Download Analyzing Rotating Magnetic Fields in 3-Phase Electrical Machines: Vector Analysis and more Assignments Mechanical Engineering in PDF only on Docsity!
Proceedings of the 2002 American Society for Engineering Education Annual Conference & Exposition
VECTOR ANALYSIS APPLICATION IN ROTATING MAGNETIC
FIELDS
Bruno Osorno
Department of Electrical And Computer Engineering California State University Northridge 18111 Nordhoff St Northridge CA 91330- Email:[email protected]
Abstract
Rotating ma gnetic fields in three-phase electrical machines has been one of the hardest topics to convey to our students in the area of energy conversion. These fields are transformed into phasors (vectors) that
rotate in space around the stator of an electrical machine. The mathematical proof of such rotation will be shown and a MATLAB simulation describing these vectors will be given. Classic textbooks approach this concept mathematically and usually the students are left to imagine the space vector rotation. This is no longer the case, now we can simulate and SEE in our desktop computers, using MatlabT M , rotating phasors.
Polyphase Analysis
Three phase induction machines are the work- horse of industry, and these machines have a “rotating magnetic field”. We will give a brief physical description; consider a sequence a-b-c and a symmetric
distribution of the phases by 120^0 electrical degrees each in space and around the air gap. The basic three-phase machine will have three coils that we consider to have the following terminals:
a a’ b b’ c c’
These coils are excited by a time dependent sinusoidal alternating current producing a sinusoidal magneto- motive force (mmf) wave at the center of the magnetic axis of particular phase. Therefore the three-space sinusoidal mmf waves are displaced 120^0 electrical degrees in space. Figure 1 shows how
we determine the magnetic axis of a coil. Figure two indicates three magnetic-axis of three coils placed in space around the stator. A three-phase system requires three coils to create three magnetic fields that will interact among each other to obtain a resulting magnetic component. Furthermore, this resultant component will rotate in space around the air gap of the electric machine.
Page 7.1294.
Proceedings of the 2002 American Society for Engineering Education Annual Conference & Exposition
Magnetic Axis
Current out Magnetic flux around the conductor
Current in Magnetic flux around the conductor
X
X'
Figure 1. Magnetic axis of a single coil
Figure 1 shows a coil with current coming out conductor X with a dot and a conductor X’ with current getting into the conductor. Using the right hand rule [3], we determine the direction of the magnetic flux.
The magnetic axis is, therefore, created perpendicular to the coil. Figure 2 shows three coils placed in space around the stator of a primitive three-phase machine. Notice that the magnetic axis of each coil is perpendicular to their corresponding plane.
a
a'
b
b'
c
c' magnetic axis coil a-a'
magnetic axis coil b-b'
magnetic axis coil c-c'
Figure 2. Magnetic axis of three coils displaced symmetrically
For balanced conditions the instantaneous currents, for each coil, are:
i (^) a = Im sin ω t (1) Page 7.1294.
Proceedings of the 2002 American Society for Engineering Education Annual Conference & Exposition
a
a'
b
b'
c
c'
Resultant magnetic axis B
magnetic axis magnetic axis coil b-b' coil c-c'
R
Figure3. Resultant magnetic flux at ωt=
At ωt =90o^ the resultant magnetic flux is equal to jo
- 5 e^0. Figure 4 shows the vector diagram for this resultant magnetic flux.
a
a'
b
b'
c
c'
Resultant
magnetic axis
B
magnetic axis
coil b-b'
magnetic axis
coil c-c'
R
Magnetic axis coil
a-a'
Page 7.1294.
Proceedings of the 2002 American Society for Engineering Education Annual Conference & Exposition
Figure4. Resultant magnetic flux for ωt = 30o
Simulation
This a program developed using MatlabT M^ that performs the calculations for Br for 360 degrees. The animation part of it and the graphics indicate very clearly the vector rotation of electric magnetic fields in an electric three-phase machine.
% CALIFORNIA STATE UNIVERSITY NORTHRIDGE
% ROTATING MAGNETIC FIELD
% JANUARY 2002
% CALCULATE NET MAGNETIC FIELD PRODUCED BY A THREE-PHASE STATOR
% BMAX = 1
% FREQUENCY = 60 HZ
% GENERATE THREE COMPONENTS MAGNETIC FIELD
hz = 60 t = 0:1/6000:1/ omega = hz23. j = sqrt(-1)
Bx = sin(omegat)(cos(0) + jsin(0)) By = sin(omega * t - 2pi/3)(cos(2pi/3)+ jsin(2pi/3))
Bz = sin(omegat + 2pi/3)(cos(-2pi/3)+ jsin(-2pi/3))
Br = Bx + By + Bz
circle = 1.5(cos(omegat)+ jsin(omegat)) % plot the magnitude and direction of the resultant magnetic field Br 1.5 * Bmax*angle
ii = 1:length(t) k = 1
%subplot(3,1,1) %plot([ 0 real(Bx(ii))],[0 j*(Bx(ii))],'k','linewidth',2)
%subplot(3,1,2)
%plot(circle, 'k')
Page 7.1294.
Proceedings of the 2002 American Society for Engineering Education Annual Conference & Exposition
1
2
30
210
60
240
90
270
120
300
150
330
(^180 )
Plot of Bx, By, Bz and Br Rotating Magnetic Field
Conclusions
MATLAB is a very powerful simulation package that we are trying to include in our energy conversion curriculum. As it can be seen in the output of the simulation the numerical part is quite clear, yet, the graphical part emphasizes the result and “brings home” the concept of having a constant vector magnitude of1.5 B for the resultant magnetic flux rotating in space around the stator.
The effect that this simulation has had in our students is great, due to the fact that they get very motivated to continue studying our electric machines and looking forward to perform simulations that would match their homework or concepts explained in class.
For further research we will look into the rotating phasors for synchronous machines operating at different power factors.
Page 7.1294.
Proceedings of the 2002 American Society for Engineering Education Annual Conference & Exposition
Bibliography
[1] Bruno Osorno “Electrical Machines Lecture Notes”, California State University Northridge, 1994. [2] Turan Gonen “Electrical Machines”, Power International Press, First Ed.
[3] A.E. Fitzgerald, et.al., “Electric Machinery. McGraw-Hill, 5th^ Ed. 1990. [4] “Syed A. Nasar, “Electric Machines and Power Systems”, McGraw-Hill, 1st^ Ed. 1995 [5] Guru and Hiziroglu, “Electric Machinery and Transformers”, Saunders College Publishing, 2nd^ Ed. 1995.
BRUNO OSORNO, is a professor of electrical and computer engineering at California State University Northridge. He is the lead faculty member in the Power Systems and Power Electronics program. Professor Osorno has written over 20 technical papers. His current interest in research is Fuzzy Logic applications in power electronics and electric motor drives.
Page 7.1294.
