BLDC

Three-winding brushless DC motor with trapezoidal flux distribution

Library:
Simscape / Electrical / Electromechanical / Permanent Magnet

Description

The BLDC block models a permanent magnet synchronous machine with a three-phase wye-wound stator. The block has four options for defining the permanent magnet flux distribution as a function of rotor angle. Two options allow for simple parameterization by assuming a perfect trapezoid for the back emf. For simple parameterization, you specify either the flux linkage or the rotor-induced back emf. The other two options give more accurate results using tabulated data that you specify. For more accurate results, you specify either the flux linkage partial derivative or the measured back emf constant for a given rotor speed.

The figure shows the equivalent electrical circuit for the stator windings.

Motor Construction

This figure shows the motor construction with a single pole-pair on the rotor.

For the axes convention in the preceding figure, the a-phase and permanent magnet fluxes are aligned when rotor angle θ_r is zero. The block supports a second rotor-axis definition. For the second definition, the rotor angle is the angle between the a-phase magnetic axis and the rotor q-axis.

Trapezoidal Rate of Change of Flux

The rotor magnetic field due to the permanent magnets create a trapezoidal rate of change of flux with rotor angle. The figure shows this rate of change of flux.

Back emf is the rate of change of flux, defined by

$\frac{d Φ}{d t} = \frac{\partial Φ}{\partial θ} \frac{d θ}{d t} = \frac{\partial Φ}{\partial θ} ω,$

where:

Φ is the permanent magnet flux linkage.
θ is the rotor angle.
ω is the mechanical rotational speed.

The height h of the trapezoidal rate of change of flux profile is derived from the permanent magnet peak flux.

Integrating $\frac{\partial Φ}{\partial θ}$ over the range 0 to π/2,

$Φ_{m a x} = \frac{h}{2} (θ_{F} + θ_{W}),$

where:

Φ_max is the permanent magnet flux linkage.
h is the rate of change of flux profile height.
θ_F is the rotor angle range over which the back emf that the permanent magnet flux induces in the stator is constant.
θ_W is the rotor angle range over which back emf increases or decreases linearly when the rotor moves at constant speed.

Rearranging the preceding equation,

$h = 2 Φ_{m a x} / (θ_{F} + θ_{W}) .$

Electrical Defining Equations

Voltages across the stator windings are defined by

$[\begin{matrix} v_{a} \\ v_{b} \\ v_{c} \end{matrix}] = [\begin{matrix} R_{s} & 0 & 0 \\ 0 & R_{s} & 0 \\ 0 & 0 & R_{s} \end{matrix}] [\begin{matrix} i_{a} \\ i_{b} \\ i_{c} \end{matrix}] + [\begin{matrix} \frac{d ψ_{a}}{d t} \\ \frac{d ψ_{b}}{d t} \\ \frac{d ψ_{c}}{d t} \end{matrix}],$

where:

v_a, v_b, and v_c are the external voltages applied to the three motor electrical connections.
R_s is the equivalent resistance of each stator winding.
i_a, i_b, and i_c are the currents flowing in the stator windings.
$\frac{d ψ_{a}}{d t},$ $\frac{d ψ_{b}}{d t},$ and $\frac{d ψ_{c}}{d t}$

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 by

$[\begin{matrix} ψ_{a} \\ ψ_{b} \\ ψ_{c} \end{matrix}] = [\begin{matrix} L_{a a} & L_{a b} & L_{a c} \\ L_{b a} & L_{b b} & L_{b c} \\ L_{c a} & L_{c b} & L_{c c} \end{matrix}] [\begin{matrix} i_{a} \\ i_{b} \\ i_{c} \end{matrix}] + [\begin{matrix} ψ_{a m} \\ ψ_{b m} \\ ψ_{c m} \end{matrix}],$

where:

ψ_a, ψ_b, and ψ_c are the total fluxes linking each stator winding.
L_aa, L_bb, and L_cc are the self-inductances of the stator windings.
L_ab, L_ac, L_ba, etc. are the mutual inductances of the stator windings.
ψ_am, ψ_bm, and ψ_cm are the permanent magnet fluxes linking the stator windings.

