Flux-Based PM Controller

Controller for a flux-based permanent magnet synchronous motor

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Libraries:
Powertrain Blockset / Propulsion / Electric Motor Controllers

Description

The Flux Based PM Controller block implements a flux-based, field-oriented controller for an interior permanent magnet synchronous motor (PMSM) with an optional outer-loop speed controller. The internal torque control implements strategies for achieving maximum torque per ampere (MTPA) and weakening the magnetic flux. You can specify either the speed or torque control type.

The Flux Based PM Controller implements equations for speed control, torque determination, regulators, transforms, and motors.

The figure illustrates the information flow in the block.

The block implements equations using these variables.

ω	Rotor speed
ω*	Rotor speed command
T*	Torque command
id i*d	d-axis current d-axis current command
iq i*q	q-axis current q-axis current command
vd, v*d	d-axis voltage d-axis voltage command
vq v*q	q-axis voltage q-axis voltage command
va, vb, vc	Stator phase a, b, c voltages
ia, ib, ic	Stator phase a, b, c currents

Speed Controller

To implement the speed controller, select the Control Type parameter Speed Control. If you select the Control Type parameter Torque Control, the block does not implement the speed controller.

The speed controller determines the torque command by implementing a state filter, and calculating the feedforward and feedback commands. If you do not implement the speed controller, input a torque command to the Flux Based PM Controller block.

State Filter

The state filter is a low-pass filter that generates the acceleration command based on the speed command. The discrete form of characteristic equation is given by:

$z+K_{sf}{T}_{sm}-1$

The filter calculates the gain using this equation.

$K_{sf}=\frac{1-\exp \left(-T_{sm}2\pi E{V}_{sf}\right)}{T_{sm}}$

The equations use these variables.

EVsf	Bandwidth of the speed command filter
Tsm	Motion controller sample time
Ksf	Speed regulator time constant

State Feedback

To generate the state feedback torque, the block uses the filtered speed error signal from the state filter. To filter the speed, the block uses a proportional integral (PI) controller.

$$T_{cmd}=K{p}_{\omega }({\omega }_m^{*}-{\omega }_m)+K{i}_{\omega }\frac{z{T}_{sm}}{z-1}({\omega }_m^{*}-{\omega }_m)$$

The equations use these variables.

ωm	Rotor speed
ω*m	Rotor speed command
Tcmd	Torque command
Kpω	Speed regulator proportional gain
Kiω	Speed regulator integral gain
Tsm	Speed regulator sample rate

Command Feedforward

To generate the state feedforward torque, the block uses the filtered speed and acceleration from the state filter. Also, the feedforward torque calculation uses the inertia, viscous damping, and static friction. To achieve zero tracking error, the torque command is the sum of the feedforward and feedback torque commands.

The feedforward torque command uses this equation.

$$T_{cmd\_ff}= J_p{\dot{\omega }}_m+F_v{\omega }_m+F_{s }\frac{{\omega }_m}{\left|{\omega }_m\right|}$$

where:

Jp	Rotor inertia
Tcmd_ff	Torque command feedforward
Fs	Static friction torque constant
Fv	Viscous friction torque constant
Fs	Static friction torque constant
ωm	Rotor speed

Current Command

The block uses lookup tables to determine the d-axis and q-axis current commands. The lookup tables are functions of mechanical speed and torque. To determine the lookup tables, you can use an external finite element analysis (FEA) models or dynamometer test results.

$$\begin{array}{l}{i}_{dref}=f(\left|{\omega }_m\right|,\left|T_{ref}\right|)\\ i_{qref}=sign(T_{ref})\ast f(\left|{\omega }_m\right|,\left|T_{ref}\right|)\end{array}$$

The equations use these variables.

