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K. Naresh, Vaddi Ramesh, CH. Punya Sekhar and P Anjappa 16
International Journal of Emerging Trends in Electrical and Electronics (IJETEE – ISSN: 2320-9569) Vol. 10, Issue. 1, Jan-2014.
Simulations for Three Phase to Two PhaseTransformation
K. Naresh, Vaddi Ramesh, CH. Punya Sekhar and P Anjappa
Abstract: This paper the model that has been developed so faris for two phase machine Three phase induction machine arecommon: [1]-[3]two phase machine are rarely used inindustrial application . a dynamic model for the three phaseinduction machine can be derived from the two phase machineif the equivalence between three and two phase is established.The equivalence is based on the equality of the MMFproduced in the two phase and three phase winding and equalcurrent magnitudes. Shows simulation results are compared.
Key word: Induction machine, Two phase transformation, Threephase transformation
I. INTRODUCTION
This transformation could also be thought of as atransformation from three (abc) axes to three new (dqo)axesfor uniqueness of the transformation from one set of axes toanother set of axes, including unbalances in the abcvariables requires three variables such as the dq0.[4]Thereason for this is that it is easy to converter from three abcvariables to to qd variables if the abc variables have aninherent relationship among themselves, such as the equalphase displacement and magnitude. Therefore, in such acase there are only two independent variables in a,b,c: thethird is a dependent variable obtained is unique under thatcircumstance
This paper the variable s have no such inherent relationship,then there are three distinct and independent variables:Hence the third variable cannot be recovered from theknowledge of the other two variables only[5] .It is alsomean that they are not recoverable from two variables qdbut require another variable such as the zero sequencecomponent, to recover the[7] abc variables from the dq0variables
II. THREE PHASE(a,b,c) to TWOPHASE(d,q,0)TRANSFORMATION
In electrical engineering, direct–quadrature–zero(abc) transformation or zero–direct–quadrature (dqo)transformation is a mathematical transformation that rotatesthe reference frame of three-phase systems in an effort tosimplify the analysis of three-phase circuits. In the case ofbalanced three-phase circuits, application of the dqo
K. Naresh1, Vaddi Ramesh2, CH. Punya Sekhar3, P Anjappa4,1Assistant Professor, Department of Electrical Engineering, K L University,Andhra Pradesh, India, knaresh@kluniversity.in, 2Ph.d Student,Department of Electrical Engineering, K L University, Andhra Pradesh,India, rameshvaddi6013@kluniversity.in, 3Assistant Professor, Departmentof Electrical Engineering, A N University, Andhra Pradesh, India,alif.shareef@gmail.com, 4Department of EEE, HOD & Associate professorin Golden valley integrated campus, Madanapalli, India,anji_abhi@yahoo.co.in
transform reduces the three AC quantities totwo DC quantities. Simplified calculations can then becarried out on these DC quantities before performing theinverse transform to recover the actual three-phase ACresults. It is often used in order to simplify the analysis ofthree-phase synchronous machines or to simplifycalculations for the control of three-phase inverters. Thepower-invariant, right-handed dqo transform applied to anythree-phase quantities (e.g. voltages, currents, flux linkages,etc.) is shown below in matrix form
.
The inverse transform is:
A. Geometric Interpretation
The dqo transformation is two sets of axis rotations insequence. We can begin with a 3D space where a, b,and c are orthogonal axes.
If we rotate about the axis by -45°, we get thefollowing rotation matrix:
which resolves to
K. Naresh, Vaddi Ramesh, CH. Punya Sekhar and P Anjappa 16
International Journal of Emerging Trends in Electrical and Electronics (IJETEE – ISSN: 2320-9569) Vol. 10, Issue. 1, Jan-2014.
Simulations for Three Phase to Two PhaseTransformation
K. Naresh, Vaddi Ramesh, CH. Punya Sekhar and P Anjappa
Abstract: This paper the model that has been developed so faris for two phase machine Three phase induction machine arecommon: [1]-[3]two phase machine are rarely used inindustrial application . a dynamic model for the three phaseinduction machine can be derived from the two phase machineif the equivalence between three and two phase is established.The equivalence is based on the equality of the MMFproduced in the two phase and three phase winding and equalcurrent magnitudes. Shows simulation results are compared.
