3.5 Analysis of asynchronous machine parameters
3.5.1 Simulation of the no load condition
or from the linkage magnetic flux by means of integral A.J:
Lσ1þLσ20¼Lσ¼lav ψ
I21N ¼lav∮A�JdV
I21N ¼1610:000239885
2:752 ¼5:1 mH The measured value of total leakage inductance is 5.2 mH (seeTable 7), which means a very good coincidence of the results.
In this FEMM program, it is possible to calculate approximately the resistance of primary and secondary winding. For more precise calculation, the 3D program would be needed.
The resistance depends on the electrical conductivity of copper from which the windings are made. In simulating the value of copper, specific electrical conductiv- ityσ= 58 MS/m is used or can be set based on the material library in the FEMM program. The calculation starts from the short circuit simulation, whereby z-coor- dinate is set on 1 mm. Now the average length of the turn of primary lavpand secondary lavswinding must be calculated. The calculation is made based on the geometrical dimensions inFigures 56and65. Then lavp= 255.2 mm and
lavs= 339.2 mm. After the calculation, in postprocessor, the blocks must be marked, which correspond to the primary winding, and by means of the command Resistive losses, the Joule losses in the primary windingΔPjare calculated. In the same way, the losses in the secondary winding are calculated. Nevertheless, it is the same value because the cross-section area of the winding and current density is the same. The resistance is then calculated at 20°C for both windings as follows:
Rp¼lavpΔPj
I1N2¼255:20:050276
2:752 ¼1:71Ω Rs¼lavsΔPj
I2N2¼339:20:050276
26:32 ¼24:6 mΩ
For comparison the measured values are Rp= 1.91Ωand Rs= 20 mΩ.
In the end the simulated and measured values of equivalent circuit parameters are summarized inTable 8. It can be proclaimed that the values obtained by simulation and measurement are in good coincidence.
or from the linkage magnetic flux by means of integral A.J:
Lσ1þLσ20¼Lσ¼lav ψ
I21N¼lav∮A�JdV
I21N ¼1610:000239885
2:752 ¼5:1 mH The measured value of total leakage inductance is 5.2 mH (seeTable 7), which means a very good coincidence of the results.
In this FEMM program, it is possible to calculate approximately the resistance of primary and secondary winding. For more precise calculation, the 3D program would be needed.
The resistance depends on the electrical conductivity of copper from which the windings are made. In simulating the value of copper, specific electrical conductiv- ityσ= 58 MS/m is used or can be set based on the material library in the FEMM program. The calculation starts from the short circuit simulation, whereby z-coor- dinate is set on 1 mm. Now the average length of the turn of primary lavpand secondary lavswinding must be calculated. The calculation is made based on the geometrical dimensions inFigures 56and65. Then lavp= 255.2 mm and
lavs= 339.2 mm. After the calculation, in postprocessor, the blocks must be marked, which correspond to the primary winding, and by means of the command Resistive losses, the Joule losses in the primary windingΔPjare calculated. In the same way, the losses in the secondary winding are calculated. Nevertheless, it is the same value because the cross-section area of the winding and current density is the same. The resistance is then calculated at 20°C for both windings as follows:
Rp¼lavp ΔPj
I1N2¼255:20:050276
2:752 ¼1:71Ω Rs¼lavsΔPj
I2N2¼339:20:050276
26:32 ¼24:6 mΩ
For comparison the measured values are Rp= 1.91Ωand Rs= 20 mΩ.
In the end the simulated and measured values of equivalent circuit parameters are summarized inTable 8. It can be proclaimed that the values obtained by simulation and measurement are in good coincidence.
3.5 Analysis of asynchronous machine parameters
Asynchronous motor parameters are simulated based on the no load test and locked rotor test in accordance with the equivalent circuit parameters [21]. Also calculation of the air gap electromagnetic torque and its ripple is made. Analysis is made for a real three-phase squirrel-cage asynchronous motor (its type symbol is 4AP90L); the nameplate and rated values are inTable 9. A picture of its geometri- cal parts and their dimensions are inFigures 67and68.
Measurement FEMM Deviation
Magnetizing inductance Lμ[H] 2.82 2.85 1.05%
Total leakage inductance Lσ[mH] 5.2 5.1 1.9%
Primary winding resistance Rp[Ω] 1.91 1.71 10.4%
Secondary winding resistance Rs[Ω] 0.02 0.0246 18.6%
Table 8.