ωm	Rotor speed
Tref	Torque command
idref, iqref	d- and q-axis reference current, respectively

Voltage Command

The block uses these equations to calculate the voltage in the motor reference frame.

$$\begin{array}{l}{v}_d=\frac{d{\psi }_d}{dt}+R_s{i}_d-{\omega }_e{\psi }_q\\ v_q=\frac{d{\psi }_q}{dt}+R_s{i}_q+{\omega }_e{\psi }_d\end{array}$$

$$\begin{array}{l}\frac{d{\psi }_d}{dt}+R_s{i}_d=K{p}_d(i_d^{*}-i_d)+K{i}_d\frac{z{T}_{st}}{z-1}(i_d^{*}-i_d)\\ \frac{d{\psi }_q}{dt}+R_s{i}_q=K{p}_q(i_q^{*}-i_q)+K{i}_q\frac{z{T}_{st}}{z-1}(i_q^{*}-i_q)\\ \\ v_d=K{p}_i(i_d^{*}-i_d)+K{i}_d\frac{z{T}_{st}}{z-1}(i_d^{*}-i_d)+{\omega }_e{\psi }_q\\ v_q=K{p}_i(i_q^{*}-i_q)+K{i}_q\frac{z{T}_{st}}{z-1}(i_q^{*}-i_q)-{\omega }_e{\psi }_d\\ \\ {\psi }_q=f(i_d,i_q)\\ {\psi }_d=f(i_d,i_q)\end{array}$$

The equations use these variables.

ωm	Rotor mechanical speed
ωe	Rotor electrical speed
Rs, Rr	Resistance of the stator and rotor windings, respectively
iq, id	q- and d-axis current, respectively
vq, vd	q- and d-axis voltage, respectively
Ψq, Ψd	q- and d-axis magnet flux, respectively
Tst	Current regulator sample rate
Kid , Kiq	d- and q- axis integral gain, respectively
Kpd , Kpq	d- and q- axis proportional gain, respectively

Transforms

To calculate the voltages and currents in balanced three-phase (a, b) quantities, quadrature two-phase (α, β) quantities, and rotating (d, q) reference frames, the block uses the Clarke and Park Transforms.

In the transform equations.

$\begin{array}{l}{\omega }_e=P{\omega }_m\\ \frac{d{\theta }_e}{dt}= {\omega }_e\end{array}$

Transform	Description	Equations
Clarke	Converts balanced three-phase quantities (a, b) into balanced two-phase quadrature quantities (α, β).	$\begin{array}{l}{x}_{\alpha }= \frac{2}{3}{x}_a- \frac{1}{3}{x}_b -\frac{1}{3}{x}_c\\ x_{\beta }= \frac{\sqrt{3}}{2}{x}_b- \frac{\sqrt{3}}{2}{x}_c\end{array}$
Park	Converts balanced two-phase orthogonal stationary quantities (α, β) into an orthogonal rotating reference frame (d, q).	$\begin{array}{l}{x}_d= x_{\alpha }\cos {\theta }_e+ x_{\beta }\sin {\theta }_e\\ x_q= -x_{\alpha }\sin {\theta }_e+ x_{\beta }\cos {\theta }_e\end{array}$
Inverse Clarke	Converts balanced two-phase quadrature quantities (α, β) into balanced three-phase quantities (a, b).	$\begin{array}{l}{x}_a= x_a\\ x_b= -\frac{1}{2}{x}_{\alpha }+ \frac{\sqrt{3}}{2}{x}_{\beta }\\ x_c= -\frac{1}{2}{x}_{\alpha }- \frac{\sqrt{3}}{2}{x}_{\beta }\end{array}$
Inverse Park	Converts an orthogonal rotating reference frame (d, q) into balanced two-phase orthogonal stationary quantities (α, β).	$\begin{array}{l}{x}_{\alpha }= x_d\cos {\theta }_e- x_q\sin {\theta }_e\\ x_{\beta }= x_d\sin {\theta }_e+ x_q\cos {\theta }_e\end{array}$

The transforms use these variables.

