IM with Saturation Section

The IM with Saturation model implements a three-phase induction (asynchronous) machine with magnetizing inductance saturation in the stationary (stator) reference frame, along with the temperature effects on the stator and rotor resistances. This model includes support for resolvers and encoders. The machine can operate in both motoring mode, when the mechanical torque is positive, and generating mode when the mechanical torque is negative.

Configuration Page

In the System Explorer window configuration tree, expand the Power Electronics Add-On custom device and select Circuit Model >> IM with Saturation to display this page. Use this page to configure the Induction Machine with Saturation model.

This page includes the following components, configurable at edit-time only:

Machine Model Settings
Name	Specifies the name of the machine model.
Description	Specifies a description for the machine model.
Machine Configuration
	Symbol	Units	Default Value	Description
Machine Type			Squirrel-Cage Induction Machine	Choose from one of the following types: Squirrel-Cage Induction Machine
Enable			True	Indicates whether the selected machine model is enabled. When a machine is enabled, it computes and generates output data at the specified Applied Solver Timestep. Because up to four machines can be simulated at once, the number of enabled machines impacts the minimum achievable time step of each machine.
Enable Advanced Channels			False	Exposes additional parameters as tunable VeriStand Channels. See the Advanced Channels section below for more details. This setting also enables the Temperature Difference configuration parameter.
Magnetizing Inductance File Path				Specifies the path to the Magnetizing Inductance JSON file defining the saturation parameters of the machine. Refer to Magnetizing Inductance File [JSON] for details regarding the file format.
Stator Resistance	R_s	Ω	0.087	Stator winding resistance of phases A, B, and C.
Stator Leakage Inductance	L_ls	H	0.00035	Stator winding leakage inductance of phases A, B, and C.
Rotor Resistance	R'_r	Ω	0.228	Equivalent rotor winding resistance of phases A, B, and C, referred to the stator.
Rotor Leakage Inductance	L'_lr	H	0.00035	Equivalent rotor winding leakage inductance of phases A, B, and C, referred to the stator.
Pole Pairs	PP		2	Number of machine pole pairs.
Magnetizing Inductance Cutoff Frequency	F_cLm	Hz	3000	Cut-off frequency associated with the frequency response of the magnetic coil, which acts as a filtering element.
Synchronous Reference Frame Orientation			Stator Flux	Configures the orientation of the synchronous (dq) reference frame. The direct-axis of the dq reference frame can be aligned in the direction of one of the following: Stator Flux Rotor Flux Magnetizing Flux
Synchronous Reference Frame Offset	θ_offset	deg	0	Offset angle applied to the estimated synchronous angle. This parameter exists to provide support for the different conventions commonly used in the dq transformation. By default, the offset is 0°, indicating that the direct-axis is aligned with phase A when the rotor angle is 0. Setting the parameter to -90° indicates that the quadrature-axis is aligned with phase A when the rotor angle is 0.
Stator Rotor Turn Ratio	N_sr		1	Stator to rotor winding turn ratio.
Temperature Difference	ΔT	ºC	0	Temperature difference with respect to the initial temperature of stator and rotor windings. This parameter is made available when Enable Advanced Channels is enabled.
Rotor Temperature Coefficient	α_r	°C^-1	0	Temperature coefficient of resistance of the rotor winding material.
Stator Temperature Coefficient	α_s	°C^-1	0	Temperature coefficient of resistance of the stator winding material.
Applied Solver Timestep	T_s	s	4.81E-7	The timestep at which the machine model executes. New outputs are computed by the FPGA machine model at each timestep. If Optimize Solver Timestep is enabled in the Circuit Model page, the Applied Solver Timestep is automatically set to an optimal value and cannot be edited.
Minimum Solver Timestep	T_sm	s	4.81E-7	The minimum achievable timestep at which the machine model can execute when all four machines are enabled. The minimum achievable timestep is a function of the number of enabled machines.
Input Mapping Configuration
Use the Input Mapping Configuration to route signals to the Voltage Phase A, Voltage Phase B, and Voltage Phase C inputs of the machine model. Available routing options may vary depending on the selected Hardware Configuration.
Group				Specifies the group that will be routed to the input voltages of the machine. The available routing options are defined by the selected Hardware Configuration, however it is typical to see the following options by default: Measurements - eHS circuit model measurements
Element				Specifies the index of the measurement in the group that has been selected as the input voltage of the machine.