Comparison of the equivalent circuit parameters.
3.5.1 Simulation of the no load condition
An ideal no load condition is defined at synchronous speed of the rotor, when rotor frequency is zero. In fact, at real no load condition, the rotor rotates at speed lower than synchronous speed, but the difference is not significant. Therefore, an ideal no load condition can be investigated without a big error. A magnetizing inductance of the equivalent circuit and also fundamental harmonic of the magnetic flux density in the air gap can be calculated by means of FEM. Here is a procedure how to do it:
• Draw a model of the investigated motor in a cross-section area in a program of FEMM (Figure 69).
• Enter the three-phase currents to the stator windings, materials, boundary conditions, and a mesh density. (The rotor currents are in the ideal no load condition zero).
• After the calculation, analyze in postprocessor distribution of the air gap magnetic flux density
• To make a Fourier series of the air gap magnetic flux density, calculate its fundamental harmonic, electromotive force (induced voltage), and from it the magnetizing inductance.
3.5.1.1 Drawing of the asynchronous motor model
FromFigure 69it is seen that the asynchronous motor cross-section area is much more complicated than the transformer. There are more possibilities how to
Rated stator voltage U1N 400 V
Stator winding connection Y
Rated power PN 1500 W
Rated frequency f 50 Hz
Rated speed n 1410 min1
Phase number m 3
Rated slip sN 6%
Number of pole pairs p 2
Rated torque TN 10.15 Nm
Number of one-phase turns Ns 282
Number of slots per phase per pole q 3
Winding factor kw 0.959
Active length of the rotor lFe 98 mm
Number of conductors in the slot zQ 47
Magnetizing current I0obtained from no load measurement 2.3 A Lμmagnetizing inductance obtained from no load measurement 0.32 H
Rated stator current IsN 3.4 A
Table 9.
Nameplate and parameters of the investigated three-phase asynchronous motor.
draw this model, either directly in FEMM or in other graphical program (e.g., AUTOCAD, CAD, with the suffix *.dxf), and then to import it into FEMM. Here a drawing of the editor of the FEMM program is explained.
The first step is the setting of the task type which is investigated. In the beginning, when a new problem is investigated, the program calls up the user to define the type of the problem. For the no load condition, the Magnetostatic Problem is set. In the block problem, the units of the geometrical dimensions are set, usually in mm. Stator and rotor frequency is zero, because only one instant is investigated.
In the block Depth, the active length of iron lFeis set. Problem Type is planar (seeFigure 69).
Figure 67.
Cross-section area of the stator sheet and detail of the stator slot with its geometrical dimensions.
The drawing starts with changing over the drawing editor to the point mode, and by means of tabulator, the points based on the x- and y-coordinates are set. It is recommended to draw the model in such a way that the center of the machine has coordinates 0,0. After the points are drawn, change the program into the line mode or arc mode, and join the points by straight lines or curves. If there is curve mode, the user is asked, which angle should have the curve, e.g., for semicircle it is 180o, and what the accuracy should be. According to the accuracy of the calculation, it is recommended to enter the number 1, higher number means lower accuracy.
If the same objects are drawn several times, e.g., stator or rotor slots, it is possible to apply copying, which is in the block Edit and Copy.
The next step is the setting of the materials and currents into the appropriate blocks. All bordered areas created during the drawing present the blocks into which it is necessary to input the materials. On the toolbar, it is necessary to change over to block label, group mode, to mark all blocks, and to define them. But before that, the materials must be designated and defined.
Figure 68.
Cross-section area of the rotor sheet and detail of the rotor slot with its geometrical dimensions.
draw this model, either directly in FEMM or in other graphical program (e.g., AUTOCAD, CAD, with the suffix *.dxf), and then to import it into FEMM. Here a drawing of the editor of the FEMM program is explained.
The first step is the setting of the task type which is investigated. In the beginning, when a new problem is investigated, the program calls up the user to define the type of the problem. For the no load condition, the Magnetostatic Problem is set. In the block problem, the units of the geometrical dimensions are set, usually in mm. Stator and rotor frequency is zero, because only one instant is investigated.
In the block Depth, the active length of iron lFeis set. Problem Type is planar (seeFigure 69).
Figure 67.