The inductances in the stator windings are functions of rotor angle, defined by

$L_{a a} = L_{s} + L_{m} cos (2 θ_{r}),$

$L_{b b} = L_{s} + L_{m} cos (2 (θ_{r} - 2 π / 3)),$

$L_{c c} = L_{s} + L_{m} cos (2 (θ_{r} + 2 π / 3)),$

$L_{a b} = L_{b a} = - M_{s} - L_{m} \cos (2 (θ_{r} + π / 6)),$

$L_{b c} = L_{c b} = - M_{s} - L_{m} \cos (2 (θ_{r} + π / 6 - 2 π / 3)),$

and

$L_{c a} = L_{a c} = - M_{s} - L_{m} \cos (2 (θ_{r} + π / 6 + 2 π / 3)),$

where:

L_s is the stator self-inductance per phase — The average self-inductance of each of the stator windings.
L_m is the stator inductance fluctuation — The fluctuation in self-inductance and mutual inductance with changing rotor angle.
M_s is the stator mutual inductance — The average mutual inductance between the stator windings.

The permanent magnet flux linking each stator winding follows the trapezoidal profile shown in the figure. The block implements the trapezoidal profile using lookup tables to calculate permanent magnet flux values.

Simplified Equations

The defining voltage and torque equations for the block are

$[\begin{matrix} v_{d} \\ v_{q} \\ v_{0} \end{matrix}] = P ([\begin{matrix} v_{a} \\ v_{b} \\ v_{c} \end{matrix}] - N ω [\begin{matrix} \frac{\partial ψ_{a m}}{\partial θ_{r}} \\ \frac{\partial ψ_{b m}}{\partial θ_{r}} \\ \frac{\partial ψ_{c m}}{\partial θ_{r}} \end{matrix}]),$

$v_{d} = R_{s} i_{d} + L_{d} \frac{d i_{d}}{d t} - N ω i_{q} L_{q},$

$v_{q} = R_{s} i_{q} + L_{q} \frac{d i_{q}}{d t} + N ω i_{d} L_{d},$

$v_{0} = R_{s} i_{0} + L_{0} \frac{d i_{0}}{d t},$

and

$T = \frac{3}{2} N (i_{q} i_{d} L_{d} - i_{d} i_{q} L_{q}) + [\begin{matrix} i_{a} & i_{b} & i_{c} \end{matrix}] [\begin{matrix} \frac{\partial ψ_{a m}}{\partial θ_{r}} \\ \frac{\partial ψ_{b m}}{\partial θ_{r}} \\ \frac{\partial ψ_{c m}}{\partial θ_{r}} \end{matrix}],$

where:

v_d, v_q, and v₀ are the d-axis, q-axis, and zero-sequence voltages.
P is Park’s Transformation, defined by
$P = 2 / 3 [\begin{matrix} \cos θ_{e} & \cos (θ_{e} - 2 π / 3) & \cos (θ_{e} + 2 π / 3) \\ - \sin θ_{e} & - \sin (θ_{e} - 2 π / 3) & - \sin (θ_{e} + 2 π / 3) \\ 0.5 & 0.5 & 0.5 \end{matrix}]$
N is the number of rotor permanent magnet pole pairs.
ω is the rotor mechanical rotational speed.
$\frac{\partial ψ_{a m}}{\partial θ_{r}},$ $\frac{\partial ψ_{b m}}{\partial θ_{r}},$ and $\frac{\partial ψ_{c m}}{\partial θ_{r}}$

are the partial derivatives of instantaneous permanent magnet flux linking each phase winding.
i_d, i_q, and i₀ are the d-axis, q-axis, and zero-sequence currents, defined by
$[\begin{matrix} i_{d} \\ i_{q} \\ i_{0} \end{matrix}] = P [\begin{matrix} i_{a} \\ i_{b} \\ i_{c} \end{matrix}] .$
L_d = L_s + M_s + 3/2 L_m. L_d is the stator d-axis inductance.
L_q = L_s + M_s − 3/2 L_m. L_q is the stator q-axis inductance.
L₀ = L_s – 2M_s. L₀ is the stator zero-sequence inductance.
T is the rotor torque. Torque flows from the motor case (block physical port C) to the motor rotor (block physical port R).

Calculating Iron Losses

Iron losses are divided into two terms, one representing the main magnetizing path, and the other representing the cross-tooth tip path that becomes active during field weakened operation. The iron losses model, which is based on the work of Mellor [3].

