# Three Phase Doubly Fed Induction Machine

Description of the Three Phase Doubly Fed Induction Machine component in Schematic Editor.

Table 1. Three Phase Doubly Fed Induction Machine component in Typhoon HIL Schematic Editor
component component dialog window component parameter
• Property tabs:
As, Bs, and Cs are the stator winding terminals. Ar, Br, and Cr are the rotor winding terminals. Both the stator and the rotor winding use the current source 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 by the turns ratio m.

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

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

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

${v}_{\alpha r}^{}=m{v}_{\alpha r}^{r}$

${v}_{\beta r}^{}=m{v}_{\beta r}^{r}$

${i}_{\alpha r}^{r}=m{i}_{\alpha r}^{}$

${i}_{\beta r}^{r}=m{i}_{\beta r}^{}$

If the saturation effects are considered, the equations have the same form, but in that case the magnetizing flux is a function of the magnetizing current:

${\psi }_{m}=f\left({i}_{m}\right)$

where:

${\psi }_{m}=\sqrt{{\left({\psi }_{\alpha s}+{\psi }_{\alpha r}\right)}^{2}+{\left({\psi }_{\beta s}+{\psi }_{\beta r}\right)}^{2}}$

${i}_{m}=\sqrt{{\left({i}_{\alpha s}+{i}_{\alpha r}\right)}^{2}+{\left({i}_{\beta s}+{i}_{\beta r}\right)}^{2}}$

If the saturation effects and stator and rotor leakage fluxes are considered, the equations also have the same form, except in that case, the magnetizing flux is a function of the magnetizing current, the stator leakage flux is a function of the stator current, and the rotor leakage flux is a function of the rotor current:

${\psi }_{ls}={f}_{s}\left({i}_{s}\right)$

${\psi }_{lr}={f}_{r}\left({i}_{r}\right)$

where:

${i}_{s}=\sqrt{{i}_{\alpha s}^{2}+{i}_{\beta s}^{2}}$

${i}_{r}=\sqrt{{i}_{\alpha r}^{2}+{i}_{\beta r}^{2}}$
Table 2. 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]
iαrr Alpha axis component of the rotor current [A]
iβrr Beta axis component of the rotor current [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]
vαrr Alpha axis component of the rotor voltage [V]
vβrr Beta axis component of the rotor voltage [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]
m Stator to rotor turns ratio
ψm Magnetizing flux [Wb]
im Magnetizing current [A]
ψls Stator leakage flux [H]
is Amplitude of the stator current [A]
ψlr Rotor leakage flux [H]
ir Amplitude of the rotor current [A]

## Mechanical sub-system model

Motion equation:

Table 3. Mechanical sub-system model variables
symbol description
ωm Rotor mechanical speed [rad/s]
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]
θm Rotor mechanical angle [rad]
Note: Motion equation is the same for all of the rotating machine models.

