Synchronous Machine With Damper Winding#

The Synchronous Machine With Damper Winding block is a model of an electric three-phase synchronous motor with distinct poles.

The motor model contains three types of windings:

stator windings, which are three windings shifted by 120\(^о\) relative to each other;
rotor winding, which is an excitation winding and rotates together with the rotor;
damper winding, which is installed on the rotor.

Model description

The main part of the calculations takes place in the d-q-0 system, for which the following transformations and calculations are carried out:

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

(1)#\[i_{sd}=\frac{2}{3}\cdot (i_A \cdot cos(\theta_e)+i_B \cdot cos(\theta_e-\frac{2\cdot \pi}{3})+i_C \cdot cos(\theta_e+\frac{2\cdot \pi}{3}));\]

(2)#\[i_{sq}=-\frac{2}{3}\cdot (i_A \cdot sin(\theta_e)+i_B \cdot sin(\theta_e-\frac{2\cdot \pi}{3})+i_C \cdot sin(\theta_e+\frac{2\cdot \pi}{3}));\]

(3)#\[i_{s0}=\frac{1}{3}\cdot (i_A +i_B +i_C),\]

where:

\(i_A, i_B, i_C\) – instantaneous current values in the respective stator phases, A;

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

main stator flux linkages (4), (5):

(4)#\[\psi_{sd}=-i_{sd}\cdot L_{sd}+i_f \cdot L_{mf}+i_{kd}\cdot L_{akd};\]

(5)#\[\psi_{sq}=-i_{sq}\cdot L_{sq}+i_{kq}\cdot L_{akq},\]

where:

\(L_{sd}, L_{sq}\) – self-inductances of stator windings in projections on d-q axis, H;

\(i_f\) – instantaneous value of current in the excitation winding, A;

\(L_{mf}\) – mutual inductance between excitation winding and stator windings in axis d, rad.

Note

The model assumes that axis d coincides with the axis of the excitation winding

\(i_{kd}, i_{kq}\) – damper winding currents in projections on axis d-q, A;

\(L_{akd}, L_{akq}\) – mutual inductance between damper winding and stator windings in projections on axis d-q, H;

EMF of rotation in stator windings (6), (7), (8):

(6)#\[e_{sd}=\frac{di_f}{dt}\cdot L_{mf}+\frac{di_{kd}}{dt}\cdot L_{akd} - \psi_{sq} \cdot \omega_e;\]

(7)#\[e_{sq}=\frac{di_{kq}}{dt}\cdot L_{akq}+\psi_{sd} \cdot \omega_e;\]

(8)#\[e_{s0}=-\frac{di_{s0}}{dt}\cdot L_{s0},\]

where:

\(L_{s0}\) – self-inductance of stator windings in zero projection, H;

\(L_{akd}, L_{akq}\) – mutual inductance between stator and damper windings in projections on axis d-q, H;

\(\omega_e\) – electric angular rotation rate, rad/s, which is determined by the formula (9):

(9)#\[\omega_e=\omega_r \cdot p,\]

where:

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

p – number of pairs of rotor poles.

Damping winding voltage equations (10), (11):

(10)#\[0=\frac{di_{kd}}{dt}\cdot L_{kkd}+\frac{di_f}{dt}\cdot L_{fkd}-\frac{di_{sd}}{dt}\cdot L_{akd}\cdot \frac{3}{2}+i_{kd} \cdot R_{kd};\]

(11)#\[0=\frac{di_{kq}}{dt}\cdot L_{kkq}-\frac{di_{sq}}{dt}\cdot L_{akq}\cdot \frac{3}{2}+i_{kq} \cdot R_{kq},\]

where:

\(L_{kkd}, L_{kkq}\) – self- inductances of damper winding in projections on axis d-q, H;

\(L_{fkd}\) – mutual inductance between damper winding and excitation winding, H;

\(R_{kd}, R_{kq}\) – active resistances of damper winding in projections on axis d-q, Ohm.

The following calculations are performed in the three-phase system of stator windings:

EMF in stator windings (12), (13), (14):

(12)#\[e_A=e_{sd}\cdot cos(\theta_e)-e_{sq}\cdot sin(\theta_e)+e_0;\]

(13)#\[e_B=e_{sd}\cdot cos(\theta_e-\frac{2\cdot \pi}{3})-e_{sq}\cdot sin(\theta_e-\frac{2\cdot \pi}{3})+e_0;\]

(14)#\[e_C=e_{sd}\cdot cos(\theta_e+\frac{2\cdot \pi}{3})-e_{sq}\cdot sin(\theta_e+\frac{2\cdot \pi}{3})+e_0,\]

voltage equations in stator windings (15), (16), (17):

(15)#\[0=u_A+e_A+i_A \cdot R_s+L_{sA}\cdot \frac{di_A}{dt};\]

(16)#\[0=u_B+e_B+i_B \cdot R_s+L_{sB}\cdot \frac{di_B}{dt};\]

(17)#\[0=u_C+e_C+i_C \cdot R_s+L_{sC}\cdot \frac{di_C}{dt},\]

where:

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

\(L_{sA}, L_{sB}, L_{sC}\) – self- inductance of the respective stator phase, H.