Key word: Induction machine, Two phase transformation, Threephase transformation
I. INTRODUCTION
This transformation could also be thought of as atransformation from three (abc) axes to three new (dqo)axesfor uniqueness of the transformation from one set of axes toanother set of axes, including unbalances in the abcvariables requires three variables such as the dq0.[4]Thereason for this is that it is easy to converter from three abcvariables to to qd variables if the abc variables have aninherent relationship among themselves, such as the equalphase displacement and magnitude. Therefore, in such acase there are only two independent variables in a,b,c: thethird is a dependent variable obtained is unique under thatcircumstance
This paper the variable s have no such inherent relationship,then there are three distinct and independent variables:Hence the third variable cannot be recovered from theknowledge of the other two variables only[5] .It is alsomean that they are not recoverable from two variables qdbut require another variable such as the zero sequencecomponent, to recover the[7] abc variables from the dq0variables
II. THREE PHASE(a,b,c) to TWOPHASE(d,q,0)TRANSFORMATION
In electrical engineering, direct–quadrature–zero(abc) transformation or zero–direct–quadrature (dqo)transformation is a mathematical transformation that rotatesthe reference frame of three-phase systems in an effort tosimplify the analysis of three-phase circuits. In the case ofbalanced three-phase circuits, application of the dqo
K. Naresh1, Vaddi Ramesh2, CH. Punya Sekhar3, P Anjappa4,1Assistant Professor, Department of Electrical Engineering, K L University,Andhra Pradesh, India, knaresh@kluniversity.in, 2Ph.d Student,Department of Electrical Engineering, K L University, Andhra Pradesh,India, rameshvaddi6013@kluniversity.in, 3Assistant Professor, Departmentof Electrical Engineering, A N University, Andhra Pradesh, India,alif.shareef@gmail.com, 4Department of EEE, HOD & Associate professorin Golden valley integrated campus, Madanapalli, India,anji_abhi@yahoo.co.in
transform reduces the three AC quantities totwo DC quantities. Simplified calculations can then becarried out on these DC quantities before performing theinverse transform to recover the actual three-phase ACresults. It is often used in order to simplify the analysis ofthree-phase synchronous machines or to simplifycalculations for the control of three-phase inverters. Thepower-invariant, right-handed dqo transform applied to anythree-phase quantities (e.g. voltages, currents, flux linkages,etc.) is shown below in matrix form
.
The inverse transform is:
A. Geometric Interpretation
The dqo transformation is two sets of axis rotations insequence. We can begin with a 3D space where a, b,and c are orthogonal axes.
If we rotate about the axis by -45°, we get thefollowing rotation matrix:
which resolves to
K. Naresh, Vaddi Ramesh, CH. Punya Sekhar and P Anjappa 16
International Journal of Emerging Trends in Electrical and Electronics (IJETEE – ISSN: 2320-9569) Vol. 10, Issue. 1, Jan-2014.
Simulations for Three Phase to Two PhaseTransformation
K. Naresh, Vaddi Ramesh, CH. Punya Sekhar and P Anjappa
Abstract: This paper the model that has been developed so faris for two phase machine Three phase induction machine arecommon: [1]-[3]two phase machine are rarely used inindustrial application . a dynamic model for the three phaseinduction machine can be derived from the two phase machineif the equivalence between three and two phase is established.The equivalence is based on the equality of the MMFproduced in the two phase and three phase winding and equalcurrent magnitudes. Shows simulation results are compared.
