3.5 Analysis of asynchronous machine parameters
3.5.2 Simulation of locked rotor condition
During the no load simulation by means of the FEMM, a magnetostatic field was employed. It is possible to do it also during the simulation of the locked rotor
Figure 76.
Marking of stator slots for magnetizing inductance calculation.
where D = 84 mm, lFe= 98 mm, Ns= 282 turns, and kw= 0.959. From the waveform of the air gap magnetic flux density inFigure 73b, the magnitudes of the harmonic components Bδνcan be calculated by means of the Fourier transforma- tion. InFigure 74, the harmonic analysis (Fourier series) of the air gap magnetic flux density for 60 harmonic components is shown. To make a Fourier transforma- tion, the program MATLAB can be employed.
The magnitude of the air gap magnetic flux density of the fundamental harmonics is Bδ1max= 0.94 T. After this value is introduced to Eq. (483), the rms value of phase induced voltage fundamental harmonic is obtained:
Ui1¼ ffiffiffi p2
πf2 πBπD
2plFeNskw¼ ffiffiffi p2
π2
π0:94π�0:084
4 0:098�282�0:959¼232:4 V The total voltage could be calculated by the sum of all harmonic components.
The terminal voltage given in the nameplate is 3 x 400 V at star connection. Then the phase value is 230 V, which is in very good coincidence with the value obtained by FEMM. For illustration inFigure 75, there is a distribution of the magnetic flux density in the cross-section area of the investigated motor at no load condition.
Figure 74.
Magnitude spectrum of the air gap magnetic flux density harmonic components in no load condition.
Figure 75.
Distribution of the magnetic flux density in no load condition.
3.5.1.4 Calculation of the magnetizing inductance at no load condition
At no load condition, also magnetizing inductance can be calculated. If the induced voltage, frequency, and no load current (from the no load measurement) are known, the magnetizing inductance is as follows:
Lμ ¼ Ui
ωI0¼ 232:4
2�π�50�2:3¼0:321 H
It was supposed that at no load condition, the magnetizing current is almost identical with no load current. The measured value of the magnetizing inductance is Lμ= 0.32 H (seeTable 9), which is in very good coincidence of the results obtained from FEMM and measurement.
A calculation of magnetizing inductance can be done also based on the linkage magnetic flux and integral A.J, because in this no load condition, the ferromagnetic circuit is in the nonlinear area. A procedure can be described as follows: After the calculation in postprocessor, all slots corresponding with the stator winding are marked, and the integral A.J is calculated (Figure 76). It must be realized that the flux is created by all three phases; therefore the value of no load current is multiplied by 3, or linkage magnetic flux is divided by 3. Then the magnetizing inductance is:
Lμ ¼ ψ
3I0¼∮A�JdV
3I20 ¼5:18345
3�2:32 ¼0:326 H
It is seen that a coincidence with the values obtained by other methods is very good.
3.5.2 Simulation of locked rotor condition
During the no load simulation by means of the FEMM, a magnetostatic field was employed. It is possible to do it also during the simulation of the locked rotor
Figure 76.
Marking of stator slots for magnetizing inductance calculation.
condition, but it is necessary to enter to the rotor bars actual rotor currents, induced there from the stator currents. Such setting of the rotor currents is time-consuming, mainly if there is bigger number of the rotor slots. For completeness this way is described in Section 5.7, where the rated condition and rated torque are calculated.
Simulation of the locked rotor condition means to solve a harmonic task. It means to enter the frequency corresponding with the actual rotor frequency at locked rotor condition. Because in FEMM program there is no possibility to enter more various frequencies in one solution, it is necessary to enter to the base settings (Figure 69) the slip frequency. At the locked rotor condition, the slip is equal to 1, and corresponding frequency is frequency of the stator current f = 50 Hz. During the analysis, the same cross-section area of the asynchronous motor as in the no load condition is employed. In stator slots, the material of the stator slots (Cu, Al) and its conductivity and permeability, which is close to the vacuum, is entered. Usually the stator conductors in asynchronous motors are copper (copper electrical conductiv- ity is 58 MS/m) and rotor cage is aluminum, the electrical conductivity of which is 24.59 MS/m. It is also the case of motor simulated here.
To define the currents, a block Circuits is employed. It is in the block below Properties, under Property definition and Circuit Property (seeFigure 77).
