Usage of an asymmetrical rotating induction motor ... - Semantic Scholar

Usage of an asymmetrical rotating induction motor to obtain a LIM dynamic model JOSÉ CARLOS QUADRADO ISEL, R. Cons. Emídio Navarro, 1949-014 LISBOA CAUTL, Av Rovisco Pais, 1600 LISBOA PORTUGAL

Abstract: - The aim of this paper is to present a new model for the linear induction motor (LIM), which is based on its equivalence with the asymmetrical rotary induction motor. The model uses the theory of positive, negative and zero sequence of symmetrical and asymmetrical components. Experimental and simulation procedures with a LIM prototype allowed the theoretical and experimental validation for this model. Key-Words: - Linear induction motor, induction motor, dynamic modelling

1 Introduction As a first step to obtain in the future a time domain modelling that allows the LIM dynamic analysis. The present work is related to the finding of an electrical machine which has its dynamic modelling already well known in order to have it extended to the LIM. As a result of this research the asymmetrical induction motor was found, since the LIM has its asymmetry also related with constructive aspects. The asymmetrical motor, considered in this work, is an ordinary rotary three-phase squirrel cage induction motor, and the stator winding has different coils number in each phase (Na ≠ Nb ≠ Nc). Taking phase A as a reference, the coil number relationships can be obtained ( b = Nb/Na and c = Nc/Na). The modelling is developed taking into consideration the values of b and c which allows the asymmetrical induction motor to be equivalent to the LIM. The equivalence between the asymmetrical motor and LIM is based in the fact that for the same feeding line voltage, we should have the same phase currents and voltages, and also the same thrust and speed in both motors.

between those components can be written as (1) and (2).

bEbp = a 2 Eap bEbn = aEan cEcp = aEap

cEcn = a 2 Ean

of the parallel association of the magnetization reactance and the corresponding sequence impedances in the secondary are obtained through the same model. The emf of the stator phases can be decomposed negative

into

positive

( Ean , Ebn , Ecn )

(E

ap , Ebp , Ecp ) and

sequence non-symmetrical

components. Through Faraday's law the relationship

(2)

Through some mathematical manipulation with equations (1) and (2), the expression Ea + bEb + cEc = 0 can be obtained, which is the origin of a system of equations from this expression's real and imaginary parts. The mentioned system gives the solution for b and c at any operating point. Due to the current circulation in the primary winding it is established in the airgap a mmf (magnetomotive force) with high order space harmonics. The mmf correspondent to the hth harmonic from the magnetic field space distribution (fh), due to the three-phase system of feeding unbalanced currents ( I a , Ib and I c ) , as a function of the sequence symmetrical components having phase A from the primary as a reference, can be written as (3).

fh =

2 Mathematical Model The present model was developed from machine's phase equations [5-9]. As described in these publications the positive and negative sequence impedances are respectively Z ap and Z an . The result

(1)

3 K h ⎡⎣ I a1h e − jhβ x + I a 2 h e jhβ x ⎤⎦ 2

(3)

where:

2 2 N e jωt kω1h π ph s I a1h = I a1h e jθa1h

Kh =

I a 2h = I a 2h e

βx =

jθ a 2 h

π x τ

where: Ns – number of turns for one of the phases in the primary winding;

(4) (5) (6) (7)

p – number of pair of poles; h – harmonic order; ω = 2πf (where f is the frequency of the mains); kω1h - primary winding factor for the hth harmonic magnetic field space distribution; I a1h – positive sequence symmetrical component magnitude for the hth harmonic magnetic field space distribution; I a 2 h – negative sequence symmetrical component magnitude for the hth harmonic magnetic field space distribution; θ a1h - phase angle for I a1h ;

