esempio pog synchronous motors - unimore
TRANSCRIPT
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Roberto Zanasi
University of Modena and Reggio EmiliaVia Vignolese 905, 41100 Modena, ItalyTel: 059 2056161; Fax: 059 2056129;
E-mail: [email protected]
Multi-phase Synchronous Motors
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Multi-phase Syncronous motor (m=5)
θ
2
3
3
2
14
4
5
5d
Φr
1p=1
q
chained total magnetic fluxes
positive-definitesymmetric matrix
where:
voltage input vector
dissipating matrix (diagonal)
system of differentialequations describing the electrical part
Electrical part
TORQUE VECTOR
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Mechanical part
= counter-electromotive forces
Energy stored in the electro-mechanical system:
differential equation describingthe mechanical part
= electromotive torque
Function φc(θ) of chained rotor flux can be developed in Fourier series of cosines with only odd harmonics, so vector Kτ(θ) can be written as:
Multi-phase Syncronous motor
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POG scheme of the multi-phase motor
energyaccumulation
energyaccumulation
energydissipa-
tion
energydissipa-
tionenergyconversion
Electrical part Mechanical part
State-Space description of the system:
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Orthonormal transformations
from Σt to Σω
Let’s consider the following orthonormal transformation and then applyit to the system:
The system dynamic equations become:
where the transformed matrices are:
diagonal matrices
and
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The torque vector
In the transformed space the structure of the torque vector is quite simple:
• The “d” and “q” terms have only harmonics with frequency multiple of 2m
This term is not present if the phases are star-connected:
“direct” terms
“quadrature” terms
• The harmonics of the last term don’t influence the “d” and “q” terms
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The torque vector
Proposition 1. The torque vector can be constant only if the flux vector can be expressed in Fourier series as follows:
All the constant components of the torque vector are obtained for n=0:
Main result. Among all the fluxes providing a constant vector, the one that minimizes the module of the current vector (and therefore the dissipated power) is given by:
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POG scheme in the transformed space
energyaccumulation
energyconversion
transfor-mation
energydissipa-
tion
energyaccumulation
energydissipa-
tion
Electrical part Mechanical part
energyconvers.
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POG Blocks in Simulink
POG m-phases Syncronous Motors
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POG Blocks in Simulink: the musk
POG m-phases Motors:
Types of rotor flux:1. Trapezoidal2. Triangular3. Squared wave4. Sinusoidal5. Cosinusoidally connected6. Sinusoidally connected7. Polynomial even8. Polynomial odd9. Trapezoidal if derived10.Fourier defined
The stator can also be “star connected”
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POG Blocks in Simulink
The POG block diagrams can be directly inserted in Simulink !!
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Simulations (1)
The POG scheme has been implemented in Simulink.
Simulation implemented with cosinusoidal interpolated rotor fluxwith 200 harmonics.
At time t=2s the load torque τe switches from 0 to 100 Nm
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Simulations (2)
Simulation implemented with even polynomial interpolation (q=2,α=π/5)
Counter-electromotive voltages are trapezoidal
This control generates high torqueswhen velocity is smallAt time t=1.5s the load torque τe
switches from 0 to 100 Nm
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Conclusions
• Power-Oriented Graphs (POG) are a simple and powerful graphical technique that can be used for modelling all types of physical systems involving power flows.
• POG are easily understandable, simple to use and suitable both for teaching and for research.