The term representing the main magnetizing path depends on the induced RMS stator voltage, $V_{m_{r m s}}^{}$ :

$P_{O C} (V_{m_{r m s}}^{}) = \frac{a_{h}}{k} V_{m_{r m s}}^{} + \frac{a_{j}}{k^{2}} V_{m_{r m s}}^{2} + \frac{a_{e x}}{k^{1.5}} V_{m_{r m s}}^{1.5}$

This is the dominant term during no-load operation. k is the back emf constant relating RMS volts per Hz. It is defined as $k = V_{m_{r m s}}^{} / f$ , where f is the electrical frequency. The first term on the right-hand side is the magnetic hysteresis loss, the second is the eddy current loss and the third is the excess loss. The three coefficients appearing on the numerators are derived from the values that you provide for the open-circuit hysteresis, eddy, and excess losses.

The term representing the cross-tooth tip path becomes important when a demagnetizing field is set up and can be determined from a finite element analysis short-circuit test. It depends on the RMS emf associated with the cross-tooth tip flux, $V_{d_{r m s}}^{*}$ :

$P_{S C} (V_{d_{r m s}}^{*}) = \frac{b_{h}}{k} V_{d_{r m s}}^{*} + \frac{b_{j}}{k^{2}} V_{d_{r m s}}^{* 2} + \frac{b_{e x}}{k^{1.5}} V_{d_{r m s}}^{* 1.5}$

The three numerator terms are derived from the values you provide for the short-circuit hysteresis, eddy, and excess losses.

Thermal Ports

The block has four optional thermal ports, one for each of the three windings and one for the rotor. These ports are hidden by default. To expose the thermal ports, right-click the block in your model, select Simscape > Block choices, and then select the desired block variant with thermal ports: Composite three-phase ports | Show thermal port or Expanded three-phase ports | Show thermal port. This action displays the thermal ports on the block icon, and exposes the Temperature Dependence and Thermal Port parameters. These parameters are described further on this reference page.

Use the thermal ports to simulate the effects of copper resistance and iron losses that convert electrical power to heat. For more information on using thermal ports in actuator blocks, see Simulating Thermal Effects in Rotational and Translational Actuators.

Variables

Use the Variables settings to specify the priority and initial target values for the block variables before simulation. For more information, see Set Priority and Initial Target for Block Variables.

Ports

Conserving

expand all

`~` — Three-phase port
electrical

Expandable three-phase port.

`n` — Neutral phase
electrical

Electrical conserving port associated with the neutral phase

`R` — Motor rotor
mechanical

Mechanical rotational conserving port associated with the motor rotor

`C` — Motor case
mechanical

Mechanical rotational conserving port associated with the motor case

`HA` — Winding A thermal port
thermal

Thermal conserving port associated with winding A. For more information, see Thermal Ports.

`HB` — Winding B thermal port
thermal

Thermal conserving port associated with winding B. For more information, see Thermal Ports.

`HC` — Winding C thermal port
thermal

Thermal conserving port associated with winding C. For more information, see Thermal Ports.

`HR` — Rotor thermal port
thermal

Thermal conserving port associated with the rotor. For more information, see Thermal Ports.

Parameters

expand all

Rotor

`Winding type` — Windings configuration
`Wye-wound` (default) | `Delta-wound`

Select the configuration for the windings:

Wye-wound — The windings are wye-wound.
Delta-wound — The windings are delta-wound. The a-phase is connected between ports a and b, the b-phase between ports b and c and the c-phase between ports c and a.

`Back EMF profile` — Back EMF profile
`Perfect trapezoid - specify maximum flux linkage` (default) | `Perfect trapezoid - specify maximum rotor-induced back emf` | `Tabulated - specify flux partial derivative with respect to rotor angle` | `Tabulated - specify rotor-induced back emf as a function of rotor angle`

Parameterization for defining the permanent magnet flux distribution as a function of rotor angle. Choose:

Perfect trapezoid - specify maximum flux linkage to specify the maximum flux linkage for the permanent magnet and the rotor angle where the back emf is constant. The block assumes a perfect trapezoid for the back emf. This is the default value.
Perfect trapezoid - specify maximum rotor-induced back emf to specify the maximum rotor-induced back emf and the corresponding rotor speed. The block assumes a perfect trapezoid for the back emf.
Tabulated - specify flux partial derivative with respect to rotor angle to specify values for the partial derivative of flux linkage and the corresponding rotor angles.
Tabulated - specify rotor-induced back emf as a function of rotor angle to specify the measured back emf constant and the corresponding rotor speed and angles.

`Maximum permanent magnet flux linkage` — Maximum permanent magnet flux linkage
`0.03` `Wb` (default)

Peak permanent magnet flux linkage with any of the stator windings.

