Three Phase Wound Rotor Synchronous Machine
Description of the Three Phase Wound Rotor Synchronous Machine component in Schematic Editor.
Component  Component dialog window  Component parameters 


A, B, and C are stator winding terminals. The stator winding uses the current source interface. R1 and R2 represent field winding terminals. The rotor winding uses the single phase variant of the current source interface.
Electrical subsystem model
The electrical part of the machine is represented by the following system of equations, modeled in the rotating dq reference frame. The dq reference frame is attached to the rotor, and the direct axis is aligned with the field winding. The modeled dynamics can be represented with the following equations:
$\left[\begin{array}{c}{v}_{ds}\\ {v}_{qs}\\ {v}_{kd}\\ {v}_{kq}\\ {v}_{fd}\\ {v}_{kq2}\end{array}\right]=\left[\begin{array}{c}{R}_{s}\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\begin{array}{c}0\\ {R}_{s}\\ 0\\ 0\\ 0\\ 0\end{array}\begin{array}{c}0\\ 0\\ {R}_{kd}\\ 0\\ 0\\ 0\end{array}\begin{array}{c}0\\ 0\\ 0\\ {R}_{kq}\\ 0\\ 0\end{array}\begin{array}{c}0\\ 0\\ 0\\ 0\\ {R}_{fd}\\ 0\end{array}\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ {R}_{kq2}\end{array}\right]\left[\begin{array}{c}{i}_{ds}\\ {i}_{qs}\\ {i}_{kd}\\ {i}_{kq}\\ {i}_{fd}\\ {i}_{kq2}\end{array}\right]+\frac{d}{dt}\left[\begin{array}{c}{\psi}_{ds}\\ {\psi}_{qs}\\ {\psi}_{kd}\\ {\psi}_{kq}\\ {\psi}_{fd}\\ {\psi}_{kq2}\end{array}\right]+{\omega}_{r}\left[\begin{array}{c}{\psi}_{qs}\\ {\psi}_{ds}\\ 0\\ 0\\ 0\\ 0\end{array}\right]$ $\left[\begin{array}{c}{\psi}_{ds}\\ {\psi}_{qs}\\ {\psi}_{kd}\\ {\psi}_{kq}\\ {\psi}_{fd}\\ {\psi}_{kq2}\end{array}\right]=\left[\begin{array}{c}{L}_{ls}+{L}_{md}\\ 0\\ {L}_{md}\\ 0\\ {L}_{md}\\ 0\end{array}\begin{array}{c}0\\ {L}_{ls}+{L}_{mq}\\ 0\\ {L}_{mq}\\ 0\\ {L}_{mq}\end{array}\begin{array}{c}{L}_{md}\\ 0\\ {L}_{c}+{L}_{lkd}+{L}_{md}\\ 0\\ {L}_{md}\\ 0\end{array}\begin{array}{c}0\\ {L}_{mq}\\ 0\\ {L}_{lkq}+{L}_{mq}\\ 0\\ {L}_{mq}\end{array}\begin{array}{c}{L}_{md}\\ 0\\ {L}_{md}\\ 0\\ {L}_{c}+{L}_{lfd}+{L}_{md}\\ 0\end{array}\begin{array}{c}0\\ {L}_{mq}\\ 0\\ {L}_{mq}\\ 0\\ {L}_{lkq2}+{L}_{mq}\end{array}\right]\left[\begin{array}{c}{i}_{ds}\\ {i}_{qs}\\ {i}_{kd}\\ {i}_{kq}\\ {i}_{fd}\\ {i}_{kq2}\end{array}\right]$ ${T}_{e}=\frac{3}{2}p({\psi}_{ds}{i}_{qs}{\psi}_{qs}{i}_{ds})$If the saturation effects are considered, the equations have the same form, but in that case fluxes are functions of stator currents:
${\psi}_{md}={f}_{d}({i}_{md},{i}_{mq})$ ${\psi}_{mq}={f}_{q}({i}_{md},{i}_{mq})$where:
${\psi}_{m}=\sqrt{{{\psi}_{md}{}_{}}^{2}+{{\psi}_{mq}{}_{}}^{2}}$ ${\psi}_{md}={{\psi}_{ds}+{\psi}_{kd}+{\psi}_{fd}}^{}{}^{}$ ${\psi}_{mq}={{\psi}_{qs}+{\psi}_{kq}+{\psi}_{kq2}}^{}{}^{}$and:
${i}_{m}=\sqrt{{{i}_{md}{}_{}}^{2}+{{i}_{mq}{}_{}}^{2}}$ ${i}_{md}={{i}_{ds}+{i}_{kd}+{i}_{fd}}^{}{}^{}$ ${i}_{mq}={{i}_{qs}+{i}_{kq}+{i}_{kq2}}^{}{}^{}$All rotor variables and parameters are referred to the stator with an appropriate turns ratio. To illustrate this, if you apply voltage v_{fd}^{r} at the field winding terminals, it is transformed to the stator value v_{fd} with the following equation:
${v}_{fd}=\frac{{N}_{s}}{{N}_{fd}}{v}_{fd}^{r}$Similarly, calculated rotor currents: field winding current i_{fd}, and damper winding currents i_{kd} and i_{kq}, are referred back to the real rotor windings values i_{fd}^{r}, i_{kd}^{r}, and i_{kq}^{r}, through the following equations:
${i}_{fd}^{r}=\frac{3}{2}\frac{{N}_{s}}{{N}_{fd}}{i}_{fd}{i}_{kd}^{r}=\frac{3}{2}\frac{{N}_{s}}{{N}_{kd}}{i}_{kd}{i}_{kq}^{r}=\frac{3}{2}\frac{{N}_{s}}{{N}_{kq}}{i}_{kq}$Symbol  Description 

ψ_{ds}  Direct axis component of the stator flux [Wb] 
ψ_{qs}  Quadrature axis component of the stator flux [Wb] 
ψ_{kd}  Direct axis component of the damper winding flux, referred to the stator [Wb] 
ψ_{kq}  Quadrature axis component of the damper winding flux, referred to the stator [Wb] 
ψ_{fd}  Field winding flux, referred to the stator [Wb] 
ψ_{kq2}  Second quadrature axis component of the damper winding flux, referred to the stator [Wb] 