Key word: Induction machine, Two phase transformation, Threephase transformation
I. INTRODUCTION
This transformation could also be thought of as atransformation from three (abc) axes to three new (dqo)axesfor uniqueness of the transformation from one set of axes toanother set of axes, including unbalances in the abcvariables requires three variables such as the dq0.[4]Thereason for this is that it is easy to converter from three abcvariables to to qd variables if the abc variables have aninherent relationship among themselves, such as the equalphase displacement and magnitude. Therefore, in such acase there are only two independent variables in a,b,c: thethird is a dependent variable obtained is unique under thatcircumstance
This paper the variable s have no such inherent relationship,then there are three distinct and independent variables:Hence the third variable cannot be recovered from theknowledge of the other two variables only[5] .It is alsomean that they are not recoverable from two variables qdbut require another variable such as the zero sequencecomponent, to recover the[7] abc variables from the dq0variables
II. THREE PHASE(a,b,c) to TWOPHASE(d,q,0)TRANSFORMATION
In electrical engineering, direct–quadrature–zero(abc) transformation or zero–direct–quadrature (dqo)transformation is a mathematical transformation that rotatesthe reference frame of three-phase systems in an effort tosimplify the analysis of three-phase circuits. In the case ofbalanced three-phase circuits, application of the dqo
K. Naresh1, Vaddi Ramesh2, CH. Punya Sekhar3, P Anjappa4,1Assistant Professor, Department of Electrical Engineering, K L University,Andhra Pradesh, India, knaresh@kluniversity.in, 2Ph.d Student,Department of Electrical Engineering, K L University, Andhra Pradesh,India, rameshvaddi6013@kluniversity.in, 3Assistant Professor, Departmentof Electrical Engineering, A N University, Andhra Pradesh, India,alif.shareef@gmail.com, 4Department of EEE, HOD & Associate professorin Golden valley integrated campus, Madanapalli, India,anji_abhi@yahoo.co.in
transform reduces the three AC quantities totwo DC quantities. Simplified calculations can then becarried out on these DC quantities before performing theinverse transform to recover the actual three-phase ACresults. It is often used in order to simplify the analysis ofthree-phase synchronous machines or to simplifycalculations for the control of three-phase inverters. Thepower-invariant, right-handed dqo transform applied to anythree-phase quantities (e.g. voltages, currents, flux linkages,etc.) is shown below in matrix form
.
The inverse transform is:
A. Geometric Interpretation
The dqo transformation is two sets of axis rotations insequence. We can begin with a 3D space where a, b,and c are orthogonal axes.
If we rotate about the axis by -45°, we get thefollowing rotation matrix:
which resolves to
K. Naresh, Vaddi Ramesh, CH. Punya Sekhar and P Anjappa 17
International Journal of Emerging Trends in Electrical and Electronics (IJETEE – ISSN: 2320-9569) Vol. 10, Issue. 1, Jan-2014.
With this rotation, the axes look like
Then we can rotate about the new b axis by
):
,
Which resolves to
.
When these two matrices are multiplied, we get the Clarketransformation matrix C:
This is the first of the two sets of axis rotations. Atthis point, we can relabel the rotated a, b, and c axes as α, β,and z. This first set of rotations places the z axis an equaldistance away from all three of the original a, b, and c axes.In a balanced system, the values on these three axes wouldalways balance each other in such a way that the z axis valuewould be zero. This is one of the core values of the dqotransformation; it can reduce the number relevant variablesin the system.
The second set of axis rotations is very simple. Inelectric systems, very often the a, b, and c values areoscillating in such a way that the net vector is spinning. In abalanced system, the vector is spinning about the z axis.Very often, it is helpful to rotate the reference frame suchthat the majority of the changes in the abc values, due to thisspinning, are canceled out and any finer variations inbecome more obvious. So, in addition to the Clarketransform, the following axis rotation is applied aboutthe z axis:
.Multiplying this matrix by the Clarke matrix results in thedqo transform:
.The dqo transformation can be thought of in geometricterms as the projection of the three separate sinusoidal phasequantities onto two axes rotating with the same angularvelocity as the sinusoidal phase quantities. The two axes arecalled the direct, or d, axis; and the quadrature or q, axis;that is, with the q-axis being at an angle of 90 degrees fromthe direct axis.
K. Naresh, Vaddi Ramesh, CH. Punya Sekhar and P Anjappa 17
International Journal of Emerging Trends in Electrical and Electronics (IJETEE – ISSN: 2320-9569) Vol. 10, Issue. 1, Jan-2014.
With this rotation, the axes look like
Then we can rotate about the new b axis by
):
,
Which resolves to
.
