Flux Observer

Compute electrical position, magnetic flux, and electrical torque of rotor

Since R2020a

• 

Libraries:
Motor Control Blockset / Sensorless Estimators

Description

The Flux Observer block computes the electrical position, magnetic flux, and electrical torque of a PMSM or an induction motor by using the per unit voltage and current values along the α- and β-axes in the stationary αβ reference frame.

The block also accepts 16-bit fixed-point data type inputs. To perform mathematical operations on 16-bit fixed-point data type signals, the block uses optimized implementation of equations for maintaining the best possible precision.

Equations

These equations describe how the block computes the electrical position, magnetic flux, and electrical torque for a PMSM.

$${\psi }_{\alpha }= {{\displaystyle \int }}^{\text{}}(V_{\alpha }-I_{\alpha }R)\text{dt}- \left(L_s\cdot I_{\alpha }\right)$$

${\psi }_{\beta }= {{\displaystyle \int }}^{\text{}}(V_{\beta }-I_{\beta }R)\text{dt}- \left(L_s\cdot I_{\beta }\right)$

If $$(V_{\alpha }-I_{\alpha }R)=v_1(t)$$ and $(V_{\beta }-I_{\beta }R)=v_2(t)$

Then, the following Laplace transforms represent the integral terms available in Ψ_α and Ψ_β:

$$\begin{array}{l}L\left({{\displaystyle \int }}^{\text{}}{\text{v}}_1\text{(t)dt}\right)=\frac{1}{s}L\left({\text{v}}_1\text{(t)}\right)=\frac{1}{s}{v}_1(s)\\ L\left({{\displaystyle \int }}^{\text{}}{\text{v}}_2\text{(t)dt}\right)=\frac{1}{s}L\left({\text{v}}_2\text{(t)}\right)=\frac{1}{s}{v}_2(s)\end{array}$$

The block uses low-pass filter (LPF) based integrator to compute these integral terms.

If the input electrical speed of the motor (ω_e) is much greater than the speed corresponding to the cut-off frequency of LPF-based integrator (ω_c), then

$$\begin{array}{l}L\left({{\displaystyle \int }}^{\text{}}{\text{v}}_1\text{(t)dt}\right)=\frac{1}{\left(s+{\omega }_c\right)}{v}_1(s)\\ L\left({{\displaystyle \int }}^{\text{}}{\text{v}}_2\text{(t)dt}\right)=\frac{1}{\left(s+{\omega }_c\right)}{v}_2(s)\end{array}$$

The LPF-based integrator that the block uses eliminates any DC offset that the block may have. For example, consider an input function X(s)=k/s. Using the preceding transfer function of $$G(s)=\frac{1}{\left(s+{\omega }_c\right)}$$, we obtain:

$$\begin{array}{l}Y(s)=G(s)X(s)=\frac{k}{s\left(s+{\omega }_c\right)}=\frac{k}{{\omega }_c}\left(\frac{1}{s}\right)-\frac{k}{{\omega }_c}\left(\frac{1}{s+{\omega }_c}\right)\\ Y(t)=\frac{k}{{\omega }_c}u(t)-\frac{k}{{\omega }_c}\exp \left(-{\omega }_{c}t\right)u(t)\end{array}$$

The preceding equation shows that the DC component in the resulting output Y(t) reduces exponentially with time. Therefore, the LPF-based integrator enables the block to eliminate any DC component available in the system.

$$\psi = \sqrt{{\psi }_{\alpha }^2+{\psi }_{\beta }^2}$$

$T_{\text{e}}=\frac{3}{2}P({\psi }_{\alpha }{I}_{\beta }-{\psi }_{\beta }{I}_{\alpha })$

${\theta }_{\text{e}}= {\tan }^{-1}\frac{{\psi }_{\beta }}{{\psi }_{\alpha }}$

These equations describe how the block computes the rotor electrical position, rotor magnetic flux, and electrical torque for an induction motor.

$${\psi }_{\alpha }=\frac{L_r}{L_m} \left({{\displaystyle \int }}^{\text{}}(V_{\alpha }-I_{\alpha }R)\text{dt}- \sigma L_s{I}_{\alpha }\right)$$

$${\psi }_{\beta }=\frac{L_r}{L_m} \left({{\displaystyle \int }}^{\text{}}(V_{\beta }-I_{\beta }R)\text{dt}- \sigma L_s{I}_{\beta }\right)$$

As explained previously, the LPF-based integrator enables the block to eliminate any DC component available in Ψ_α and Ψ_β.

$$\sigma =1-\frac{L_m^2}{L_r\cdot L_s}$$

$$\psi = \sqrt{{\psi }_{\alpha }^2+{\psi }_{\beta }^2}$$

$T_{\text{e}}=\frac{3}{2}\cdot P\cdot \frac{L_m}{L_r}({\psi }_{\alpha }{I}_{\beta }-{\psi }_{\beta }{I}_{\alpha })$

${\theta }_{\text{e}}= {\tan }^{-1}\frac{{\psi }_{\beta }}{{\psi }_{\alpha }}$

where:

• $V_{\alpha }$ and $V_{\beta }$ are the α-axis and β-axis voltages (volts).

