Three Phase Permanent Magnet Synchronous Machine (VBR)

Table 1. Three Phase Permanent Magnet Synchronous Machine (VBR) component in Typhoon HIL Schematic Editor
component	component dialog window	component parameters
		Property tabs: Electrical Mechanical Load Feedback Advanced Output

A, B, and C are stator winding terminals. The stator winding uses the voltage behind reactance 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 rotor magnets. The modeled dynamics can be represented with the following equations:

[\begin{matrix} v_{d s} \\ v_{q s} \end{matrix}] = [\begin{matrix} R_{s} & 0 \\ 0 & R_{s} \end{matrix}] [\begin{matrix} i_{d s} \\ i_{q s} \end{matrix}] + \frac{d}{d t} [\begin{matrix} ψ_{d s} \\ ψ_{q s} \end{matrix}] + ω_{r} [\begin{matrix} {- ψ}_{q s} \\ ψ_{d s} \end{matrix}]

[\begin{matrix} ψ_{d s} \\ ψ_{q s} \end{matrix}] = [\begin{matrix} L_{d} & 0_{} \\ 0_{} & L_{q} \end{matrix}] [\begin{matrix} i_{d s} \\ i_{q s} \end{matrix}] + [\begin{matrix} ψ_{P M} \\ 0 \end{matrix}]

T_{e} = \frac{3}{2} p (ψ_{d s} i_{q s} - ψ_{q s} i_{d s})

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

ψ_{d s} = f_{d} (i_{d s}, i_{q s})

ψ_{q s} = f_{q} (i_{d s}, i_{q s})

Table 2. Three Phase Permanent Magnet Synchronous Machine electrical subsystem model variables
symbol	description
ψ_ds	Direct axis component of the stator flux [Wb]
ψ_qs	Quadrature axis component of the stator flux [Wb]
ψ_PM	Flux amplitude established in stator phases by rotor permanent magnets [Wb]
i_ds	Direct axis component of the stator current [A]
i_qs	Quadrature axis component of the stator current [A]
v_ds	Direct axis component of the stator voltage [V]
v_qs	Quadrature axis component of the stator voltage [V]
R_s	Stator phase resistance [Ω]
L_d	Direct axis inductance [H]
L_q	Quadrature axis inductance [H]
ω_r	Rotor electrical speed [rad/s] ( $= p ω_{m}$ )
p	Machine number of pole pairs
T_e	Machine developed electromagnetic torque [Nm]

Mechanical sub-system model

Motion equation:

\frac{{d ω}_{m}}{d t} = \frac{1}{J_{m}} (T_{e} - T_{l} - b ω_{m})

θ_{m} = \int ω_{m} d t

Table 3. Mechanical sub-system model variables
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]

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
absolute inductance vs current
incremental inductance vs current

Table 4. Electrical parameters
symbol	description
R_s	Stator phase resistance [Ω]
L_d	Direct axis inductance [H]
L_q	Quadrature axis inductance [H]
Psi_pm	Flux amplitude established in stator phases by rotor permanent magnets [Wb]
id vector	List of stator current direct axis component values [A]
iq vector	List of stator current quadrature axis component values [A]
psid table	Table of stator flux direct axis component values [Wb]
psiq table	Table of stator flux quadrature axis component values [Wb]
Ld table	Table of stator inductance direct axis component values [H]
Lq table	Table of stator inductance quadrature axis component values [H]

The permanent magnet synchronous machine VBR model can include magnetic saturation effects. In that case, fluxes or inductances are defined as functions of stator currents i_ds and i_qs. 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:

fluxes vs stator currents
absolute inductances vs stator currents
incremental inductances vs stator currents

In each of these cases the direct axis and quadrature axis flux or inductance can depend on both stator currents or only on the corresponding stator current. When fluxes or inductances are functions of both direct axis and quadrature axis currents, it is important to define the lookup tables as nested Python lists where the number of inner lists must be equal to the number of elements in the current id vector and the number of elements of each inner list must also be equal to the number of elements in the current iq vector.

