This block implements a squirrel-cage induction machine (SCIM) with magnetizing inductance saturation in the stationary (stator) reference frame along with the temperature effects on the stator and rotor resistances.

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The SCIM block implements a three-phase induction machine (asynchronous machine) with a squirrel-cage 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

Induction Machine Electrical Model

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Fig.1. Implementation block diagram (Squirrel-cage/ Doubly-fed induction motor)

Motor modeling equations

The stator and rotor voltage equations of a induction motor in the stationary (stator reference frame) reference frame can be written a equations (1)-(4).

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V_{sd}=R_{s}i_{sd}+\frac{d\psi_{sd}}{dt}\qquad\qquad\quad(1)\\
V_{sq}=R_{s}i_{sq}+\frac{d\psi_{sq}}{dt}\qquad\qquad\quad(2)\\
V_{rd}'=R_{s}'i_{rd'}+\frac{d\psi_{rd}'}{dt}+\omega_{r}\psi_{rq}'\quad(3) \\
V_{rq}'=R_{s}'i_{rq'}+\frac{d\psi_{rq}'}{dt}-\omega_{r}\psi_{rd}'\quad(4)

Equations (1)-(4) can be represented as equation (5).

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\left[\begin{array}{c}
{V_{sd}} \\
{V_{sq}} \\
{V_{rd}'} \\
{V_{rq}'}
\end{array}\right]=\left[\begin{array}{cccc}
{R_{s}} & {0} & {0} & {0} \\
{0} & {R_{s}} & {0} & {0} \\
{0} & {0} & {R_{r}'} & {0} \\
{0} & {0} & {0} & {R_{r}'}
\end{array}\right]\left[\begin{array}{c}
{i_{sd}} \\
{i_{sq}}\\
{i_{rd}'}\\
{i_{rq}'}
\end{array}\right]+\frac{d}{dt}\left[\begin{array}{c}
{\psi_{sd}} \\
{\psi_{sq}}\\
{\psi_{rd}'}\\
{\psi_{rq}'}
\end{array}\right]+\left[\begin{array}{cccc}
{0} & {0} & {0} & {0} \\
{0} & {0} & {0} & {0} \\
{0} & {0} & {0} & {\omega_{r}} \\
{0} & {0} & {-\omega_{r}} & {0}
\end{array}\right]\left[\begin{array}{c}
{\psi_{sd}} \\
{\psi_{sq}}\\
{\psi_{rd}'}\\
{\psi_{rq}'}
\end{array}\right]\qquad(5)

The stator and rotor flux linkages can be expressed as equation (6).

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\left[\begin{array}{c}
{\psi_{sd}} \\
{\psi_{sq}}\\
{\psi_{rd}'}\\
{\psi_{rq}'}
\end{array}\right]=\left[\begin{array}{cccc}
{L_{s}} & {0} & {L_{m}} & {0} \\
{0} & {L_{s}} & {0} & {L_{m}} \\
{L_{m}} & {0} & {L_{r}'} & {0} \\
{0} & {{L_{m}} & {0} & {L_{r}'}
\end{array}\right]\left[\begin{array}{c}
{i_{sd}} \\
{i_{sq}}\\
{i_{rd}'}\\
{i_{rq}'}
\end{array}\right]\qquad(6)

Note: The stator and rotor flux linkages are calculated by using equation (5) and the stator and rotor currents are calculated by using equation (6).

2. Magnetizing flux linkage calculation

The magnetizing flux linkages are calculated from the stator and rotor flux linkages by using equations (7) and (8).

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\psi_{mq}=\psi_{sq}x\frac{L_{aq}}{L_{ls}}+\psi_{rq}'x\frac{L_{aq}}{L_{lr}'}}\qquad(7) \\
\psi_{md}=\psi_{sd}x\frac{L_{ad}}{L_{ls}}+\psi_{rd}'x\frac{L_{ad}}{L_{lr}'}}\qquad(8)

where

ψ_mq & ψ_md = q & d-axis magnetizing flux linkages

L_aq = q-axis inductance (= (L_m L_ls L_lr^‘)/(L_m L_ls + L_m L_lr^‘+L_lsL_lr^‘))

L_ad = d-axis inductance (= (L_m L_ls L_lr^‘)/(L_m L_ls + L_m L_lr^‘+L_lsL_lr^‘))

L_ls = Stator leakage inductance

L_lr^’ = Rotor leakage inductance (referred to stator side)