θ a 2 h - phase angle for I a 2 h ; x – linear coordinates with reference to a Cartesian axis fixed in the primary in the longitudinal length direction; τ - pole step relative to the magnetic field fundamental harmonic space distribution; µ0 - free space magnetic permeability. From equation (3), considering some physical boundary conditions, the harmonic order magnetization inductance (Lgh) can be obtained through (8). ( N ⋅ k ) 2 ⋅ L1 ⋅ τ (8) Lgh = 6 ⋅ µ0 ⋅ s2 ω12h h ⋅ π ⋅ p ⋅ ge where the effective airgap g, can be obtained through the Carter factor (Kc) times the mechanical airgap (gm), and whit L1, being the tickness of the primary core. Therefore the magnetization reactance (Xgh) can be described through (9)

X gh − ω ⋅ Lgh

c ⋅ E cnh = a 2 h ⋅ E anh

The asymmetrical motor considered in this work has a squirrel cage rotor, and can be represented through an equivalent three-phase winding mathematical model. Adopting the same routine in the LIM case, it will be possible to obtain the secondary impedance to be used in the sequence lumped parameters equivalent circuits. Whit short-circuited rotor phase circuits we have in the rotor a current that produce a leakage flux with the appropriate e.m.f. Due to the existence of resistance (R’2h) in the mentioned circuit it will occur also the resistive voltage drop. The airgap’s magnetic flux density generate an ⎛ ⎞ induced e.m.f. ⎜ E'ap 2 hs ⎟ in the secondary given ⎝ ⎠ by(14). ⎛ ⎞ (14) ⎜ E'ap 2 hs = sh ⋅ E'ap 2 h ⎟ ⎝ ⎠

Therefore, based in the Kirchhoff’s second law, and knowing that X 'd 2 h = ω ⋅ L 'd 2 h and making s ph = sh

which is the slip referred to the forward travelling magnetic field wave the equation (15) is obtained. 1 ⋅ ( R '2 h + jsph ⋅ X 'd 2 h ) ⋅ I 'ap 2 h (15) E'ap 2 h = s ph where: R '2 h = arh ⋅ R2 (16) X 'd 2 h = arh . X d 2 (17) arh =

(9)

The airgap mmf space harmonics distribution establish magnetic fields with a corresponding harmonic order, which are responsible for the appearance of primary induced voltages. The induced voltage phasors referred to the hth field space harmonic will have only positive and negative sequence components respectively ( E aph , E bph , E cph ) and ( E anh , E bnh , E cnh ). According to Faraday’s law and having phase A as a reference phase, the above mentioned induced voltage components have a mutual relationship according to the equations (10)(13). b ⋅ E bph = a 2 h ⋅ E bph

(10)

b ⋅ E bnh = a h ⋅ E anh

(11)

c ⋅ E cph = a h ⋅ E aph

(12)

(13)

m1 ⋅ ( N s ⋅ kω1h )

2

m2 ⋅ ( N 2 ⋅ kω 2 h )

2

(18)

where: m1- primary winding phase number; m2 – secondary winding phase number; N2 – secondary phases turns number; kω2h- secondary winding factor. The secondary impedance referred to the primary ⎛ ⎞ ⎜ Z '2 ph ⎟ is obtained from equation (15) being made ⎝ ⎠ equal to(19). 1 Z '2 ph = ⋅ ( R '2 h + jsph ⋅ X 'd 2 h ) (19) sh

Neglecting iron losses the e.m.f. E'aph = E'ap 2 h is applied to the magnetization reactance and at the secondary impedance. Therefore the two elements make a parallel association in the resulting positive sequence circuit.

Following procedures developed for the backward travelling magnetic field wave, equation (20) shows the way to obtain the secondary negative sequence impedance. 1 ⋅ ( R '2 h − jsnh ⋅ X 'd 2 h ) Z '2 nh = (20) snh Being snh the backward travelling magnetic field slip. Therefore, it is possible to obtain the negative sequence circuit as a parallel association of jXgh and

equations (27) - (33), (34) is obtained.

Z '2 nh .

The active power transferred to the secondary through the airgap, referred to the space harmonics of order h are given by (35-37).