ωm	Rotor speed
P	Rotor pole pairs
ωe	Rotor electrical speed
Θe	Rotor electrical angle
x	Phase current or voltage

Motor

The block uses the phase currents and phase voltages to estimate the DC bus current. Positive current indicates battery discharge. Negative current indicates battery charge.

The block uses these equations.

Load power	$L{d}_{Pwr}=v_a i_a+ v_b i_b+ v_c i_c$
Source power	$Sr{c}_{Pwr}= L{d}_{Pwr}+Pw{r}_{Loss}$
DC bus current	$i_{bus}= \frac{Sr{c}_{Pwr}}{v_{bus}}$
Estimated rotor torque	$$T_e=1.5P[{\psi }_d{i}_q-{\psi }_q{i}_d]$$
Power loss for single efficiency source to load	$$Pw{r}_{Loss}=\frac{100-Eff}{Eff}\cdot L{d}_{Pwr}$$
Power loss for single efficiency load to source	$$Pw{r}_{Loss}=\frac{100-Eff}{100}\cdot \left|L{d}_{Pwr}\right|$$
Power loss for tabulated efficiency	$Pw{r}_{Loss}= f({\omega }_m,MtrTr{q}_{est})$

The equations use these variables.

va, vb, vc	Stator phase a, b, c voltages
vbus	Estimated DC bus voltage
ia, ib, ic	Stator phase a, b, c currents
ibus	Estimated DC bus current
Eff	Overall inverter efficiency
ωm	Rotor mechanical speed
Lq, Ld	q- and d-axis winding inductance, respectively
Ψq, Ψd	q- and d-axis magnet flux, respectively
iq, id	q- and d-axis current, respectively
λ	Permanent magnet flux linkage
P	Rotor pole pairs

Electrical Losses

To specify the electrical losses, on the Electrical Losses tab, for Parameterize losses by, select one of these options.

Setting	Block Implementation
Single efficiency measurement	Electrical loss calculated using a constant value for inverter efficiency.
Tabulated loss data	Electrical loss calculated as a function of motor speeds and load torques.
Tabulated efficiency data	Electrical loss calculated using inverter efficiency that is a function of motor speeds and load torques. Converts the efficiency values you provide into losses and uses the tabulated losses for simulation. Ignores efficiency values you provide for zero speed or zero torque. Losses are assumed zero when either torque or speed is zero. Uses linear interpolation to determine losses. Provide tabulated data for low speeds and low torques, as required, to get the desired level of accuracy for lower power conditions. Does not extrapolate loss values for speed and torque magnitudes that exceed the range of the table.

For best practice, use Tabulated loss data instead of Tabulated efficiency data:

• Efficiency becomes ill defined for zero speed or zero torque.

• You can account for fixed losses that are still present for zero speed or torque.

Ports

Input

SpdReq — Rotor speed command

scalar

Rotor speed command, ω*_m, in rad/s.

Dependencies

To create this port, select Speed Control for the Control Type parameter.

TrqCmd — Torque command

scalar

Torque command, T*, in N·m.

Dependencies

To create this port, select Torque Control for the Control Type parameter.

BusVolt — DC bus voltage

scalar

DC bus voltage, v_bus, in V.

PhaseCurrA — Current

scalar

Stator current phase a, i_a, in A.

PhaseCurrB — Current

scalar

Stator current phase b, i_b, in A.

SpdFdbk — Rotor speed

scalar

Rotor speed, ω_m, in rad/s.

PosFdbk — Rotor electrical angle

scalar

Rotor electrical angle, Θ_m, in rad.

Output

Info — Bus signal

bus

Bus signal containing these block calculations.

Signal	Description	Units
SrcPwr	Source power	W
LdPwr	Load power	W
PwrLoss	Power loss	W
MtrTrqEst	Estimated motor torque	N·m

BusCurr — Bus current

scalar

Estimated DC bus current, i_bus, in A.

PhaseVolt — Stator terminal voltages

array

Stator terminal voltages, V_a, V_b, and V_c, in V.

