Multi-phase PMSM Model

Important:

• The multi-phase PMSM IP-cores are based on the three-phase PMSM Model IP-core, where the basics (e.g. working principle of the integrations is explained)

• There are two mulit-phase IP-cores, a six-phase and a nine-phase one

Differences to the three-phase PMSM model IP-core:

• IP-Cores model a six-phase and nine-phase PMSM

• Sample frequency of the integrator is \(T_s=\frac{1}{1\,MHz}\)

• IP-Core clock frequency must be \(f_{clk}=100\,MHz\)!

• All calculations in the IP-Core are done in double precision

• Interfaces to PL are realized in fixedpoint, while interfaces to the PS use single precision float values

System description

The modelling of the multi-phase machine is based on [[1]]. The general idea in this work and also the common approach to model multi-phase machines is, to transform the phase variables with the VSD and Park transformations, as shown and done in Multi-phase VSD and Park transformation IP-Core and Coordinate Transformation. Transformed voltages are used as input for this IP-core and the outputs will also be in the rotary or stationary reference frame ant not phase variables. This IP-core contains the electric differential equations and the mechanical part, where the torque is calculated. Additionally, either a torque load or a fixed speed can be set to test the machine model. While the equations are mostly similar to the ones of the three-phase PMSM Model, they will be shown here again for the sake of completeness. To obtain more details about the equations and integrators, reading PMSM Model is advised.

dq equations

\[\begin{split}\begin{align} \psi_{d}(k+1) &= T_s \big( v_{d}(k) - R_{1} i_{d}(k) + \omega_{el} \psi_{q}(k) \big) + \psi_{d}(k)\\ \psi_{q}(k+1) &= T_s \big( v_{q}(k) - R_{1} i_{q}(k) - \omega_{el} \psi_{d}(k) \big) + \psi_{q}(k)\\ \psi_{d}(k) &= \psi_{PM} + L_{d} i_{d}(k) \\ \psi_{q}(k) &= L_{q} i_{q}(k) \label{eq:psiq_ld} \end{align}\end{split}\]

Mechanical equations

\[\begin{split}\begin{align} M_M &= \frac{9}{2}\cdot{p}\cdot{(\Psi_{PM}\cdot{i_q}+(L_d-L_q)\cdot{i_d}\cdot{i_q})}\\ \theta_{el}(k+1) &= T_s \big(\omega_{mech}(k)\cdot{p}) + \theta_{el}(k)\\ \omega_{mech}(k+1) &= T_s \bigg( \frac{M_M(k)-M_L(k)}{J} \bigg) + \omega_{mech}(k) \end{align}\end{split}\]

Note that the integrator for \(\theta_{el}\) is limited to \(\pm \pi\) to avoid overflow (wrapping integrator).

Additional system equations

Six-phase model

\[\begin{split}\begin{align} \psi_{x}(k+1) &= T_s \big( v_{x}(k) - R_{1} i_{x}(k))+\psi_{x}(k)\\ \psi_{x}(k) &= L_{ls} i_{x}(k) \\ \psi_{y}(k+1) &= T_s \big( v_{y}(k) - R_{1} i_{y}(k))+\psi_{y}(k)\\ \psi_{y}(k) &= L_{ls} i_{y}(k) \\ \psi_{zi}(k+1) &= T_s \big( v_{zi}(k) - R_{1} i_{zi}(k))+\psi_{zi}(k)\\ \psi_{zi}(k) &= L_{ls} i_{zi}(k) \\ \nonumber\textrm{with } i&=1,2 \end{align}\end{split}\]

Nine-phase model

\[\begin{split}\begin{align} \psi_{xi}(k+1) &= T_s \big( v_{xi}(k) - R_{1} i_{xi}(k))+\psi_{xi}(k)\\ \psi_{xi}(k) &= L_{ls} i_{xi}(k) \\ \psi_{yi}(k+1) &= T_s \big( v_{yi}(k) - R_{1} i_{yi}(k))+\psi_{yi}(k)\\ \psi_{yi}(k) &= L_{ls} i_{yi}(k) \\ \psi_{0}(k+1) &= T_s \big( v_{0}(k) - R_{1} i_{0}(k))+\psi_{0}(k) \\ \psi_{0}(k) &= L_{ls} i_{0}(k) \\ \nonumber\textrm{with } i&=1,2,3 \end{align}\end{split}\]