InFigure 78, there are phasors of stator currents Isfor corresponding phases, eventually circuits. The circuits can be set on the Parallel or Series (Figure 77). Here the circuit Series is employed, and then the currents definition is in coincidence with Figure 78. It is a product of the complex figure which corresponds to the current phasor position, number of the conductors in the slot, and magnitude of the cur- rent. In this case a current is set on its rated value IN= 3.4 A, and then its magnitude is INmax= 4.8 A. The phases are excited as follows:
Phase A+: 1ð þj0ÞzQINmax= 1ð þj0Þ47 ffiffiffiffi p2�
3:4¼ð226þj0Þ Phase A�:ð�1þj0ÞzQINmax¼ �1ð þj0Þ47 ffiffiffiffi
p2�
3:4¼ �226ð þj0Þ Phase B+: �21�jpffiffi3
2
� �
47� ffiffiffiffi p2�
3:4¼ �113ð �j195:7Þ Phase B�: 12þjpffiffi3
2
� �
47� ffiffiffiffi p2�
3:4¼ð113þj195:7Þ Phase C+: �21þjpffiffi3
2
� �
47� ffiffiffiffi p2�
3:4¼ �113ð þj195:7Þ Phase C�: 12�jpffiffi3
2
� �
47� ffiffiffiffi p2�
3:4¼ð113�j195:7Þ From the winding theory, it is known:
q¼ QS
2 pm¼ 36
2�2�3¼3 (484)
where q is number of slots per phase per pole and p is number of pole pairs.
It means that the three adjacent stator slots are defined by the same phase current,
Figure 77.
Currents definition in locked rotor condition.
e.g., A+. These triplets of the slots are changed in the order, A+, A+, A+, C-, C-, C-, B+, B+, B+, A-, A-, A-, C+, C+, C+, B-, B-, B-, which is seen also in theFigure 78, where this changing of the phases creates clockwise rotating magnetic field.
As the number of the pole pairs p = 2, this order is repeated two times around the stator inner circumference. If there is double-layer winding, it must be taken into account. In here investigated motor, there is a single-layer winding.
In rotor conductors, the aluminum with its electrical conductivity is set, because at the harmonic task the currents are induced in the rotor conductors.
3.5.2.1 Calculation of the equivalent circuit parameters from the locked rotor simulation The equivalent circuit parameters needed for the calculation of its parameters in locked rotor condition are inFigure 79. The resistance and leakage inductance
Figure 78.
The current phasors in complex plain for the locked rotor condition.
Figure 79.
Modified equivalent circuit of the asynchronous motor.
condition, but it is necessary to enter to the rotor bars actual rotor currents, induced there from the stator currents. Such setting of the rotor currents is time-consuming, mainly if there is bigger number of the rotor slots. For completeness this way is described in Section 5.7, where the rated condition and rated torque are calculated.
Simulation of the locked rotor condition means to solve a harmonic task. It means to enter the frequency corresponding with the actual rotor frequency at locked rotor condition. Because in FEMM program there is no possibility to enter more various frequencies in one solution, it is necessary to enter to the base settings (Figure 69) the slip frequency. At the locked rotor condition, the slip is equal to 1, and corresponding frequency is frequency of the stator current f = 50 Hz. During the analysis, the same cross-section area of the asynchronous motor as in the no load condition is employed. In stator slots, the material of the stator slots (Cu, Al) and its conductivity and permeability, which is close to the vacuum, is entered. Usually the stator conductors in asynchronous motors are copper (copper electrical conductiv- ity is 58 MS/m) and rotor cage is aluminum, the electrical conductivity of which is 24.59 MS/m. It is also the case of motor simulated here.
To define the currents, a block Circuits is employed. It is in the block below Properties, under Property definition and Circuit Property (seeFigure 77).
InFigure 78, there are phasors of stator currents Isfor corresponding phases, eventually circuits. The circuits can be set on the Parallel or Series (Figure 77). Here the circuit Series is employed, and then the currents definition is in coincidence with Figure 78. It is a product of the complex figure which corresponds to the current phasor position, number of the conductors in the slot, and magnitude of the cur- rent. In this case a current is set on its rated value IN= 3.4 A, and then its magnitude is INmax= 4.8 A. The phases are excited as follows:
Phase A+: 1ð þj0ÞzQINmax= 1ð þj0Þ47 ffiffiffiffi p2�
3:4¼ð226þj0Þ Phase A�:ð�1þj0ÞzQINmax¼ �1ð þj0Þ47 ffiffiffiffi
p2�
3:4¼ �226ð þj0Þ Phase B+: �12�jpffiffi3
2
� �
47� ffiffiffiffi p2�
3:4¼ �113ð �j195:7Þ Phase B�: 12þjpffiffi3
2
� �
47� ffiffiffiffi p2�
3:4¼ð113þj195:7Þ Phase C+: �12þjpffiffi3
2
� �
47� ffiffiffiffi p2�
3:4¼ �113ð þj195:7Þ Phase C�: 12�jpffiffi3
2
� �
47� ffiffiffiffi p2�
3:4¼ð113�j195:7Þ From the winding theory, it is known:
q¼ QS
2 pm¼ 36
2�2�3¼3 (484)
where q is number of slots per phase per pole and p is number of pole pairs.
