is made to the publication QINGFANG TENGI et al., “Current Sensorless Model Predictive Torque Control Based on Adaptive Backstepping Observer for PMSM Drives”, Department of Automation and Electrical Engineering, Lanzhou Jiaotong University, Lanzhou, Gansu, 730070 PRCHINA.

The actual phase currents laist, Ibist, Icist are stored in a phase current estimation block 44 according to a machine model from the current terminal potential specifications S _u , S _v , S _{w of} the phase strings 23 estimated.

Furthermore, an actual torque Mist is made from the three estimated actual phase currents laist, Ibist and Icist in a torque calculation block 45 estimated. Using a flow detection block 46 the actual motor flows are determined. For this, reference is made to the publication QINGFANG TENGI et al., “Current Sensorless Model Predictive Torque Control Based on Adaptive Backstepping Observer for PMSM Drives”, Department of Automation and Electrical Engineering, Lanzhou Jiaotong University, Lanzhou, Gansu, 730070 PRCHINA.

For calculating the actual phase currents laist, Ibist, Icist in the phase current estimation block 44 the modified machine model is used.

In the context of a model predictive speed control, the modified machine model is used as part of an optimization in order to obtain the optimized values of the terminal potentials for controlling the inverter 3 can be found using the terminal potential information Su, Sv, Sw. It is used as a cost function a distance between the setpoint and the actual torque or the setpoint motor fluxes and the actual motor fluxes is minimized.

The calculation of the machine model is carried out according to the description, among other things.

Using the flow chart of the 2 The following is an exemplary method of operating the engine system 1 described.

In step S1 a set speed n set _is specified. N _to the set speed can for example be a command value of a position or speed control.

In step S2 a setpoint torque and / or setpoint motor flows are determined from the control deviation between the setpoint speed and the measured actual speed. The target torque represents a manipulated variable of a speed control.

In step S3 actual phase currents I aist, I _bist , I _{cist are} estimated based on the last applied terminal _{potential information S u} , S _v , S _w using the modified machine model described below.

In step S4 an estimated actual torque and estimated actual motor _{fluxes are} determined from the phase currents I aist, I bist, I _cist.

In step S5 To solve a continuously operated optimization process, a cost function is minimized in order to obtain the terminal potential information for controlling the inverter. The cost function can evaluate a distance between the setpoint torque and the actual torque and / or a distance between the setpoint motor flows and the actual motor flows.

The terminal potential information resulting from the optimization process can be found in step S6 to the inverter 3 be created.

During the optimization, the optimized values for the terminal potentials Vu, Vv, Vw are sought, which lead to the minimization of the cost function. The modified machine model is solved in each iteration. If the cost function is minimal, the torque of the machine or the motor flows reaches the value of the setpoint torque or the setpoint motor flows, which means that the actual speed should have reached the setpoint speed.

The conventional machine model is: $$\left[\begin{array}{c}Vu-Vv\\ Vv-Vw\\ Vv-Vu\end{array}\right]=\mathrm{R.}\cdot \left[\begin{array}{c}\mathrm{I.}a\\ \mathrm{I.}b\\ \mathrm{I.}c\end{array}\right]+\frac{d}{dt}\left[\left[\begin{array}{c}\mathrm{L.}a\left(\phi \right)\mathrm{M.}ab\left(\phi \right)\mathrm{M.}ac\left(\phi \right)\\ \mathrm{M.}ab\left(\phi \right)\mathrm{L.}b\left(\phi \right)\mathrm{M.}bc\left(\phi \right)\\ \mathrm{M.}ac\left(\phi \right)\mathrm{M.}bc\left(\phi \right)\mathrm{L.}c\left(\phi \right)\end{array}\right]\cdot \left[\begin{array}{c}\mathrm{I.}a\\ \mathrm{I.}b\\ \mathrm{I.}c\end{array}\right]\right]+\frac{d}{dt}\left[\begin{array}{c}{\psi }_{ma}\\ {\psi }_{mb}\\ {\psi }_{mc}\end{array}\right]$$

The conventional inversion of the above inductance matrix is computationally expensive and is only an approximation that only takes harmonics up to the 2nd order into account.