The

positive

and

negative

sequence

⎛ ⎞ ⎛ ⎞ impedances respectively ⎜ Z aph ⎟ and ⎜ Z anh ⎟ , relative ⎝ ⎠ ⎝ ⎠ to hth space harmonic order, is a result of the parallel association of the magnetization reactance with the corresponding secondary sequence impedances, taking phase A as the reference phase. Therefore, they can relate the electromotive forces with currents through equations (21) and (22). E aph = Z aph ⋅ I aph

(21)

E anh = Z anh ⋅ I anh

(22)

I aph and I anh currents are positive and negative non symmetric components respectively. Those components are related to the feeding primary winding phase currents. The fact makes possible to obtain the following expressions: I ap1 = I an 5 = I ap 7 = I an11 = I ap13 = ... = I ap (23) I an1 = I ap 5 = I an 7 = I ap11 = I an13 = ... = I an

Z ap a2 Z ap b a Z ap c

⎤ Z an ⎥ ⎥ ⎡ I al ⎤ a Z an ⎥ ⎢⎢ I ap ⎥⎥ (34) ⎥ b ⎥ ⎢⎣ I an ⎥⎦ a2 Z an ⎥ ⎥⎦ c

2 Pph = 3I aph ℜ( Z aph ) 2 Pnh = 3I anh ℜ( Z anh ) vsx vsxh = h

(35) (36) (37)

where vsx is the linear synchronous speed. The positive (Fuph) and negative (Funh) sequence and resulting (FR) propelling forces, are given by (38-40).

Fuph =

Funh

Pph

vsx P = − nh vsx

FR = ∑ ( Fuph + Funh )

(38) (39) (40)

h

(24)

where the numerical indexes are the representation of harmonic order. Taking the two previous equations as a reference we have: E ap = E ap1 + E an 5 + E ap 7 + E an11 + ...

(25)

E an = E an1 + E ap 5 + E an 7 + E ap11 + ...

(26)

where: E a = E ap + E an

⎡ ⎢ Z a (1 + fd ) ⎡Va ⎤ ⎢ ⎢V ⎥ = ⎢ Z a 2 + afd ) ⎢ b⎥ ⎢ b( ⎢⎣Vc ⎥⎦ ⎢ ⎢ Z ( a + a 2 fd ) ⎢⎣ c

3 Prototype Implementation In this work, a three-phase linear induction motor (LIM), two poles, short bilateral primary, was built and tested in the laboratory. The primary packages (stator) with core made of silicon steel sheets, and the secondary (linor) built of a non-magnetic material, the aluminium, with a flat disc shape, as illustrated in Figure 1.

(27)

using the same concept we have:

Eb = Ebp + Ebn

(28)

Ec = Ecp + Ecn

(29)

Eap = I ap ( Z ap1 + Z an5 + Z ap 7 + Z an11 + ...)

(30)

Ean = I an ( Z an1 + Z ap 5 + Z an 7 + Z ap11 + ...)

(31)

Z ap = Z ap1 + Z an 5 + Z ap 7 + Z an11 + ...

(32)

Z an = Z an1 + Z ap 5 + Z an 7 + Z ap11 + ...

(33)

With some algebraic manipulations in the electrical

Figure 1 – Illustration of the linear induction motor used, (1) - aluminium linor; (2) – primary package.

The primary copper winding in each package is threephase concentric, step series 1:8 and 2:7, single-layer, 200 turns/phase with the winding scheme presented in Figure 2.

