# Three Phase Wound Rotor Synchronous Machine

Description of the Three Phase Wound Rotor Synchronous Machine component in Schematic Editor.

Table 1. Three Phase Wound Rotor Synchronous Machine component in Typhoon HIL Schematic Editor
Component Component dialog window Component parameters
• Property tabs:

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 sub-system 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:

${T}_{e}=\frac{3}{2}p\left({\psi }_{ds}{i}_{qs}-{\psi }_{qs}{i}_{ds}\right)$

If the saturation effects are considered, the equations have the same form, but in that case fluxes are functions of stator currents:

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 vfdr at the field winding terminals, it is transformed to the stator value vfd with the following equation:

${v}_{fd}=\frac{{N}_{s}}{{N}_{fd}}{v}_{fd}^{r}$

Similarly, calculated rotor currents: field winding current ifd, and damper winding currents ikd and ikq, are referred back to the real rotor windings values ifdr, ikdr, and ikqr, through the following equations:

Table 2. Three Phase Wound Rotor Synchronous Machine electrical subsystem model variables
Symbol Description
ψds Direct axis component of the stator flux [Wb]
ψqs Quadrature axis component of the stator flux [Wb]
ψkd Direct axis component of the damper winding flux, referred to the stator [Wb]
ψkq Quadrature axis component of the damper winding flux, referred to the stator [Wb]
ψfd Field winding flux, referred to the stator [Wb]
ψkq2 Second quadrature axis component of the damper winding flux, referred to the stator [Wb]
ids Direct axis component of the stator current [A]
iqs Quadrature axis component of the stator current [A]
ikd Direct axis component of the damper winding current, referred to the stator [A]
ikdr Direct axis component of the damper winding current [A]
ikq Quadrature axis component of the damper winding current, referred to the stator [A]
ikqr Quadrature axis component of the damper winding current [A]
ifd Field winding current, referred to the stator [A]
ifdr Field winding current [A]
ikq2 Second quadrature axis component of the damper winding current, referred to the stator [A]
vds Direct axis component of the stator voltage [V]
vqs Quadrature axis component of the stator voltage [V]
vkd Direct axis component of the damper winding voltage, referred to the stator [V]
vkq Quadrature axis component of the damper winding voltage, referred to the stator [V]
vfd Field winding voltage, referred to the stator [V]
vfdr Field winding voltage [V]
vkq2 Second quadrature axis component of the damper winding voltage, referred to the stator [V]
imd Direct axis component of the magnetizing current [A]
imq Quadrature axis component of the magnetizing current [A]
ψmd Direct axis component of the magnetizing flux [Wb]
ψmq Quadrature axis component of the magnetizing flux [Wb]
im Magnetizing current [A]
ψm Magnetizing flux [Wb]
Rs Stator phase resistance [Ω]
Rkd Direct axis damper winding resistance, referred to the stator [Ω]
Rkq Quadrature axis damper winding resistance, referred to the stator [Ω]
Rkq2 Second quadrature axis damper winding resistance, referred to the stator [Ω]
Rfd Field winding resistance, referred to the stator [Ω]
Lls Stator phase leakage inductance [H]
Lmd Direct axis magnetizing (mutual, main) inductance [H]
Lmq Quadrature axis magnetizing (mutual, main) inductance [H]
Llkd Direct axis damper winding leakage inductance, referred to the stator [H]
Llkq Quadrature axis damper winding leakage inductance, referred to the stator [H]
Llkq2 Second quadrature axis damper winding leakage inductance, referred to the stator [H]
Llfd Direct axis field winding leakage inductance, referred to the stator [H]
Lc Canay leakage inductance, referred to the stator [H]
ωr Rotor electrical speed [rad/s] ( $=p{\omega }_{m}$ )
p Machine number of pole pairs
Te Machine developed electromagnetic torque [Nm]
Ns/Nfd Turns ratio between the stator phase winding and the field winding used for transforming field winding variables to the stator-side
Ns/Nkd Turns ratio between the stator phase winding and the direct axis damper winding used for transforming damper winding variables to the stator-side
Ns/Nkq Turns ratio between the stator phase winding and the quadrature axis damper winding used for transforming damper winding variables to the stator-side

## Mechanical sub-system model

Motion equation:

Table 3. Mechanical sub-system model variables
symbol description
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]
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
Table 4. Electrical parameters
symbol description
Number of damper windings Changes the number of damper windings on the q-axis
Rs Stator phase resistance [Ω]
Lls Stator phase leakage inductance [H]
Lmd Direct axis magnetizing (mutual, main) inductance [H]
Lmq Quadrature axis magnetizing (mutual, main) inductance [H]
Rf Field winding resistance, referred to the stator [Ω]
Llfd Direct axis field winding leakage inductance, referred to the stator [H]
Rkd Direct axis damper winding resistance, referred to the stator [Ω]
Rkq Quadrature axis damper winding resistance, referred to the stator [Ω]
Rkq2 Second quadrature axis damper winding resistance, referred to the stator [Ω]
Llkd Direct axis damper winding leakage inductance, referred to the stator [H]
Llkq Quadrature axis damper winding leakage inductance, referred to the stator [H]
Llkq2 Second quadrature axis damper winding leakage inductance, referred to the stator [H]
Ns/Nfd Turns ratio between the stator phase winding and the field winding used for transforming field winding variables to the stator-side
Ns/Nkd Turns ratio between the stator phase winding and the direct axis damper winding used for transforming damper winding variables to the stator-side
Ns/Nkq Turns ratio between the stator phase winding and the quadrature axis damper winding used for transforming damper winding variables to the stator-side
Lc 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 line-to-line 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 imd and imq. These functions are represented in the form of lookup tables. The lookup tables use linear interpolation and linear extrapolation.

Saturation can be parametrized in the following ways:
2. flux vs current
In the first case when the magnetizing curve is defined as no load curve, variables must be defined as Python lists. In the second case, when the magnetizing curve is defined as flux vs current, the magnetizing current components imd vector and imq vector must be defined as Python lists, while the magnetizing flux components psimd table and psimq table should be defined as nested Python lists. The number of inner lists must be equal to the number of elements in imd vector and the number of elements in each inner list must also be equal to the number of elements in the imq vector.
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

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

symbol description
Load source Load can be set from SCADA/external or from model (in model case, one signal processing input will appear)
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)
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:

$res_carr_src=res_ai_gain·\left(AI\left(res_ai_pin_1\right)+res_ai_offset\right)$

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:

$res_carr_src=res_ai_gain·\left(\left(AI\left(res_ai_pin_1\right)-AI\left(res_ai_pin_2\right)\right)+res_ai_offset\right)$
Note: Differential external resolver signal support is available from the 2023.1 software version. Only the single ended external resolver option was available in previous software versions.
Note: In order to get a resolver signal 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, the sinusoidal signal used to generate external resolver carrier source is fed to HIL analog input 1. The 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).

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
Field inputspecifies the physical quantity which will be applied to the field winding:
1. voltage
2. current

In either case, the physical quantity refers to the rotor-side.

In order for the input field current to be referred to the stator parameter, Ns/Nfd 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.

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.

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 rotor Rotor snubber resistance value [Ω]
Rsnb 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.

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]
Stator d-axis current Direct axis component of the stator current [A]
Stator q-axis 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 d-axis flux Direct axis component of the stator flux [Wb]
Stator q-axis flux Quadrature axis component of the stator flux [Wb]