2.3.4 Compound machines
As it is known, compound machines are fitted with both series and shunt field windings. Therefore, also simulations of transients and steady-state conditions are made on the basis of combinations of appropriate equations discussed in the previous chapters.
2.3.5 Single-phase commutator series motors
These motors, known as universal motor, can work on DC as well as AC network. Their connection is identical with series DC motors, even though there are
Figure 23.
Simulations of the series dynamo and time waveforms of (a) armature current, (b) induced voltage, (c) terminal voltage, and (d) terminal voltage vs. load current in steady-state conditions at the changing of the load resistance.
Figure 22.
Measured magnetizing curve for the investigated series machine Ui= f(If).
some differences in their design. At the simulations, it is necessary to take into account that there are alternating variables of voltage and current; it means that winding’s parameters act as impedances, not only resistances.
2.4 Transformation of the three-phase system abc to the system dq0
2.4.1 IntroductionUp to now we have dealt with DC machines, the windings of which are arranged in two perpendicular axes to each other. However, alternating rotating machines obviously have three-phase distributed windings on the stator, which must be transformed into two perpendicular axes, to be able to employ equations derived in the previous chapters.
In history, it can be found that principles of the variable projections into two perpendicular axes were developed for synchronous machine with salient poles.
A different air gap in the axis that acts as field winding and magnetic flux is created and, in the axis perpendicular to that magnetic flux, was linked with a different magnetic permeance of the circuit, which resulted in different reactances of armature reaction and therefore different synchronous reactances. It was shown that this projection into two perpendicular axes and variables can be employed much wider and can be applied for investigation of transients on the basis of the general theory of electrical machines.
On the other side, it is necessary to realize that phase values transformed into dq0 system have gotten into a fictitious system with fictitious parameters, where investigation is easier, but the solution does not show real values. Therefore, an inverse transformation into the abc system must be done to gain real values of voltages, currents, torques, powers, speed, etc. This principle is not unknown in the other investigation of electrical machines. For example, the rotor variables referred to the stator in the case of asynchronous machines mean investigation in a fictitious system, where 29 the calculation and analysis is more simple, but to get real values in the rotor winding a reverse transformation must be done.
Therefore, we will deal with a transformation of the phase variables abc into the fictitious reference k-system dq0 with two perpendicular axes which rotate by angular speedωkwith regard to the stator system. The axis “0” is perpendicular to the plain given by two axes d, q. As it will be shown, the investigation of the machine properties in this system is simpler because the number of equations is reduced, which is a big advantage. However, to get values of the real variables, it will be necessary to make an inverse transformation, as it will be shown gradually in the next chapters.
A graphical interpretation of the transformation abc into the system dq0 is shown inFigure 24. This arrangement is formed according to the original letters given by the papers of R.H. Park and his co-authors (around 1928 and later), e.g., [14], although nowadays it is possible to find various other figures, corresponding to the different position of the axes d, q, and corresponding equations.
According to the original approach, if the three-phase system is symmetrical, the d-axis is shifted from the axis of the a-phase about the angleϑk, and the q-axis is ahead of the d-axis by about 90°; then the components in the d-axis and q-axis are the projections of the phase variables of voltage, linkage magnetic flux, or currents, generally marked as x-variable, into those axes. In the given papers, there are derived equations of the abc into dq0 transformation as well as the equations of the inverse transformation dq0 into abc, because of the investigation of the
synchronous reactances of the synchronous machine with salient poles. Also con- stants of proportions are given. Today this transformation is called “Park’s trans- formation” (see equations given below), even though this name is not given in the original papers. Next equations will be derived, and the constants of proportions kd, kq, and k0will be employed. Later these constants will be selected according to how the reference system will be positioned, to apply the most profitable solutions.
Employment of Park’s transformation equations is today very widespread, and they are used for all kinds of electrical machines, frequency convertors, and other three- phase circuits.
2.4.2 Equations of Park’s transformations abc into dq0 system
According toFigure 24, the d-component of the x-variable is a sum of a-, b-, and c-phase projections:
xd¼xdaþxdbþxdc, (120)
where
xda¼xacosϑk, (121)
xdb¼xbcosðϑk�120°Þ, (122) xdc¼xccosðϑkþ120°Þ: (123) Also, projections into the q-axis are made in a similar way. It is seen that the projections to the q-axis are expressed by sinusoidal function of the phase variable with a negative sign, at the given +q-axis (see Eq. (125)).
Figure 24.
Graphical interpretation of the three-phase variable transformation abc into the reference k-system dq0, rotating by the speedωk.
The zero component is a sum of the instantaneous values of the phase variables.
If the three-phase system is symmetrical, the sum of the instantaneous values is zero; therefore also the zero component is zero (see Eq. (126)). The zero component can be visualized in such a way that the three-phase variable projection is made in the 0-axis perpendicular to the plain created by the d-axis and q-axis, whereby the 0-axis is conducted through the point 0.
Then the equation system for the Park transformation from the abc to the dq0 system is created by Eqs. (124)–(126). To generalize the expressions, proportional constants kd, kq, and k0are employed:
xd¼kd xacosϑkþxbcos ϑk�2π 3
� �
þxccos ϑkþ2π 3
� �
� �
, (124)
xq¼ �kq xasinϑkþxbsin ϑk�2π 3
� �
þxcsin ϑkþ2π 3
� �
� �
, (125)
x0¼k0ðxaþxbþxcÞ: (126) It is true that R.H. Park does not mention such constants in the original paper, because he solved synchronous machine, which will be explained later (Sections 8, 10, and 16). For the purposes of this textbook, it is suitable to start as general as possible and gradually adapt the equations to the individual kinds of electrical machines to get a solution as advantageous as possible. Therefore, the constants can be whichever except zero, though of such, that the equation determinant is not zero (see Eq. (127)). Then the inverse transformation will be possible to do and to find the real phase variables.