Parameters

Block Options

Control Type — Select control

Torque Control (default) | Speed Control

If you select Torque Control, the block does not implement the speed controller.

This table summarizes the port configurations.

Port Configuration	Creates Ports
Speed Control	SpdReq
Torque Control	TrqCmd

Motor Parameters

Number of pole pairs, PolePairs — Poles

4 (default) | scalar

Motor pole pairs, P.

Vector of d-axis current breakpoints, id_index — Current

vector

d-axis current, i_{d_index}, in A.

Vector of q-axis current breakpoints, iq_index — current

vector

q-axis current, i_{q_index}, in A.

Corresponding d-axis flux, lambda_d — Flux

vector

d-axis flux, λ_d, in Wb.

Corresponding q-axis flux, lambda_q — Flux

vector

q-axis flux, λ_q, in Wb.

Current Controller

Sample time for the torque control, Tst — Time

1e-4 (default) | scalar

Torque control sample time, T_st, in s.

D-axis proportional gain, Kp_d — Gain

2.4056 (default) | scalar

d-axis proportional gain, Kp_d, in V/A.

Q-axis proportional gain, Kp_q — Gain

2.4056 (default) | scalar

q-axis proportional gain, Kp_q, in V/A.

D-axis integral gain, Ki_d — Gain

192.45 (default) | scalar

d-axis integral gain, Ki_d, in V/A·s.

Q-axis integral gain, Ki_q — Gain

192.45 (default) | scalar

q- axis integral gain, Ki_q, in V/A·s.

Vector of speed breakpoints, wbp — Breakpoints

vector

Speed breakpoints, ω_bp, in rad/s.

Vector of torque breakpoints, tbp — Breakpoints

vector

Torque breakpoints, T_bp, in N·m.

Corresponding d-axis current reference, id_ref — Current

vector

d-axis reference current, i_dref, in A.

Corresponding q-axis current reference, iq_ref — Current

vector

q-axis reference current, i_qref, in A.

Speed Controller

Speed regulation time constant, Ksf — Time

.1 (default) | scalar

Speed regulator time constant, K_sf, in 1/s.

Dependencies

To enable this parameter, for the Control Type parameter, select Speed Control.

Proportional gain, Kp_w — Gain

0.40475 (default) | scalar

Proportional gain, Kp_ω, in N·m/(rad/s).

Dependencies

To enable this parameter, for the Control Type parameter, select Speed Control.

Integral gain, Ki_w — Gain

10.1615 (default) | scalar

Integral gain, Ki_ω N·m/rad.

Dependencies

To enable this parameter, for the Control Type parameter, select Speed Control.

Inertia compensation, Jcomp — Inertia

0.0027 (default) | scalar

Inertia compensation, in kg·m^2.

Dependencies

To enable this parameter, for the Control Type parameter, select Speed Control.

Static friction, Fs — Friction

0 (default) | scalar

Static friction, in N·m.

Dependencies

To enable this parameter, for the Control Type parameter, select Speed Control.

Viscous damping compensation, Fv — Dampint

0.0004924 (default) | scalar

Viscous damping compensation, in N·m/(rad/s).

Dependencies

To enable this parameter, for the Control Type parameter, select Speed Control.

Electrical Losses

Parameterize losses by — Select type

Single efficiency measurement (default) | Tabulated loss data | Tabulated efficiency data

Setting	Block Implementation
Single efficiency measurement	Electrical loss calculated using a constant value for inverter efficiency.
Tabulated loss data	Electrical loss calculated as a function of motor speeds and load torques.
Tabulated efficiency data	Electrical loss calculated using inverter efficiency that is a function of motor speeds and load torques. Converts the efficiency values you provide into losses and uses the tabulated losses for simulation. Ignores efficiency values you provide for zero speed or zero torque. Losses are assumed zero when either torque or speed is zero. Uses linear interpolation to determine losses. Provide tabulated data for low speeds and low torques, as required, to get the desired level of accuracy for lower power conditions. Does not extrapolate loss values for speed and torque magnitudes that exceed the range of the table.

