SPACE HARMONIC THEREOF

TECHNICAL FIELD

The invention relates to shaded-pole asynchronous motors used in small power applications requiring low startup moment such as electrical household appliances, toys and ventilation systems, as well as novel mathematical model and space harmonics for these motors.

STATE OF THE ART

Shaded-pole asynchronous motors are used in small house appliances, ventilation systems, toys, and small workbenches requiring single-phase low torque. Only in Europe, about 10 million shaded-pole asynchronous motors are manufactured in a year. This figure is quite high compared to the other single-phase motors used in the similar applications. For example, 700-800 thousand single-phase capacitor motors are manufactured in a year in Europe. 100 thousand other alternatives to the efficient motors are manufactured in a year in Europe. Thus, shaded-pole asynchronous motors have a considerably wide usage area.

Shaded-pole asynchronous motors are mostly manufactured in accordance with the practical knowledge or trial-and-error method, because their theoretical analysis is difficult. Despite their simple structure, the fact that they have magnetically coupled coils enabling the formation of the rotating magnetic field and their air gap is variable make the analysis of these machines difficult. In the designs, mathematical modeling and realization of the motor performance analyses for such motors, there is a standard procedure. The difficulty in the theoretical analysis derives from the presence of strong space harmonics arising from the fact that their stator air gap circumference is not uniform, presence of non-symmetrical coils in the stator, the effect of the saturation in the magnetic circuit, and difficult calculations of inductances. Computer mediated analyses before the manufacture and resulting performance analyses for these machines may not be performed easily due to these difficulties. Therefore, it may not be determined before the manufacture stage whether the performed design is an optimum design. Consequently, more materials than the required are used in the manufacture, which increases the cost of the motors.

DESCRIPTION OF THE INVENTION

The present invention relates to asynchronous motors and novel mathematical model and space harmonics for these motors developed for eliminating the disadvantages mentioned above and providing the related technical field with novel advantages. Strong moment components are obtained which may not be comprised in the performance analyses in the literature, by means of the invention. Dominant space harmonics and space harmonics whose effect is ignorable are found by the obtained data. A new equivalent circuit model where performance analysis may be directly carried out by ignoring the space harmonics having little effect and considering dominant harmonics. In this way, obtaining more accurate results is aimed in the mathematical analyses and motor performance analyses.

The motors to be manufactured by means of the new Analysis Model and Space Harmonic equivalent circuit is going to be able to proceed to the actual manufacture after obtaining the most economical design by computer-mediated design and performance analysis carried out before the actual manufacture stage, which will provide saving for the manufacturer due to the usage of more economical amount of material. An object of the invention is that the motor is modeled such that two rotor bars of the motor whose squirrel-cage rotors are side by side and ring pieces of the squirrel cage between thereof will be a closed circuit, thus the space harmonics formed due to the geometry of the machine are modeled delicately within the machine, and therefore they are able to modeled basing the complex effects of the air gap on the performance on a concrete ground.

Another object of the invention is that new analysis method and space harmonic equivalent circuit allows for carrying out the design, modeling and optimization of the shaded-pole motors which were manufactured by trial-and-error method before, by means of scientific methods and for more economical motor manufactures. Drawings

The applications of the present invention summarized briefly above and addressed in more detail below may be understood by referring to the exemplary applications defined in the appended drawings of the invention. However, it has to be mentioned that the appended drawings illustrates the typical applications of this invention only, thus it cannot be assumed that they limit the scope of the invention, because it may allow for equally effective applications.

Figure 1 is an equivalent circuit of the invention.

Figure 2 is the view of the rotor orifice.

Figure 3 is a flow chart of the experiment arrangement of the invention. For the ease of understanding, identical reference numbers are used in the possible cases in order to define the identical members common in the figures. The figures have not been drawn to scale and they may be simplified for clarity. It has to be considered that the members and properties of an application may be included in the other applications usefully without any need for further explanation.