Dependencies

To enable this parameter, set Back EMF profile to Perfect trapezoid - specify maximum flux linkage.

`Rotor angle over which back emf is constant` — Rotor angle over which back emf is constant
`pi / 12` `rad` (default)

Rotor angle range over which the permanent magnet flux linking the stator winding is constant. This angle is θ_F in the figure that shows the Trapezoidal Rate of Change of Flux.

Dependencies

To enable this parameter, set Back EMF profile 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
`9.6` `V` (default)

Peak rotor-induced back emf into the stator windings.

Dependencies

To enable this parameter, set Back EMF profile to Perfect trapezoid - specify maximum rotor-induced back emf.

`Rotor-induced back emf` — Rotor-induced back emf
`[0, -9.6, -9.6, 9.6, 9.6, 0]` `V` (default)

Vector of values for the rotor-induced back emf as a function of rotor angle. The first and last values must be the same, and are normally both zero. For more information, see the Corresponding rotor angles parameter. First and last values are the same because flux is cyclic with period $2 π / N$ , where N is the number of permanent magnet pole pairs.

Dependencies

To enable this parameter, set Back EMF profile to Tabulated - specify rotor-induced back emf as a function of rotor angle.

`Flux linkage partial derivative with respect to rotor angle` — Flux linkage partial derivative with respect to rotor angle
`[0, -.1528, -.1528, .1528, .1528, 0]` `Wb/rad` (default)

Vector of values for the partial derivative of flux linkage (where flux linkage is flux times number of winding turns) with respect to rotor angle. The first and last values must be the same, and are normally both zero. For more information, see the Corresponding rotor angles parameter. First and last values are the same because flux is cyclic with period $2 π / N$ , where N is the number of permanent magnet pole pairs.

`Corresponding rotor angles` — Corresponding rotor angles
`[0, 7.5, 22.5, 37.5, 52.5, 60]` `deg` (default)

Vector of rotor angles where the flux linkage partial derivative or rotor-induced back emf is defined. Rotor angle is defined as the angle between the a-phase magnetic axis and the d-axis. That is, when the angle is zero, the magnetic fields due to the rotor and the a-phase winding align. This definition is used regardless of your block setting for rotor angle definition. The first value is zero, and the last value is $2 π / N$ , where N is the number of permanent magnet pole pairs.

Dependencies

To enable this parameter, set Back EMF profile to Tabulated - specify flux partial derivative with respect to rotor angle or Tabulated - specify rotor-induced back emf as a function of rotor angle.

`Rotor speed used for back emf measurement` — Rotor speed used for back emf measurement
`600` `rpm` (default)

Specify the rotor speed corresponding to the maximum rotor-induced back emf.

Dependencies

To enable this parameter, set Back EMF profile to Perfect 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
`6` (default)

Number of permanent magnet pole pairs on the rotor.

`Rotor angle definition` — Reference point for the rotor angle measurement
`Angle between the a-phase magnetic axis and the d-axis` (default) | `Angle between the a-phase magnetic axis and the q-axis`

Reference point for the rotor angle measurement. The default value is Angle between the a-phase magnetic axis and the d-axis. This definition is shown in the Motor Construction figure. When you select this value, the rotor and a-phase fluxes are aligned when the rotor angle is zero.

The other value you can choose for this parameter is Angle between the a-phase magnetic axis and the q-axis. When you select this value, the a-phase current generates maximum torque when the rotor angle is zero.

Stator

`Modeling fidelity` — Modeling fidelity
`Constant Ld and Lq` (default) | `Tabulated Ld and Lq`

Select the modeling fidelity:

Constant Ld and Lq — Ld and Lq values are constant and defined by their respective parameters.
Tabulated Ld and Lq — Ld and Lq values are computed online from DQ currents look-up tables as follows:

$L_{d} = f_{1} (i_{d}, i_{q})$

$L_{d} = f_{2} (i_{d}, i_{q})$

Dependencies

To enable this parameter, set Stator parameterization to Specify Ld, Lq, and L0.

`Stator parameterization` — Stator parameterization
`Specify Ld, Lq, and L0` (default) | `Specify Ls, Lm, and Ms`

Choose Specify Ld, Lq, and L0 or Specify Ls, Lm, and Ms.

`Stator d-axis inductance, Ld` — Stator d-axis inductance
`0.00022` `H` (default)

D-axis inductance.

