Three-winding brushless DC motor with trapezoidal flux distribution.
Description
The BLDC unit simulates a permanent magnet synchronous machine with a three-phase stator with star connection. The block has two options to determine the flux distribution of the permanent magnets as a function of the rotor angle. A simplified parameterization is presented, assuming that the counter-EMF has a perfect trapezoidal shape. For the simplified one, either the flux-coupling or the rotor induced counter-EMF is specified.
The figure shows the equivalent circuit diagram for the stator windings.
Motor design
This figure shows the motor design with one pair of poles on the rotor.
For the adopted axis arrangement corresponding to the figure, the phase flux and the permanent magnet flux are aligned when the rotor angle is zero. The block supports a second variant of the rotor axis definition. For the second definition, the rotor angle is the angle between the magnetic phase axis and the rotor axis .
Trapezoidal flux change rate
The rotor magnetic field due to permanent magnets produces a trapezoidal rate of flux change as a function of rotor angle, as shown in the figure.
The counter-EMF is the rate of change of flow, which is defined as follows
where
- is the flux-coupling of the permanent magnet;
- rotor rotation angle;
- mechanical speed of rotation.
The height of the trapezoidal flux change rate profile is determined from the maximum flux value of the permanent magnet.
Integrating in the range from to , we obtain
where
- flux-coupling of the permanent magnet;
- is the height of the flux change velocity profile;
- the range of rotor angles in which the counter-EMF induced by the flux of permanent magnets in the stator is constant;
- the range of rotor angles in which the counter-EMF linearly increases or decreases when the rotor moves with constant speed.
From the previous equation we can obtain
Defining electrical equations
The stator winding voltages are defined as
where
, , and are the external voltages applied to the three electrical connections of the motor;
- is the equivalent active resistance of each stator winding;
, , and are the currents flowing in the stator windings;
, and are the rates of change of magnetic flux in each stator winding.
The permanent magnet and the three windings contribute to the total flux linking each winding. The total flux is defined as follows
where
, , and are the total fluxes connecting each stator winding;
, , and are the stator winding inductances;
, , etc. - mutual inductances of the stator windings;
, , and - fluxes of permanent magnets connecting the stator windings.
The inductances in the stator windings are functions of the rotor rotation angle and are defined as follows
where
- is the stator phase eigeninductance, i.e. the average eigeninduction of each stator winding;
- amplitude of stator inductance change, i.e. amplitude of change of own and mutual inductance at change of rotor rotation angle;
- stator mutual inductance, i.e. the average mutual inductance between the stator windings.
The permanent magnet flux linking each stator winding corresponds to the trapezoidal profile shown in the figure above. The block implements the trapezoidal profile using lookup tables to calculate the permanent magnet flux values.
Simplified equations
The defining stress and torque equations for the block are as follows
where
, , are the axis voltages , and zero sequence voltage, respectively;
- Park’s transformation, defined as follows:
;
- number of pole pairs of the permanent magnet of the rotor;
- mechanical speed of rotor rotation;
, and are the partial derivatives of the instantaneous fluxes of the permanent magnets linking each phase winding;
, and are the axis currents , and zero sequence, respectively, defined as follows:
.
- axis stator inductance ;
- axis stator inductance ;
- stator zero-sequence inductance;
- rotor torque. The torque is transferred from the motor housing (port C) to the motor rotor (port R).
Modelling thermal effects
You can open thermal ports to simulate thermal losses. To do this, select the Enable thermal port checkbox.
Ports
Non-directional
~ - three-phase port `three-phase electricity
Three-phase port.
Dependencies
To enable this port, set Electrical connection to Composite three-phase ports and Winding type to Wye-wound or Delta-wound.
n - neutral electricity
A non-directional port connected to the neutral.
Dependencies
To enable this port, set Winding type to Wye-wound' and Zero sequence to `on.
R is the rotor of the machine `rotational mechanics
A mechanical port associated with the rotor of a machine.
C - the machine housing `rotational mechanics
A mechanical port associated with the machine housing.
HA - thermal port of the winding A heat
Thermal port associated with winding .
Dependencies
To use this port, select the checkbox for Enable thermal port.
HB - winding thermal port B heat
Thermal port associated with winding .
Dependencies
To use this port, select the checkbox for Enable thermal port.
HC - winding thermal port C heat
Thermal port associated with winding .
Dependencies
To use this port, select the checkbox for Enable thermal port.
HR - rotor thermal port heat
The heat port associated with the rotor.
Dependencies
To use this port, select the Enable thermal port check box.
The value of the parameter determines whether the thermal ports of the block are available and whether thermal loss modelling will be performed.
Rotor
Back EMF profile - counter EMF profile Perfect trapezoid - specify maximum flux linkage (By default) | Perfect trapezoid - specify maximum rotor-induced back emf
Parameterization to define the flux distribution of a permanent magnet as a function of rotor angle. Available values:
Perfect trapezoid - specify maximum flux linkage - use this value to specify the maximum flux linkage for the permanent magnet and the rotor angle at which the counter-EMF is constant. The shape of the counter-EMF is assumed to be a perfect trapezoid.