• $I_{\alpha }$ and $I_{\beta }$ are the α-axis and β-axis current (amperes).

• $$R$$ is the stator resistance of the motor (ohms).

• $$L_s$$ is the stator inductance of the motor (henry).

• $$L_r$$ is the rotor inductance of the motor (henry).

• $$L_m$$ is the magnetizing inductance of the motor (henry).

• $\sigma$ is the total leakage factor of the induction motor.

• $$P$$ is the number of motor pole pairs.

• $$\psi$$ is the rotor magnetic flux (weber).

• ${\psi }_{\alpha }$ and ${\psi }_{\beta }$ are the rotor magnetic fluxes along the α- and β-axes (weber).

• $$T_e$$ is the electrical torque of the rotor (Nm).

• ${\theta }_e$ is the electrical position of the rotor (radians).

Examples

Sensorless Field-Oriented Control of PMSM

Implements the field-oriented control (FOC) technique to control the speed of a three-phase permanent magnet synchronous motor (PMSM). For details about FOC, see Field-Oriented Control (FOC).

Open Model

Sensorless Field-Oriented Control of Induction Motor

Uses sensorless position estimation to implement the field-oriented control (FOC) technique to control the speed of a three-phase AC induction motor (ACIM). For details about FOC, see Field-Oriented Control (FOC).

Open Model

Ports

Input

V_α — α-axis voltage

scalar

Voltage component along the α-axis in the stationary αβ reference frame.

Data Types: single | double | fixed point

V_β — β-axis voltage

scalar

Voltage component along the β-axis in the stationary αβ reference frame.

Data Types: single | double | fixed point

I_α — α-axis current

scalar

Current along the α-axis in the stationary αβ reference frame.

Data Types: single | double | fixed point

I_β — β-axis current

scalar

Current along the β-axis in the stationary αβ reference frame.

Data Types: single | double | fixed point

Rst — Reset block

scalar

The pulse (true value) that resets the block algorithm.

Data Types: single | double | fixed point

Output

θ_e — Electrical position of motor

scalar

The electrical position of the rotor as estimated by the block.

Dependencies

To enable this port, set Block output to Position.

Data Types: single | double | fixed point

Ψ — Rotor flux of motor

scalar

The magnetic flux of the rotor as estimated by the block.

Dependencies

To enable this port, set Block output to Flux.

Data Types: single | double | fixed point

T_e — Electrical torque of motor

scalar

The electrical torque of the rotor as estimated by the block.

Dependencies

To enable this port, set Block output to Torque.

Data Types: single | double | fixed point

Parameters

Parameters

Motor selection — Type of motor

PMSM (default) | ACIM

Select the type of motor from the types that the block supports.

Support 16-bit fixedpoint datatype — Support the data type of 16-bit fixed-point

on (default) | off

To process 16-bit fixed-point inputs and perform computation using 16-bit fixed-point data type, select this parameter. To use a data type other than 16-bit fixed point, clear this parameter.

If you select this parameter:

• The block sets the Input unit parameter to Per-unit because you cannot use the 16-bit data type with SI units.

• The block selects the Position option in the Block output parameter because it cannot generate flux or torque output using the 16-bit data type.

Input unit — Unit of voltage and current inputs

SI unit (default) | Per-unit

Select the unit of the α- and β-axes voltage and current input values.

Base voltage (V) — Nominal voltage corresponding to one per unit

13.86 (default) | scalar

The maximum phase voltage applied to the PMSM. For details, see Per-Unit System.

Dependencies

To enable this parameter, set Input unit to Per-unit.

Base current (A) — Nominal current corresponding to one per unit

19.3 (default) | scalar

The maximum measurable current supplied to the PMSM. For details, see Per-Unit System.

Dependencies

To enable this parameter, set Input unit to Per-unit.

Block output — Select block outputs

Position (default) | Flux | Torque

Select one or more quantities that you want the block to compute and display in the block output.

Note

You must select at least one value. The block displays an error message if you click Ok or Apply without selecting any value.

Pole pairs — Number of pole pairs available in motor

4 (default) | scalar

Number of pole pairs available in the motor.

Dependencies

To enable this parameter, set Block output to Torque.

Stator resistance (ohm) — Stator winding resistance

0.36 (default) | scalar

Stator phase winding resistance of the motor in ohms.

Stator d-axis inductance (H) — Stator winding inductance along d-axis

0.2e-3 (default) | scalar

Stator winding inductance of the motor along the d-axis in henries.

Dependencies

To enable this parameter, set Motor selection to PMSM.

Stator leakage inductance (H) — Leakage inductance of stator winding

0.0068 (default) | scalar

Leakage inductance of the induction motor stator winding in henries.

Dependencies

To enable this parameter, set Motor selection to ACIM.