Dependencies

To enable this parameter, set Stator parameterization to Specify Ld, Lq, and L0 and Modeling fidelity to Constant Ld and Lq.

`Stator q-axis inductance, Lq` — Stator q-axis inductance
`0.00022` `H` (default)

Q-axis inductance.

Dependencies

To enable this parameter, set Stator parameterization to Specify Ld, Lq, and L0 and Modeling fidelity to Constant Ld and Lq.

`Direct-axis current vector, iD` — Direct-axis current vector
`[-200, 0, 200]` `A` (default)

Direct-axis current vector, iD.

Dependencies

To enable this parameter, set Stator parameterization to Specify Ld, Lq, and L0 and Modeling fidelity to Tabulated Ld and Lq.

`Quadrature-axis current vector, iQ` — Quadrature-axis current vector
`[-200, 0, 200]` `A` (default)

Quadrature-axis current vector, iQ.

Dependencies

To enable this parameter, set Stator parameterization to Specify Ld, Lq, and L0 and Modeling fidelity to Tabulated Ld and Lq.

`Ld matrix, Ld(id,iq)` — Ld matrix
`0.00022 * ones(3, 3)` `H` (default)

Ld matrix.

Dependencies

To enable this parameter, set Stator parameterization to Specify Ld, Lq, and L0 and Modeling fidelity to Tabulated Ld and Lq.

`Lq matrix, Lq(id,iq)` — Lq matrix
`0.00022 * ones(3, 3)` `H` (default)

Lq matrix.

Dependencies

To enable this parameter, set Stator parameterization to Specify Ld, Lq, and L0 and Modeling fidelity to Tabulated Ld and Lq.

`Stator zero-sequence inductance, L0` — Stator zero-sequence inductance
`0.00016` `H` (default)

Zero-sequence inductance.

Dependencies

To enable this parameter either:

Set Winding Type to Wye-wound, Zero sequence to Include, and Stator parameterization to Specify Ld, Lq, and L0.
Set Winding Type to Delta-wound and Stator parameterization to Specify Ld, Lq, and L0.

`Stator self-inductance per phase, Ls` — Stator self-inductance per phase
`0.0002` `H` (default)

Average self-inductance of each of the three stator windings.

Dependencies

To enable this parameter, set Stator parameterization to Specify Ls, Lm, and Ms.

`Stator inductance fluctuation, Lm` — Stator inductance fluctuation
`0` `H` (default)

Fluctuation in self-inductance and mutual inductance of the stator windings with rotor angle.

Dependencies

To enable this parameter, set Stator parameterization to Specify Ls, Lm, and Ms.

`Stator mutual inductance, Ms` — Stator mutual inductance
`0.00002` `H` (default)

Average mutual inductance between the stator windings.

Dependencies

To enable this parameter, set Stator parameterization to Specify Ls, Lm, and Ms.

`Stator resistance per phase, Rs` — Stator resistance per phase
`0.013` `Ohm` (default)

Resistance of each of the stator windings.

`Zero sequence` — Zero sequence option
`Include` (default) | `Exclude`

Option to include or exclude zero-sequence terms.

Include — Include zero-sequence terms. To prioritize model fidelity, use this default setting. Using this option:
- Results in an error for simulations that use the Partitioning solver. For more information, see Increase Simulation Speed Using the Partitioning Solver.
- Exposes a zero-sequence parameter in the Impedances settings.
Exclude — Exclude zero-sequence terms. To prioritize simulation speed for desktop simulation or real-time deployment, select this option.

Dependencies

This parameter is visible only when you set the Winding Type parameter to Wye-wound.

Iron Losses

`Iron-loss` — Enable Iron losses computation
`None` (default) | `Empirical`

Specify the iron losses computational model.

`Open-circuit iron losses, [P_hysteresis P_eddy P_excess]` — Open-circuit iron losses
`[0, 0, 0]` `W` (default)

Row vector, of length 3, of the open-circuit iron losses due to hysteresis, Eddy, and excess losses, respectively, at the frequency specified by Electrical frequency at which losses determined.

Dependencies

To enable this parameter, set Iron-loss to Empirical.

`Short-circuit iron losses, [P_hysteresis P_eddy P_excess]` — Short-circuit iron losses
`[0, 0, 0]` `W` (default)

Row vector, of length 3, of the short-circuit iron losses due to hysteresis, Eddy, and excess losses, respectively, at the frequency specified by Electrical frequency at which losses determined.

Dependencies

To enable this parameter, set Iron-loss to Empirical.