Section Channels

This section includes the following custom device channels:

Channel Name	Symbol	Type	Units	Default Value	Description

Channel Name	Symbol	Type	Units	Description
Stator Current Phase A	I_sa	Output	A	Phase A stator current.
Stator Current Phase B	I_sb	Output	A	Phase B stator current.
Stator Current Phase C	I_sc	Output	A	Phase C stator current.
Stator Direct Axis Current	I_sd	Output	A	Direct-axis stator current, as defined by the synchronous reference frame.
Stator Quadrature Axis Current	I_sq	Output	A	Quadrature-axis stator current, as defined by the synchronous reference frame.
Stator Alpha Voltage	V_sα	Output	V	Stator voltage in the alpha direction defined by the stationary reference frame.
Stator Beta Voltage	V_sβ	Output	V	Stator voltage in the beta direction defined by the stationary reference frame.
Stator Alpha Flux	Φ_sα	Output	Wb	Stator flux in the alpha direction defined by the stationary reference frame.
Stator Beta Flux	Φ_sβ	Output	Wb	Stator current in the beta direction defined by the stationary reference frame.
Stator Alpha Current	I_sα	Output	A	Stator current in the alpha direction defined by the stationary reference frame.
Stator Beta Current	I_sβ	Output	A	Stator voltage in the beta direction defined by the stationary reference frame.
Rotor Current Phase A	I'_ra	Output	A	Phase A rotor current referred to the stator.
Rotor Current Phase B	I'_rb	Output	A	Phase B rotor current referred to the stator.
Rotor Current Phase C	I'_rc	Output	A	Phase C rotor current referred to the stator.
Rotor Direct Axis Current	I'_rd	Output	A	Direct-axis rotor current referred to the stator.
Rotor Quadrature Axis Current	I'_rq	Output	A	Quadrature-axis rotor current referred to the stator.
Rotor Alpha Voltage	V'_rα	Output	V	Rotor voltage in the alpha direction defined by the stationary reference frame.
Rotor Beta Voltage	V'_rβ	Output	V	Rotor voltage in the beta direction defined by the stationary reference frame.
Rotor Alpha Flux	Φ'_rα	Output	Wb	Rotor flux in the alpha direction defined by the stationary reference frame.
Rotor Beta Flux	Φ'_rβ	Output	Wb	Rotor flux in the beta direction defined by the stationary reference frame.
Rotor Alpha Current	I'_rα	Output	A	Rotor current in the alpha direction defined by the stationary reference frame.
Rotor Beta Current	I'_rβ	Output	A	Rotor current in the beta direction defined by the stationary reference frame.

Advanced Channels

The following VeriStand channels are displayed under the Advanced section when the Enable Advanced Channels option is enabled on the IM with Saturation configuration page. Channel values can be modified dynamically at execution time.

Channel Name	Symbol	Type	Units	Default Value	Description

Channel Name	Symbol	Type	Units	Default Value	Description
Stator Resistance Override	R_s	Input	Ω	0.087	Overrides the Stator Resistance parameter defined in the Configuration page for the machine. When the simulation is started, the value of this channel is automatically updated to reflect the value of the Stator Resistance parameter defined in the machine Configuration page. Changes made to the value of this channel during the simulation take effect immediately.
Rotor Resistance Override	R'_r	Input	Ω	0.228	Overrides the Rotor Resistance parameter defined in the Configuration page for the machine. When the simulation is started, the value of this channel is automatically updated to reflect the value of the Rotor Resistance parameter defined in the machine Configuration page. Changes made to the value of this channel during the simulation take effect immediately.
Temperature Difference Override	ΔT	Input	ºC	0	Overrides the Temperature Difference parameter defined in the Configuration page for the machine. When the simulation is started, the value of this channel is automatically updated to reflect the value of the Temperature Difference parameter defined in the machine Configuration page. Changes made to the value of this channel during the simulation take effect immediately.
Stator Leakage Inductance Override	L_ls	Input	H	0.00035	Overrides the Stator Leakage Inductance parameter defined in the Configuration page for the machine. When the simulation is started, the value of this channel is automatically updated to reflect the value of the Stator Leakage Inductance parameter defined in the machine Configuration page. Changes made to the value of this channel during the simulation take effect immediately.
Rotor Leakage Inductance Override	L'_lr	Input	H	0.00035	Overrides the Rotor Leakage Inductance parameter defined in the Configuration page for the machine. When the simulation is started, the value of this channel is automatically updated to reflect the value of the Rotor Leakage Inductance parameter defined in the machine Configuration page. Changes made to the value of this channel during the simulation take effect immediately.

Model Description

Induction Machines are common electrical machines in the the automotive and transportation industry. AC Induction Motors are usually chosen because they are relatively low cost in terms of production and maintenance, and are self-starting. However, compared to Permanent Magnet Synchronous Machines, they are typically less efficient and larger in size.