Note

This inductance varies in accordance with the angle of rotor rotation and is determined by the formulas (18), (19), (20):

(18)#\[L_{sA}=L_{aa0}+L_{aa2}\cdot cos(2\cdot \theta_e);\]

(19)#\[L_{sB}=L_{aa0}+L_{aa2}\cdot cos(2\cdot (\theta_e-\frac{2\cdot \pi}{3}));\]

(20)#\[L_{sC}=L_{aa0}+L_{aa2}\cdot cos(2\cdot (\theta_e+\frac{2\cdot \pi}{3})),\]

where:

\(L_{aa0}\) – constant component of stator winding intrinsic inductance, H;

\(L_{aa2}\) – variable component of stator winding intrinsic inductance, H.

The self-inductance components of the stator winding are determined by the following formulas (21), (22):

(21)#\[L_{aa2}=\frac{L_{sd}-L_{sq}}{3};\]

(22)#\[L_{aa0}=\frac{L_{s0}+2 \cdot L_{sq} +3 \cdot L_{aa2}}{3}.\]

The excitation winding voltage equation is as follows (23):

(23)#\[U_f=R_f \cdot i_f+\frac{di_{kd}}{dt}\cdot L_{fkd}-\frac{di_{sd}}{dt}\cdot L_{mf} \cdot \frac{3}{2}+L_f \cdot \frac{di_f}{dt}\]

where:

\(R_f\) – active resistance of the excitation winding, Ohm;

\(L_f\) – self-inductance of excitation winding, H.

The electromagnetic torque on the motor shaft is determined by the formula (24):

(24)#\[M_e=\frac{3}{2}\cdot p \cdot(\psi_{sd}\cdot i_{sq}+ \psi_{sq}\cdot i_{sd})\]

Motor mechanical equation is as follows (25):

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

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 torque, determined by friction in bearings and brushes, Н\(\cdot\)m. This torque is determined by the formula (26):

(26)#\[M_{тр}=\omega \cdot K_{тр}\]

where \(K_{fr}\) – friction factor.

1. Properties#

Property	Default Value	Description	Units of measurement
Number of pole pairs	4.0	–	un.
Active resistance of stator winding	0.08	–	Ohm
Stator inductance of the d-axis	0.00015	–	H
Stator inductance of the q-axis	0.00021	–	H
Mutual inductance between the field winding and the d-axis stator winding	0.007	–	H
Stator inductance of zero sequence	0.000012	–	H
Main field resistance	3.0	–	Ohm
Inductance of the main field winding	0.5	–	H
Mutual inductance between the d-axis stator winding and the d-axis damper winding	0.0002	–	H
Mutual inductance between the q-axis stator winding and the q-axis damper winding	0.0002	–	H
Mutual inductance between the field winding and the d-axis stator winding	0.0002	–	H
Damper windings inductance of the d-axis	0.0001	–	H
Damper windings inductance of the q-axis	0.0001	–	H
Damper windings resistance of the d-axis	0.02	–	Ohm
Damper windings resistance of the q-axis	0.02	–	Ohm
Moment of inertia of rotor	0.01	–	kg\(\cdot\)m\(^2\)
Rotor friction coefficient	0.01	–	Н\(\cdot\)m/(rad/sec)
Array of initial conditions [IF, w, \( heta\)]	0.0; 0.0; 0.0	–	A; rad/s; rad
Star connection of stator windings	On	–	–

2. Inputs and outputs#

Parameters	Input/Output	Description	Units of measurement	Port type
Load torque	Input	Value of shaft moment due to load	N\(\cdot\) m	Automation
N, available when `Star connection of stator windings` - `On`	Input	Zero point of stator windings	-	Electronics
+ОВ	Input	Excitation winding positive lead	-	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
-ОВ	Output	Excitation winding negative lead	-	Electronics
A-, available when `Star connection of stator windings` - `Off`	Input	Phase A winding negative lead	-	Electronics
B-, available when `Star connection of stator windings` - `Off`	Input	Phase B winding negative lead	-	Electronics
C-, available when `Star connection of stator windings` - `Off`	Input	Phase C winding negative lead	-	Electronics

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