`Perfect trapezoid - specify maximum rotor-induced back emf' - use this value to specify the maximum rotor-induced back EMF and the corresponding rotor speed. The shape of the back EMF is assumed to be a perfect trapezoid.
Maximum permanent magnet flux linkage - maximum value of permanent magnet flux linkage 0.03 Vb (by default)
Maximum permanent magnet flux linkage with either stator winding.
Dependencies
To use this parameter, set Back EMF profile to `Perfect trapezoid - specify maximum flux coupling'.
Rotor angle over which back emf is constant - rotor angle over which back EMF is constant `pi/12 (by default)
The range of rotor angles over which the permanent magnetic flux linking the stator winding is constant. In the figure labelled as .
Dependencies
To use this parameter, set the Back EMF profile parameter to Perfect trapezoid - specify maximum flux linkage or Perfect trapezoid - specify maximum rotor-induced back emf.
Maximum rotor-induced back emf - maximum rotor-induced back emf value 9.6 V (by default).
Maximum rotor-induced back emf in the stator windings.
Dependencies
To use this parameter, set Back EMF profile to `Perfect trapezoid - specify maximum rotor-induced back emf'.
Rotor speed used for back emf measurement - rotor speed used for back emf measurement 600 rpm (By default)
The rotor speed corresponding to the maximum back EMF induced by the rotor.
Dependencies
To use this parameter, set the Back EMF profile parameter to Terfect trapezoid - specify maximum rotor-induced back emf or Tabulated - specify rotor-induced back emf as a function of rotor angle.
Number of pole pairs - number of pole pairs of the machine 6 (By default).
Number of pole pairs of permanent magnet on the rotor.
Rotor angle definition - the reference point for measuring the rotor angle Angle between the a-phase magnetic axis and the d-axis (by default) | Angle between the a-phase magnetic axis and the q-axis.
The reference point for measuring the rotor angle. If Angle between the a-phase magnetic axis and the d-axis' is selected, the rotor and phase fluxes are aligned when the rotor angle is zero. If `Angle between the a-phase magnetic axis and the q-axis is selected, the phase current produces maximum torque when the rotor angle is zero.
Stator
Winding type - winding connection Wye-wound (by default) | Delta-wound
Select the winding connection:
Wye-wound - star connection.
Delta-wound - delta connection. Phase is connected between ports a and b, phase is connected between ports b and c, phase is connected between ports c and a.
Stator parameterization - stator parameterization Specify Ld, Lq, and L0 (by default) | `Specify Ls, Lm, and Ms `
Method of stator parameterization.
Select Specify Ld, Lq, and L0 or Specify Ls, Lm, and Ms.
Wye-wound for the Winding Type parameter, Enabled for the Zero sequence parameter, and Specify Ld, Lq, and L0 for the Stator parameterization parameter.
For the Winding Type parameter the value is Delta-wound and for the Stator parameterization parameter the value is Specify Ld, Lq, and L0.
Stator self-inductance per phase, Ls - stator self-inductance per phase 0.0002 Gn (by default)
Average self-inductance of each of the three stator windings.
Dependencies
To use this parameter, set the Stator parameterization parameter to Specify Ls, Lm and Ms.
These parameters appear only for units with open thermal ports.
Resistance temperature coefficient - resistance temperature coefficient 3.93e-3 1/K (By default).
The coefficient in the equation relating active resistance to temperature. The winding resistance is assumed to depend linearly on temperature and is defined as:
,
where:
- active resistance at temperature ;
- active resistance at measurement temperature ;
- temperature coefficient of resistance. The value for copper is 3.93e-3 1/K.
Permanent magnet flux temperature coefficient - permanent magnet flux temperature coefficient -0.001 1/K (By default).
The partial derivative of the permanent magnet flux density from temperature. Used to linearly reduce torque and induced EMF as temperature increases.
Measurement temperature - measurement temperature `298.15 (By default).
The temperature for which the motor parameters are given.
Thermal Port
These parameters appear only for units with open thermal ports.
Thermal mass for each stator winding - heat capacity for each stator winding 100 J/K (By default).
Heat capacity value for each stator winding. Heat capacity is the energy required to raise the temperature by one degree.
Rotor heat capacity. Heat capacity is the energy required to raise the temperature by one degree.
References
[1] Kundur, P. PowerSystemStabilityandControl. New York, NY: McGraw Hill, 1993.
[2] Anderson, P. M. AnalysisofFaultedPowerSystems. Hoboken, NJ: Wiley-IEEE Press, 1995.
[3] Mellor, P. H., R. Wrobel, and D. Holliday. "A computationally efficient iron loss model for brushless AC machines that caters for rated flux and field weakened operation." IEEEElectricMachinesandDrivesConference. May 2009.