Nine Phase Squirrel Cage Induction Machine
Description of the Nine Phase Squirrel Cage Induction Machine component in Schematic Editor.
Component  Component dialog window  Component parameters 


In this component, terminals A, B, C, D, E, F, G, H, and I are stator winding terminals. The stator winding uses the current source interface. The stator consists of nine phase windings, displaced by 40 degrees. Each phase is considered equidistant to both the previous and next phase. Using a Clarke transformation, all 9 phases map into a single αβ domain.
Electrical subsystem model
The electrical part of the machine is represented by the following system of equations, modeled in the stationary αβ reference frame. In this model, leakage inductances between different phases are neglected and only selfleakage inductance is included. The first two $\alpha \beta $ stator components define variables that will lead to fundamental flux and torque production. Equations for $\mathrm{\alpha \_n}\mathrm{\beta \_n}$ components are completely decoupled from all other $\alpha \beta $ components. These correspond to certain voltage, flux, and current harmonics and do not contribute to torque production.
$\left[\begin{array}{c}{v}_{\mathrm{\alpha}\mathrm{s}}\\ {v}_{\beta s}\\ {v}_{\alpha \mathrm{s}1}\\ {v}_{\beta s1}\\ {v}_{\alpha s2}\\ {v}_{\beta s2}\\ {v}_{\alpha s3}\\ {v}_{\beta s3}\\ {v}_{\alpha r}\\ {v}_{\beta r}\end{array}\right]=\left[\begin{array}{cccccccccc}{R}_{s}& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& {R}_{s}& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& {R}_{s}& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& {R}_{s}& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& {R}_{s}& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& {R}_{s}& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& {R}_{s}& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& {R}_{s}& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& {R}_{r}& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& {R}_{r}\end{array}\right]\left[\begin{array}{c}{i}_{\mathrm{\alpha}\mathrm{s}}\\ {i}_{\beta s}\\ {i}_{\alpha s1}\\ {i}_{\beta s1}\\ {i}_{\alpha s2}\\ {i}_{\beta s2}\\ {i}_{\alpha s3}\\ {i}_{\beta s3}\\ {i}_{\alpha r}\\ {i}_{\beta r}\end{array}\right]+\frac{d}{dt}\left[\begin{array}{c}{\psi}_{\mathrm{\alpha}\mathrm{s}}\\ {\psi}_{\beta s}\\ {\psi}_{\alpha s1}\\ {\psi}_{\beta s1}\\ {\psi}_{\alpha s2}\\ {\psi}_{\beta s2}\\ {\psi}_{\alpha s3}\\ {\psi}_{\beta s3}\\ {\psi}_{\alpha r}\\ {\psi}_{\beta r}\end{array}\right]+\left[\begin{array}{cccccccccc}0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& {\omega}_{r}\\ 0& 0& 0& 0& 0& 0& 0& 0& {\omega}_{r}& 0\end{array}\right]\left[\begin{array}{c}{\psi}_{\mathrm{\alpha}\mathrm{s}}\\ {\psi}_{\beta s}\\ {\psi}_{\alpha s1}\\ {\psi}_{\beta s1}\\ {\psi}_{\alpha s2}\\ {\psi}_{\beta s2}\\ {\psi}_{\alpha s3}\\ {\psi}_{\beta s3}\\ {\psi}_{\alpha r}\\ {\psi}_{\beta r}\end{array}\right]$ $\left[\begin{array}{c}{\psi}_{\mathrm{\alpha}\mathrm{s}}\\ {\psi}_{\beta s}\\ {\psi}_{\alpha s1}\\ {\psi}_{\beta s1}\\ {\psi}_{\alpha s2}\\ {\psi}_{\beta s2}\\ {\psi}_{\alpha s3}\\ {\psi}_{\beta s3}\\ {\psi}_{\alpha r}\\ {\psi}_{\beta r}\end{array}\right]=\left[\begin{array}{cccccccccc}{L}_{s}& 0& 0& 0& 0& 0& 0& 0& {L}_{m}& 0\\ 0& {L}_{s}& 0& 0& 0& 0& 0& 0& 0& {L}_{m}\\ 0& 