Figure 2 – Winding illustrative scheme for each primary package

The design for the primary packages has, finite length, open structure relatively to the magnetic circuit, when compared with the closed structure for the three-phase squirrel cage rotor induction motor, and the primary winding spread along the direction of movement. The mentioned factors, the solid secondary structure, and relative movement between primary and secondary are the elements responsible for the phenomena called border effects [3-4], and therefore for the unbalance between phase voltages and currents. In order to obtain the LIM parameters firstly the stator leakage inductances are obtained experimentally [1]. The primary phase resistances are obtained through direct measurements. Magnetization reactances are obtained at noload test, and it corresponds to a condition without the linor or with the linor being driven at synchronous speed [2]. The secondary impedance determination is made with locked linor. In both cases the building of experimental procedures follows the scheme in Figure 3. By applying to the machine a three-phase balanced sinusoidal voltage system, the data acquisition system stores the voltage and current waveforms referred to each test. The acquired voltage and current waveforms shows a high order time harmonic content. Therefore, through the use of Fast Fourier Transform (FFT) it is obtained the magnitude of harmonic content up to a certain specified order, in this work it is considered up to 50th harmonic, and it will be possible to estimate the Total Harmonic Distortion (THD) with a reasonable accuracy.

Figure 3 - Assembly diagram to obtain voltage and current waveforms.

Since it was observed that the voltage and current waveforms THD in time is less than 0.5% only the fundamental components in time are considered. Hence, it will be obtained the corresponding phasors to voltage and current waveforms. From the mathematical development shown at section 2, the parameters from the secondary will be found, considering harmonics up to the 11th order.

3.1 No load data Table 1 illustrates percentage values related to the fundamental component from time harmonics content up to 7th order. Table 1 – Percent values related to the fundamental component, for harmonic content up to 7th order, from phase voltage and current waveforms. Vb Vc Ia Ib Ic -----Va Fund 100 100 100 100 100 100 2º 0.0701 0.0594 0.0958 0.0293 0.0126 0.0543 3º 0.1430 0.0505 0.0447 0.0599 0.0342 0.0278 4º 0.0324 0.0213 0.0296 0.0049 0.0034 0.0211 5º 0.0337 0.0229 0.0642 0.0094 0.0145 0.0185 6º 0.0590 0.0049 0.0446 0.0153 0.0032 0.0336 7º 0.0072 0.0166 0.0181 0.0255 0.0257 0.0113

Balanced voltage (V) from the source: VL = 77.0171 Resulting values (rms):

Phase voltages (V): Va=46.9678; Vb=46.6967; Vc = 39.9741 Phase currents (A): Ia = 1.4900; Ib = 1.5297; Ic = 0.9897.

3.2 Locked linor data Table 2 illustrate the percent values related to the fundamental component of time harmonic content up to 7th one. Table 2 – Percent values related to the fundamental component, for harmonic content up to 7th order, from phase voltage and current waveforms. Vb Vc Ia Ib Ic -----Va Fund 100 100 100 100 100 100 2nd 0.0616 0.0240 0.0470 0.0589 0.0285 0.0332 3rdº 0.1680 0.1443 0.0758 0.1042 0.0755 0.0640 4thº 0.0868 0.0424 0.0535 0.0458 0.0244 0.0279 5th 0.1004 0.0701 0.1164 0.0288 0.0190 0.0586 6th 0.1288 0.0407 0.0779 0.0467 0.0201 0.0431 7th 0.0141 0.0412 0.0066 0.0295 0.0119 0.0380

Balanced voltage (V) from the source: VL = 75.6990 Resulting values (rms): Phase voltages (V): Va=45,0288; Vb=45.3695; Vc = 40.7952 Phase currents (A): Ia = 3.2980; Ib = 3.2923; Ic = 2.5687. Parameters obtained without taking in consideration the space harmonics are illustrated in Table 3. In Table 4 are presented parameters referred to the fundamental space harmonic. Table 3. LIM parameters (Ohm), values obtained experimentally. Magnetization Impedances Reactance Phase Primary Secondary A 5.348+j5.729 11.608+j1.234 35.054

Table 4. LIM parameters (Ohm), values obtained experimentally referred to fundamental space harmonic. Magnetization Impedances Reactance Phase Primary Secondary A 5.348+j5.729 11.608+j1.232 34.736