The figure below illustrates the equivalent circuits of the IM with Saturation model in the synchronous (dq) reference frame.

General Equations in the Stationary Reference Frame

The stationary reference frame includes the three-phase quantities of the machine in the abc coordinate system, as well as the projection of these quantities onto the α and β axes, as obtained through the Clarke transformation.

Voltage Equations in the abc Stationary Reference Frame

The voltages of the machine stator and rotor are described by the following set of general equations in the abc reference frame, where the variable Ψ represents the flux linkage. Note that all rotor parameters are referred to the stator, as distinguished by a prime sign.

(1)

(2)

For the Squirrel-Cage Induction Machine type, the rotor is short-circuited and it is always the case that:

Transformation from abc to αβ

The Clarke transformation, shown below in equation , is applied to obtain the α and β components of the three-phase voltages and currents.

Voltage Equations in the αβ Stationary Reference Frame

Using the Clarke transformation, the voltage equations for the machine stator and rotor are rewritten in the αβ reference frame.

We consolidate equations and into matrix form to obtain equation , where the flux linkage matrices can be further simplified into equation . This set of equations is used to calculate the stator and rotor currents generated by the machine.

Note that for the Squirrel-Cage Induction Machine type, the rotor is short-circuited and it is once again the case that:

General Equations in the Synchronous Reference Frame

The synchronous reference frame is a rotating reference frame obtained through the Park transformation and denoted by the axes d and q. The orientation of the reference frame is dependent upon the configuration of the Synchronous Reference Frame Orientation and Synchronous Reference Frame Offset parameters. The user can choose to align the d-axis of the reference frame in the direction of the Stator Flux, the Rotor Flux, or the Magnetizing Flux.

Transformation from αβ to dq

The transformation below is performed on the αβ components in the stationary reference frame to obtain the dq components in the synchronous reference frame. See Reference Frame Transformations for a complete list of transformation equations.

When computing the dq quantities output from the machine model channels, the angle θ represents the sum of the estimated flux angle, θ_e, and the Synchronous Reference Frame Offset, θ_offset. The value of the estimated flux angle θ_e is dependent upon the configuration of the Synchronous Reference Frame Orientation parameter:

if the reference frame is aligned with the Stator Flux

if the reference frame is aligned with the Rotor Flux

if the reference frame is aligned with the Magnetizing Flux

where Ψ is the flux linkage.

Electrical Angle

For all orientations of the synchronous reference frame, the value of the electrical angle is equivalent to the estimated flux angle. The quantity can also be expressed as follows:

where θ_m is the mechanical angle of the machine and ω_slip is the slip speed.

Torque Equation

The electromagnetic torque of the machine is described by equation , where Ψ is the flux linkage.

Modeling Machine Saturation

To model the effects of saturation in the machine, the α and β components of the magnetizing flux linkage, ψ_mβ and ψ_mα, are calculated from the stator and rotor flux linkages using equations and . We use these quantities to compute the magnetizing flux linkage, Ψ_m , so that it can be passed to a look-up table generated from the parameters defined in the Magnetizing Inductance File [JSON].

where Lα and Lβ are the α- and β- axis inductances such that:

Thermal Modeling of Stator and Rotor Resistances

When the Temperature Difference parameter ΔT is configured, the stator and rotor resistances of the machine are calculated using equations and .

where:

R_s is the stator resistance at final temperature T_f

R'_r is the stator resistance at final temperature T_f

R_snom is the stator resistance at initial temperature T_i_.This value is set by the Stator Resistance machine parameter.

R'_rnom is the rotor resistance at initial temperature T_i. This value is set by the Rotor Resistance machine parameter.

Reference Frame Transformations

The transformations between the stationary three-phase reference frame (abc), the stationary reference frame (αβ), and the synchronous reference frame (dq) are described in the equations below. All equations are referred to the stator.

The angle θ represents the sum of the estimated flux angle, θ_e , and the Synchronous Reference Frame Offset, θ_offset. The value of the estimated flux angle θ_e is dependent upon the configuration of the Synchronous Reference Frame Orientation parameter. See General Equations in the Synchronous Reference Frame for more information.

abc to dq

The Park transformation and its inverse, shown below, are applied to transform the stationary three-phase reference frame (abc) to the synchronous reference frame (dq) and vice-versa.

abc to αβ

We can set θ = 0 to further reduce equations and . The resulting equations allow us to project the three-phase abc components onto a stationary two-axis reference frame, αβ.

αβ to dq

The transformations shown below are used to transform the αβ components in the stationary reference frame to dq components in the synchronous reference frame, and vice versa.