Cross-section area of the stator sheet and detail of the stator slot with its geometrical dimensions.
The drawing starts with changing over the drawing editor to the point mode, and by means of tabulator, the points based on the x- and y-coordinates are set. It is recommended to draw the model in such a way that the center of the machine has coordinates 0,0. After the points are drawn, change the program into the line mode or arc mode, and join the points by straight lines or curves. If there is curve mode, the user is asked, which angle should have the curve, e.g., for semicircle it is 180o, and what the accuracy should be. According to the accuracy of the calculation, it is recommended to enter the number 1, higher number means lower accuracy.
If the same objects are drawn several times, e.g., stator or rotor slots, it is possible to apply copying, which is in the block Edit and Copy.
The next step is the setting of the materials and currents into the appropriate blocks. All bordered areas created during the drawing present the blocks into which it is necessary to input the materials. On the toolbar, it is necessary to change over to block label, group mode, to mark all blocks, and to define them. But before that, the materials must be designated and defined.
Figure 68.
Cross-section area of the rotor sheet and detail of the rotor slot with its geometrical dimensions.
Materials are defined in Properties – Add properties, where the constants for materials are entered (Figure 70). In asynchronous machine, there is air in the air gap and ventilating channels; ferromagnetic circuit, which is defined by a nonlinear magnetizing B-H curve (in this case the employed sheet is Ei70, the thickness of the sheet is 0.5 mm, block name in the program is core); and material, from which the stator and rotor conductors are produced. It is usually copper or aluminum. These materials can be copied from the program library. For stator slots, the current density, corresponding to no load condition, must be entered. A calculation of the current density for stator slots is as follows:
Think one instant of the three-phase no load current for all three phases. For example, if phase A crosses to zero, then phases B and C have the values equaled to sin60ofrom the magnitude. If the next phase sequence is assumed around the stator circumference: +A, -C, +B, -A, +C, -B, etc., then A = 0,�A = 0, -C = -Jmaxsin 60o, C = +Jmaxsin 60o, B = -Jmaxsin 60o, and�B = +Jmaxsin 60o, where Jmaxis the magnitude of the current density. It can be calculated as follows:
Jmax ¼zQImax
Sd ¼47� ffiffiffi p2
�2:3
49:6 ¼3:082 MA=m2 (482)
where Imaxis the magnitude of the no load current, which flows along the conductors, zQis a number of the conductors in the slot, and Sdis a cross-section area of the stator slot (Figure 67). In calculation there is a no load current I0= 2.3 A used (seeTable 9).
Figure 69.
The 1/4 of the cross-section area of the squirrel-cage asynchronous motor together with basic settings.
The calculated current densities are entered to the appropriate slots. In this case the number of slots per phase per pole is q = 3.
If all materials are defined, then it is necessary to allocate them to the appropri- ate blocks. The demanded block is designated by the right mouse button, and by pushing the space key, it is possible to allocate the material to the block. In this way, all blocks are defined. If it happens that a block is forgotten, it is not possible to make calculation until the block is not designated.
3.5.1.2 Setting of the boundary conditions
The last task before the calculation is launched, which is definition of the boundary conditions. Because the whole cross-section area of the asynchronous motor is analyzed, the Dirichlet boundary condition on the outer circumference, which is zero magnetic vector A = 0, can be applied (Figure 71).
In the block Properties, click on the Boundary and define the boundary condition according toFigure 71. Then change over to the curve mode, choose the stator external circle by right mouse button, and by means of space key the appropriate boundary condition is allocated to the circle.
Before starting the calculation, click on the icon Mesh, and create the demanded mesh of the finite elements. If it looks in some parts to be widely spaced, it can be densified in the next way: It is necessary to change over to the block mode, by means of the right mouse button, choose the demanded block, mark Let triangle choose Mesh size, and set the required value of the finite elements. Then the very calculation can be started by means of the icon solve.
3.5.1.3 Calculation of the air gap magnetic flux density
After the calculation is finished, a distribution of the magnetic flux lines in the cross-section area of the investigated motor appears on the screen (seeFigure 72). For illustration there is the whole cross-section area of the four-pole asynchronous motor.
Figure 70.
Material designation and definition.