i_{ds}  Direct axis component of the stator current [A] 
i_{qs}  Quadrature axis component of the stator current [A] 
i_{kd}  Direct axis component of the damper winding current, referred to the stator [A] 
i_{kd}^{r}  Direct axis component of the damper winding current [A] 
i_{kq}  Quadrature axis component of the damper winding current, referred to the stator [A] 
i_{kq}^{r}  Quadrature axis component of the damper winding current [A] 
i_{fd}  Field winding current, referred to the stator [A] 
i_{fd}^{r}  Field winding current [A] 
i_{kq2}  Second quadrature axis component of the damper winding current, referred to the stator [A] 
v_{ds}  Direct axis component of the stator voltage [V] 
v_{qs}  Quadrature axis component of the stator voltage [V] 
v_{kd}  Direct axis component of the damper winding voltage, referred to the stator [V] 
v_{kq}  Quadrature axis component of the damper winding voltage, referred to the stator [V] 
v_{fd}  Field winding voltage, referred to the stator [V] 
v_{fd}^{r}  Field winding voltage [V] 
v_{kq2}  Second quadrature axis component of the damper winding voltage, referred to the stator [V] 
i_{md}  Direct axis component of the magnetizing current [A] 
i_{mq}  Quadrature axis component of the magnetizing current [A] 
ψ_{md}  Direct axis component of the magnetizing flux [Wb] 
ψ_{mq}  Quadrature axis component of the magnetizing flux [Wb] 
i_{m}  Magnetizing current [A] 
ψ_{m}  Magnetizing flux [Wb] 
R_{s}  Stator phase resistance [Ω] 
R_{kd}  Direct axis damper winding resistance, referred to the stator [Ω] 
R_{kq}  Quadrature axis damper winding resistance, referred to the stator [Ω] 
R_{kq2}  Second quadrature axis damper winding resistance, referred to the stator [Ω] 
R_{fd}  Field winding resistance, referred to the stator [Ω] 
L_{ls}  Stator phase leakage inductance [H] 
L_{md}  Direct axis magnetizing (mutual, main) inductance [H] 
L_{mq}  Quadrature axis magnetizing (mutual, main) inductance [H] 
L_{lkd}  Direct axis damper winding leakage inductance, referred to the stator [H] 
L_{lkq}  Quadrature axis damper winding leakage inductance, referred to the stator [H] 
L_{lkq2}  Second quadrature axis damper winding leakage inductance, referred to the stator [H] 
L_{lfd}  Direct axis field winding leakage inductance, referred to the stator [H] 
L_{c}  Canay leakage inductance, referred to the stator [H] 
ω_{r}  Rotor electrical speed [rad/s] ( $=p{\omega}_{m}$ ) 
p  Machine number of pole pairs 
T_{e}  Machine developed electromagnetic torque [Nm] 
N_{s}/N_{fd}  Turns ratio between the stator phase winding and the field winding used for transforming field winding variables to the statorside 
N_{s}/N_{kd}  Turns ratio between the stator phase winding and the direct axis damper winding used for transforming damper winding variables to the statorside 
N_{s}/N_{kq}  Turns ratio between the stator phase winding and the quadrature axis damper winding used for transforming damper winding variables to the statorside 
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
This component offers two levels of model fidelity, designated by the Model Type property. The following options are available: linear
 nonlinear
 no load curve
 flux vs current
symbol  description 

Number of damper windings  Changes the number of damper windings on the qaxis 
R_{s}  Stator phase resistance [Ω] 
L_{ls}  Stator phase leakage inductance [H] 
L_{md}  Direct axis magnetizing (mutual, main) inductance [H] 
L_{mq}  Quadrature axis magnetizing (mutual, main) inductance [H] 
R_{f}  Field winding resistance, referred to the stator [Ω] 
L_{lfd}  Direct axis field winding leakage inductance, referred to the stator [H] 
R_{kd}  Direct axis damper winding resistance, referred to the stator [Ω] 
R_{kq}  Quadrature axis damper winding resistance, referred to the stator [Ω] 
R_{kq2}  Second quadrature axis damper winding resistance, referred to the stator [Ω] 
L_{lkd}  Direct axis damper winding leakage inductance, referred to the stator [H] 
L_{lkq}  Quadrature axis damper winding leakage inductance, referred to the stator [H] 