`Electrical frequency at which losses determined` — Electrical frequency at which losses determined
`60` `Hz` (default)

Electrical frequency at which the open-circuit and short-circuit iron losses were measured.

Dependencies

To enable this parameter, set Iron-loss to Empirical.

`Short-circuit RMS current for short-circuit iron losses` — Short-circuit RMS current for short-circuit iron losses
`95` `A` (default)

The resulting short-circuit RMS phase current when measuring the short-circuit losses.

Dependencies

To enable this parameter, set Iron-loss to Empirical.

Mechanical

`Rotor inertia` — Rotor inertia
`0.01` `kg*m^2` (default)

Inertia of the rotor attached to mechanical translational port R. The value can be zero.

`Rotor damping` — Rotor damping
`0` `N*m/(rad/s)` (default)

Rotary damping.

Temperature Dependence

These parameters appear only for blocks with exposed thermal ports. For more information, see Thermal Ports.

`Measurement temperature` — Measurement temperature
`298.15` `K` (default)

The temperature for which motor parameters are quoted.

`Resistance temperature coefficient` — Resistance temperature coefficient
`3.93e-3` `1/K` (default)

Coefficient α in the equation relating resistance to temperature, as described in Thermal Model for Actuator Blocks. The default value is for copper.

`Permanent magnet flux temperature coefficient` — Permanent magnet flux temperature coefficient
`-0.001` `1/K` (default)

The fractional rate of change of permanent magnet flux density with temperature. It is used to linearly reduce the torque and the induced back EMF as temperature rises.

Thermal Port

These parameters appear only for blocks with exposed thermal ports. For more information, see Thermal Ports.

`Thermal mass for each stator winding` — Thermal mass for each stator winding
`100` `J/K` (default)

The thermal mass value for the A, B, and C windings. The thermal mass is the energy required to raise the temperature by one degree.

`Rotor thermal mass` — Rotor thermal mass
`200` `J/K` (default)

The thermal mass of the rotor, that is, the energy required to raise the temperature of the rotor by one degree.

`Percentage of main flux path iron losses associated with the rotor` — Percentage of main flux path iron losses associated with the rotor
`90` (default)

The percentage of the main flux path iron losses associated with the magnetic path through the rotor. It determines how much of the iron loss heating is attributed to the rotor thermal port HR, and how much is attributed to the three winding thermal ports HA, HB, and HC.

Dependencies

To enable this parameter, set Iron-loss to Empirical.

`Percentage of cross-tooth flux path iron losses associated with the rotor` — Percentage of cross-tooth flux path iron losses associated with the rotor
`30` (default)

The percentage of the cross-tooth flux path iron losses associated with the magnetic path through the rotor. It determines how much of the iron loss heating is attributed to the rotor thermal port HR, and how much is attributed to the three winding thermal ports HA, HB, and HC.

Dependencies

To enable this parameter, set Iron-loss to Empirical.

Model Examples

BLDC Speed Control

Control the rotor speed in a BLDC based electrical drive. An ideal torque source provides the load. The Control subsystem uses a PI-based cascade control structure with an outer speed control loop and an inner dc-link voltage control loop. The dc-link voltage is adjusted through a DC-DC buck converter. The BLDC is fed by a controlled three-phase inverter. The gate signals for the inverter are obtained from hall signals. The simulation uses speed steps. The Scopes subsystem contains scopes that allow you to see the simulation results.

BLDC Position Control with Thermal Model

Control the rotor angle in a BLDC based electrical drive. The BLDC includes a thermal model and empirical iron losses. An ideal torque source provides the load. The Control subsystem uses a PI-based cascade control structure with three control loops: an outer position control loop, a speed control loop, and an inner current control loop. The BLDC is fed by a controlled three-phase inverter. The gate signals for the inverter are obtained from hall signals. The simulation uses step references. The initial temperature of the stator windings and rotor is set to 25 degrees Celsius. Ambient temperature is 27 degrees Celsius. The Scopes subsystem contains scopes that allow you to see the simulation results.

References

[1] Kundur, P. Power System Stability and Control. New York, NY: McGraw Hill, 1993.

[2] Anderson, P. M. Analysis of Faulted Power Systems. 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.” IEEE Electric Machines and Drives Conference. May 2009.