Materials are defined in Properties – Add properties, where the constants for materials are entered (Figure 70). In asynchronous machine, there is air in the air gap and ventilating channels; ferromagnetic circuit, which is defined by a nonlinear magnetizing B-H curve (in this case the employed sheet is Ei70, the thickness of the sheet is 0.5 mm, block name in the program is core); and material, from which the stator and rotor conductors are produced. It is usually copper or aluminum. These materials can be copied from the program library. For stator slots, the current density, corresponding to no load condition, must be entered. A calculation of the current density for stator slots is as follows:
Think one instant of the three-phase no load current for all three phases. For example, if phase A crosses to zero, then phases B and C have the values equaled to sin60ofrom the magnitude. If the next phase sequence is assumed around the stator circumference: +A, -C, +B, -A, +C, -B, etc., then A = 0,�A = 0, -C = -Jmaxsin 60o, C = +Jmaxsin 60o, B = -Jmaxsin 60o, and�B = +Jmaxsin 60o, where Jmaxis the magnitude of the current density. It can be calculated as follows:
Jmax¼zQImax
Sd ¼47� ffiffiffi p2
�2:3
49:6 ¼3:082 MA=m2 (482)
where Imaxis the magnitude of the no load current, which flows along the conductors, zQis a number of the conductors in the slot, and Sdis a cross-section area of the stator slot (Figure 67). In calculation there is a no load current I0= 2.3 A used (seeTable 9).
Figure 69.
The 1/4 of the cross-section area of the squirrel-cage asynchronous motor together with basic settings.
The calculated current densities are entered to the appropriate slots. In this case the number of slots per phase per pole is q = 3.
If all materials are defined, then it is necessary to allocate them to the appropri- ate blocks. The demanded block is designated by the right mouse button, and by pushing the space key, it is possible to allocate the material to the block. In this way, all blocks are defined. If it happens that a block is forgotten, it is not possible to make calculation until the block is not designated.
3.5.1.2 Setting of the boundary conditions
The last task before the calculation is launched, which is definition of the boundary conditions. Because the whole cross-section area of the asynchronous motor is analyzed, the Dirichlet boundary condition on the outer circumference, which is zero magnetic vector A = 0, can be applied (Figure 71).
In the block Properties, click on the Boundary and define the boundary condition according toFigure 71. Then change over to the curve mode, choose the stator external circle by right mouse button, and by means of space key the appropriate boundary condition is allocated to the circle.
Before starting the calculation, click on the icon Mesh, and create the demanded mesh of the finite elements. If it looks in some parts to be widely spaced, it can be densified in the next way: It is necessary to change over to the block mode, by means of the right mouse button, choose the demanded block, mark Let triangle choose Mesh size, and set the required value of the finite elements. Then the very calculation can be started by means of the icon solve.
3.5.1.3 Calculation of the air gap magnetic flux density
After the calculation is finished, a distribution of the magnetic flux lines in the cross-section area of the investigated motor appears on the screen (seeFigure 72). For illustration there is the whole cross-section area of the four-pole asynchronous motor.
Figure 70.
Material designation and definition.
Figure 71.
Definition of the boundary conditions.
Figure 72.
Distribution of the magnetic flux lines in four-pole asynchronous motor in no load condition.
If a distribution of the magnetic flux density in the whole cross-section area is needed, it is possible to see its value in each point of the cross section and accord these values to optimize construction of the motor.
Most important is to know the shape of the waveform of the air gap magnetic flux density. Therefore, after the calculation in postprocessor, it is necessary to mark a circle in the middle of air gap (seeFigure 73a) and on this circle in dialog window, to mark calculation of the normal component of the magnetic flux density. These values can be saved in data file and their elaboration in the other program continued.
In such a way based on the Fourier series, the fundamental harmonic component can be calculated. InFigure 73b, a waveform of the air gap magnetic flux density is shown. After the Fourier series is made, it is possible to calculate electromotive force (induced voltage) and from this value the magnetizing inductance.
Based on the geometrical dimensions, a calculation of the induced voltage can be made. The induced voltage in the three-phase alternating rotating machines is given by an expression as follows:
Ui¼ ffiffiffi p2
πfΦavNskw¼ ffiffiffi p2
πf2 πBπD
2p lFeNskw (483)
Figure 73.
(a) Marking of the circle in the middle of the air gap to calculate a normal component of the magnetic flux density and (b) distribution of the normal component of the air gap magnetic flux density in the investigated asynchronous motor.