Six Phase Doubly Fed Induction Machine (Double Stator)
Description of the Six Phase Doubly Fed Induction Machine (Double Stator) component in Schematic Editor.
component  component dialog window  component parameters 

Property tabs:

A1, B1, and C1 are the stator winding 1 terminals. Similarly, A2, B2, and C2 are the stator winding 2 terminals. Ar, Br, and Cr represent the rotor winding terminals. All windings use the current source interface.
Electrical subsystem 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.
$\left[\begin{array}{c}{v}_{\alpha s1}\\ {v}_{\beta s1}\\ {v}_{\alpha s2}\\ {v}_{\beta s2}\\ {v}_{\alpha r}\\ {v}_{\beta r}\end{array}\right]=\left[\begin{array}{cccccc}{R}_{s}& 0& 0& 0& 0& 0\\ 0& {R}_{s}& 0& 0& 0& 0\\ 0& 0& {R}_{s}& 0& 0& 0\\ 0& 0& 0& {R}_{s}& 0& 0\\ 0& 0& 0& 0& {R}_{r}& 0\\ 0& 0& 0& 0& 0& {R}_{r}\end{array}\right]\left[\begin{array}{c}{i}_{\alpha s1}\\ {i}_{\beta s1}\\ {i}_{\alpha s2}\\ {i}_{\beta s2}\\ {i}_{\alpha r}\\ {i}_{\beta r}\end{array}\right]+\frac{d}{dt}\left[\begin{array}{c}\begin{array}{c}{\psi}_{\alpha s1}\\ {\psi}_{\beta s1}\end{array}\\ {\psi}_{\alpha s2}\\ {\psi}_{\beta s2}\\ {\psi}_{\alpha r}\\ {\psi}_{\beta r}\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ \begin{array}{c}{\omega}_{r}{\psi}_{\beta r}\\ {\omega}_{r}{\psi}_{\alpha r}\end{array}\end{array}\right]$
$\left[\begin{array}{c}{\psi}_{\alpha s1}\\ {\psi}_{\beta s1}\\ {\psi}_{\alpha s2}\\ {\psi}_{\beta s2}\\ \begin{array}{c}{\psi}_{\alpha r}\\ {\psi}_{\beta r}\end{array}\end{array}\right]=\left[\begin{array}{cccccc}{L}_{ls}+{L}_{lm}+{L}_{m}& 0& {L}_{lm}+{L}_{m}& {L}_{l\alpha \beta}& {L}_{m}& 0\\ 0& {L}_{ls}+{L}_{lm}+{L}_{m}& {L}_{l\alpha \beta}& {L}_{lm}+{L}_{m}& 0& {L}_{m}\\ {L}_{lm}+{L}_{m}& {L}_{l\alpha \beta}& {L}_{ls}+{L}_{lm}+{L}_{m}& 0& {L}_{m}& 0\\ {L}_{l\alpha \beta}& {L}_{lm}+{L}_{m}& 0& {L}_{ls}+{L}_{lm}+{L}_{m}& 0& {L}_{m}\\ {L}_{m}& 0& {L}_{m}& 0& {L}_{lr}+{L}_{m}& 0\\ 0& {L}_{m}& 0& {L}_{m}& 0& {L}_{lr}+{L}_{m}\end{array}\right]\left[\begin{array}{c}\begin{array}{c}{i}_{\alpha s1}\\ {i}_{\beta s1}\end{array}\\ {i}_{\alpha s2}\\ {i}_{\beta s2}\\ {i}_{\alpha r}\\ {i}_{\beta r}\end{array}\right]$
${T}_{e}=\frac{3}{2}p{L}_{m}\left(\left({i}_{\beta s1}+{i}_{\beta s2}\right){i}_{\alpha r}\left({i}_{\alpha s1}+{i}_{\alpha s2}\right){i}_{\beta r}\right)$
${v}_{\alpha r}^{}=m{v}_{\alpha r}^{r}$ ${v}_{\beta r}^{}=m{v}_{\beta r}^{r}$ ${i}_{\alpha r}^{r}=m{i}_{\alpha r}^{}$ ${i}_{\beta r}^{r}=m{i}_{\beta r}^{}$In the equations above L_{lm} and L_{lαβ} are the mutual leakage inductance terms, obtained from the following expressions:
${L}_{lm}={L}_{la1a2}\mathrm{cos}\zeta +{L}_{la1b2}\mathrm{cos}\left(\zeta +\frac{2\pi}{3}\right)+{L}_{la1c2}\mathrm{cos}\left(\zeta \frac{2\pi}{3}\right)$
${L}_{l\alpha \beta}={L}_{la1a2}\mathrm{sin}\zeta +{L}_{la1b2}\mathrm{sin}\left(\zeta +\frac{2\pi}{3}\right)+{L}_{la1c2}\mathrm{sin}\left(\zeta \frac{2\pi}{3}\right)$
where L_{la1a2}, L_{la1b2}, and L_{la1c2} are the mutual leakage inductances between the two stator winding sets and $\zeta $ is the displacement angle between the two stator windings. It is assumed that ${L}_{la1a2}={L}_{lb1b2}={L}_{lc1c2}$ , ${L}_{la1b2}={L}_{lb1c2}={L}_{lc1a2}$ , and ${L}_{la1c2}={L}_{lb1a2}={L}_{lc1b2}$ .
symbol  Description 
ψ_{αs1}  Alpha axis component of stator winding 1 flux [Wb] 
ψ_{βs1}  Beta axis component of stator winding 1 flux [Wb] 
ψ_{αs2}  Alpha axis component of stator winding 2 flux [Wb] 
ψ_{βs2}  Beta axis component of stator winding 2 flux [Wb] 
ψ_{αr}  Alpha axis component of rotor winding flux, referred to the stator [Wb] 
ψ_{βr}  Beta axis component of rotor winding flux, referred to the stator [Wb] 
i_{αs1}  Alpha axis component of stator winding 1 current [A] 
i_{βs1}  Beta axis component of stator winding 1 current [A] 
i_{αs2}  Alpha axis component of stator winding 2 current [A] 
i_{βs2}  Beta axis component of stator winding 2 current [A] 
i_{αr}  Alpha axis component of rotor winding current, referred to the stator [A] 
i_{βr}  Beta axis component of rotor winding current, referred to the stator [A] 
i_{αr}^{r}  Alpha axis component of the rotor current [A] 
i_{βr}^{r}  Beta axis component of the rotor current [A] 
v_{αs1}  Alpha axis component of stator winding 1 voltage [V] 
v_{βs1}  Beta axis component of stator winding 1 voltage [V] 
v_{αs2}  Alpha axis component of stator winding 2 voltage [V] 
v_{βs2}  Beta axis component of stator winding 2 voltage [V] 
v_{αr}  Alpha axis component of rotor winding voltage, referred to the stator [V] 
v_{βr}  Beta axis component of rotor winding voltage, referred to the stator [V] 
v_{αr}^{r}  Alpha axis component of the rotor voltage [V] 
v_{βr}^{r}  Beta axis component of the rotor voltage [V] 
R_{s}  Stator phase resistance [Ω] 
R_{r}  Rotor phase resistance, referred to the stator [Ω] 
L_{ls}  Stator leakage inductance [H] 
L_{la1a2}  Mutual leakage inductance between phase a of stator winding 1 and phase a of stator winding 2 [H] 
L_{la1b2}  Mutual leakage inductance between phase a of stator winding 1 and phase b of stator winding 2 [H] 
L_{la1c2}  Mutual leakage inductance between phase a of stator winding 1 and phase c of stator winding 2 [H] 
L_{lr}  Rotor leakage inductance, referred to the stator [H] 
L_{m}  Magnetizing (mutual, main) inductance [H] 
ω_{r}  Rotor electrical speed [rad/s] ( $=p{\omega}_{m}^{}$ ) 
ζ  Displacement angle between stator windings 1 and 2 [rad] 
p  Machine number of pole pairs 
T_{e}  Machine developed electromagnetic torque [Nm] 
m  Stator to rotor turns ratio 
Mechanical subsystem model
Motion equation:
$\frac{{d\omega}_{m}}{dt}=\frac{1}{{J}_{m}}({T}_{e}{T}_{l}b{\omega}_{m})$ ${\theta}_{m}=\int {\omega}_{m}dt$symbol  description 