mit $$dLMa\left(\phi \right)=La\left(\phi \right)-Mac\left(\phi \right)$$

$$dLMb\left(\phi \right)=Mab\left(\phi \right)-Mac\left(\phi \right)$$

$$dLMc\left(\phi \right)=Mab\left(\phi \right)-Mbc\left(\phi \right)$$

$$dLMd\left(\phi \right)=Lb\left(\phi \right)-Mbc\left(\phi \right)$$

Instead, a machine model is now used that only uses a 2x2 matrix. $$\frac{d}{dt}\left(\left[\begin{array}{c}d\mathrm{L.}\mathrm{M.}a\left(\phi \right)d\mathrm{L.}\mathrm{M.}b\left(\phi \right)\\ d\mathrm{L.}\mathrm{M.}c\left(\phi \right)d\mathrm{L.}\mathrm{M.}d\left(\phi \right)\end{array}\right]\cdot \left[\begin{array}{c}\mathrm{I.}a\\ \mathrm{I.}b\end{array}\right]\right)=\left[\begin{array}{c}Vu-Vv\\ Vv-Vw\end{array}\right]-\mathrm{R.}\cdot \left[\begin{array}{c}\mathrm{I.}a\\ \mathrm{I.}b\end{array}\right]-\frac{d}{dt}\left[\begin{array}{c}{\psi }_{ma}\\ {\psi }_{mb}\end{array}\right]$$

With $$d\mathrm{L.}\mathrm{M.}a\left(\phi \right)=\mathrm{L.}a\left(\phi \right)-\mathrm{M.}ac\left(\phi \right)$$

$$d\mathrm{L.}\mathrm{M.}b\left(\phi \right)=\mathrm{M.}ab\left(\phi \right)-\mathrm{M.}ac\left(\phi \right)$$

$$d\mathrm{L.}\mathrm{M.}c\left(\phi \right)=\mathrm{M.}ab\left(\phi \right)-\mathrm{M.}bc\left(\phi \right)$$

$$d\mathrm{L.}\mathrm{M.}d\left(\phi \right)=\mathrm{L.}b\left(\phi \right)-\mathrm{M.}bc\left(\phi \right)$$

Under the condition that l _k = m _k , k ∉ {0, 3, 6, ...} and a _j = 0, each {3, 6, ...}

This can then be converted to determine the phase currents: $$\begin{array}{l}\frac{d}{dt}\left(\left[\begin{array}{c}\mathrm{I.}aist\\ \mathrm{I.}bist\end{array}\right]\right)=\frac{1}{d\mathrm{L.}\mathrm{M.}a\left(\phi \right)\cdot d\mathrm{L.}\mathrm{M.}d\left(\phi \right)-d\mathrm{L.}\mathrm{M.}b\left(\phi \right)\cdot d\mathrm{L.}\mathrm{M.}c\left(\phi \right)}\left[\begin{array}{c}d\mathrm{L.}\mathrm{M.}d\left(\phi \right)-d\mathrm{L.}\mathrm{M.}b\left(\phi \right)\\ -d\mathrm{L.}\mathrm{M.}c\left(\phi \right)d\mathrm{L.}\mathrm{M.}a\left(\phi \right)\end{array}\right]\\ \cdot \left\{\left[\begin{array}{c}Vu-Vv\\ Vv-Vw\end{array}\right]-\mathrm{R.}\cdot \left[\begin{array}{c}\mathrm{I.}a\\ \mathrm{I.}b\end{array}\right]-\frac{d}{dt}\left[\begin{array}{c}{\psi }_{ma}\\ {\psi }_{mb}\end{array}\right]-\frac{d}{dt}\left(\left[\begin{array}{c}d\mathrm{L.}\mathrm{M.}a\left(\phi \right)d\mathrm{L.}\mathrm{M.}b\left(\phi \right)\\ d\mathrm{L.}\mathrm{M.}c\left(\phi \right)d\mathrm{L.}\mathrm{M.}d\left(\phi \right)\end{array}\right]\right)\cdot \left[\begin{array}{c}\mathrm{I.}aist\\ \mathrm{I.}bist\end{array}\right]\right\}\end{array}$$

The missing actual phase current Icist can be calculated using the equation laist + Ibist + lcist = 0.

mit $$\begin{array}{c}dLM\alpha \left(\phi \right)=\left(l_0-m_0\right)+\left(l_1+2\cdot m_1\right)\cdot cos\left(\phi \right)+\left(l_2+2\cdot m_2\right)\cdot cos\left(2\cdot \phi \right)\\ +\left(l_3-m_3\right)\cdot cos\left(3\cdot \phi \right)+\cdots \end{array}$$

$$\begin{array}{c}dLM\beta \left(\phi \right)=\left(l_0-m_0\right)+\left(l_1+2\cdot m_1\right)\cdot cos\left(\phi -\frac{2\pi }{3}\right)+\left(l_2+2\cdot m_2\right)\\ \cdot cos\left[2\cdot \left(\phi -\frac{2\pi }{3}\right)\right]+\left(l_3-m_3\right)\cdot cos\left[3\cdot \left(\phi -\frac{2\pi }{3}\right)\right]+\cdots \end{array}$$

$$\begin{array}{c}dLM\gamma \left(\phi \right)=\left(l_0-m_0\right)+\left(l_1+2\cdot m_1\right)\cdot cos\left(\phi +\frac{2\pi }{3}\right)+\left(l_2+2\cdot m_2\right)\\ \cdot cos\left[2\cdot \left(\phi +\frac{2\pi }{3}\right)\right]+\left(l_3-m_3\right)\cdot cos\left[3\cdot \left(\phi +\frac{2\pi }{3}\right)\right]+\cdots \end{array}$$

