This block implements a doubly-fed induction machine (DFIM)

The DFIM block implements a three-phase induction machine (asynchronous machine) with an accessible wound rotor model with 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.

Model Formulation

Q-d Transformation

The 3-phase to q-d transformation and the inverse used for the model are:

The induction machine model uses the rotor as a reference, thus the angle of theta_r.

Induction Machine Electrical Model

Induction machine models in state-space framework are based on magnetic fluxes. The state variables and winding currents (as the outputs) can be represented as follows:

where the coefficient matrices are as follows:

and

The Induction Machine is modeled in rotor reference frame so ω=ωr.

In the model, all the rotor parameters and variables are seen from the stator distinguished by a prime sign. The stator to rotor turn ratio (Nsr) is applied as follows to transfer input three-phase rotor voltage to the stator side, and transfer back the output three-phase rotor current measurements to the rotor side:

The electrical torque is calculated as follows:

Mechanical Model

The equation of the mechanical model in torque mode is expressed as follows:

where ω_m is the rotor speed, T_e is the electromagnetic torque, T_m is the torque command, F_v is the viscous friction coefficient, J is the inertia and T_s the time step. There is a dead-zone implementation with the static friction torque, if the electromagnetic doesn't exceed the static friction torque, the speed remains zero.

In speed mode, the rotor speed is directly set to the speed command ω_rc.

Resolver Encoder Model

The equations of the resolver encoder can be expressed as follows:

where θ_resis the resolver angle, θ_mecis the mechanical angle of the machine, θ_offsetis the angle offset, R_pp is the Number of pole pairs of the resolver and R_k are the resolver sine cosine gains.

Parameters and Measurements

The DFIM's parameters and measurements are separated in 4 different tabs, Electrical, Mechanical, Resolver and Encoder.

Electrical Parameters and Measurements

Symbol	Name	Description	Unit	Type
R_s	Stator resistance	Stator winding resistance of phase a, b, and c	Ω	Edit-input
L_ls	Stator leakage inductance	Stator winding leakage inductance of phase a, b, and c	H	Edit-input
R_r'	Rotor resistance	Equivalent rotor winding resistance referred to the stator of phase a, b, and c	Ω	Edit-input
L_lr'	Rotor leakage inductance	Equivalent rotor winding leakage inductance referred to the stator of phase a, b, and c	H	Edit-input
L_m	Mutual inductance	Stator-rotor mutual (magnetizing) inductance of phase a, b, and c	H	Edit-input
pp	Number of pole pairs	Number of pole pairs	N/A	Edit-input
i_s	Stator phase currents	Stator currents measured at phases a, b and c	A	Measurement
i_sdq	Stator dq currents	Stator currents in dq frame	A	Measurement
Φ_sdq	Stator dq fluxes	Stator fluxes in dq frame	Wb	Measurement
V_sdq	Stator dq voltages	Stator voltages in dq frame	V	Measurement
i_r'	Rotor phase currents	Rotor equivalent phase a, b, and c currents, referred to the stator	A	Measurement
i_rdq	Rotor dq currents	Rotor currents in dq frame, referred to the stator	A	Measurement
Φ_rdq'	Rotor dq fluxes	Rotor fluxes in dq frame, referred to the stator	Wb	Measurement
V_rdq	Rotor dq voltages	Rotor voltages in dq frame, referred to the stator	V	Measurement
R_sn	Snubber resistance	Resistances of the snubber on phase A, B and C	Ω	Input
C_sn	Snubber capacitance	Capacitance of the snubber on phase A, B and C	F	Input

Mechanical Parameters and Measurements

Symbol	Name	Description	Unit	Type
J	Rotor inertia	Moment of inertia of the rotor	kg*m²	Input
F_v	Viscous friction coefficient	Viscous friction	Nms/rad	Input
F_s	Static friction torque	Static friction	N*m	Input
ctrl	Mechanical control mode	Control mode of the mechanical model. Has two possible values: speed or torque. In speed mode, the mechanical model is bypassed and the speed command is sent directly. In torque mode, the torque command is used to measure the speed using the mechanical parameters of the machine.		Input
T	Torque command	Torque command sent to the mechanical model	N*m	Input
ω_rc	Rotor speed command	Speed command sent to the mechanical model	rpm	Input
ω_r	Rotor speed	Speed of the rotor	rpm	Measurement
T_e	Electromagnetic torque	Torque measured at the rotor	N*m	Measurement
θ₀	Initial rotor angle	Rotor position at time t = 0	°	Input
θ	Rotor angle	Rotor position from 0 to 360 degrees	°	Measurement

