# Three Phase Squirrel Cage Induction Machine (VBR)

Description of the Three Phase Squirrel Cage Induction Machine (VBR) component in Schematic Editor.

Table 1. Three Phase Squirrel Cage Induction Machine (VBR) component in Typhoon HIL Schematic Editor
component component dialog window component parameters  • Property tabs:

A, B, and C are the stator winding terminals. The stator winding uses the voltage behind reactance interface.

## Electrical sub-system model

The electrical part of the machine is represented by the following system of equations, modeled in the stationary αβ reference frame. All rotor variables and parameters are referred to the stator.

$\left[\begin{array}{c}{v}_{\alpha s}\\ {v}_{\beta s}\\ \begin{array}{c}{v}_{\alpha r}^{}\\ {v}_{\beta r}^{}\end{array}\end{array}\right]=\left[\begin{array}{cccc}{R}_{s}& 0& 0& 0\\ 0& {R}_{s}& 0& 0\\ 0& 0& {R}_{r}^{}& 0\\ 0& 0& 0& {R}_{r}^{}\end{array}\right]\left[\begin{array}{c}{i}_{\alpha s}\\ {i}_{\beta s}\\ \begin{array}{c}{i}_{\alpha r}^{}\\ {i}_{\beta r}^{}\end{array}\end{array}\right]+\frac{d}{dt}\left[\begin{array}{c}{\psi }_{\alpha s}\\ {\psi }_{\beta s}\\ \begin{array}{c}{\psi }_{\alpha r}^{}\\ {\psi }_{\beta r}^{}\end{array}\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ \begin{array}{c}{}_{}{\omega }_{r}{\psi }_{\beta r}{}_{}\\ {}_{}{-\omega }_{r}{\psi }_{\alpha r}{}_{}\end{array}\end{array}\right]$

$\left[\begin{array}{c}{\psi }_{\alpha s}\\ {\psi }_{\beta s}\\ \begin{array}{c}{\psi }_{\alpha r}^{}\\ {\psi }_{\beta r}^{}\end{array}\end{array}\right]=\left[\begin{array}{cccc}{L}_{s}& 0& {L}_{m}& 0\\ 0& {L}_{s}& 0& {L}_{m}\\ {L}_{m}& 0& {L}_{r}& 0\\ 0& {L}_{m}& 0& {L}_{r}\end{array}\right]\left[\begin{array}{c}{i}_{\alpha s}\\ {i}_{\beta s}\\ \begin{array}{c}{i}_{\alpha r}^{}\\ {i}_{\beta r}^{}\end{array}\end{array}\right]$

${T}_{e}=\frac{3}{2}p\left({\psi }_{\alpha s}{i}_{\beta s}-{\psi }_{\beta s}{i}_{\alpha s}\right)$
Table 2. Three Phase Squirrel Cage Induction Machine (VBR) electrical subsystem model variables
symbol description
ψαs Alpha axis component of the stator flux [Wb]
ψβs Beta axis component of the stator flux [Wb]
ψαr Alpha axis component of the rotor flux, referred to the stator [Wb]
ψβr Beta axis component of the rotor flux, referred to the stator [Wb]
iαs Alpha axis component of the stator current [A]
iβs Beta axis component of the stator current [A]
iαr Alpha axis component of the rotor current, referred to the stator [A]
iβr Beta axis component of the rotor current, referred to the stator [A]
vαs Alpha axis component of the stator voltage [V]
vβs Beta axis component of the stator voltage [V]
vαr Alpha axis component of the rotor voltage, referred to the stator [V]
vβr Beta axis component of the rotor voltage, referred to the stator [V]
Rs Stator phase resistance [Ω]
Rr Rotor phase resistance, referred to the stator [Ω]
Lm Magnetizing (mutual, main) inductance [H]
Ls Stator phase inductance [H] ( $={L}_{ls}+{L}_{m}$ )
Lr Rotor phase inductance, referred to the stator [H] ( $={L}_{lr}+{L}_{m}$ )
ωr Rotor electrical speed [rad/s] ( $=p{\omega }_{m}$ )
p Machine number of pole pairs
Te Machine developed electromagnetic torque [Nm]

## Mechanical sub-system model

Motion equation:

Table 3. Mechanical sub-system model variables
symbol description
Jm Combined rotor and load moment of inertia [kgm2]
Te Machine developed electromagnetic torque [Nm]
Tl Shaft mechanical load torque [Nm]
b Machine viscous friction coefficient [Nms]
Note: Motion equation is the same for all of the rotating machine models.