ω_{m}  Rotor mechanical speed [rad/s] 
J_{m}  Combined rotor and load moment of inertia [kgm2] 
T_{e}  Machine developed electromagnetic torque [Nm] 
T_{l}  Shaft mechanical load torque [Nm] 
b  Machine viscous friction coefficient [Nms] 
θ_{m}  Rotor mechanical angle [rad] 
Electrical
symbol  description 

R_{s}  Stator phase resistance [Ω] 
L_{ls}  Stator leakage inductance [H] 
L_{la1a2}  Mutual leakage inductance between phase a of stator winding 1 and phase a of stator winding 2 [H] 
L_{la1b2}  Mutual leakage inductance between phase a of stator winding 1 and phase b of stator winding 2 [H] 
L_{la1c2}  Mutual leakage inductance between phase a of stator winding 1 and phase c of stator winding 2 [H] 
R_{r}  Rotor phase resistance, referred to the stator [Ω] 
L_{lr}  Rotor leakage inductance, referred to the stator [H] 
L_{m}  Magnetizing (mutual, main) inductance [H] 
m  Stator to rotor winding turns ratio 
Displacement  Displacement angle between stator windings 1 and 2 [rad] 
Mechanical
symbol  description 

pms  Machine number of pole pairs 
J_{m}  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
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:
${T}_{l}=load\_ai\_gain\xb7\left(AI\right(load\_ai\_pin)+load\_ai\_offset)$
Feedback
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:
$res\_carr\_src=res\_ai\_gain\xb7\left(AI\right(res\_ai\_pin\_1)+res\_ai\_offset)$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:
$res\_carr\_src=res\_ai\_gain\xb7\left(\right(AI(res\_ai\_pin\_1)AI(res\_ai\_pin\_2))+res\_ai\_offset)$The following expression must hold in order to properly generate the encoder signals:
$4\xb7enc\_ppr\xb7{f}_{m}{\xb7T}_{s}\le 1$symbol  description 

enc_ppr  Encoder number of pulses per revolution 
f_{m}  Rotor mechanical frequency [Hz] 
T_{s}  Simulation time step [s] 
Advanced
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.
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.
symbol  description 

R_{snb} rotor  Rotor snubber resistance value [Ω] 
R_{snb} stator w1  Stator winding 1 snubber resistance value [Ω] 
R_{snb} stator w2  Stator winding 2 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.
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 1 alpha axis current  Alpha axis component of the stator 1 current [A] 
Stator 1 beta axis current  Beta axis component of the stator 1 current [A] 
Stator 1 alpha axis flux  Alpha axis component of the stator 1 flux [A] 
Stator 1 beta axis flux  Beta axis component of the stator 1 flux [Wb] 
Stator 2 alpha axis current  Alpha axis component of the stator 2 current [A] 
Stator 2 beta axis current  Beta axis component of the stator 2 current [A] 
Stator 2 alpha axis flux  Alpha axis component of the stator 2 flux [Wb] 
Stator 2 beta axis flux  Beta axis component of the stator 2 flux [Wb] 