Resolver Parameters and Measurements

Symbol	Name	Description	Unit	Type
R_en	Enable resolver	Whether or not to enable the resolver	N/A	Input
R_sc	Resolver feedback signals	The two two-phase windings producing a sine and cosine feedback current proportional to the sine and cosine of the angle of the motor	N/A	Measurement
R_pp	Number of resolver pole pairs	Number of pole pairs of the resolver	N/A	Input
R_dir	Direction of the sensor rotation	Direction in which the sensor is turning, either clockwise or counterclockwise	N/A	Input
R_θ	Angle offset Δθ ( Sensor- Rotor )	Angle offset between the resolver and the rotor position from 0 to 360 degrees	°	Input
R_k	Resolver sine cosine gains	The sine/cosine modulation output sine/cosine component amplitude. Default value are 1, 0, 0 and 1	N/A	Input
E_type	Excitation source type	The source from which the excitation of the resolver is generated. Can either be AC, which is generated inside the FPGA with the specified frequency, DC, which is generated with a 90° from the rotor and External, which is generated from outside the model	N/A	Input
E_f	Excitation frequency	Frequency of the excitation when in AC mode	Hz	Input
E_src	Excitation source	Source of the external excitation source when in External mode	N/A	Input
E_ts	Excitation time shift	This parameter is used to compensate the time offset between the carrier generation's input in the system and modulated signals' output	s	Input

OPAL-RT's resolver models are based off of the following sets of equations:

(1)

(2)

Where Sin.Sin, Sin.Cos, Cos.Sin, and Cos.Cos represent gains that are applied to simulate a non-ideal resolver. To simulate an ideal resolver, set the Sin.Sin and Cos.Cos gains to 1, set the Sin.Cos and Cos.Sin gains to 0, set the pp to 1, and set the θ_Offset to 0. This results in the following equations:

(3)

(4)

Encoder Parameters and Measurements

Symbol	Name	Description	Unit	Type
_Encen	Enable encoder	Whether or not to enable the encoder	N/A	Input
Q_ABZ	A B Z encoder signals	A B and Z signals of the encoder	N/A	Measurement
Q_ppr	Number of pulses per revolution	Number of pulses in one full revolution of the encoder	N/A	Input
Q_d_ir	Direction of the sensor rotation	Direction in which the sensor is turning, either clockwise or counterclockwise	N/A	Input
θ_offset	Angle offset Δθ ( Sensor - Rotor )	Angle offset between the encoder and the rotor position from 0 to 360 degrees	°	Input

Visualization of Resolver Encoder Parameters effects

Number of resolver pole pairs affects the number of electrical turns per mechanical turns. On the left figure, the number of resolver pole pairs is 2, on the right figure, the number of resolver pole pairs is 4.

Resolver sine cosine gains affect the sine ( first axe ) and cosine ( second axe ) modulation output. Default values set to 1, 0, 0, 1 make it so the sine modulation has a sine form and the cosine modulation has a cosine form. If set to 0, 1, 1, 0, the sine modulation would have a cosine and the cosine modulation would have a sine form.

Excitation frequency, in AC excitation source type, affects the frequency of the carrier signal. We can see the time step highlighted in red. ( Figure 3 is a zoomed in view of Figure 2 )

Number of pulses per revolution (Q_ppr) defines how many times signals A and B pulse between two Z pulses ( one full rotation ).

Direction of the sensor rotation describes if A leads B ( Clockwise ) or if B leads A ( Counterclockwise )

Electrical ports

This block has six electrical ports, the three terminals of the stator on the left side and three terminals of the rotor on the right side.

Doubly-Fed Induction Machine