Machine model

1

Consider the machine as two sets of abc windings, one on the stator and one on the rotor.

θm

ωm

Machine model

2

The voltage equation for each phase will have the form:That is, we can write them all in the following form: dt

tdtritv )()()(

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iiiiii

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vvvvvv

000000000000000000000000000000

All rotor terms are given on the rotor side in these equations.

We can write the flux terms as functions of the currents, via an equation for each flux of the form λ=ΣLkik, where the summation is over all six winding currents. However, we must take note that there are four kinds of terms in each summation.

Machine model

3

• Stator-stator terms: These are terms which relate a stator winding flux to a stator winding current. Because the positional relationship between any pair of stator windings does not change with rotor position, these inductances are not a function of rotor position; they are constants.

• Rotor-rotor terms: These are terms which relate a rotor winding flux to a rotor winding current. As in stator-stator-terms, these are constants.

• Rotor-stator terms: These are terms which relate a rotor winding flux to a stator winding current. As the rotor turns, the positional relationship between the rotor winding and the stator winding will change, and so the inductance will change. Therefore the inductance will be a function of rotor position, characterized by rotor angle θ.

• Stator-rotor terms: These are terms which relate a stator winding flux to a rotor winding current. As described for the rotor-stator terms, the inductance will be a function of rotor position, characterized by rotor angle θ.

Machine model

4

There are two more comments to make about the flux-current relations:•Because the rotor motion is periodic, the functional dependence of each rotor-stator or stator-rotor inductance on θ is cosinusoidal. •Because θ changes with time as the rotor rotates, the inductances are functions of time. We may now write down the flux equations for the stator and the rotor windings.

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rrs

srs

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iiiiii

LLLL

Each of the submatrices in the inductance matrix is a 3x3, as given on the next slide…

Note here that all quantities are now referred to the stator. The effect of referring is straight-forward, given in the book by P. Krause, “Analysis of Electric Machinery,” 1995, IEEE Press, pp. 167-168. I will not go through it here.

Machine model

5

msmm

mmsm

mmms

s

LLLL

LLLL

LLLL

L

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mrmm

mmrm

mmmr

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LLLL

LLLL

LLLL

L

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mmm

mmm

mmm

msr LL

cos120cos120cos120coscos120cos120cos120coscos

Diagonal elements are the self-inductance of each winding and include leakage plus mutual. Off-diagonal elements are mutual inductances between windings and are negative because 120° axis offset between any pair of windings results in flux contributed by one winding to have negative component along the main axis of another winding.

Tsr

mmm

mmm

mmm

mrs LLL

cos120cos120cos120coscos120cos120cos120coscos

θm

ωm

Machine model

6

msmm

mmsm

mmms

s

LLLL

LLLL

LLLL

L

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mrmm

mmrm

mmmr

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LLLL

LLLL

LLLL

L

21

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21

21

Summarizing….

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000000000000000000000000000000

cr

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rrs

srs

cr

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iiiiii

LLLL

Tsr

mmm

mmm

mmm

mrs LLL

cos120cos120cos120coscos120cos120cos120coscos

mmm

mmm

mmm

msr LL

cos120cos120cos120coscos120cos120cos120coscos

Machine model

7

msmm

mmsm

mmms

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LLLL

LLLL

LLLL

L

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mrmm

mmrm

mmmr

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LLLL

LLLL

LLLL

L

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cr

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srs

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LLLL

dtd

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000000000000000000000000000000Combining….

It is here that we observe a difficulty – that the stator-rotor and rotor-stator terms, Lsr and Lrs, because they are functions of θr, and thus functions of time, will also need to be differentiated. Therefore differentiation of fluxes results in expressions likeThe differentiation with respect to L, dL/dt, will result in time-varyingcoefficients on the currents. This will make our set of state equations difficult to solve.

dtdiLi

dtdL

dtd

Tsr

mmm

mmm

mmm

mrs LLL

cos120cos120cos120coscos120cos120cos120coscos

mmm

mmm

mmm

msr LL

cos120cos120cos120coscos120cos120cos120coscos