When these two matrices are multiplied, we get the Clarketransformation matrix C:
This is the first of the two sets of axis rotations. Atthis point, we can relabel the rotated a, b, and c axes as α, β,and z. This first set of rotations places the z axis an equaldistance away from all three of the original a, b, and c axes.In a balanced system, the values on these three axes wouldalways balance each other in such a way that the z axis valuewould be zero. This is one of the core values of the dqotransformation; it can reduce the number relevant variablesin the system.
The second set of axis rotations is very simple. Inelectric systems, very often the a, b, and c values areoscillating in such a way that the net vector is spinning. In abalanced system, the vector is spinning about the z axis.Very often, it is helpful to rotate the reference frame suchthat the majority of the changes in the abc values, due to thisspinning, are canceled out and any finer variations inbecome more obvious. So, in addition to the Clarketransform, the following axis rotation is applied aboutthe z axis:
.Multiplying this matrix by the Clarke matrix results in thedqo transform:
.The dqo transformation can be thought of in geometricterms as the projection of the three separate sinusoidal phasequantities onto two axes rotating with the same angularvelocity as the sinusoidal phase quantities. The two axes arecalled the direct, or d, axis; and the quadrature or q, axis;that is, with the q-axis being at an angle of 90 degrees fromthe direct axis.
K. Naresh, Vaddi Ramesh, CH. Punya Sekhar and P Anjappa 17
International Journal of Emerging Trends in Electrical and Electronics (IJETEE – ISSN: 2320-9569) Vol. 10, Issue. 1, Jan-2014.
With this rotation, the axes look like
Then we can rotate about the new b axis by
):
,
Which resolves to
.
When these two matrices are multiplied, we get the Clarketransformation matrix C:
This is the first of the two sets of axis rotations. Atthis point, we can relabel the rotated a, b, and c axes as α, β,and z. This first set of rotations places the z axis an equaldistance away from all three of the original a, b, and c axes.In a balanced system, the values on these three axes wouldalways balance each other in such a way that the z axis valuewould be zero. This is one of the core values of the dqotransformation; it can reduce the number relevant variablesin the system.
The second set of axis rotations is very simple. Inelectric systems, very often the a, b, and c values areoscillating in such a way that the net vector is spinning. In abalanced system, the vector is spinning about the z axis.Very often, it is helpful to rotate the reference frame suchthat the majority of the changes in the abc values, due to thisspinning, are canceled out and any finer variations inbecome more obvious. So, in addition to the Clarketransform, the following axis rotation is applied aboutthe z axis:
.Multiplying this matrix by the Clarke matrix results in thedqo transform:
.The dqo transformation can be thought of in geometricterms as the projection of the three separate sinusoidal phasequantities onto two axes rotating with the same angularvelocity as the sinusoidal phase quantities. The two axes arecalled the direct, or d, axis; and the quadrature or q, axis;that is, with the q-axis being at an angle of 90 degrees fromthe direct axis.
K. Naresh, Vaddi Ramesh, CH. Punya Sekhar and P Anjappa 18
International Journal of Emerging Trends in Electrical and Electronics (IJETEE – ISSN: 2320-9569) Vol. 10, Issue. 1, Jan-2014.
Shown above is the dqo transform as applied to the stator ofa synchronous machine. There are three windings separatedby 120 physical degrees. The three phase currents are equalin magnitude and are separated from one another by 120electrical degrees. The three phase currents lag theircorresponding phase voltages by . The d-q axis is shownrotating with angular velocity equal to , the same angularvelocity as the phase voltages and currents. The d axismakes an angle with the A winding which has
been chosen as the reference. The currents and areconstant DC quantities.
III. SIMULATION RESULTS
Fig 1. Simulate three phase to two phase transformation
Fig 2. Simulate the voltage ¤t
K. Naresh, Vaddi Ramesh, CH. Punya Sekhar and P Anjappa 18
International Journal of Emerging Trends in Electrical and Electronics (IJETEE – ISSN: 2320-9569) Vol. 10, Issue. 1, Jan-2014.
Shown above is the dqo transform as applied to the stator ofa synchronous machine. There are three windings separatedby 120 physical degrees. The three phase currents are equalin magnitude and are separated from one another by 120electrical degrees. The three phase currents lag theircorresponding phase voltages by . The d-q axis is shownrotating with angular velocity equal to , the same angularvelocity as the phase voltages and currents. The d axismakes an angle with the A winding which has
been chosen as the reference. The currents and areconstant DC quantities.