_{The resulting torque equation} using the inductance information to determine the actual torque Mist from the estimated actual phase currents I aist, I _bist , I _cist then results in: $$\begin{array}{c}\mathrm{M.}ist=\frac{d{\psi }_{ma}}{d\phi }\cdot {\mathrm{I.}}_{aist}+\frac{d{\psi }_{mb}}{d\phi }\cdot {\mathrm{I.}}_{bist}+\frac{d{\psi }_{mc}}{d\phi }\cdot {\mathrm{I.}}_{cist}+\frac{1}{2}\\ \cdot \left(\frac{dd\mathrm{L.}\mathrm{M.}\alpha \left(\phi \right)}{d\phi }\cdot {\mathrm{I.}}_{ais\mathrm{<figure-callout\; id="2"\; label="t"\; filenames="DE102019220103A1\_0064.png"\; state="\{\{state\}\}">t</figure-callout>}}^2+\frac{dd\mathrm{L.}\mathrm{M.}\beta \left(\phi \right)}{d\phi }\cdot {\mathrm{I.}}_{bis\mathrm{<figure-callout\; id="2"\; label="t"\; filenames="DE102019220103A1\_0064.png"\; state="\{\{state\}\}">t</figure-callout>}}^2+\frac{dd\mathrm{L.}\mathrm{M.}\gamma \left(\phi \right)}{d\phi }\cdot {\mathrm{I.}}_{cist}^2\right)\end{array}$$

With $$\begin{array}{c}d\mathrm{L.}\mathrm{M.}\alpha \left(\phi \right)=\left(l_0-m_0\right)+\left({\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="DE102019220103A1\_0065.png"\; state="\{\{state\}\}">l</figure-callout>}}_1+2\cdot m_1\right)\cdot cOs\left(\phi \right)+\left({\mathrm{<figure-callout\; id="2"\; label="l"\; filenames="DE102019220103A1\_0064.png"\; state="\{\{state\}\}">l</figure-callout>}}_2+2\cdot m_2\right)\cdot cOs\left(2\cdot \phi \right)\\ +\left(l_3-m_3\right)\cdot cOs\left(3\cdot \phi \right)+\cdots \end{array}$$

$$\begin{array}{c}d\mathrm{L.}\mathrm{M.}\beta \left(\phi \right)=\left(l_0-m_0\right)+\left({\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="DE102019220103A1\_0065.png"\; state="\{\{state\}\}">l</figure-callout>}}_1+2\cdot m_1\right)\cdot cOs\left(\phi -\frac{2\pi }{3}\right)+\left({\mathrm{<figure-callout\; id="2"\; label="l"\; filenames="DE102019220103A1\_0064.png"\; state="\{\{state\}\}">l</figure-callout>}}_2+2\cdot m_2\right)\\ \cdot cOs\left[2\cdot \left(\phi -\frac{2\pi }{3}\right)\right]+\left(l_3-m_3\right)\cdot cOs\left[3\cdot \left(\phi -\frac{2\pi }{3}\right)\right]+\cdots \end{array}$$

$$\begin{array}{c}d\mathrm{L.}\mathrm{M.}\gamma \left(\phi \right)=\left(l_0-m_0\right)+\left({\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="DE102019220103A1\_0065.png"\; state="\{\{state\}\}">l</figure-callout>}}_1+2\cdot m_1\right)\cdot cOs\left(\phi +\frac{2\pi }{3}\right)+\left({\mathrm{<figure-callout\; id="2"\; label="l"\; filenames="DE102019220103A1\_0064.png"\; state="\{\{state\}\}">l</figure-callout>}}_2+2\cdot m_2\right)\\ \cdot cOs\left[2\cdot \left(\phi +\frac{2\pi }{3}\right)\right]+\left(l_3-m_3\right)\cdot cOs\left[3\cdot \left(\phi +\frac{2\pi }{3}\right)\right]+\cdots \end{array}$$

Claims (12)

