Synchronous reluctance machine#

The Synchronous reluctance machine block is a model of a three-phase electronic synchronous RM. RM model comprises stator windings representing three windings shifted by 120\(^о\) relative to each other. The rotor can be represented in two configurations:

without permanent magnets;
with permanent magnets.

Note

Rotor design is selected in the block properties.

The model is described as follows. The main part of the calculations takes place in d-q-0 system using the following formulas:

stator winding voltage projections (1), (2), (3):

(1)#\[u_{sd}=\frac{2}{3}\cdot (u_A \cdot cos(\theta_e)+u_B \cdot cos(\theta_e-\frac{2\cdot \pi}{3})+u_C \cdot cos(\theta_e+\frac{2\cdot \pi}{3}));\]

(2)#\[u_{sq}=-\frac{2}{3}\cdot (u_A \cdot sin(\theta_e)+u_B \cdot sin(\theta_e-\frac{2\cdot \pi}{3})+u_C \cdot sin(\theta_e+\frac{2\cdot \pi}{3}));\]

(3)#\[u_{s0}=\frac{1}{3}\cdot (u_A +u_B +u_C),\]

where:

\(u_A, u_B, u_C\) – instantaneous voltage values at the respective stator phases, A;

\( heta_e\) – electric angle of rotor rotation, rad.

stress equations (4), (5), (6):

(4)#\[u_{sd}=R_s\cdot i_{sd}+L_d\cdot \frac{i_{sd}}{dt}-\omega_e \cdot i_{sq}\cdot L_q\]

(5)#\[u_{sq}=R_s\cdot i_{sq}+L_q\cdot \frac{i_{sq}}{dt}+\omega_e \cdot i_{sd}\cdot L_d\]

(6)#\[u_{s0}=R_s\cdot i_{s0}+L_0\cdot \frac{i_{s0}}{dt}\]

where:

\(R_s\) – active resistance of one stator phase, Ohm;

\(L_d, L_q\) – self-inductance of stator windings in projections on axis d-q, H;

\(L_0\) – self-inductance of stator windings in zero projection, H;

\(\omega_e\) – electric angular rotation rate, rad/s, which is determined by the formula ElectronicSynRM7:

(7)#\[\omega_e=\omega_r\cdot p\]

where:

\(\omega_r\) – angular rotor rotation rate, rad/s;

p – number of pairs of rotor poles.

After calculations in d-q-0 system, the block returns instantaneous values of phase currents using the formulas (8), (9), (10):

(8)#\[i_A=i_{sd}\cdot cos(\theta_e)-i_{sq}\cdot sin(\theta_e)+i_0;\]

(9)#\[i_B=i_{sd}\cdot cos(\theta_e-\frac{2\cdot \pi}{3})-i_{sq}\cdot sin(\theta_e-\frac{2\cdot \pi}{3})+i_0;\]

(10)#\[i_C=i_{sd}\cdot cos(\theta_e+\frac{2\cdot \pi}{3})-i_{sq}\cdot sin(\theta_e+\frac{2\cdot \pi}{3})+i_0\]

The electromagnetic torque on the shaft is calculated using the formula (11):

(11)#\[M_e=\frac{3}{2}\cdot p \cdot(i_{sq}\cdot i_{sd}\cdot L_d - i_{sd}\cdot i_{sq}\cdot L_q)\]

The above equations are valid for a rotor without permanent magnets. When selecting a permanent magnet rotor design, the formulas (4), (5) and (11) take the following form:

stress equations over axes d-q (12), (13):

(12)#\[u_{sd}=R_s\cdot i_{sd}+L_d\cdot \frac{i_{sd}}{dt}-\omega_e \cdot (i_{sq}\cdot L_q+\psi_q)\]

(13)#\[u_{sq}=R_s\cdot i_{sq}+L_q\cdot \frac{i_{sq}}{dt}+\omega_e \cdot (i_{sd}\cdot L_d+\psi_d)\]

electromagnetic torque equation {eq} ElectronicSynRM14:

(14)#\[M_e=\frac{3}{2}\cdot p \cdot(i_{sq} \cdot (i_{sd} \cdot L_d+\psi_d) - i_{sd} \cdot (i_{sq} \cdot L_q+\psi_q))\]

where \(\psi_d, \psi_q\) - flux linkages from permanent magnets of the rotor in projections on axes d-q, Wb.

RM mechanical equation is as follows (15):

(15)#\[\frac{d\omega}{dt}\cdot J_Я=M_e-M_{тр}-M_{сн}\]

where:

\(J_Я\) – moment of inertia of the rotor, Н\(\cdot\)m\(^2\);

\(M_{res}\) – moment of resistance created by mechanical load on the rotor shaft, Н\(\cdot\)m;

\(M_{fr}\) – friction moment, determined by friction in bearings and brushes, Н\(\cdot\)m. This moment is determined by the formula (16):

(16)#\[M_{тр}=\omega \cdot K_{тр}\]

where \(K_{fr}\) – friction factor.

1. Properties#

Property	Default Value	Description	Units of measurement
Number of pairs of poles	6.0	–	un.
Active resistance of the stator winding	0.7	–	Ohm
Stator inductance on the d axis	0.004	–	H
Stator inductance on the q axis	0.0031	–	H
Stator inductance on the zero sequence	0.0005	–	H
Moment of inertia of the rotor	0.01	–	kg\(\cdot\)m\(^2\)
Rotor Coefficient of friction	0.01	–	Н\(\cdot\)m/(rad/s)
Array of initial conditions [w, \( heta\)]	0.0; 0.0	–	rad/s; rad
Permanent magnets in the rotor	Off	–	-
Flux linkage of rotors permanent magnets on the d axis, available when selecting `Permanent magnets in the rotor` - `On`	0.0	–	Wb
Flux linkage of rotors permanent magnets on the q axis, available when selecting `Permanent magnets in the rotor` - `On`	0.02	–	Wb

2. Inputs and outputs#

Parameters	Input/Output	Description	Units of measurement	Port type
External torque	Input	Значение момента на валу от нагрузки	N\(\cdot\) m	Automation
N	Input	Zero point of stator windings	-	Electronics
Angular frequency	Output	Gets connected to angular velocity ports of blocks from the library Mechanics/Rotational motion	rad/s	Angular velocity
+A	Output	Phase A winding positive lead	-	Electronics
+B	Output	Phase B winding positive lead	-	Electronics
+C	Output	Phase C winding positive lead	-	Electronics

In order to see the connection capabilities of the block, you need to go to page Description of block ports