Description of the Details in Drawings

The equivalences of the reference numbers shown in the figures are presented below: 1- Rotor orifice

10- Start

11- Defining microprocessor and selecting design parameters

12- Change the rotor position

13- Informing the operator by the help of LCD

14- Changing position

DETAILED DESCRIPTION OF THE INVENTION In this detailed description, preferred alternatives of the asynchronous motor and novel mathematical model and space harmonics of this motor of the invention are described for better understanding the subject and without any limiting effect.

The presence of the non-symmetrical coils of the shaded-pole asynchronous motors (SPAM) causes the voltage equation to have a complex structure. Equivalent circuit belonging to the invention is shown in the figure 1. The stator, split pole ring and rotor voltage equations shown in the equivalent circuit belonging to the invention is presented in Equation 4 and Equation 15 in detail.

Stator coil voltage equation is stated by Equation 1 below.

Equation 1

The harmonic current to be induced by the rotor current should be expressed here. Nth harmonic flux to be induced by ith harmonic of the rotor current in the stator coil is expressed by Equation 2 below. It is stated in this equation that from which components the voltage applied on the stator coil is comprised. The most significant point in this equation is that mutual inductance as much as the rotor groove number is added to the voltage calculation. Concerning the components of the equations, R_a i_a component is the ohmic voltage drop on the stator coil resistance. Second component is the component formed by stator self-inductance, the third component is the component formed by the stator-shaded-pole inductance and the last component is the component formed by the stator-rotor orifice mutual inductance. Thus, the effect of the each coil on the stator is removed.

Equation 2

The flux generation the voltage expressed in the Equation 1 to be induced on the stator coil by the harmonic components of the rotor current is expressed by the equation 2. “Ge” term in the equations refers to the actual component. The first component in the expression expresses the flow of the direct harmonic current of the rotor current. The second component is the value of the components of the stator-rotor mutual inductance in the amount of N1. In this study, N1 value is until its 9th Harmonic component. 3rd and 4th components are the expressions of the reverse harmonic component. Ge expresses the real part of the expression here. While examining Equation 2, harmonic currents one of which is in the right direction (given by the d subscript) and the other one of which is in the reverse direction (given by t subscript) may be mentioned. While carrying out the required intermediary processes in the Equation 1 and 2, stator coil voltage equation may be stated in effective value phasors as in Equation 3.

Equation 3

The obtained equation is shown in the equivalent circuit in Figure 1. Split pole ring voltage equation bears exact similarity to the stator coil voltage equation. Lettering given here is presented with their explanations at the end of the part. Equation 3 is the clearly written form of the expression given in Equation 1 . The effect of the harmonic components of the stator-rotor inductance from 1 st to 9th harmonic for each harmonic component is given in the equation for 26, in other words S2 amount of grooves. The components of the voltage applied on the stator shall be obtained by this equation. The importance of this equation is that it comprises harmonic components unlike conventional stator voltage equations, which is the object of the patent.

The parameters of the harmonic equivalent circuit model given in Figure 1 are clearly presented in Equation 4 - Equation 15.

It is the component located on the stator portion of the equivalent circuit. jooM_ab l ^" b component in this equation is the voltage component induced by stator-shaded pole mutual inductance in the stator portion. Other five components are voltage components consisting of 26 grooves mutual inductance for each harmonic component (1st harmonic, 3rd harmonic, 5th harmonic, 7th harmonic, and 9th harmonic).

It is the portion forming the shaded-pole component in the equivalent circuit. Voltage values formed in the shaded-pole is modeled in this variable. Eb voltage is the voltage induced by the stator mutual inductance and rotor groove mutual inductances harmonic components (1th harmonic, 3rd harmonic, 5th harmonic, 7th harmonic, and 9th harmonic). For example, to explain the components:

component of the stator-shaded pole mutual inductance

1st harmonic component of the mutual inductance of shaded pole-rotor grooves. Here, d subscript expresses direct harmonic component and t subscript expresses reverse subcomponent. S2 indicated the presence of mutual inductance in the amount of the groove number.