III. SIMULATION RESULTS
Fig 1. Simulate three phase to two phase transformation
Fig 2. Simulate the voltage ¤t
K. Naresh, Vaddi Ramesh, CH. Punya Sekhar and P Anjappa 18
International Journal of Emerging Trends in Electrical and Electronics (IJETEE – ISSN: 2320-9569) Vol. 10, Issue. 1, Jan-2014.
Shown above is the dqo transform as applied to the stator ofa synchronous machine. There are three windings separatedby 120 physical degrees. The three phase currents are equalin magnitude and are separated from one another by 120electrical degrees. The three phase currents lag theircorresponding phase voltages by . The d-q axis is shownrotating with angular velocity equal to , the same angularvelocity as the phase voltages and currents. The d axismakes an angle with the A winding which has
been chosen as the reference. The currents and areconstant DC quantities.
III. SIMULATION RESULTS
Fig 1. Simulate three phase to two phase transformation
Fig 2. Simulate the voltage ¤t
K. Naresh, Vaddi Ramesh, CH. Punya Sekhar and P Anjappa 19
International Journal of Emerging Trends in Electrical and Electronics (IJETEE – ISSN: 2320-9569) Vol. 10, Issue. 1, Jan-2014.
Fig 3. Simulate the Vd & Vq voltage
Fig 4. Simulate the three phase voltage Vabc
Fig 5.Simulate the three phase voltage Vabc1
K. Naresh, Vaddi Ramesh, CH. Punya Sekhar and P Anjappa 20
International Journal of Emerging Trends in Electrical and Electronics (IJETEE – ISSN: 2320-9569) Vol. 10, Issue. 1, Jan-2014.
IV. CONCLUSION
This paper presented two phase machine are rarely used inindustrial application. a dynamic model for the three phaseinduction machine can be derived from the two phasemachine if the equivalence between three and two phase isestablished .the equivalence is based on the equality of theMMF produced in the two phase and three phase windingand equal current magnitudes. Shows the simulation resultsare compared.
V. REFERENCES
[1] S. Halász, A. A. M. Hassan, and B. T. Huu, “Optimal control ofthree level PWM inverters,” IEEE Trans. Ind. Electron., vol. 44, pp.96–107,Feb. 1997.
[2] F. Bauer and H. D. Hening, “Quick response space vector control foa high power three-level inverter drive system,” in Proc. EPEConf.,Aachen, Germany, 1989, pp. 417–421.
[3] B. T. Huu, S. Halász, and G. Csonka, “Three-level inverters withsinusoidal PWM techniques,” in Proc. 5th Int. Conf. Optimization ofElectric and Electronic Equipments, Brassov, Romania, 1996, pp.1261–1264.
[4] B. Velaerts, P. Mathys, and G. Bingen, “New developments of 3-leve PWM strategies,” in Proc. EPE Conf., Aachen, Germany, 1989,pp 411–416.
[5] M. Depenbrock, “Pulse width control of a 3-phase inverter with nosinusoidal phase voltages,” in Proc. IEEE Int. SemiconductorPowerConf., 1977, pp. 399–403.
[6] C. L. Fortescue, “Method of symmetrical co-ordinates applied to thesolution of polyphase networks,” Trans. Amer. Inst. Elect. Eng., vol.37, no. 2, pp. 1027–1140, 1918.
[7] P. M. Anderson, Analysis of Faulted Power Systems. New York:IEEE Press, 1995
[8] J. L. Blackburn, Symmetrical Components for Power Systems Engi-neering. Boca Raton, FL: CRC, 1993, vol. 85.
[9] R. G. Wasley and M. A. Shlash, “Newton-Raphson algorithm fo 3-phase load flow,” Proc. Inst. Elect. Eng., Gen., Transm., Distrib., vol.121, no. 7, pp. 630–638, 1974.
[10] Generalized theory of electrical machines:Dr.P.S.Bimbhra[11] Analysis of Electric Machinery And Drive Systems:Second Edition
:PC krause,oleg wasynczuk,scottd.sudhoff[12] Electric motor drives machines modeling:R.KRISHNAN
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