Figure 1. Nonlinear Model Type: fluxes as functions of stator currents, ψ_d=f_d(i_d) , ψ_q=f_q(i_q)

id_vector = [-40.0, -20.0, 0.0, 20.0, 40.0]
   iq_vector = [-40.0, -20.0, 0.0, 20.0, 40.0]
   psid_table = [-0.0492472, -0.0433668, -0.0425532, -0.0433464, -0.0484104]
   psiq_table = [-0.1330824, -0.0838922, 0.0, 0.0838828, 0.133098]

Figure 2. Nonlinear Model Type: fluxes as functions of stator currents, ψ_d=f_d(i_d , i_q) , ψ_q=f_q(i_d , i_q)

id_vector = [-40.0, -20.0, 0.0, 20.0, 40.0]
   iq_vector = [-40.0, -20.0, 0.0, 20.0, 40.0]
   psid_table = [[-0.0492472, -0.0433668, -0.0425532, -0.0433464, -0.0484104],
   [-0.0115952, -0.0274476, -0.0330376, -0.02771, -0.0126918],
   [0.032, 0.032, 0.032, 0.032, 0.032],
   [0.064706, 0.0662274, 0.0593586, 0.0677826, 0.0649068],
   [0.0805368, 0.0705448, 0.05448328, 0.070713, 0.0812716]]
   psiq_table = [[-0.1330824, -0.0838922, 0.0, 0.0838828, 0.133098],
   [-0.1313616, -0.1041012, 0.0, 0.1041148, 0.1282268],
   [-0.1286288, -0.1076058, 0.0, 0.107, 0.1278272],
   [-0.1175936, -0.084391, 0.0, 0.0839394, 0.1162836],
   [-0.1092448, -0.0588548, 0.0, 0.0585804, 0.1084576]]

Figure 3. Nonlinear Model Type: inductances as functions of stator currents, L_d=f_d(i_d) , L_q=f_q(i_q)

id_vector = [-40.0, -20.0, 0.0, 20.0, 40.0]
   iq_vector = [-40.0, -20.0, 0.0, 20.0, 40.0]
   Ld_table = [0.00186383, 0.00325188, 0.00399657, 0.00136793, 0.000562082]
   Lq_table = [0.00321572, 0.00538029, 0.00779154, 0.00535, 0.00319568]

Figure 4. Nonlinear Model Type: inductances as functions of stator currents, L_d=f_d(i_d , i_q) , L_q=f_q(i_d , i_q)

id_vector = [-40.0, -20.0, 0.0, 20.0, 40.0]
   iq_vector = [-40.0, -20.0, 0.0, 20.0, 40.0]
   Ld_table = [[0.00203118, 0.00188417, 0.00186383, 0.00188366, 0.00201026],
   [0.00217976, 0.00297238, 0.00325188, 0.0029855, 0.00223459],
   [0.00226518, 0.00283656, 0.00399657, 0.00280727, 0.00218666],
   [0.0016353, 0.00171137, 0.00136793, 0.00178913, 0.00164534],
   [0.00121342, 0.00096362, 0.000562082, 0.000967825, 0.00123179]]
   Lq_table = [[0.00332706, 0.00419461, 0.0049565, 0.00419414, 0.00332745],
   [0.00328404, 0.00520506, 0.00635444, 0.00520574, 0.00320567],
   [0.00321572, 0.00538029, 0.00779154, 0.00535, 0.00319568],
   [0.00293984, 0.00421955, 0.00547829, 0.00419697, 0.00290709],
   [0.00273112, 0.00294274, 0.00323358, 0.00292902, 0.00271144]]

Mechanical

Table 5. Mechanical parameters
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

Table 6. Load parameters
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:

$T_{l} = l o a d_a i_g a i n \cdot (A I (l o a d_a i_p i n) + l o a d_a i_o f f s e t)$

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
Encoder Z pulse length	Z digital signal pulse length in periods. Can be Quarter length or Full period (default)
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:

r e s_c a r r_s r c = r e s_a i_g a i n \cdot (A I (r e s_a i_p i n_1) + r e s_a i_o f f s e t)

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:

r e s_c a r r_s r c = r e s_a i_g a i n \cdot ((A I (r e s_a i_p i n_1) - A I (r e s_a i_p i n_2)) + r e s_a i_o f f s e t)

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 \cdot e n c_p p r \cdot f_{m} {\cdot T}_{s} \leq 1

Table 8. Variables in the encoder limitation expression
symbol	description
enc_ppr	Encoder number of pulses per revolution
f_m	Rotor mechanical frequency [Hz]
T_s	Simulation time step [s]

Figure 6. Machine encoder signals when *Full period* option is selected

Figure 7. Machine encoder signals when *Quarter period* option is selected

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).

Note: More information about absolute encoder protocol could be found here.

Advanced

Table 9. Advanced parameters
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 6. In the second one, the alpha axis is aligned with the stator phase a axis (indicated in b) Figure 6. The user can select between these two situations.

Figure 8. Selection of a stationary reference frame position: a) Theta_ab= -pi/2; b) Theta_ab= 0

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.

Figure 9. Example of model containing abc to dq transformation component and chosen stationary reference frame position Theta_ab= -pi/2

Note: This property is available only on certain machine components.

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 10. 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]