IP-core interfaces

Table 111 Interface of nine-phase PMSM Model IP-Core

Port Name	Port Type	Data Type	Interface	Range	Description
reset_integrators	Input	boolean	AXI 0x100		resets all integrators
use_axi_input	Input	boolean	AXI 0x104		true: use AXI voltage values as input; false: use PL voltage values as input
simulate_mechanical	Input	boolean	AXI 0x108		true: load simulation; false: fix PMSM speed
load_torque	Input	single	AXI 0x10C		load torque if simulate_mechanical is true
omega_mech	Input	single	AXI 0x110		fixed speed if simulate_mechanical is false
physical_parameters	Input	single bus (12/15)	AXI 0x114-0x170		set machine parameters; description see “uz_pmsm_model9ph_dq_config_t”/ “uz_pmsm_model9ph_dq_config_t” struct
voltage_input_dq_axi	Input	single vector (6/9)	AXI 0x180-0x1A0		input voltages as transformed values in the form of “uz_6ph_dq_t”/ “uz_9ph_dq_t” from AXI
voltage_input_dq	Input	sfix27_En16 vector (6/9)	External Signal	-1024 to 1023	input voltages as transformed values in the form of “uz_6ph_dq_t”/ “uz_9ph_dq_t” from PL
theta_el_out_axi	Output	single	AXI 0x14C	\(\pm \pi\)	actual electric rotor angle
theta_el_out	Output	sfix18_En14	External Signal	\(\pm \pi\)	actual electric rotor angle
M_Mi_out_axi	Output	single	AXI 0x174		actual machine torque
omega_mech_out_axi	Output	single	AXI 0x178		actual machine speed
omega_mech_out	Output	sfix24_En11	External Signal	-4096 to 4095	actual machine speed
currents_dq_out_axi	Output	single vector (6/9)	AXI 0x200-0x220		actual currents in the rotary or stationary reference frame to AXI in the form of “uz_6ph_dq_t”/ “uz_9ph_dq_t”
currents_dq_out	Output	sfix27_En18 vector (6/9)	External Signal	-256 to +255	actual currents in the rotary or stationary reference frame to PL in the form of “uz_6ph_dq_t”/ “uz_9ph_dq_t”
voltage_input_dq_out_axi_fb	Output	single vector (6/9)	AXI 0x280-2A0		feedback actual input voltages to AXI (available for use_axi_input true and false)

Driver reference

The set and get functions for voltage and currents are implemented as normal and unsafe version. In addition to the regular functions, unsafe versions of the driver exist (_unsafe). These functions are considerably faster than their safe counterparts but violate the software rules outlined in Software Development Guidelines. It is strongly advised to manually test by comparing the safe and unsafe versions before using _unsafe!””

Six-phase model

typedef struct uz_pmsm_model6ph_dq_t uz_pmsm_model6ph_dq_t

Object data type definition of the PMSM model IP-Core driver.

struct uz_pmsm_model6ph_dq_config_t

Configuration struct for the PMSM model IP-Core driver.

Public Members

uint32_t base_address

Base address of the IP-Core instance to which the driver is coupled

uint32_t ip_core_frequency_Hz

Clock frequency of IP-Core

float polepairs

Polepairs of the PMSM

float r_1

Stator resistance in ohm

uz_6ph_dq_t inductance

Subsystem inductances

float psi_pm

PM flux linkage

float friction_coefficient

Linear coefficient of friction

float coulomb_friction_constant

Static friction constant

float inertia

Inertia of the PMSM

bool simulate_mechanical_system

Determine if mechanical system is simulated or speed is an input

bool switch_pspl

true: inputs from PS, false: inputs from PL

struct uz_pmsm_model6ph_dq_outputs_general_t

Struct to return and read the general outputs of the PMSM Model.

Public Members

float torque

Inner torque of PMSM in Nm

float omega_mech

Rotational speed of PMSM in 1/s

float theta_el

Rotor electrical angle in rad

uz_pmsm_model6ph_dq_t *uz_pmsm_model6ph_dq_init(struct uz_pmsm_model6ph_dq_config_t config)

Initialize an instance of the driver.

Parameters:

• config – Config struct

Returns:

uz_pmsm_model6ph_dq_t* Pointer to an initialized instance of the driver

void uz_pmsm_model6ph_trigger_voltage_input_strobe(uz_pmsm_model6ph_dq_t *self)

Takes the values of the AXI shadow register and pass them to the actual input.

Parameters:

• self –

void uz_pmsm_model6ph_trigger_voltage_output_strobe(uz_pmsm_model6ph_dq_t *self)

Takes the values of the shadow register and pass them to the actual AXI register.

Parameters:

• self –

void uz_pmsm_model6ph_trigger_current_output_strobe(uz_pmsm_model6ph_dq_t *self)

Takes the values of the shadow register and pass them to the actual AXI register.