L_m = Magnetizing inductance

3. Thermal modelling of stator and rotor resistances

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R_{s} = R_{snom} * \left\{1+\alpha_{cond}*{\theta_{f}-\theta_{i})\right\}\qquad(9) \\
R_{r}' = R_{rnom}' * \left\{1+\alpha_{cond}*{\theta_{f}-\theta_{i})\right\}\qquad(10)

where

R_snom = stator resistance at temp ϴ_i

R_rnom ^‘ = rotor resistance at temp ϴ_i

R_s = stator resistance at temp ϴ_f

R_r^‘ = rotor resistance at temp ϴ_f

α_cond = temperature co-efficient of resistance of the material

abc to dq0 (stator side)

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\left(\begin{array}{c}
{d} \\
{q} \\
{0}
\end{array}\right)=\left(\frac{2}{3}\right)*\left(\begin{array}{ccc}
{\cos (\theta_{e}) & \cos \left(\theta_{e}-\frac{2\pi}{3}\right) & \cos \left(\theta_{e}+\frac{2\pi}{3}\right) \\
{-\sin(\theta_{e}) & -\sin\left(\theta_{e}-\frac{2\pi}{3}\right) & -\sin\left(\theta_{e}+\frac{2\pi}{3}\right) \\
{\frac{1}{2} & \frac{1}{2} & \frac{1}{2}
\end{array}\right)*\left(\begin{array}{c}
{a} \\
{b} \\
{c}
\end{array}\right)

Stationary reference frame (ϴ_e=0)

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\left(\begin{array}{c}
{d} \\
{q} \\
{0}
\end{array}\right)=\left(\frac{2}{3}\right)*\left(\begin{array}{ccc}
{1 & -\frac{1}{2} & -\frac{1}{2} \\
{0 & \frac{\sqrt{3}}{2} & -\frac{\sqrt{3}}{2} \\
{\frac{1}{2} & \frac{1}{2} & \frac{1}{2}
\end{array}\right)*\left(\begin{array}{c}
{a} \\
{b} \\
{c}
\end{array}\right)

dq0 to abc (stator side)

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\left(\begin{array}{c}
{a} \\
{b} \\
{c}
\end{array}\right)=\left(\begin{array}{ccc}
{\cos (\theta_{e}) & \-sin (\theta_{e}) & 1 \\
{\cos\left(\theta_{e}-\frac{2\pi}{3}\right) & -\sin\left(\theta_{e}-\frac{2\pi}{3}\right) & 1 \\
{\cos\left(\theta_{e}+\frac{2\pi}{3}\right) & -\sin\left(\theta_{e}+\frac{2\pi}{3}\right) & 1
\end{array}\right)*\left(\begin{array}{c}
{d} \\
{q} \\
{0}
\end{array}\right)

Stationary reference frame (ϴ_e=0)

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\left(\begin{array}{c}
{a} \\
{b} \\
{c}
\end{array}\right)=\left(\begin{array}{ccc}
{1 & 0 & 1 \\
{-\frac{1}{2}& \frac{\sqrt{3}}{2} & 1 \\
{-\frac{1}{2} & -\frac{\sqrt{3}}{2} & 1
\end{array}\right)*\left(\begin{array}{c}
{d} \\
{q} \\
{0}
\end{array}\right)

abc to dq0 (rotor side)

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\left(\begin{array}{c}
{d} \\
{q} \\
{0}
\end{array}\right)=\left(\frac{2}{3}\right)*\left(\begin{array}{ccc}
{\cos (\theta_{e}-\theta_{r}) & \cos\left(\theta_{e}-\theta_{r}-\frac{2\pi}{3}\right) & \cos\left(\theta_{e}-\theta_{r}+\frac{2\pi}{3}\right) \\
{-\sin(\theta_{e}-\theta_{r}) & -\sin\left(\theta_{e}-\theta_{r}-\frac{2\pi}{3}\right) & -\sin\left(\theta_{e}-\theta_{r}+\frac{2\pi}{3}\right) \\
{\frac{1}{2} & \frac{1}{2}  & \frac{1}{2} }
\end{array}\right)*\left(\begin{array}{c}
{a} \\
{b} \\
{c}
\end{array}\right)

Stationary reference frame (ϴ_e=0)