The determinant of the system is as follows:
kdcosϑk kdcos ϑk�2π 3
� �
kdcos ϑkþ2π 3
� �
�kqsinϑk �kqsin ϑk�2π 3
� �
�kqsin ϑkþ2π 3
� �
k0 k0 k0
��
��
��
��
��
�
��
��
��
��
��
�
¼kdkqk03 ffiffiffi p3
2 cos ϑk�2π 3
� �
: (127)
2.4.3 Equations for the m-phase system transformation
Equations for the three-phase system transformation can be spread to the m-phase system. Now the phases will be marked by 1, 2, 3, etc., to be able to express the mth phase and to see how the argument of the functions is created:
xd¼kd x1cosϑkþx2cos ϑk�2π m
� �
þx3cos ϑk�4π m
� �
þ … þxmcos ϑk�2 mð �1Þπ m
� �
� �
: (128) Similarly, equations for the q- and 0-components are written. If a proportional constant 2/3 will be used for the three-phase system, then the corresponding con- stant for the m-phase system is 2/m [2].
xd¼ 2
m x1cosϑkþx2cos ϑk�2π m
� �
þx3cos ϑk�4π m
� �
þ… þxmcos ϑk�2 mð �1Þπ m
� �
� �
: (129)
synchronous reactances of the synchronous machine with salient poles. Also con- stants of proportions are given. Today this transformation is called “Park’s trans- formation” (see equations given below), even though this name is not given in the original papers. Next equations will be derived, and the constants of proportions kd, kq, and k0will be employed. Later these constants will be selected according to how the reference system will be positioned, to apply the most profitable solutions.
Employment of Park’s transformation equations is today very widespread, and they are used for all kinds of electrical machines, frequency convertors, and other three- phase circuits.
2.4.2 Equations of Park’s transformations abc into dq0 system
According toFigure 24, the d-component of the x-variable is a sum of a-, b-, and c-phase projections:
xd¼xdaþxdbþxdc, (120)
where
xda¼xacosϑk, (121)
xdb¼xbcosðϑk�120°Þ, (122) xdc¼xccosðϑkþ120°Þ: (123) Also, projections into the q-axis are made in a similar way. It is seen that the projections to the q-axis are expressed by sinusoidal function of the phase variable with a negative sign, at the given +q-axis (see Eq. (125)).
Figure 24.
Graphical interpretation of the three-phase variable transformation abc into the reference k-system dq0, rotating by the speedωk.
The zero component is a sum of the instantaneous values of the phase variables.
If the three-phase system is symmetrical, the sum of the instantaneous values is zero; therefore also the zero component is zero (see Eq. (126)). The zero component can be visualized in such a way that the three-phase variable projection is made in the 0-axis perpendicular to the plain created by the d-axis and q-axis, whereby the 0-axis is conducted through the point 0.
Then the equation system for the Park transformation from the abc to the dq0 system is created by Eqs. (124)–(126). To generalize the expressions, proportional constants kd, kq, and k0are employed:
xd¼kd xacosϑkþxbcos ϑk�2π 3
� �
þxccos ϑkþ2π 3
� �
� �
, (124)
xq¼ �kq xasinϑkþxbsin ϑk�2π 3
� �
þxcsin ϑkþ2π 3
� �
� �
, (125)
x0¼k0ðxaþxbþxcÞ: (126) It is true that R.H. Park does not mention such constants in the original paper, because he solved synchronous machine, which will be explained later (Sections 8, 10, and 16). For the purposes of this textbook, it is suitable to start as general as possible and gradually adapt the equations to the individual kinds of electrical machines to get a solution as advantageous as possible. Therefore, the constants can be whichever except zero, though of such, that the equation determinant is not zero (see Eq. (127)). Then the inverse transformation will be possible to do and to find the real phase variables.
The determinant of the system is as follows:
kdcosϑk kdcos ϑk�2π 3
� �
kdcos ϑkþ2π 3
� �
�kqsinϑk �kqsin ϑk�2π 3
� �
�kqsin ϑkþ2π 3
� �
k0 k0 k0
��
��
��
��
��
�
��
��
��
��
��
�
¼kdkqk03 ffiffiffi p3
2 cos ϑk�2π 3
� �
: (127)
2.4.3 Equations for the m-phase system transformation
Equations for the three-phase system transformation can be spread to the m-phase system. Now the phases will be marked by 1, 2, 3, etc., to be able to express the mth phase and to see how the argument of the functions is created:
xd¼kd x1cosϑkþx2cos ϑk�2π m
� �
þx3cos ϑk�4π m
� �
þ …þxmcos ϑk�2 mð �1Þπ m
� �
� �
: (128) Similarly, equations for the q- and 0-components are written. If a proportional constant 2/3 will be used for the three-phase system, then the corresponding con- stant for the m-phase system is 2/m [2].
xd¼ 2
m x1cosϑkþx2cos ϑk�2π m
� �
þx3cos ϑk�4π m
� �
þ … þxmcos ϑk�2 mð �1Þπ m
� �
� �
: (129)