Documentation

BLDC

Description

Motor Construction

Trapezoidal Rate of Change of Flux

Electrical Defining Equations

Simplified Equations

Calculating Iron Losses

Thermal Ports

Variables

Ports

Conserving

~ — Three-phase port electrical

n — Neutral phase electrical

R — Motor rotor mechanical

C — Motor case mechanical

HA — Winding A thermal port thermal

HB — Winding B thermal port thermal

HC — Winding C thermal port thermal

HR — Rotor thermal port thermal

Parameters

Rotor

Winding type — Windings configuration Wye-wound (default) | Delta-wound

Maximum permanent magnet flux linkage — Maximum permanent magnet flux linkage 0.03 Wb (default)

Dependencies

Rotor angle over which back emf is constant — Rotor angle over which back emf is constant pi / 12 rad (default)

Dependencies

Maximum rotor-induced back emf — Maximum rotor-induced back emf 9.6 V (default)

Dependencies

Rotor-induced back emf — Rotor-induced back emf [0, -9.6, -9.6, 9.6, 9.6, 0] V (default)

Dependencies

Flux linkage partial derivative with respect to rotor angle — Flux linkage partial derivative with respect to rotor angle [0, -.1528, -.1528, .1528, .1528, 0] Wb/rad (default)

Corresponding rotor angles — Corresponding rotor angles [0, 7.5, 22.5, 37.5, 52.5, 60] deg (default)

Dependencies

Rotor speed used for back emf measurement — Rotor speed used for back emf measurement 600 rpm (default)

Dependencies

Number of pole pairs — Number of pole pairs 6 (default)

Rotor angle definition — Reference point for the rotor angle measurement Angle between the a-phase magnetic axis and the d-axis (default) | Angle between the a-phase magnetic axis and the q-axis

Stator

Modeling fidelity — Modeling fidelity Constant Ld and Lq (default) | Tabulated Ld and Lq

Dependencies

Stator parameterization — Stator parameterization Specify Ld, Lq, and L0 (default) | Specify Ls, Lm, and Ms

Stator d-axis inductance, Ld — Stator d-axis inductance 0.00022 H (default)

Dependencies

Stator q-axis inductance, Lq — Stator q-axis inductance 0.00022 H (default)

Dependencies

Direct-axis current vector, iD — Direct-axis current vector [-200, 0, 200] A (default)

Dependencies

Quadrature-axis current vector, iQ — Quadrature-axis current vector [-200, 0, 200] A (default)

Dependencies

Ld matrix, Ld(id,iq) — Ld matrix 0.00022 * ones(3, 3) H (default)

Dependencies

Lq matrix, Lq(id,iq) — Lq matrix 0.00022 * ones(3, 3) H (default)

Dependencies

Stator zero-sequence inductance, L0 — Stator zero-sequence inductance 0.00016 H (default)

Dependencies

Stator self-inductance per phase, Ls — Stator self-inductance per phase 0.0002 H (default)

Dependencies

Stator inductance fluctuation, Lm — Stator inductance fluctuation 0 H (default)

Dependencies

Stator mutual inductance, Ms — Stator mutual inductance 0.00002 H (default)

Dependencies

Stator resistance per phase, Rs — Stator resistance per phase 0.013 Ohm (default)

Zero sequence — Zero sequence option Include (default) | Exclude

Dependencies

Iron Losses

Iron-loss — Enable Iron losses computation None (default) | Empirical

Open-circuit iron losses, [P_hysteresis P_eddy P_excess] — Open-circuit iron losses [0, 0, 0] W (default)

Dependencies

Short-circuit iron losses, [P_hysteresis P_eddy P_excess] — Short-circuit iron losses [0, 0, 0] W (default)

Dependencies

Electrical frequency at which losses determined — Electrical frequency at which losses determined 60 Hz (default)

Dependencies

Short-circuit RMS current for short-circuit iron losses — Short-circuit RMS current for short-circuit iron losses 95 A (default)

Dependencies

Mechanical

Rotor inertia — Rotor inertia 0.01 kg*m^2 (default)

Rotor damping — Rotor damping 0 N*m/(rad/s) (default)

Temperature Dependence

Measurement temperature — Measurement temperature 298.15 K (default)

`~` — Three-phase port
electrical

`n` — Neutral phase
electrical

`R` — Motor rotor
mechanical

`C` — Motor case
mechanical

`HA` — Winding A thermal port
thermal

`HB` — Winding B thermal port
thermal

`HC` — Winding C thermal port
thermal

`HR` — Rotor thermal port
thermal

`Winding type` — Windings configuration
`Wye-wound` (default) | `Delta-wound`

`Maximum permanent magnet flux linkage` — Maximum permanent magnet flux linkage
`0.03` `Wb` (default)