For best practice, use Tabulated loss data instead of Tabulated efficiency data:

• Efficiency becomes ill defined for zero speed or zero torque.

• You can account for fixed losses that are still present for zero speed or torque.

Overall inverter efficiency, eff — Constant

98 (default) | scalar

Overall inverter efficiency, Eff, in %.

Dependencies

To enable this parameter, for Parameterize losses by, select Tabulated loss data.

Vector of speeds (w) for tabulated loss, w_loss_bp — Breakpoints

[0 200 400 600 800 1000] (default) | 1-by-M vector

Speed breakpoints for lookup table when calculating losses, in rad/s.

Dependencies

To enable this parameter, for Parameterize losses by, select Tabulated loss data.

Vector of torques (T) for tabulated loss, T_loss_bp — Breakpoints

[0 25 50 75 100] (default) | 1-by-N vector

Torque breakpoints for lookup table when calculating losses, in N·m.

Dependencies

To enable this parameter, for Parameterize losses by, select Tabulated loss data.

Corresponding losses, losses_table — Table

[100 100 100 100 100;100 150 200 250 300;100 200 300 400 500;100 250 400 550 700;100 300 500 700 900;100 350 600 850 1100] (default) | M-by-N array

Array of values for electrical losses as a function of M speeds and N torques, in W. Each value specifies the losses for a specific combination of speed and torque. The matrix size must match the dimensions defined by the speed and torque vectors.

Dependencies

To enable this parameter, for Parameterize losses by, select Tabulated loss data.

Vector of speeds (w) for tabulated efficiency, w_eff_bp — Breakpoints

[200 400 600 800 1000] (default) | 1-by-M vector

Speed breakpoints for lookup table when calculating efficiency, in rad/s.

Dependencies

To enable this parameter, for Parameterize losses by, select Tabulated efficiency data.

Vector of torques (T) for tabulated efficiency, T_eff_bp — Breakpoints

[25 50 75 100] (default) | 1-by-N vector

Torque breakpoints for lookup table when calculating efficiency, in N·m.

Dependencies

To enable this parameter, for Parameterize losses by, select Tabulated efficiency data.

Corresponding efficiency, efficiency_table — Table

[96.2 98.1 98.7 99;98.1 99 99.4 99.5;98.7 99.4 99.6 99.7;99 99.5 99.7 99.8;99.2 99.6 99.7 99.8] (default) | M-by-N array

Array of efficiency as a function of M speeds and N torque, in %. Each value specifies the efficiency for a specific combination of speed and torque. The matrix size must match the dimensions defined by the speed and torque vectors.

The block ignores efficiency values for zero speed or zero torque. Losses are zero when either torque or speed is zero. The block uses linear interpolation.

To get the desired level of accuracy for lower power conditions, you can provide tabulated data for low speeds and low torques.

Dependencies

To enable this parameter, for Parameterize losses by, select Tabulated efficiency data.

References

[1] Hu, Dakai, Yazan Alsmadi, and Longya Xu. “High-Fidelity Nonlinear IPM Modeling Based on Measured Stator Winding Flux Linkage.” IEEE^® Transactions on Industry Applications 51, no. 4 (July/August 2015).

[2] Chen, Xiao, Jiabin Wang, Bhaskar Sen, Panagiotis Lasari, and Tianfu Sun. “A High-Fidelity and Computationally Efficient Model for Interior Permanent-Magnet Machines Considering the Magnetic Saturation, Spatial Harmonics, and Iron Loss Effect.” IEEE Transactions on Industrial Electronics 62, no. 7 (July 2015).

[3] Ottosson, J., and M. Alakula. “A Compact Field Awakening Controller Implementation.” International Symposium on Power Electronics, Electrical Drives, Automation and Motion, July 2006.

Extended Capabilities

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

Version History

Introduced in R2017b