## Electrical

Table 4. Electrical parameters
symbol description
Rs Stator phase resistance [Ω]
Rr Rotor phase resistance, referred to the stator [Ω]
Lls Stator leakage inductance [H]
Llr Rotor leakage inductance, referred to the stator [H]
Lm Magnetizing (mutual, main) inductance [H]

In the Electrical block tab, you can enable the self-excitation of the induction machine. By enabling self-excitation, a new property for setting remanent flux is available. The remanent flux in the rotor of the machine induces back electromotive force in the stator windings at a frequency proportional to the rotor speed. The induction machine is then capable of operating as a generator without a grid supply.

Electrical sub-system model for the machine with enabled self-excitation:

$\left[\begin{array}{c}{v}_{\alpha s}\\ {v}_{\beta s}\\ \begin{array}{c}{v}_{\alpha r}^{}\\ {v}_{\beta r}^{}\end{array}\end{array}\right]=\left[\begin{array}{cccc}{R}_{s}& 0& 0& 0\\ 0& {R}_{s}& 0& 0\\ 0& 0& {R}_{r}^{}& 0\\ 0& 0& 0& {R}_{r}^{}\end{array}\right]\left[\begin{array}{c}{i}_{\alpha s}\\ {i}_{\beta s}\\ \begin{array}{c}{i}_{\alpha r}^{}\\ {i}_{\beta r}^{}\end{array}\end{array}\right]+\frac{d}{dt}\left[\begin{array}{c}{\psi }_{\alpha s}\\ {\psi }_{\beta s}\\ \begin{array}{c}{\psi }_{\alpha r}^{}\\ {\psi }_{\beta r}^{}\end{array}\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ \begin{array}{c}{}_{}{\omega }_{r}{\psi }_{\beta r}{}_{}\\ {}_{}{-\omega }_{r}{\psi }_{\alpha r}{}_{}\end{array}\end{array}\right]$

$\left[\begin{array}{c}{\psi }_{\alpha s}\\ {\psi }_{\beta s}\\ \begin{array}{c}{\psi }_{\alpha r}^{}\\ {\psi }_{\beta r}^{}\end{array}\end{array}\right]=\left[\begin{array}{cccc}{L}_{s}& 0& {L}_{m}& 0\\ 0& {L}_{s}& 0& {L}_{m}\\ {L}_{m}& 0& {L}_{r}& 0\\ 0& {L}_{m}& 0& {L}_{r}\end{array}\right]\left[\begin{array}{c}{i}_{\alpha s}\\ {i}_{\beta s}\\ \begin{array}{c}{i}_{\alpha r}^{}\\ {i}_{\beta r}^{}\end{array}\end{array}\right]+\left[\begin{array}{c}{}_{}{\psi }_{r}\mathrm{cos}{\theta }_{r}{}_{}\\ {}_{}{\psi }_{r}\mathrm{sin}{\theta }_{r}{}_{}\\ \begin{array}{c}0\\ 0\end{array}\end{array}\right]$

${T}_{e}=\frac{3}{2}p\left({\psi }_{\alpha s}{i}_{\beta s}-{\psi }_{\beta s}{i}_{\alpha s}\right)$

where: ψr is remanent flux in the machine rotor and θr is rotor electrical angle.

Note: This model doesn't include hysteresis, that is, initial magnetization curve is used.

## Mechanical

Table 5. Mechanical parameters
symbol description
pms Machine number of pole pairs
Star/delta Stator winding connection (star or delta)
Jm Combined rotor and load moment of inertia [kgm2]
Friction coefficient Machine viscous friction coefficient [Nms]
Unconstrained mechanical angle Limiting mechanical angle between 0 and 2π

symbol description
Load source Load can be set from SCADA/external or from model (in model case, one signal processing input will appear)
Load ai offset Assigned offset value to the input signal representing external torque command
Load ai gain Assigned gain value to the input signal representing external torque command

External load enables you to use an analog input signal from a HIL analog channel with the load_ai_pin address as an external torque/speed load, and to assign offset (V) and gain (Nm/V) to the input signal, according to the formula:

${T}_{l}=load_ai_gain·\left(AI\left(load_ai_pin\right)+load_ai_offset\right)$

Note: An analog input pin may be overwritten if another component uses the same analog input pin. If another property (from the same or a different component) uses the same analog input pin, the input signal value will be applied to only one of those properties. For instance, if both the load and the resolver carrier signal use the same analog input pin, the signal value will only be applied only to one of these.