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\left(\begin{array}{c}
{d} \\
{q} \\
{0}
\end{array}\right)=\left(\frac{2}{3}\right)*\left(\begin{array}{ccc}
{\cos (\theta_{r}) & \cos\left(\theta_{r}+\frac{2\pi}{3}\right) & \cos\left(\theta_{r}-\frac{2\pi}{3}\right) \\
{\sin(\theta_{r}) & \sin\left(\theta_{r}+\frac{2\pi}{3}\right) & \sin\left(\theta_{r}-\frac{2\pi}{3}\right) \\
{\frac{1}{2} & \frac{1}{2}  & \frac{1}{2} }
\end{array}\right)*\left(\begin{array}{c}
{a} \\
{b} \\
{c}
\end{array}\right)

dq0 to abc (rotor side)

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\left(\begin{array}{c}
{a} \\
{b} \\
{c}
\end{array}\right)=\left(\begin{array}{ccc}
{\cos (\theta_{e}-\theta_{r}) & -\sin (\theta_{e}-\theta_{r}) & 1 \\
{\cos \left(\theta_{e}-\theta_{r} - \frac{2\pi}{3}\right) & -\sin \left(\theta_{e}-\theta_{r} - \frac{2\pi}{3}\right) & 1 \\
{\cos \left(\theta_{e}-\theta_{r} + \frac{2\pi}{3}\right) & -\sin \left(\theta_{e}-\theta_{r} + \frac{2\pi}{3}\right) & 1 }
\end{array}\right)*\left(\begin{array}{c}
{d} \\
{q} \\
{0}
\end{array}\right)

Stationary reference frame (ϴ_e=0)

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\left(\begin{array}{c}
{a} \\
{b} \\
{c}
\end{array}\right)=\left(\begin{array}{ccc}
{\cos (\theta_{r}) & \sin (\theta_{r}) & 1 \\
{\cos \left(\theta_{r} + \frac{2\pi}{3}\right) & \sin \left(\theta_{r} + \frac{2\pi}{3}\right) & 1 \\
{\cos \left(\theta_{r} - \frac{2\pi}{3}\right) & \sin \left(\theta_{r} - \frac{2\pi}{3}\right) & 1 }
\end{array}\right)*\left(\begin{array}{c}
{d} \\
{q} \\
{0}
\end{array}\right)

Mechanical Model

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

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\omega_{m}(t)=\left(\frac{1}{F_{v}}\right)*(a*\omega_{m}\left(t-T_{s}\right)+b*\left(T_{e}-T_{m}\right))

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a=\left(1-T_{s}*(\frac{F_{v}}{J})\right)

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b={T_{s}}*(\frac{F_{v}}{J})

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:

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\theta_{\text {res}}=\left(\theta_{\text {mec}}+\theta_{\text {offset}}\right) \times R_{p p}

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\text {cosine}=\text {Excitation } \times\left(\cos \left(\theta_{\text {res}}\right) * R_{k_{c s}}+\sin \left(\theta_{\text {res}}\right) * R_{k_{s s}}\right)

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\text { sine }=\text { Excitation } \times\left(\cos \left(\theta_{\text {res}}\right) * R_{k_{c c}}+\sin \left(\theta_{\text {res}}\right) * R_{k_{s c}}\right)

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.This block implements a squirrel-cage induction machine (SCIM) with magnetizing inductance saturation in the stationary (stator) reference frame along with the temperature effects on the stator and rotor resistances.

...

The SCIM block implements a three-phase induction machine (asynchronous machine) with a squirrel-cage 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

The SCIM electrical model formulation is decribed in the sections below.

Include Page

	INT:SCIM/DFIM with saturation - Model Formulation
	INT:SCIM/DFIM with saturation - Model Formulation

Parameters and Measurements

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

Electrical Parameters and Measurements

Symbol	Name	Description	Unit	Type
R_snom	Stator resistance	Stator winding resistance of phase a, b, and c	Ω	Input
R_rnom'	Rotor resistance	Equivalent rotor winding resistance referred to the stator of phase a, b, and c	Ω	Input
Δ_θ	Temperature difference	Temperature difference w.r.t the initial temperature of stator and rotor windings	°C	Input
α_condstator	Stator temperature coefficient of resistance	Temperature coefficient of resistance of stator winding	°C^-1	Input
α_condrotor	Rotor temperature coefficient of resistance	Temperature coefficient of resistance of rotor winding	°C^-1	Input
L_ls	Stator leakage inductance	Stator winding leakage inductance of phase a, b, and c	H	Input
L_lr'	Rotor leakage inductance	Equivalent rotor winding leakage inductance referred to the stator of phase a, b, and c	H	Input
E_sat	Electrical saturation profile input	Choose your saturation profile between (

