induction machine model
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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 )()()(
cr
br
ar
cs
bs
as
cr
br
ar
cs
bs
as
r
r
r
s
s
s
cr
br
ar
cs
bs
as
dtd
iiiiii
rr
rr
rr
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.
cr
br
ar
cs
bs
as
rrs
srs
cr
br
ar
cs
bs
as
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
21
21
21
21
21
21
mrmm
mmrm
mmmr
r
LLLL
LLLL
LLLL
L
21
21
21
21
21
21
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
21
21
21
21
21
21
mrmm
mmrm
mmmr
r
LLLL
LLLL
LLLL
L
21
21
21
21
21
21
Summarizing….
cr
br
ar
cs
bs
as
cr
br
ar
cs
bs
as
r
r
r
s
s
s
cr
br
ar
cs
bs
as
dtd
iiiiii
rr
rr
rr
vvvvvv
000000000000000000000000000000
cr
br
ar
cs
bs
as
rrs
srs
cr
br
ar
cs
bs
as
iiiiii
LLLL
Tsr
mmm
mmm
mmm
mrs LLL
cos120cos120cos120coscos120cos120cos120coscos
mmm
mmm
mmm
msr LL
cos120cos120cos120coscos120cos120cos120coscos
Machine model
7
msmm
mmsm
mmms
s
LLLL
LLLL
LLLL
L
21
21
21
21
21
21
mrmm
mmrm
mmmr
r
LLLL
LLLL
LLLL
L
21
21
21
21
21
21
cr
br
ar
cs
bs
as
rrs
srs
cr
br
ar
cs
bs
as
r
r
r
s
s
s
cr
br
ar
cs
bs
as
iiiiii
LLLL
dtd
iiiiii
rr
rr
rr
vvvvvv
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
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