`Rotor angle over which back emf is constant` — Rotor angle over which back emf is constant
`pi / 12` `rad` (default)

`Maximum rotor-induced back emf` — Maximum rotor-induced back emf
`9.6` `V` (default)

`Rotor-induced back emf` — Rotor-induced back emf
`[0, -9.6, -9.6, 9.6, 9.6, 0]` `V` (default)

`Flux linkage partial derivative with respect to rotor angle` — Flux linkage partial derivative with respect to rotor angle
`[0, -.1528, -.1528, .1528, .1528, 0]` `Wb/rad` (default)

`Corresponding rotor angles` — Corresponding rotor angles
`[0, 7.5, 22.5, 37.5, 52.5, 60]` `deg` (default)

`Rotor speed used for back emf measurement` — Rotor speed used for back emf measurement
`600` `rpm` (default)

`Number of pole pairs` — Number of pole pairs
`6` (default)

`Rotor angle definition` — Reference point for the rotor angle measurement
`Angle between the a-phase magnetic axis and the d-axis` (default) | `Angle between the a-phase magnetic axis and the q-axis`

`Modeling fidelity` — Modeling fidelity
`Constant Ld and Lq` (default) | `Tabulated Ld and Lq`

`Stator parameterization` — Stator parameterization
`Specify Ld, Lq, and L0` (default) | `Specify Ls, Lm, and Ms`

`Stator d-axis inductance, Ld` — Stator d-axis inductance
`0.00022` `H` (default)

`Stator q-axis inductance, Lq` — Stator q-axis inductance
`0.00022` `H` (default)

`Direct-axis current vector, iD` — Direct-axis current vector
`[-200, 0, 200]` `A` (default)

`Quadrature-axis current vector, iQ` — Quadrature-axis current vector
`[-200, 0, 200]` `A` (default)

`Ld matrix, Ld(id,iq)` — Ld matrix
`0.00022 * ones(3, 3)` `H` (default)

`Lq matrix, Lq(id,iq)` — Lq matrix
`0.00022 * ones(3, 3)` `H` (default)

`Stator zero-sequence inductance, L0` — Stator zero-sequence inductance
`0.00016` `H` (default)

`Stator self-inductance per phase, Ls` — Stator self-inductance per phase
`0.0002` `H` (default)

`Stator inductance fluctuation, Lm` — Stator inductance fluctuation
`0` `H` (default)

`Stator mutual inductance, Ms` — Stator mutual inductance
`0.00002` `H` (default)

`Stator resistance per phase, Rs` — Stator resistance per phase
`0.013` `Ohm` (default)

`Zero sequence` — Zero sequence option
`Include` (default) | `Exclude`

`Iron-loss` — Enable Iron losses computation
`None` (default) | `Empirical`

`Open-circuit iron losses, [P_hysteresis P_eddy P_excess]` — Open-circuit iron losses
`[0, 0, 0]` `W` (default)

`Short-circuit iron losses, [P_hysteresis P_eddy P_excess]` — Short-circuit iron losses
`[0, 0, 0]` `W` (default)

`Electrical frequency at which losses determined` — Electrical frequency at which losses determined
`60` `Hz` (default)

`Short-circuit RMS current for short-circuit iron losses` — Short-circuit RMS current for short-circuit iron losses
`95` `A` (default)

`Rotor inertia` — Rotor inertia
`0.01` `kg*m^2` (default)

`Rotor damping` — Rotor damping
`0` `N*m/(rad/s)` (default)

`Measurement temperature` — Measurement temperature
`298.15` `K` (default)

`Resistance temperature coefficient` — Resistance temperature coefficient
`3.93e-3` `1/K` (default)

`Permanent magnet flux temperature coefficient` — Permanent magnet flux temperature coefficient
`-0.001` `1/K` (default)

`Thermal mass for each stator winding` — Thermal mass for each stator winding
`100` `J/K` (default)

`Rotor thermal mass` — Rotor thermal mass
`200` `J/K` (default)

`Percentage of main flux path iron losses associated with the rotor` — Percentage of main flux path iron losses associated with the rotor
`90` (default)

`Percentage of cross-tooth flux path iron losses associated with the rotor` — Percentage of cross-tooth flux path iron losses associated with the rotor
`30` (default)

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.