L_{lkq2}  Second quadrature axis damper winding leakage inductance, referred to the stator [H] 
N_{s}/N_{fd}  Turns ratio between the stator phase winding and the field winding used for transforming field winding variables to the statorside 
N_{s}/N_{kd}  Turns ratio between the stator phase winding and the direct axis damper winding used for transforming damper winding variables to the statorside 
N_{s}/N_{kq}  Turns ratio between the stator phase winding and the quadrature axis damper winding used for transforming damper winding variables to the statorside 
L_{c}  Canay leakage inductance, referred to the stator [H] 
rated speed  Machine mechanical rated speed used in the nonlinear model for the magnetizing curve defined as the no load curve [rpm] 
if vector  List of no load excitation currents, referred to the field winding [A] 
vs vector  List of no load stator linetoline RMS voltages [V] 
imd vector  List of magnetizing current direct axis component values [A] 
imq vector  List of magnetizing current quadrature axis component values [A] 
psimd table  Table of magnetizing flux direct axis component values [Wb] 
psimq table  Table of magnetizing flux quadrature axis component values [Wb] 
The Three Phase Wound Rotor Synchronous Machine machine model can include magnetic saturation effects. In that case, fluxes are defined as functions of magnetizing currents i_{md} and i_{mq}. These functions are represented in the form of lookup tables. The lookup tables use linear interpolation and linear extrapolation.
 no load curve
 flux vs current
if_vector = [0, 4514.0, 9498.0, 13260.0, 15260.0, 16710.0, 18200.0, 19210.0, 21340.0, 23650.0, 25930.0]
vs_vector = [0.0, 4986.55, 10388.65, 14313.256, 16298.64, 17637.6, 18884.26, 19623, 20915.82, 22116.28, 23224.4]
imd_vector = [9498.0, 8548.2, 7598.4, 6648.6, 5698.8]
imq_vector = [9498.0, 8548.2, 7598.4, 6648.6, 5698.8]
psimd_table = [22.46306805, 20.2273073, 17.99014399, 15.75121391, 13.51008229]
psimq_table = [11.07784855, 9.9724961, 8.86656821, 7.76004375, 6.65291642]
imd_vector = [9498.0, 8548.2, 7598.4, 6648.6, 5698.8]
imq_vector = [9498.0, 8548.2, 7598.4, 6648.6, 5698.8]
psimd_table = [[22.46306805, 22.46854837, 22.47394023, 22.47914448, 22.48404604],
[20.2273073, 20.23340111, 20.23948657, 20.24544982, 20.25115112],
[17.99014399, 17.99688273, 18.00373418, 18.01057435, 18.01723878],
[15.75121391, 15.75858553, 15.7662438, 15.77406724, 15.78187459],
[13.51008229, 13.51799327, 13.52642503, 13.53528506, 13.54440031]]
psimq_table = [[11.07784855, 9.9724961, 8.86656821, 7.76004375, 6.65291642],
[11.08362731, 9.97826979, 8.87224078, 7.765498, 6.65801557],
[11.08998614, 9.9847262, 8.87869102, 7.77180627, 6.6640132],
[11.09691706, 9.99189941, 8.88600464, 7.77911226, 6.67111073],
[11.10434879, 9.99976596, 8.89422511, 7.7875446, 6.67953349]]
Mechanical
symbol  description 

pms  Machine number of pole pairs 
Star/delta  Stator winding connection (star or delta) 
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] 
Field Input  Physical quantity (voltage/current) applied to the field winding 
 voltage
 current
In either case, the physical quantity refers to the rotorside.
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 $\frac{3}{2}$ .
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. You can select between these two situations.
It is important to know the value of Theta_ab when the rotor position feedback is necessary. For 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 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  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 current  Alpha axis component of the stator current [A] 
Stator beta axis current  Beta axis component of the stator current [A] 
Stator daxis current  Direct axis component of the stator current [A] 
Stator qaxis current  Quadrature axis component of the stator current [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] 
Stator daxis flux  Direct axis component of the stator flux [Wb] 
Stator qaxis flux  Quadrature axis component of the stator flux [Wb] 