Six Phase Doubly Fed Induction Machine (Double Stator)

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 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.

$\left[\begin{array}{c}{v}_{\alpha s1}\\ v_{\beta s1}\\ v_{\alpha s2}\\ v_{\beta s2}\\ v_{\alpha r}\\ v_{\beta r}\end{array}\right]=\left[\begin{array}{cccccc}{R}_s& 0& 0& 0& 0& 0\\ 0& R_s& 0& 0& 0& 0\\ 0& 0& R_s& 0& 0& 0\\ 0& 0& 0& R_s& 0& 0\\ 0& 0& 0& 0& R_r& 0\\ 0& 0& 0& 0& 0& R_r\end{array}\right]\left[\begin{array}{c}{i}_{\alpha s1}\\ i_{\beta s1}\\ i_{\alpha s2}\\ i_{\beta s2}\\ i_{\alpha r}\\ i_{\beta r}\end{array}\right]+\frac{d}{dt}\left[\begin{array}{c}\begin{array}{c}{\psi }_{\alpha s1}\\ {\psi }_{\beta s1}\end{array}\\ {\psi }_{\alpha s2}\\ {\psi }_{\beta s2}\\ {\psi }_{\alpha r}\\ {\psi }_{\beta r}\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ \begin{array}{c}{\omega }_r{\psi }_{\beta r}\\ {-\omega }_r{\psi }_{\alpha r}\end{array}\end{array}\right]$

$\left[\begin{array}{c}{\psi }_{\alpha s1}\\ {\psi }_{\beta s1}\\ {\psi }_{\alpha s2}\\ {\psi }_{\beta s2}\\ \begin{array}{c}{\psi }_{\alpha r}\\ {\psi }_{\beta r}\end{array}\end{array}\right]=\left[\begin{array}{cccccc}{L}_{ls}+L_{lm}+L_m& 0& L_{lm}+L_m& L_{l\alpha \beta }& L_m& 0\\ 0& L_{ls}+L_{lm}+L_m& {-L}_{l\alpha \beta }& L_{lm}+L_m& 0& L_m\\ L_{lm}+L_m& {-L}_{l\alpha \beta }& L_{ls}+L_{lm}+L_m& 0& L_m& 0\\ L_{l\alpha \beta }& L_{lm}+L_m& 0& L_{ls}+L_{lm}+L_m& 0& L_m\\ L_m& 0& L_m& 0& L_{lr}+L_m& 0\\ 0& L_m& 0& L_m& 0& L_{lr}+L_m\end{array}\right]\left[\begin{array}{c}\begin{array}{c}{i}_{\alpha s1}\\ i_{\beta s1}\end{array}\\ i_{\alpha s2}\\ i_{\beta s2}\\ i_{\alpha r}\\ i_{\beta r}\end{array}\right]$

$T_e=\frac{3}{2}p{L}_m\left(\left(i_{\beta s1}+i_{\beta s2}\right)i_{\alpha r}-\left(i_{\alpha s1}+i_{\alpha s2}\right)i_{\beta r}\right)$

In the equations above L_lm and L_lαβ are the mutual leakage inductance terms, obtained from the following expressions:

$L_{lm}=L_{la1a2}\cos \zeta +L_{la1b2}\cos \left(\zeta +\frac{2\pi }{3}\right)+L_{la1c2}\cos \left(\zeta -\frac{2\pi }{3}\right)$

$L_{l\alpha \beta }=L_{la1a2}\sin \zeta +L_{la1b2}\sin \left(\zeta +\frac{2\pi }{3}\right)+L_{la1c2}\sin \left(\zeta -\frac{2\pi }{3}\right)$

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 $L_{la1c2}=L_{lb1a2}=L_{lc1b2}$ .

Mechanical sub-system model

Motion equation:

Ports

• A1 (electrical)
  • Stator winding 1 phase A port.

• B1 (electrical)
  • Stator winding 1 phase B port.

• C1 (electrical)
  • Stator winding 1 phase C port.

• A2 (electrical)
  • Stator winding 2 phase A port.

• B2 (electrical)
  • Stator winding 2 phase B port.

• C2 (electrical)
  • Stator winding 2 phase C port.

• Ar (electrical)
  • Rotor winding phase A port.

• Br (electrical)
  • Rotor winding phase B port.

• Cr (electrical)
  • Rotor winding phase C port.

• in
  • Available if Model Load source is selected.

Electrical (Tab)

Mechanical (Tab)

• pms
  • Machine number of pole pairs
• 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 (Tab)

External load enables you to use an analog input signal from a HIL/TyphoonSim (internal virtual IO bus in TyphoonSim) 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:

$T_l=load\_ai\_gain·\left(AI\right(load\_ai\_pin)+load\_ai\_offset)$

Feedback (Tab)

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/TyphoonSim (internal virtual IO bus in TyphoonSim) 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:

The differential external resolver carrier source type enables use of two analog input signals from the HIL/TyphoonSim (internal virtual IO bus in TyphoonSim) analog channels with the res_ai_pin_1 and the res_ai_pin_2 addresses. Analog signals from these HIL/TyphoonSim (internal virtual IO bus in TyphoonSim) 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:

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

Table 3. 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]

Advanced (Tab)

• Theta_ab
  • Position of the stationary αβ reference frame, in respect to the stator phase a axis [rad]
• Field Input
  • Physical quantity applied to the field winding - voltage or current. It refers to the rotor-side.

  • In order for the input field current to be referred to the stator parameter, N_s/N_fd in the Electrical Tab should be set to $\frac{3}{2}$

    Note: If field input is defined as current, the field winding equation is removed from the mathematical model. Instead, the field input current is used as the model's input. There is no support for 2 q-axis damper windings with current input as of yet. For linear machine models, you can choose whether to apply voltage or current to the field winding. Nonlinear machine models only allows voltage as a field input.
  • Field Input option is not supported in TyphoonSim. Changing its value will not affect TyphoonSim simulation at all

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 (Tab)

• Rsnb stator
  • Stator snubber resistance value [Ω]

• Rsnb stator w1
  • Stator winding 1 snubber resistance value [Ω]

• Rsnb stator w2
  • Stator winding 2 snubber resistance value [Ω]

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 7 and Figure 8 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.

Output (Tab)

• 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]

Extras (Tab)

The Extras tab gives you the opportunity to set Signal Access Management for the component.