## Feedback

Table 7. Feedback parameters
symbol description
Encoder ppr Incremental encoder number of pulses per revolution
Resolver pole pairs Resolver number of pole pairs
Resolver carrier source Resolver carrier signal source selection (internal or external)
External resolver carrier source type External resolver carrier signal source type selection (single ended or differential); available only if the Resolver carrier source property is set to external
Resolver carrier frequency Resolver carrier signal frequency (internal carrier) [Hz]
Resolver ai pin 1 Resolver carrier input channel 1 address (external carrier)
Resolver ai pin 2 Resolver carrier input channel 2 address (external carrier); available only if the External resolver carrier source type property is set to differential
Resolver ai offset Resolver carrier input channel offset (external carrier)
Resolver ai gain Resolver carrier input channel gain (external carrier)
Absolute encoder protocol Standardized protocol providing the absolute machine encoder position

If an external resolver carrier source is selected, the source signal type can be set as either single ended or differential. The single ended external resolver carrier source type enables use of an analog input signal from the HIL analog channel with the res_ai_pin_1 address as the external carrier source. Additionally, offset (V) and gain (V/V) values can be assigned to the input signal, according to the formula:

$res_carr_src=res_ai_gain·\left(AI\left(res_ai_pin_1\right)+res_ai_offset\right)$

The differential external resolver carrier source type enables use of two analog input signals from the HIL analog channels with the res_ai_pin_1 and the res_ai_pin_2 addresses. Analog signals from these HIL analog inputs are subtracted, and the resulting signal is used as the external differential carrier source. Additionally, offset (V) and gain (V/V) values can be assigned to the input signal (similarly to the single ended case), according to the formula:

$res_carr_src=res_ai_gain·\left(\left(AI\left(res_ai_pin_1\right)-AI\left(res_ai_pin_2\right)\right)+res_ai_offset\right)$
Note: Differential external resolver signal support is available from the 2023.1 software version. Only the single ended external resolver option was available in previous software versions.
Note: In order to get a resolver signal with an amplitude of 1 when using an external carrier signal, the offset and the gain should be chosen in such a way that the resolver carrier signal has an amplitude of 1 after the adjustment. As shown in Figure 1, the sinusoidal signal used to generate external resolver carrier source is fed to HIL analog input 1. The analog input signal is scaled in order to get the resolver signals with an amplitude of 1.
Note: An analog input pin may be overwritten if another component uses the same analog input pin. If another property (from the same or a different component) uses the same analog input pin, the input signal value will be applied to only one of those properties. E.g. if both the load and the resolver carrier signal use the same analog input pin, the signal value will be applied only to one of these.

The following expression must hold in order to properly generate the encoder signals:

$4·enc_ppr·{f}_{m}{·T}_{s}\le 1$
Table 8. Variables in the encoder limitation expression
symbol description
enc_ppr Encoder number of pulses per revolution
fm Rotor mechanical frequency [Hz]
Ts Simulation time step [s]
Note: While the machine speed is positive, the encoder channel B signal leads the encoder channel A signal.
Note: Absolute encoder protocol is not supported on HIL402 (configurations: 1, 2, 3, and 4).

symbol description
Theta_ab Position of the stationary αβ reference frame, in respect to the stator phase a axis [rad]

The machine model output variables (currents, voltages and fluxes) can be observed from a stationary reference frame. There are two widely used approaches in electrical machine modeling: in the first, the alpha axis of the stationary reference frame lags by 90 degrees in regard to the stator phase a axis (used by default, and indicated in a) Figure 5. In the second one, the alpha axis is aligned with the stator phase a axis (indicated in b) Figure 5. The user can select between these two situations. Figure 4. Selection of a stationary reference frame position: a) Theta_ab= -pi/2; b) Theta_ab= 0

It is important to know the value of Theta_ab when the rotor position feedback is necessary. As an example, if a model uses the mechanical angle as a feedback signal and feeds it to one of the abc to dq, alpha beta to dq, dq to abc, or dq to alpha beta transformation blocks, the same transformation angle offset value should be used in both components to ensure the expected simulation results. Figure 5. Example of model containing abc to dq transformation component and chosen stationary reference frame position Theta_ab= -pi/2
Note: This property is available only on certain machine components.

## Output

This block tab enables a single, vectorized signal output from the machine. The output vector contains selected machine mechanical and/or electrical variables in the same order as listed in this tab.

Note: All machine components have the Execution rate, Electrical torque, Mechanical speed, and Mechanical angle, but the rest of the signals differ from component to component.
Table 10. Output parameters
symbol description
Execution rate Signal processing output execution rate [s]
Electrical torque Machine electrical torque [Nm]
Mechanical speed Machine mechanical angular speed [rad/s]
Mechanical angle Machine mechanical angle [rad]
Stator alpha axis emf Alpha axis component of the stator VBR equivalent back emf [V]
Stator beta axis emf Beta axis component of the stator VBR equivalent back emf [V]
Rotor alpha axis current Alpha axis component of the rotor current, referred to the stator [A]
Rotor beta axis current Beta axis component of the rotor current, referred to the stator [A]
Stator alpha axis flux Alpha axis component of the stator flux [Wb]
Stator beta axis flux Beta axis component of the stator flux [Wb]
Rotor alpha axis flux Alpha axis component of the rotor flux, referred to the stator [Wb]
Rotor beta axis flux Beta axis component of the rotor flux, referred to the stator [Wb]