## Electrical

This component offers two levels of model fidelity, designated by the Model Type property. The following options are available:
• linear
• nonlinear
If the selected Model Type is nonlinear, the following saturation types can be specified:
• flux vs current
• absolute inductance vs current
Note: If the Saturable leakage inductance checkbox is checked, the only available saturation type is absolute inductance vs current.
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]
Saturable leakage inductance If enabled, stator leakage inductance is a function of the stator current and rotor leakage inductance is a function of the rotor current
is vector List of instantaneous values of the stator current [A]
Lls vector List of the stator leakage inductance values [H]
ir vector List of instantaneous values of the rotor current [A]
Llr vector List of the rotor leakage inductance values [H]
im vector List of instantaneous values of the magnetizing current [A]
psim vector List of instantaneous values of the magnetizing flux [Wb]
Lm vector List of the magnetizing inductance values [H]
m Stator to rotor phase voltage ratio (turns ratio)
The Three phase doubly fed induction machine can include magnetic saturation effects, according to the functions described in Electrical sub-system model. Each of these functions are represented in the model in the form of lookup tables. Lookup tables use linear interpolation and linear extrapolation. This means that if magnetic saturation effects are considered, lookup tables defining either magnetizing flux (Figure 1) or magnetizing inductance (Figure 2) as a function of the magnetizing current should be used. If the Saturable leakage inductance property is enabled, then lookup tables for the saturation effects of magnetizing inductance as a function of magnetizing current (Figure 2) and stator (Figure 3) and rotor (Figure 4) leakage fluxes as a function of their respective currents are considered.
Saturation can be parametrized in the following ways:
1. magnetizing flux vs magnetizing current
2. magnetizing inductance vs magnetizing current
im_vector = [0.0, 0.661428, 0.957988, 1.224002, 1.527775, 1.836498, 2.485056, 3.197537, 4.162313, 5.57879, 8.211348, 12.342407, 22.172606]
psim_vector = [0.0, 0.125279, 0.192308, 0.25488, 0.318532, 0.382499, 0.511695, 0.635623, 0.76725, 0.885866, 1.007544, 1.097936, 1.186302]
im_vector = [0.00, 2.50, 5.00, 6.75, 7.50, 8.75, 11.50, 15.00, 20.00, 25.00]
Lm_vector = [0.00, 0.060648, 0.059024, 0.0561556, 0.054150667, 0.0495085714, 0.0404947826, 0.033212, 0.0265335, 0.0220932]
is_vector = [0.00, 1.88, 5.00, 10.00, 15.00, 20.00, 25.00, 30.00, 35.00, 40.00, 45.00, 50.00, 55.00, 60.00, 65.00, 70.00]
Lls_vector = [0.00, 0.0036010638, 0.00325, 0.002653, 0.00234667, 0.002166, 0.0020576, 0.00196733, 0.0018797142, 0.0018275, 0.001744889, 0.0016786, 0.001624545, 0.001579334, 0.00152030769, 0.0014698571]
ir_vector = [0.00, 1.88, 5.00, 10.00, 15.00, 20.00, 25.00, 30.00, 35.00, 40.00, 45.00, 50.00, 55.00, 60.00, 65.00, 70.00]
Llr_vector = [0.00, 0.0036010638, 0.00325, 0.002653, 0.00234667, 0.002166, 0.0020576, 0.00196733, 0.0018797142, 0.0018275, 0.001744889, 0.0016786, 0.001624545, 0.001579334, 0.00152030769, 0.0014698571]

## 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

## Load

Table 6. Load parameters
symbol description
Load source Load can be set from SCADA/external or from model (in model case, one signal processing input will appear)
External/Model load type External/Model load type: torque or speed
Load ai pin HIL analog input address for external torque command
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)
Resolver carrier frequency Resolver carrier signal frequency (internal carrier) [Hz]
Resolver ai pin Resolver carrier input channel address (external carrier)
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

External resolver carrier source enables the user to use an analog input signal from a HIL analog channel with the res_ai_pin address as an external carrier source, and to assign offset (V) and gain (V/V) to the input signal, according to the formula:

$res_carr_src=res_ai_gain·\left(AI\left(res_ai_pin\right)+res_ai_offset\right)$
Note: In order to get the resolver signals 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, sinusoidal signal used to generate external resolver carrier source is fed to HIL's analog input 1. 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).
Note: More information about absolute encoder protocol could be found here.

## Advanced

Table 9. Advanced parameters
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.

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.

Note: This property is available only on certain machine components.

## Snubber

All machines with current source based circuit interfaces have the Snubber tab in the properties window where the value of snubber resistance can be set. Snubbers are necessary in the cases when an inverter or a contactor is directly connected to the machine terminals. This value can be set to infinite (inf), but it is not recommended when a machine is directly connected to the inverter since there will be a current source directly connected to an open switch. In this case, one of each switch pairs S1 and S2, S3 and S4, and S5 and S6 will be forced closed by the circuit solver in order to avoid the topological conflicts. On the other hand, with finite snubber values, there's always a path for the currents Ia and Ib, so all inverter switches can be open in this case. Circuit representations of this circuit without and with snubber resistors are shown in Figure 3 and Figure 4 respectively. Snubbers are connected across the current sources.

Note: Snubbers exist only in the machine components that have the current source based circuit interface.
Note: Snubbers are dynamic, which means that the snubber is dynamically added to circuit modes where topological conflicts are detected.
Table 10. Snubber parameters
symbol description
Rsnb stator Stator snubber resistance value [Ω]
Rsnb rotor Rotor snubber resistance value [Ω]

## 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 11. 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 current Alpha axis component of the stator current [A]
Stator beta axis current Beta axis component of the stator current [A]
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]