Dr. Amr AbdAllah* Electric Machines IIIACOURSE EPM 405A FOR 4th Year Power and MachinesELECTRICAL DEPARTMENT

Lecture 01

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ContentsReference Frame TheoryThree Phase Induction Machine ModelingSynchronous Machine ModelingDynamic Preformance of Synchronous machines.

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Grading System30 grade for term activitiesMid Term Exam 15 gradesSection evaluation 10 gradesLecture and Matlab Reports 5 grades70 grade for final examTeaching assistant: Eng. MokhtarLectures: Saturdays 2nd slot

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ReferencesAnalysis of Electric Machinery and Drive Systems, Paul C. Krause.

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Dr. Amr AbdAllah*Reference Frame TheoryIntroductionThe voltage equations that describes the performance of the induction and synchronous machines are based on the differential equation

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Dr. Amr AbdAllah*Reference Frame TheoryIntroductionSome of the machine inductances are function of the rotor speed (position) whereupon the coefficients of the differential equations that describe the behavior of these machine are time varying.

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Dr. Amr AbdAllah*Reference Frame TheoryIntroductionA change of variables is often used to reduce the complexity of the differential equations.There are several changes of variables that are used.All changes of variables used to transform real variables are contained in one transformation.

This general transformation refers machine variables to a frame of reference that rotates at an arbitrary angular velocity.All known real transformations are obtained from this transformation by simply assigning the speed of the rotation of the reference frame.

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Dr. Amr AbdAllah*Reference Frame TheoryBackground:Parks transformationIt formulates a change of variables which, in effect, replaced the variables (voltages, current and flux linkages) associated with the stator winding of a synchronous machine with variables associated with fictitious windings rotating with the rotor.The synchronous machine has inductances which are time varying due to 1- Electric circuits in relative motion2- Electric circuit with varying magnetic reluctance (due to rotor saliency).This transformation has the unique property of eliminating the time varying inductances from the voltage equations of the synchronous machine.

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Dr. Amr AbdAllah*Reference Frame TheoryBackground:Stanleys transformationIt formulates a change of variables to the time varying inductances in the voltage equations of the induction machines (due to electric circuits in relative motion).

It eliminates the time varying inductances of the induction machine by transforming the variables associated with the rotor windings (rotor variables) to variables associated with fictitious stationary windings.

This means that the rotor variables are transformed to a frame reference fixed in the stator.

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Dr. Amr AbdAllah*Reference Frame TheoryBackground:Krons transformation

It formulates a change of variables to the time varying inductances in the voltage equations of the induction machines (due to electric circuits in relative motion) by transforming both the stator and rotor variables to a reference frame rotating in synchronism with the rotating magnetic field.

This reference frame is commonly referred to as the synchronously rotating reference frame.

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Dr. Amr AbdAllah*Reference Frame TheoryBackground:Brereton's transformationEmployed a change of variables that eliminate the time varying inductances of a symmetrical induction machine by transforming the stator variables to a reference frame fixed in the rotor. This is essentially Parks transformation applied to induction machines.

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Dr. Amr AbdAllah*Reference Frame TheoryBackground:All in One; arbitrary reference frame:Park, Stanley, Kron and Brereton transformations are actually a derivation -as previously mentioned- from one general transformation that eliminates all time varying inductances by referring the stator and rotor variables to a reference frame that may rotate at any arbitrary angular velocity or remain stationary.

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Dr. Amr AbdAllah*Reference Frame TheoryEquations of Transformation:Change of Variables:

where

It can be shown that the inverse transformation is given by:

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Dr. Amr AbdAllah*Reference Frame TheoryEquations of Transformation:Change of Variables:

where

It can be shown that the inverse transformation is given by: or

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Dr. Amr AbdAllah*Reference Frame TheoryEquations of Transformation:Change of Variables:Although the transformation of the arbitrary reference frame is a change of variables and needs no physical connotation, it often convenient to visualize the transformation equations as trigonometric relationships between variables.fas, fbs and fcs, can be thought as the direction of magnetic axes of the stator windings.fds and fqs can be considered as the direction of the magnetic axes of the new windings created by the change of variables.It is important to notice that f0s variables are not associated with the arbitrary reference frame. It is related to the abc variables independent on .

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Dr. Amr AbdAllah*Reference Frame TheoryEquations of Transformation:Change of Variables:The total instantaneous power may be expressed in abc variables as:The total power expressed in the dq0 variables must equal the total power expressed in the abc variables; Applying the transformations yields for:

The 3/2 factor comes due to the choice of the constant in the transformation. It will be noticed that the waveform of the total power is the same regardless of the reference frame in which it is evaluated

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Dr. Amr AbdAllah*Reference Frame TheoryEquations of Transformation:Example 1:

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Dr. Amr AbdAllah*Reference Frame TheoryEquations of Transformation:Example 1: Contd

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Dr. Amr AbdAllah*Reference Frame TheoryEquations of Transformation:Example 1: Contd

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Dr. Amr AbdAllah*Reference Frame TheoryEquations of Transformation:Example 1: Contd

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Dr. Amr AbdAllah*Next Lecture

CHAPTER I Reference Frame Theory Contd

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