Parameters:

• self –

void uz_pmsm_model6ph_dq_set_inputs_general(uz_pmsm_model6ph_dq_t *self, float omega_mech, float load_torque)

Set general inputs of the model and write them to the PMSM model.

Parameters:

• self – Pointer to driver instance

• omega_mech – value for fixed omega if uz_pmsm_model6ph_dq_config_t.simulate_mechanical_system is false

• load_torque – value for load torque if uz_pmsm_model6ph_dq_config_t.simulate_mechanical_system is true

struct uz_pmsm_model6ph_dq_outputs_general_t uz_pmsm_model6ph_dq_get_outputs_general(uz_pmsm_model6ph_dq_t *self)

Returns general outputs of PMSM model.

Parameters:

• self – Pointer to driver instance

Returns:

struct uz_pmsm_model6ph_dq_outputs_general_t output values

void uz_pmsm_model6ph_dq_set_voltage(uz_pmsm_model6ph_dq_t *self, uz_6ph_dq_t voltages)

Safe function to set the input voltages of PMSM if PS is selected as source for voltages.

Parameters:

• self –

• voltages – struct of type uz_6ph_dq_t containing the voltages to set

void uz_pmsm_model6ph_dq_set_voltage_unsafe(uz_pmsm_model6ph_dq_t *self, uz_6ph_dq_t voltages)

Unsafe function to set the input voltages of PMSM if PS is selected as source for voltages.

Parameters:

• self –

• voltages – struct of type uz_6ph_dq_t containing the voltages to set

uz_6ph_dq_t uz_pmsm_model6ph_dq_get_input_voltages(uz_pmsm_model6ph_dq_t *self)

Safe function to read out the set input voltages of PMSM. Works for voltages coming from PS and PL.

Parameters:

• self –

Returns:

struct of type uz_6ph_dq_t containing the set voltages

uz_6ph_dq_t uz_pmsm_model6ph_dq_get_input_voltages_unsafe(uz_pmsm_model6ph_dq_t *self)

Unsafe function to read out the set input voltages of PMSM. Works for voltages coming from PS and PL.

Parameters:

• self –

Returns:

struct of type uz_6ph_dq_t containing the set voltages

uz_6ph_dq_t uz_pmsm_model6ph_dq_get_output_currents(uz_pmsm_model6ph_dq_t *self)

Safe function to read out the actual currents of PMSM.

Parameters:

• self –

Returns:

struct of type uz_6ph_dq_t containing the actual currents

uz_6ph_dq_t uz_pmsm_model6ph_dq_get_output_currents_unsafe(uz_pmsm_model6ph_dq_t *self)

Unsafe function to read out the actual currents of PMSM.

Parameters:

• self –

Returns:

struct of type uz_6ph_dq_t containing the actual currents

void uz_pmsm_model6ph_dq_reset(uz_pmsm_model6ph_dq_t *self)

Resets the PMSM model by writing zero to all PS inputs and sets integrators to zero.

Parameters:

• self – Pointer to driver instance

void uz_pmsm_model6ph_dq_set_use_axi_input(uz_pmsm_model6ph_dq_t *self, bool use_axi)

Change input source for voltages (PL or PS) during runtime.

Parameters:

• self – Pointer to driver instance

• use_axi – true: voltages from PS, false: voltages from PL

Nine-phase model

typedef struct uz_pmsm_model9ph_dq_t uz_pmsm_model9ph_dq_t

Object data type definition of the PMSM model IP-Core driver.

struct uz_pmsm_model9ph_dq_config_t

Configuration struct for the PMSM model IP-Core driver.

Public Members

uint32_t base_address

Base address of the IP-Core instance to which the driver is coupled

uint32_t ip_core_frequency_Hz

Clock frequency of IP-Core

float polepairs

Polepairs of the PMSM

float r_1

Stator resistance in ohm

uz_9ph_dq_t inductance

Subsystem inductances

float psi_pm

PM flux linkage

float friction_coefficient

Linear coefficient of friction

float coulomb_friction_constant

Static friction constant

float inertia

Inertia of the PMSM

bool simulate_mechanical_system

Determine if mechanical system is simulated or speed is an input

bool switch_pspl

true: inputs from PS, false: inputs from PL

struct uz_pmsm_model9ph_dq_outputs_general_t

Struct to return and read the general outputs of the PMSM Model.