Rotor leakage inductance (H) — Leakage inductance of rotor winding

0.0068 (default) | scalar

Leakage inductance of the induction motor rotor winding in henries.

Dependencies

To enable this parameter, set Motor selection to ACIM.

Magnetizing inductance (H) — Magnetizing inductance of induction motor

0.0300 (default) | scalar

Magnetizing inductance of the induction motor in henries.

Dependencies

To enable this parameter, set Motor selection to ACIM.

Cutoff frequency (Hz) — Cutoff frequency of internal highpass filter

3.1863 (default) | scalar

Cutoff frequency of the internal highpass filter (that filters noise) in hertz.

The Flux Observer block uses an internal first-order IIR highpass filter. Set the Cutoff frequency (Hz) for this filter to a value that is lower than the lowest frequency corresponding to the minimum speed of the motor. For example, you can enter a value that is one-tenth of the lowest electrical frequency of the stator voltages and currents. However, you can adjust this value to determine a more accurate cutoff frequency that generates the desired block output.

Discrete step size (s) — Interval between consecutive block executions

50e-6 (default) | scalar

The fixed time interval in seconds between two consecutive instances of block execution.

Datatypes

Position unit — Unit of electrical position output

Radians (default) | Degrees | Per-unit

Unit of the electrical position output.

Dependencies

To enable this parameter, set Block output to Position.

Position datatype — Data type of electrical position output

single (default) | double | fixed point

Data type of the electrical position output.

Dependencies

To enable this parameter, set Block output to Position.

Flux unit — Unit of magnetic flux output

Weber (default) | Per-unit

Unit of the magnetic flux output.

Dependencies

To enable this parameter, set Block output to Flux.

Flux datatype — Data type of magnetic flux output

single (default) | double | fixed point

Data type of the magnetic flux output.

Dependencies

To enable this parameter, set Block output to Flux.

Torque unit — Unit of electrical torque output

Nm (default) | Per-unit

Unit of the electrical torque output.

Dependencies

To enable this parameter, set Block output to Torque.

Torque datatype — Data type of electrical torque output

single (default) | double | fixed point

Data type of the electrical torque output.

Dependencies

To enable this parameter, set Block output to Torque.

References

[1] A. Podder and D. Pandit, "Study of Sensorless Field-Oriented Control of SPMSM Using Rotor Flux Observer & Disturbance Observer Based Discrete Sliding Mode Observer," 2021 IEEE 22nd Workshop on Control and Modelling of Power Electronics (COMPEL), 2021, pp. 1-8. (doi: 10.1109/COMPEL52922.2021.9645939)

[2] O. Sandre-Hernandez, J. J. Rangel-Magdaleno and R. Morales-Caporal, "Simulink-HDL cosimulation of direct torque control of a PM synchronous machine based FPGA," 2014 11th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE), Campeche, 2014, pp. 1-6. (doi: 10.1109/ICEEE.2014.6978298)

[3] Y. Inoue, S. Morimoto and M. Sanada, "Control method suitable for direct torque control based motor drive system satisfying voltage and current limitations," The 2010 International Power Electronics Conference - ECCE ASIA -, Sapporo, 2010, pp. 3000-3006. (doi: 10.1109/IPEC.2010.5543698)

Extended Capabilities

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

HDL Code Generation

Generate VHDL, Verilog and SystemVerilog code for FPGA and ASIC designs using HDL Coder™.

HDL Coder™ provides additional configuration options that affect HDL implementation and synthesized logic.

HDL Architecture

This block has one default HDL architecture.

HDL Block Properties

ConstrainedOutputPipeline	Number of registers to place at the outputs by moving existing delays within your design. Distributed pipelining does not redistribute these registers. The default is 0. For more details, see ConstrainedOutputPipeline (HDL Coder).
InputPipeline	Number of input pipeline stages to insert in the generated code. Distributed pipelining and constrained output pipelining can move these registers. The default is 0. For more details, see InputPipeline (HDL Coder).
OutputPipeline	Number of output pipeline stages to insert in the generated code. Distributed pipelining and constrained output pipelining can move these registers. The default is 0. For more details, see OutputPipeline (HDL Coder).
SharingFactor	Number of functionally equivalent resources to map to a single shared resource. The default is 0. See also Resource Sharing (HDL Coder).

Restrictions

• The block supports HDL code generation only when you set the Motor selection parameter to PMSM.

• The block does not support HDL code generation when:

  • Input or output signals have mixed datatypes.

  • The parameter Support 16-bit fixedpoint datatype is selected.

  • Input or output signals have fixed point datatype of size less than 16 bits.

• The block needs native floating point library to support fixed point datatype with HDL code generation.

• If you run third-party (3p) simulation, the block may show a minor mismatch (of the order 1e-7) in the simulation results because of an internal trigonometric function that uses the floating point datatype.

Fixed-Point Conversion
Design and simulate fixed-point systems using Fixed-Point Designer™.

Version History

Introduced in R2020a