Method for the controlled operation of a multiphase electrical machine (2) with several phase strings (23) connected in a delta circuit, with the following steps: - Provision of a control deviation, - Control of the operation of the electrical machine (2) depending on computationally determined actual string currents in order to provide terminal potential information (Su, Sv, Sw) for controlling an inverter (3), which applies terminal potentials (Vu, Vv, Vw) to the electrical machine (2), the actual _phase currents (I aist, I _bist , I _cist) ) can be determined with the help of a machine model depending on the terminal potentials (Vu, Vv, Vw), whereby the machine model depends on an inductance matrix that determines the magnetic properties of the electrical machine in such a way that these voltage differences between the terminal potentials (Vu, Vv, Vw) the actual line currents (I _aist , I _bist , I _cist ) are mapped. Procedure according to Claim 1 , wherein the regulation comprises a speed regulation, a torque regulation or a flux regulation. Procedure according to Claim 1 or 2 , the target terminal potentials being specified as pulse-width modulated signals for controlling an inverter, the terminal potentials (Vu, Vv, Vw) being measured in order to determine the actual _phase currents (I aist, I _bist , I _cist ) using the machine model. Method according to one of the Claims 1 to 3 Wherein the actual phase currents (Ia, Ib, Ic) are used to an actual engine torque _(M) or actual motor flow to determine and /. Procedure according to Claim 4 , the actual engine torque (M _ist ) and / or the actual engine fluxes being used for a torque control or a flux control. Method according to one of the Claims 1 to 5 , wherein the inductance matrix has terms of inductances which are formed from phase inductances and coupling inductances, which are each dependent on the rotor position, and the inductances are defined by coefficients of harmonics of a rotation of the electrical position angle. Procedure according to Claim 6 , wherein the inductances take into account all or at least the first two, in particular the first five harmonics of the phase inductances and coupling inductances between the phase phases (23), in particular the third harmonics and multiples thereof being neglected. Method according to one of the Claims 1 to 7th , whereby for the regulation actual _phase currents (I aist, I _bist , I _cist ) are determined with the help of the modified machine model with inverted inductance matrix, which are determined in an optimization problem for the determination of the terminal potential information (S _u , S _v , S _w ). Method according to one of the Claims 1 to 8th , where the machine model used has the form: $$\frac{d}{dt}\left(\left[\begin{array}{c}d\mathrm{L.}\mathrm{M.}a\left(\phi \right)d\mathrm{L.}\mathrm{M.}b\left(\phi \right)\\ d\mathrm{L.}\mathrm{M.}c\left(\phi \right)d\mathrm{L.}\mathrm{M.}d\left(\phi \right)\end{array}\right]\cdot \left[\begin{array}{c}\mathrm{I.}a\\ \mathrm{I.}b\end{array}\right]\right)=\left[\begin{array}{c}Vu-Vv\\ Vv-Vw\end{array}\right]-\mathrm{R.}\cdot \left[\begin{array}{c}\mathrm{I.}a\\ \mathrm{I.}b\end{array}\right]-\frac{d}{dt}\left[\begin{array}{c}{\psi }_{ma}\\ {\psi }_{mb}\end{array}\right]$$

With $$d\mathrm{L.}\mathrm{M.}a\left(\phi \right)=\mathrm{L.}a\left(\phi \right)-\mathrm{M.}ac\left(\phi \right)$$

$$d\mathrm{L.}\mathrm{M.}b\left(\phi \right)=\mathrm{M.}ab\left(\phi \right)-\mathrm{M.}ac\left(\phi \right)$$

$$d\mathrm{L.}\mathrm{M.}c\left(\phi \right)=\mathrm{M.}ab\left(\phi \right)-\mathrm{M.}bc\left(\phi \right)$$

$$d\mathrm{L.}\mathrm{M.}d\left(\phi \right)=\mathrm{L.}b\left(\phi \right)-\mathrm{M.}bc\left(\phi \right)$$

where Vu, Vv, Vw the terminal potentials of the phase strands, la, lb the phase currents, R the electrical phase resistance, La, Lb, Lc the phase inductances, Mab, Mac, Mbc the coupling inductances, φ the rotor position, and Ψma, Ψmb the rotation of the runner correspond to the respective induced flows. Device, in particular control unit, for the regulated operation of a multiphase electrical machine (2) with several phase strands (23) connected in a delta circuit, the device being designed to: - determine a control deviation, and - regulate the operation of the electrical machine (2) depending on calculated actual string currents to provide terminal potential information (Su, Sv, Sw) for controlling an inverter (3), which applies terminal potentials (Vu, Vv, Vw) to the electrical machine (2), whereby the actual string currents ( I aist, I _bist , I _{cist ) can be determined with the help of} a machine model depending on the terminal potentials (Vu, Vv, Vw), the machine model being dependent on an inductance matrix, the magnetic one Properties of the electrical machine are determined in such a way that these voltage differences between the terminal potentials (Vu, Vv, Vw) _{are mapped onto the actual phase} currents (I aist, I bist, I _cist ). Computer program which is set up to carry out all the steps of a method according to one of the Claims 1 to 9 to execute. Electronic storage medium on which a computer program is based Claim 12 is stored.

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