Public Members

float torque

Inner torque of PMSM in Nm

float omega_mech

Rotational speed of PMSM in 1/s

float theta_el

Rotor electrical angle in rad

uz_pmsm_model9ph_dq_t *uz_pmsm_model9ph_dq_init(struct uz_pmsm_model9ph_dq_config_t config)

Initialize an instance of the driver.

Parameters:

• config – Config struct

Returns:

uz_pmsm_model9ph_dq_t* Pointer to an initialized instance of the driver

void uz_pmsm_model9ph_trigger_voltage_input_strobe(uz_pmsm_model9ph_dq_t *self)

Takes the values of the AXI shadow register and pass them to the actual input.

Parameters:

• self –

void uz_pmsm_model9ph_trigger_voltage_output_strobe(uz_pmsm_model9ph_dq_t *self)

Takes the values of the shadow register and pass them to the actual AXI register.

Parameters:

• self –

void uz_pmsm_model9ph_trigger_current_output_strobe(uz_pmsm_model9ph_dq_t *self)

Takes the values of the shadow register and pass them to the actual AXI register.

Parameters:

• self –

void uz_pmsm_model9ph_dq_set_inputs_general(uz_pmsm_model9ph_dq_t *self, float omega_mech, float load_torque)

Set general inputs of the model and write them to the PMSM model.

Parameters:

• self – Pointer to driver instance

• omega_mech – value for fixed omega if uz_pmsm_model9ph_dq_config_t.simulate_mechanical_system is false

• load_torque – value for load torque if uz_pmsm_model9ph_dq_config_t.simulate_mechanical_system is true

struct uz_pmsm_model9ph_dq_outputs_general_t uz_pmsm_model9ph_dq_get_outputs_general(uz_pmsm_model9ph_dq_t *self)

Returns general outputs of PMSM model.

Parameters:

• self – Pointer to driver instance

Returns:

struct uz_pmsm_model9ph_dq_outputs_general_t output values

void uz_pmsm_model9ph_dq_set_voltage(uz_pmsm_model9ph_dq_t *self, uz_9ph_dq_t voltages)

Safe function to set the input voltages of PMSM if PS is selected as source for voltages.

Parameters:

• self –

• voltages – struct of type uz_9ph_dq_t containing the voltages to set

void uz_pmsm_model9ph_dq_set_voltage_unsafe(uz_pmsm_model9ph_dq_t *self, uz_9ph_dq_t voltages)

Unsafe function to set the input voltages of PMSM if PS is selected as source for voltages.

Parameters:

• self –

• voltages – struct of type uz_9ph_dq_t containing the voltages to set

uz_9ph_dq_t uz_pmsm_model9ph_dq_get_input_voltages(uz_pmsm_model9ph_dq_t *self)

Safe function to read out the set input voltages of PMSM. Works for voltages coming from PS and PL.

Parameters:

• self –

Returns:

struct of type uz_9ph_dq_t containing the set voltages

uz_9ph_dq_t uz_pmsm_model9ph_dq_get_input_voltages_unsafe(uz_pmsm_model9ph_dq_t *self)

Unsafe function to read out the set input voltages of PMSM. Works for voltages coming from PS and PL.

Parameters:

• self –

Returns:

struct of type uz_9ph_dq_t containing the set voltages

uz_9ph_dq_t uz_pmsm_model9ph_dq_get_output_currents(uz_pmsm_model9ph_dq_t *self)

Safe function to read out the actual currents of PMSM.

Parameters:

• self –

Returns:

struct of type uz_9ph_dq_t containing the actual currents

uz_9ph_dq_t uz_pmsm_model9ph_dq_get_output_currents_unsafe(uz_pmsm_model9ph_dq_t *self)

Unsafe function to read out the actual currents of PMSM.

Parameters:

• self –

Returns:

struct of type uz_9ph_dq_t containing the actual currents

void uz_pmsm_model9ph_dq_reset(uz_pmsm_model9ph_dq_t *self)

Resets the PMSM model by writing zero to all PS inputs and sets integrators to zero.

Parameters:

• self – Pointer to driver instance

void uz_pmsm_model9ph_dq_set_use_axi_input(uz_pmsm_model9ph_dq_t *self, bool use_axi)

Change input source for voltages (PL or PS) during runtime.

Parameters:

• self – Pointer to driver instance

• use_axi – true: voltages from PS, false: voltages from PL

Example usage (standalone)

The IP-core has two intended use cases:

• Using the model in the dq domain only with inputs coming from the PS

• Simulating a complete multi-phase drive system including the Inverter Model, Multi-phase VSD and Park transformation IP-Core (see the CIL examples in CIL PMSM)

Using the IP-core in PS only is similar to the use cases shown in PMSM Model open loop example which is recreated here. The placement of the IP-core for the use from PS only is straight forward as only the default PL interfaces have to be connected. For the example the nine-phase model is used, but the same can also be applied for the six-phase model.