Harmonic component of the

mutual inductance of shaded pole-rotor grooves mutual inductance.

The others are accordingly the values of 5th, 7th, and 9th harmonic components.

Equation 6 to Equation 15

They comprise the expressions of the rotor portion of the equivalent circuit. Only 1st component will now be explained because there are components from the 1 st harmonic to the 9th harmonic. The same explanation is applicable for the other components. 1st harmonic component of the rotor voltage consists of right direction component E ^" .d1 and reverse direction componen The others are respectively given in the equations.

The portion of the 3th harmonic component in the right direction by Equation 8, the portion of the 3th harmonic component in the reverse direction by Equation 9;

The portion of the 5th harmonic component in the right direction by Equation 10, the portion of the 5th harmonic component in the reverse direction by Equation 11 ;

The portion of the 7th harmonic component in the right direction by Equation 12, the portion of the 7th harmonic component in the reverse direction by Equation 13;

The portion of the 9th harmonic component in the right direction by Equation 14, the portion of the 9th harmonic component in the reverse direction by Equation 15

Also, the explanations about the components in the right direction are applicable for the components in the reverse direction. Thus, the equation whose subscript is d will be explained and the one whose subscript is t will be the component of the same expression as this equation in the reverse direction. In this equation, transformer voltage and

rotor movement equation values are given for the component of the 1st Harmonic in the right direction. this expression is the harmonic voltage induced by the stator in the rotor

orifices, is the harmonic voltage induced by the shaded-pole in the rotor orifices, and E _r1d1 and E _rrd1 are harmonic voltages induced by the rotor orifice from the self and mutual inductances.

Stator, shaded-pole and rotor voltage equations for the shaded-pole asynchronous motors (SPAM) are formed by the obtained equations. In this way, mathematical model comprising the effect of the space harmonics having a dominant effect also on the motor performance has been obtained in the analysis of such motors. When the mathematical model is examined, it is seen that the effective harmonics are 1 st, 3rd, 5th, 7th, and 9th harmonics and these are located in the equivalent circuit as it is seen in Figure 1.

The meanings of the abbreviations used in the equations are given below:

a : Electrical angle within the sight of the rotor orifice

S_2 : Rotor groove number

l_a : Effective value of the stator current

\~_a : Phasor of the stator current

f_1 : Frequency of the stator current

s : Shear f_2k : The frequency of the kth harmonic of rotor current

j _ The flux to be induced by the split pole ring on the mth orifice of the rotor n : Harmonic number

3_mk : Harmonic phase difference coming from the development of the mutual inductances between the 1 st rotor orifice and the other orifices to Fourier series k : Degree of the harmonics of the mutual inductances

R_a : Effective value of the stator resistance

: Mutual flux between stator coils and rotor coils

M_ar : Mutual inductance between stator coil and a rotor orifice

M_ab : Mutual inductance between stator coil and a split pole ring

L_bk : Total leakage inductance of the split pole ring

L_b : Self-inductance of the split pole ring

M_br : Mutual inductance between split pole ring and a rotor orifice

M_r1 k : Mutual inductance between the 1 st rotor orifice and kth rotor orifice

L_ob : Split pole ring groove leakage inductance

Ge : It shows the real part of a complex number

d_h : Phase difference

L_a : Stator coil self-inductance

M_ar : Mutual inductance between stator coil and a rotor orifice

M_ab : Mutual inductance between stator coil and a split pole ring

L_bk : Total leakage inductance of the split pole ring

L_b : Self-inductance of the split pole ring

M_br : Mutual inductance between split pole ring and a rotor orifice

The presence of magnetically coupled asymmetric coils in these motors, enabling the formation of the rotating magnetic field makes the analysis of the machine more difficult. In the designs, mathematical modeling and realization of the motor performance analyses for such motors, there is a standard procedure. The difficulty in the theoretical analysis derives from the presence of strong space harmonics arising from the fact that their stator air gap circumference is not uniform, presence of non- symmetrical coils in the stator, the effect of the saturation in the magnetic circuit, and difficult calculations of inductances.