4 Experimental Validation The theoretical and experimental validation is made through the comparison of measured and calculated values of currents and propelling forces. In the present investigation, it is verified the thrust characteristic as a function of linear speed, for different operating points. With this purpose, in the developed thrust measurement, a DC generator was used coupled to the axis of the secondary disk, with the generator feeding a resistive load bank. The speed

measurements were made with an encoder with an optical active sensor. The theoretical and experimental results are presented in Tables 5 to 7 and Figures 4 and 5. In these tables the currents measured and calculated values are shown as well as the percentage differences referred to measured values for a range of speed values (1-s), with s being the rotor slip. Table 5. Speed (1-s) and corresponding measured (Iam), and calculated (Iac) currents and percent differences (differ.%) related to measured currents at phase A. (1-s) Iam (A) Iac (A) Differ. % 0 2.7603 2.7491 0.4058 0.2133 2.5800 2.5705 0.3686 0.2770 2.5010 2.4914 0.3830 0.4256 2.3919 2.3867 0.2159 0.5486 2.2988 2.2290 3.0374 0.5913 2.2460 2.1956 2.2406 0.6167 2.2404 2.1975 1.9131 0.6383 2.2329 2.2326 0.0145 0.7108 2.2312 2.3218 -4.0605

Table 6. Speed (1-s) and corresponding measured (Iam), and calculated (Iac) currents and percent differences (differ.%) related to measured currents at phase B. (1-s) Ibm (A) Ibc (A) Differ. % 0 2.7609 2.7397 0.7679 0.2133 2.5692 2.5484 0.8131 0.2770 2.4784 2.4596 0.7576 0.4256 2.3271 2.3262 0.0388 0.5486 2.1976 2.1404 2.6014 0.5913 2.1371 2.0952 1.9637 0.6167 2.1235 2.0885 1.6458 0.6383 2.1124 2.1142 -0.0861 0.7108 2.1009 2.1693 -3.2560

Table 7. Speed (1-s) and corresponding measured (Iam), and calculated (Iac) currents and percent differences (differ.%) related to measured currents at phase B. (1-s) Icm (A) Icc (A) Differ. % 0 2.1639 2.1441 0.9150 0.2133 1.9917 1.9854 0.3159 0.2770 1.9105 1.9086 0.0968 0.4256 1.7818 1.7744 0.4150 0.5486 1.6520 1.5980 3.2723 0.5913 1.5898 1.5505 2.4749 0.6167 1.5640 1.5371 1.7211 0.6383 1.5505 1.5484 0.1371 0.7108 1.4959 1.5618 -4.4053

The percent deviation presented by current calculated values in the three phases, related to measured values, are never superior to 5%, therefore this can be seen as the validation of the model applicability under the current point of view.

Figure 4 – Parameters b and c as a function of 1-s

Figure 4 shows the behaviour of measured and calculated values for parameters “b” and “c” as s function of (1-s), where s is the rotor slip. In Figure 4 stars and crosses show measured values (tests), and dotted and continuous lines indicate simulated values.

Figure 5 – Thrust force as a function of 1-s

Figure 5 shows the behaviour of measured and calculated values of propelling force as a function of (1-s). In this figure the stars represents measured values (tests), and the continuous line is representing the calculated values considering space harmonics up to 11th. The dotted line refers to the situation where are considered the space harmonics. Figure 5 shows also that taking into consideration the propelling thrust in the range observed, deviations presented between measured and calculated values when considering the space harmonics are perfectly acceptable. However, when these harmonics are not considered the model can be applied only for no speed or near zero values for the speed.

5 Conclusions The equivalence between the two machines referred is very promising and it can lead to future dynamic studies for the linear induction motor.

Experimental results show acceptable percent differences between measured and calculated values. This makes the model applicable under the current point of view. The current unbalance in the Linear Induction Motor is caused not only by its design characteristic, but also due to the inherent border effects. Parameter determination for each phase of the LIM was made through the use of quantities that were measured from the machine’s terminals. The philosophy adopted in the development of this research was to implement new characteristics in our model. This first attempt, the inclusion of space harmonics show promising results. The follow up of this research is oriented through this path, and the next step will be the inclusion of other characteristics pursuing the minimization of differences between practical and theoretical results.

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