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Flux-Based PM Controller

Controller for a flux-based permanent magnet synchronous motor

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
i_d i_d*	d-axis current d-axis current command
i_q i_q*	q-axis current q-axis current command
v_d, v_d*	d-axis voltage d-axis voltage command
v_q v_q*	q-axis voltage q-axis voltage command
v_a, v_b, v_c	Stator phase a, b, c voltages
i_a, i_b, i_c	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:

$z + K_{s f} T_{s m} - 1$

The filter calculates the gain using this equation.

$K_{s f} = \frac{1 - \exp (- T_{s m} 2 π E V_{s f})}{T_{s m}}$

The equations use these variables.

EV_sf	Bandwidth of the speed command filter
T_sm	Motion controller sample time
K_sf	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.

$T_{c m d} = K p_{ω} (ω_{m}^{*} - ω_{m}) + K i_{ω} \frac{z T_{s m}}{z - 1} (ω_{m}^{*} - ω_{m})$

The equations use these variables.

ω_m	Rotor speed
ω_m*	Rotor speed command
T_cmd	Torque command
Kp_ω	Speed regulator proportional gain
Ki_ω	Speed regulator integral gain
T_sm	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.

$T_{c m d_f f} = J_{p} {\dot{ω}}_{m} + F_{v} ω_{m} + F_{s} \frac{ω_{m}}{| ω_{m} |}$

where:

J_p	Rotor inertia
T_{cmd_ff}	Torque command feedforward
F_s	Static friction torque constant
F_v	Viscous friction torque constant
F_s	Static friction torque constant
ω_m	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.

$\begin{array}{l} i_{d r e f} = f (| ω_{m} |, | T_{r e f} |) \\ i_{q r e f} = s i g n (T_{r e f}) * f (| ω_{m} |, | T_{r e f} |) \end{array}$

The equations use these variables.

ω_m	Rotor speed
T_ref	Torque command
i_dref, i_qref	d- and q-axis reference current, respectively

Voltage Command

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

$\begin{array}{l} v_{d} = \frac{d ψ_{d}}{d t} + R_{s} i_{d} - ω_{e} ψ_{q} \\ v_{q} = \frac{d ψ_{q}}{d t} + R_{s} i_{q} + ω_{e} ψ_{d} \end{array}$

$\begin{array}{l} \frac{d ψ_{d}}{d t} + R_{s} i_{d} = K p_{d} (i_{d}^{*} - i_{d}) + K i_{d} \frac{z T_{s t}}{z - 1} (i_{d}^{*} - i_{d}) \\ \frac{d ψ_{q}}{d t} + R_{s} i_{q} = K p_{q} (i_{q}^{*} - i_{q}) + K i_{q} \frac{z T_{s t}}{z - 1} (i_{q}^{*} - i_{q}) \\ v_{d} = K p_{i} (i_{d}^{*} - i_{d}) + K i_{d} \frac{z T_{s t}}{z - 1} (i_{d}^{*} - i_{d}) + ω_{e} ψ_{q} \\ v_{q} = K p_{i} (i_{q}^{*} - i_{q}) + K i_{q} \frac{z T_{s t}}{z - 1} (i_{q}^{*} - i_{q}) - ω_{e} ψ_{d} \\ ψ_{q} = f (i_{d}, i_{q}) \\ ψ_{d} = f (i_{d}, i_{q}) \end{array}$

The equations use these variables.

ω_m	Rotor mechanical speed
ω_e	Rotor electrical speed
R_s, R_r	Resistance of the stator and rotor windings, respectively
i_q, i_d	q- and d-axis current, respectively
v_q, v_d	q- and d-axis voltage, respectively
Ψ_q, Ψ_d	q- and d-axis magnet flux, respectively
T_st	Current regulator sample rate
Ki_d , Ki_q	d- and q- axis integral gain, respectively
Kp_d , Kp_q	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.

$\begin{array}{l} ω_{e} = P ω_{m} \\ \frac{d θ_{e}}{d t} = ω_{e} \end{array}$

Transform	Description	Equations
Clarke	Converts balanced three-phase quantities (a, b) into balanced two-phase quadrature quantities (α, β).	$\begin{array}{l} x_{α} = \frac{2}{3} x_{a} - \frac{1}{3} x_{b} - \frac{1}{3} x_{c} \\ x_{β} = \frac{\sqrt{3}}{2} x_{b} - \frac{\sqrt{3}}{2} x_{c} \end{array}$
Park	Converts balanced two-phase orthogonal stationary quantities (α, β) into an orthogonal rotating reference frame (d, q).	$\begin{array}{l} x_{d} = x_{α} \cos θ_{e} + x_{β} \sin θ_{e} \\ x_{q} = - x_{α} \sin θ_{e} + x_{β} \cos θ_{e} \end{array}$
Inverse Clarke	Converts balanced two-phase quadrature quantities (α, β) into balanced three-phase quantities (a, b).	$\begin{array}{l} x_{a} = x_{a} \\ x_{b} = - \frac{1}{2} x_{α} + \frac{\sqrt{3}}{2} x_{β} \\ x_{c} = - \frac{1}{2} x_{α} - \frac{\sqrt{3}}{2} x_{β} \end{array}$
Inverse Park	Converts an orthogonal rotating reference frame (d, q) into balanced two-phase orthogonal stationary quantities (α, β).	$\begin{array}{l} x_{α} = x_{d} \cos θ_{e} - x_{q} \sin θ_{e} \\ x_{β} = x_{d} \sin θ_{e} + x_{q} \cos θ_{e} \end{array}$

The transforms use these variables.

ω_m	Rotor speed
P	Rotor pole pairs
ω_e	Rotor electrical speed
Θ_e	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	$L d_{P w r} = v_{a} i_{a} + v_{b} i_{b} + v_{c} i_{c}$
Source power	$S r c_{P w r} = L d_{P w r} + P w r_{L o s s}$
DC bus current	$i_{b u s} = \frac{S r c_{P w r}}{v_{b u s}}$
Estimated rotor torque	$T_{e} = 1.5 P [ψ_{d} i_{q} - ψ_{q} i_{d}]$
Power loss for single efficiency source to load	$P w r_{L o s s} = \frac{100 - E f f}{E f f} \cdot L d_{P w r}$
Power loss for single efficiency load to source	$P w r_{L o s s} = \frac{100 - E f f}{100} \cdot \| L d_{P w r} \|$
Power loss for tabulated efficiency	$P w r_{L o s s} = f (ω_{m}, M t r T r q_{e s t})$

The equations use these variables.

v_a, v_b, v_c	Stator phase a, b, c voltages
v_bus	Estimated DC bus voltage
i_a, i_b, i_c	Stator phase a, b, c currents
i_bus	Estimated DC bus current
Eff	Overall inverter efficiency
ω_m	Rotor mechanical speed
L_q, L_d	q- and d-axis winding inductance, respectively
Ψ_q, Ψ_d	q- and d-axis magnet flux, respectively
i_q, i_d	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

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.

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

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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

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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

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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