It means that the three adjacent stator slots are defined by the same phase current,
Figure 77.
Currents definition in locked rotor condition.
e.g., A+. These triplets of the slots are changed in the order, A+, A+, A+, C-, C-, C-, B+, B+, B+, A-, A-, A-, C+, C+, C+, B-, B-, B-, which is seen also in theFigure 78, where this changing of the phases creates clockwise rotating magnetic field.
As the number of the pole pairs p = 2, this order is repeated two times around the stator inner circumference. If there is double-layer winding, it must be taken into account. In here investigated motor, there is a single-layer winding.
In rotor conductors, the aluminum with its electrical conductivity is set, because at the harmonic task the currents are induced in the rotor conductors.
3.5.2.1 Calculation of the equivalent circuit parameters from the locked rotor simulation The equivalent circuit parameters needed for the calculation of its parameters in locked rotor condition are inFigure 79. The resistance and leakage inductance
Figure 78.
The current phasors in complex plain for the locked rotor condition.
Figure 79.
Modified equivalent circuit of the asynchronous motor.
depend also on the frequency. In locked rotor condition, the rotor frequency is identical with the stator frequency; therefore in starting settings of the FEMM program, (problem) f = 50 Hz is entered.
In the 2D program, it is not possible to calculate resistance and leakage induc- tance of the end connectors and rotor end rings. For a correct calculation, these parameters must be calculated in another way or employ a 3D program.
InFigure 79there is an adapted equivalent circuit, where the dashed line shows which parameters can be calculated by 2D program.
Parameters out of dashed line are caused by 3D effect. They are the following:
Rs, stator winding resistance; Lσs,3D, stator leakage inductance of end windings;
Lσ,2D, stator and rotor leakage inductance without end windings and rotor rings;R0r,2Ds , rotor resistance referred to the stator without rotor rings;R0r,3Ds , resistance of rotor rings referred to the stator; and Lσr,3D, leakage inductance of the rotor rings. Sup- plied current in the simulation is rated current which corresponds also to locked rotor measurement. Magnetic flux lines distribution in locked rotor state is shown inFigure 80. The following calculations are carried out in accordance with [11].
During locked rotor simulation, the following parameters can be obtained: Lσ,2D
and R’r,2D. Usually, locked rotor test is done with low supply voltage, so no saturation effect is present. Then, leakage inductance Lσ,2Dcan be calculated from stored energy:
Lσ,2D¼2W
3I2sN¼2�0:367
3�3:42 ¼21:1 mH (485)
From (485), it can be seen that the current is multiplied by 3. It is caused by three-phase supplying and all three phases create energy of magnetic field.
Figure 80.
Distribution of the magnetic flux lines of the asynchronous motor in the locked rotor condition.
The value of energy W can be obtained from whole cross-section area of the motor in postprocessor. To obtain total leakage inductance of the stator and rotor, analytical calculation of stator end winding leakage inductance and rotor ring leakage induc- tance must be taken into account, e.g., from [20], which is 26.8 mH. The value obtained from the locked rotor measurement is 29.5 mH, which is appropriate coincidence of the results.
The value of the rotor resistance is calculated from the losses in the rotor bars.
In postprocessor, all blocks belonging to the rotor bars are marked (Figure 81) and by means of the integral, theΔPjrare calculated. Then the resistance of the rotor bars without the end rings is calculated as follows:
R0r,2D¼ΔPjr
3I2sN¼128:755
3�3:42 ¼3:71Ω (486)
The resistance of the rotor end rings is calculated from the expressions known from the design of electrical machines according to [1]. The total rotor resistance, which includes a bar and corresponding part of the end rings, referred to the stator side is 3.812Ω. The measured value from the locked rotor test is 3.75Ω, which is an appropriate coincidence of the parameters.