Fig. 348 Test setup for IP-core PS test in Vivado

The following code is used in main.c (initialization) and isr.c (application):

Listing 194 initialization in main.c (R5)

#include "IP_Cores/uz_pmsm_model_9ph_dq/uz_pmsm_model9ph_dq.h"
uz_pmsm_model9ph_dq_t *pmsm=NULL;
struct uz_pmsm_model9ph_dq_config_t pmsm_config = {   // example config values
  .base_address=XPAR_UZ_PMSM_MODEL_0_BASEADDR,
  .ip_core_frequency_Hz = 100000000U,
  .polepairs = 3.0f,
  .r_1 = 31.3f,
  .inductance.d = 0.46f,
  .inductance.q = 0.46f,
  .inductance.x1 = 0.08f,
  .inductance.y1 = 0.08f,
  .inductance.x2 = 0.08f,
  .inductance.y2 = 0.08f,
  .inductance.x3 = 0.08f,
  .inductance.y3 = 0.08f,
  .inductance.zero = 0.08f,
  .psi_pm = 0.072f,
  .friction_coefficient = 0.001f,
  .coulomb_friction_constant = 0.001f,
  .inertia = 0.001f,
  .simulate_mechanical_system = false,
  .switch_pspl = true};

// .. rest of the code in main.c before loop
int main(void)
// ..
  case init_ip_cores: // default line from main.c
    pmsm = uz_pmsm_model9ph_dq_init(pmsm_config);

Listing 195 usage in isr.c

#include "../IP_Cores/uz_pmsm_model_9ph_dq/uz_pmsm_model9ph_dq.h"
extern uz_pmsm_model9ph_dq_t *pmsm;                               // pointer to PMSM object
struct uz_pmsm_model9ph_dq_outputs_general_t out_general = {0};   // stores general outputs
uz_9ph_dq_t in_voltages = {                                       // stores input voltages (set random voltages for testing)
              .d = 1.0f,
              .q = 2.0f,
              .x1 = 3.0f,
              .y1 = 4.0f,
              .x2 = 5.0f,
              .y2 = 6.0f,
              .x3 = 7.0f,
              .y3 = 8.0f,
              .zero = 9.0f};
uz_9ph_dq_t out_currents = {0};                                   // stores output currents
float omega_mech = 10.0f;                                         // fixed speed can be set from Expressions with this variable
int reset = 0;                                                    // use reset variable to reset integrators from Expressions

// .. rest of the code in isr.c before loop
void ISR_Control(void *data)
// ..
  update_speed_and_position_of_encoder_on_D5(&Global_Data);       // default line from isr.c

  if(reset)
    uz_pmsm_model9ph_dq_reset(pmsm);                              // use reset variable to reset integrators from Expressions

  uz_pmsm_model9ph_dq_set_inputs_general(pmsm,omega_mech,0.0f);   // set fixed speed, because load simulation is disabled by pmsm_config.simulate_mechanical_system
  uz_pmsm_model9ph_dq_set_voltage(pmsm,in_voltages);              // set input voltage
  out_general = uz_pmsm_model9ph_dq_get_outputs_general(pmsm);    // read out resulting general outputs
  out_currents = uz_pmsm_model9ph_dq_get_output_currents(pmsm);   // read out actual currents

To prove functionality, the output currents of the shown example are evaluated. The resulting machine torque is \(-0.01562337\,Nm\) an the resulting currents are shown in the following equation. The results were recreated with the Simulink model.

\[\begin{split}\begin{align} out-currents = \begin{bmatrix} i_{d} \\ i_{q} \\ i_{x_1} \\ i_{y_1} \\ i_{x_2} \\ i_{y_2} \\ i_{x_3} \\ i_{y_3} \\ i_{zero} \end{bmatrix} = \begin{bmatrix} 0.03166196\\ -0.006507777 \\ 0.09584665 \\ 0.1277955 \\ 0.1597444 \\ 0.1916933 \\ 0.2236422 \\ 0.2555911 \\ 0.2875399 \end{bmatrix} \end{align}\end{split}\]

Sources

[1]

13. Slunjski, “Control of a ninephase symmetrical pmsm with reduced rare earth material,” Dissertation, Liverpool John Moores University, 2020. [Online]. Available: https://researchonline.ljmu.ac.uk/id/eprint/14732/1/2021MarkoPhD.pdf