In designs of electrical machines and in determining their working properties, it is significant to know the distribution of the magnetic field through the machine. Accuracy degree in calculating the working magnitudes in the alternating current machines is directly related to accurately knowing the inductances. The calculation of the shaded- pole motor inductances with varying air gaps is very complex. The inductance values are non-linear functions changing in accordance with the rotor position and current due to the reluctance change in the electrical machines and saturation effect of the material used in the core. Firstly coil end inductance value of the stator coil of the motor was calculated; stator coil self-inductance, stator coil leakage inductance, stator-rotor mutual inductance, and mutual values between split pole-rotor orifice are determined by the experiments. Mutual inductances were carried out by means of measuring the induced voltage in the measurement bobbins by reeling up measuring bobbins on the neighboring two rotor groove and split pole ring grooves in the motors where cage for the rotor and split pole ring for the stator are not attached. Positioning manner of the measurement bobbin is shown in Figure 2. The experiment arrangement in Figure 3 was installed in order to obtain the inductances by certain steps and without disrupting the air gap value. A step motor with 200 steps was coupled to the shaded-pole motor in the arrangement. The step motor works in 1.8° mechanic and 3.6° electrical angle sensitivity. The step motor was triggered by a microprocessor in every 10 seconds. Induced voltage and stator coil current in the measurement bobbin was read for each rotor position. There are a DC source providing the power of the control card in the experiment arrangement and voltmeters for determining voltage values. Flow chart of the established system is shown in Figure 3. According to the flow chart firstly start (10) command is issued for the process and settings of in-out pins and changing assignments of the micro controller was carried out. Defining the microprocessor and selecting (1 1 ) the variables and design parameters are done, and time=0. Then rotor position is changed (12). The operator was enabled to be informed (13) about the position of the motor by means of LCD. The microprocessor controlling the system enables the motor to move a step forward in each 10 seconds and thus provides 3.6° (electrical) position change (14). In this way, the process goes back to the change the rotor position (12) stage.

The changes of the mutual inductances between stator coil-rotor orifice (1 ), shaded- pole ring-rotor orifice (1 ) and mutual inductances between the rotor orifices (1 ) by the position of the rotor were determined by means of the experiment arrangement mentioned above. These inductances should be expressed in accordance with the rotor position in order to be used in the calculations such as current, voltage, etc. These inductance wave shapes changing according to the rotor position are mathematically expressed by Fourier series. These Fourier components also constituting the basis of this patent study shows which harmonic components the mutual inductances comprise. The harmonics comprised by the mutual inductances and dominant harmonics are presented in the Table 1.

Table 1. Effective harmonics.

The relations to be present between the harmonic degrees of the mutual inductances and the harmonic degrees of the rotor current were determined. Voltage equation of stator coil, rotor for the steady-situation and split pole ring voltage equations are expressed by Equation 4-15. Space harmonic equivalent circuit presented in Figure 1 above by the equations was designed.

Harmonic components in the equivalent circuit being the most important part of the patent study were determined by the Fourier analyses performed. In general terms;

• Mutual inductances were determined according to the rotor position by means of the experiment arrangement established. Thus, a significant progress was accomplished on the accurate determination of the inductances being the issue in the theoretical analysis of the shaded-pole machines.

• Because there are harmonic components in the obtained equivalent circuit, the equivalent circuit may be stated to be more accurate circuit.

• The design, modeling and optimization of the shaded-pole motors manufactured by the trial-and-error method previously were enabled to be realized by means of the scientific methods by using the analysis and experimental methods in the patent study.

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