More

more details) Stator Voltage Line to Line vs. Stator Current Magnetizing Inductance vs. Magnetizing Flux	N/A	Dropdown
V_sll	Stator voltage line to line table	No Load saturation curve parameters	V	File (more details)
L_m	Magnetizing inductance table	Stator-rotor mutual (magnetizing) inductance of phase a, b, and c	H	File (more details)
F_cLm	Cut-off frequency magnetizing inductance	Cut-off frequency associated with the frequency response of the magnetic coil, which acts as a filtering element. The default value is 3 kHz.	Hz	Input
N_sr	Turns ratio	Stator to rotor windings turns ratio	N/A	Input
pp	Number of pole pairs	Number of pole pairs	N/A	Input
i_s	Stator phase currents	Stator currents measured at phases a, b and c	A	Measurement
i_r	Rotor phase currents	Rotor equivalent phase a, b, and c currents, referred to the stator	A	Measurement
i_sα_β	Stator αβcurrents	Stator currents in αβ (alpha-beta) reference frame	A	Measurement
i_sqd	Stator qd currents	Stator currents in dq reference frame	A	Measurement
i_rα_β	Rotor αβcurrents	Rotor currents in αβ reference frame, referred to the stator	A

Measurementi_sqdStator qd currentsStator currents in qd frame (αβ frame)AMeasurementi_rαβRotor αβ currentsRotor currents in αβ frame, referred to the statorAMeasurementi_rqdRotor qd currentsRotor currents in qd frame, referred to the stator (αβ frame)A

Measurement
i_rqd	Rotor qd currents	Rotor currents in dq reference frame, referred to the stator	A	Measurement
Frame_sel	Synchronous reference frame orientation	Selection of the synchronous reference frame among: Stator flux-oriented reference frame (default value); Rotor flux-oriented reference frame; Air gap flux-oriented reference frame.	N/A	Dropdown
Frame_offset	Synchronous reference offset	Offset Angle for Synchronous Reference Frame	degree	Input
θ_est	Synchronous angle	Estimated Synchronous angle	degree	Measurement
V_s_α_β	Stator αβvoltages	Stator voltages in αβ reference frame	V	Measurement
Φ_s_α_β	Stator αβ fluxes	Stator fluxes in

αβ

αβ reference frame	Wb	Measurement
Φ_r_α_β	Rotor αβ fluxes	Rotor fluxes in

αβ

αβ reference frame, referred to the stator	Wb	Measurement
R_sn	Snubber resistance	Resistances of the snubber on phase

A

a,

B

b and

C

c	Ω	Input
C_sn	Snubber capacitance	Capacitance of the snubber on phase

A

a,

B

b and

C

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

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

...

Encoder Parameters and Measurements

Symbol	Name	Description	Unit	Type
Enc_en	Enable encoder	Whether or not to enable the encoder	N/A	Input
Enc_type	Encoder type	Encoder type, either Quadrature or Hall Effect	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_dir	Direction of the sensor rotation	Direction in which the sensor is turning, either A leads B or B leads A	N/A	Input
Q_θ	Angle offset Δθ ( Sensor - Rotor )	Angle offset between the encoder and the rotor position from 0 to 360 degrees	°	Input
Q_rat	Encoder speed ratio ( sensor to mechanical position )	Mechanical to encoder ratio. Angle of Encoder = Qrat * machine mechanical angle.	N/A	Input
H_θ	Hall effect sensor position	Position of sensor phases A, B and C in Hall effect mode	°	Input

...

Visualization of Resolver Encoder Parameters Effects

...

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.

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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 )Image Removed

...

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 )Image Removed

...

Versions Compared

Old Version 2

New Version 3

Key

Model Formulation

Induction Machine Electrical Model

Mechanical Model

Resolver Encoder Model

Model Formulation

Parameters and Measurements

Electrical Parameters and Measurements

Δ_θ

Mechanical Parameters and Measurements

Resolver Parameters and Measurements

Encoder Parameters and Measurements

Visualization of Resolver Encoder Parameters Effects

Page Comparison

Versions Compared

Old Version 2

New Version 3

Key

Model Formulation

Induction Machine Electrical Model

Mechanical Model

Resolver Encoder Model

Model Formulation

Parameters and Measurements

Electrical Parameters and Measurements

Δθ

Mechanical Parameters and Measurements

Resolver Parameters and Measurements

Encoder Parameters and Measurements

Visualization of Resolver Encoder Parameters Effects

Δ_θ