Title: Lecture 2A
1Lecture 2A
Synchronous Machine Modeling
- Derivation of the dq0 Equations of an Idealized
Three-Phase Synchronous Machine
Professor Ali Keyhani
2Derivation of the dq0 Equations of an Idealized
Three-Phase Synchronous Machine
3Derivation of the dq0 Equations of an Idealized
Three-Phase Synchronous Machine
- Assumptions
- 1) A stator inner periphery has uniform
radius (Not a function of position around the
gap) - 2) A rotor outer periphery has non-uniform
radius. The rotor, specifically, is shaped such
that when the motor is excited by a single, full
pitched, concentrated winding located on the
stator, then the flux per unit length which
exists from the stator into the air gap is a
sinusoidal function of position around the gap. - 3) Two sets of shorted rotor coils or bars
(amortisseur windings) are located on the two
axes of rotor, which, although not sinusoidally
distributed, have induced in them sinusoidal mmf
(current) distribution due to the coupling with
the stator circuits.
4Derivation of the dq0 Equations of an Idealized
Three-phase Synchronous Machine
- A third winding is located on one axis of rotor
(d-axis, see Figure 1). Although this winding is
generally concentrated, it is assumed that the
winding produces same fundamental component of
mmf around the gap. This is permissible since it
was shown for two-phase case that the harmonic
components of mmf give rise merely to a
differential leakage flux component. - Linear magnetic current (no saturation). This
assumption will be relaxed later. Since the
circuit is linear, it is assumed that the stator
and rotor iron (finite) can be replaced by
material having infinite permeability. It is
assumed that the gap can be increased to account
for this effect. - Constant electrical parameters (i.e. R, L, C)
independent of temperature or frequency. - The stator is connected as a 4-wire system
5Derivation of the dq0 Equations of an Idealized
Three-phase Synchronous Machine
- Axes of Reference
-
- Again axes of reference are chosen for each of
the six windings. An axis of reference id
developed along the direction of maximum positive
mmf produced by each winding taken individually.
Positive mmf is assumed to be directed across the
gap from rotor to stator. -
- Machine Equations- Phase Variables
-
- The stator and rotor voltages can be expressed in
vector-matrix form as - Stator
6Derivation of the dq0 Equations of an Idealized
Three-phase Synchronous Machine
7Derivation of the dq0 Equations of an Idealized
Three-phase Synchronous Machine
- In general, the flux linkages for any orientation
of the rotor are
8Derivation of the dq0 Equations of an Idealized
Three-phase Synchronous Machine
- Stator Inductances
- The expressions for the inductances of the
machine can be written down by plausibility
arguments similar to those given in the book by
Majmudar - pages 224- 234 and 408 415. Also, in
the book by B. Adkins pages 58- 61.
9Derivation of the dq0 Equations of an Idealized
Three-phase Synchronous Machine
- Stator Rotor Mutual Inductances
- Stator Field Windings
10Derivation of the dq0 Equations of an Idealized
Three-phase Synchronous Machine
- Stator Windings and dr windings
11Derivation of the dq0 Equations of an Idealized
Three-phase Synchronous Machine
- Stator Windings and qr windings
- Rotor Inductances
- It may be noted that both the stator and rotor
self and mutual inductances are constants
independent of .
12Derivation of the dq0 Equations of an Idealized
Three-phase Synchronous Machine
- Self Inductances
- Mutual Inductances
13Derivation of the dq0 Equations of an Idealized
Three-phase Synchronous Machine
- Matrix Notation
- The following inductance matrices may be
defined - Note that represents the leakage inductance
due to the leakage flux.
14Derivation of the dq0 Equations of an Idealized
Three-phase Synchronous Machine
- Transformation from phase quantities to dq0 axes
- Field on the rotor
- The machine dynamic equations are
- where,
15Derivation of the dq0 Equations of an Idealized
Three-phase Synchronous Machine
- Matrices and are defined by
(26), (27), (28) and (29). The matrix is - Observations
- The two circuits on the rotor, that is dr and
qr, have resistances and self-inductances which
are not necessarily equal. - The three stator circuits have identical
resistances and self-inductances which are
symmetrical. Hence, it may be possible to
simplify the stator equations by a change in
variable.
16Derivation of the dq0 Equations of an Idealized
Three-phase Synchronous Machine
- It may be apparent that a transformation of
variables to any reference frame other than a
reference frame fixed on the rotor will result in
the differential equations associated with these
circuits to become more complicated rather than
simplified. -
- Transformation of Stator Variables to the dq0
axes fixed on the rotor - The dq0 axes are fixed on the rotor (see Fig.
2) and they are rotating at angular velocity of
. The stator variables as seen from dq0
axes are
17Derivation of the dq0 Equations of an Idealized
Three-phase Synchronous Machine
- where, for , the is
- Since the rotor windings are rotating at rotor
speed, the rotor variables do not need to be
transformed. - To transform equation (30) to the dq0s reference
frame, first multiply this equation by
. -
-
18Derivation of the dq0 Equations of an Idealized
Three-phase Synchronous Machine
- Using (35) and (36) in (39), we will have
- From equation (37),
- Note that
-
-
19Derivation of the dq0 Equations of an Idealized
Three-phase Synchronous Machine
- The equation (42) can be rewritten as
- Since the rotor rotates at the same speed as the
reference frame, the rotor equation remains
unchanged. -
- Transformation of the Stator Flux linkage
Equation - Multiplying (32) by ,
20Derivation of the dq0 Equations of an Idealized
Three-phase Synchronous Machine
- Note that
-
- The term can
be written as - The second term can be written
as
21Derivation of the dq0 Equations of an Idealized
Three-phase Synchronous Machine
-
- Homework problem show that equation (48) and (49)
are correct. - The stator flux equation is
- which results in
22Derivation of the dq0 Equations of an Idealized
Three-phase Synchronous Machine
-
- Transformation of the Rotor Flux linkage Equation
to the d-q axes - The rotor flux linkage equation is
- Rewrite the above as
- Equation (55) can be written as
-
23Derivation of the dq0 Equations of an Idealized
Three-phase Synchronous Machine
- After hard labor, you may obtain the following
-
- Using (56) and (29), is
24Derivation of the dq0 Equations of an Idealized
Three-phase Synchronous Machine
- The scalar forms of (57) are
-
25Derivation of the dq0 Equations of an Idealized
Three-phase Synchronous Machine
- Summary of the Synchronous Machine Equations in
the dq0 reference frame rotating at rotor speed -
26Derivation of the dq0 Equations of an Idealized
Three-phase Synchronous Machine
27Derivation of the dq0 Equations of an Idealized
Three-phase Synchronous Machine
- Note that the flux linking the qr circuit from
a unit current in the qs circuit is not the
same as the flux linking the qs circuit due to
unit current in the qr circuit. - Equations (61) (72) suggest the equivalent
circuit shown below -
-
d- axis circuit
28Derivation of the dq0 Equations of an Idealized
Three-phase Synchronous Machine
-
- q- axis circuit
- 0-axis circuit