0& {L}_{ls}& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& {L}_{ls}& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& {L}_{ls}& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& {L}_{ls}& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& {L}_{ls}& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& {L}_{ls}& 0& 0\\ {L}_{m}& 0& 0& 0& 0& 0& 0& 0& {L}_{r}& 0\\ 0& {L}_{m}& 0& 0& 0& 0& 0& 0& 0& {L}_{r}\end{array}\right]\left[\begin{array}{c}{i}_{\mathrm{\alpha}\mathrm{s}}\\ {i}_{\beta s}\\ {i}_{\alpha s1}\\ {i}_{\beta s1}\\ {i}_{\alpha s2}\\ {i}_{\beta s2}\\ {i}_{\alpha s3}\\ {i}_{\beta s3}\\ {i}_{\alpha r}\\ {i}_{\beta r}\end{array}\right]$ ${T}_{e}=\frac{9p}{2}({\psi}_{\alpha s}{i}_{\beta s}{\psi}_{\beta s}{i}_{\alpha s})$Symbol  Description 

ψ_{αs}  Alpha axis component of the stator flux [Wb] 
ψ_{βs}  Beta axis component of the stator flux [Wb] 
ψ_{αs1}  Alpha 1 axis component of the stator flux [Wb] 
ψ_{βs1}  Beta 1 axis component of the stator flux [Wb] 
ψ_{αs2}  Alpha 2 axis component of the stator flux [Wb] 
ψ_{βs2}  Beta 2 axis component of the stator flux [Wb] 
ψ_{αs3}  Alpha 3 axis component of the stator flux [Wb] 
ψ_{βs3}  Beta 3 axis component of the stator flux [Wb] 
ψ_{αr}  Alpha axis component of the rotor flux, referred to the stator [Wb] 
ψ_{βr}  Beta axis component of the rotor flux, referred to the stator [Wb] 
i_{αs}  Alpha axis component of the stator current [A] 
i_{βs}  Beta axis component of the stator current [A] 
i_{αs1}  Alpha 1 axis component of the stator current [A] 
i_{βs1}  Beta 1 axis component of the stator current [A] 
i_{αs2}  Alpha 2 axis component of the stator current [A] 
i_{βs2}  Beta 2 axis component of the stator current [A] 
i_{αs3}  Alpha 3 axis component of the stator current [A] 
i_{βs3}  Beta 3 axis component of the stator current [A] 
i_{αr}  Alpha axis component of the rotor current, referred to the stator [A] 
i_{βr}  Beta axis component of the rotor current, referred to the stator [A] 
v_{αs}  Alpha axis component of the stator voltage [V] 
v_{βs}  Beta axis component of the stator voltage [V] 
v_{αs1}  Alpha 1 axis component of the stator voltage [V] 
v_{βs1}  Beta 1 axis component of the stator voltage [V] 
v_{αs2}  Alpha 2 axis component of the stator voltage [V] 
v_{βs2}  Beta 2 axis component of the stator voltage [V] 
v_{αs3}  Alpha 3 axis component of the stator voltage [V] 
v_{βs3}  Beta 3 axis component of the stator voltage [V] 
v_{αr}  Alpha axis component of the rotor voltage, referred to the stator [V] 
v_{βr}  Beta axis component of the rotor voltage, referred to the stator [V] 
R_{s}  Stator phase resistance [Ω] 
R_{r}  Rotor phase resistance, referred to the stator [Ω] 
L_{m}  Magnetizing (mutual, main) inductance [H] 
L_{s}  Stator phase inductance [H] ( $={L}_{ls}+{L}_{\mathrm{m}}$ ) 
L_{r}  Rotor phase inductance, referred to the stator [H] ( $={L}_{lr}+{L}_{\mathrm{m}}$ ) 
ω_{r}  Rotor electrical speed [rad/s] 
p  Machine number of pole pairs 
T_{e}  Machine developed electromagnetic torque [Nm] 
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] 
R_{r}  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] 
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] 
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} stator  Stator 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 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] 
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] 