Double excitation theory-Induction Motor

4.1 MTRODUCnrON

In chapter-111, it has been ma~t iod that t& time hrmonics will be g d

in tbc supply due to switching device& Wbcn clcdic motors m givcn wch a wpply,

the iron lows would increase wnsidmbly, especially if the rotor is a solid one. The

iron losses in thc solid-iron rotor caa be estimated by representing thc mn-linear

excitation as thc sum of fundamental and the harmonic of highest magnitude. But thc

test bas been performed on single-phase induction motor by feubg thc fundamental

excitation al 50 Hz and the harmonic excitation at 450 Hz. This type of problem can

be called as rmnxcitation problem or dual excitation problem. One would come

rrrou two excitation pmblem in the feedbeck control systems, where to stabilize the

main signal, a high Freqwncy signal is ycacd in to h e system, of-course the

frequency spatation be~wecn thc IW signals is high. The numerical solution of the

tmrsxcilrtion field problem is achieved by W o - S p t r a l Method [13].

A canpulcnrd gnphiul method is rlso dcvclopd to fd the field distribution.

The theory developed in chap&-Ill, to find h e various quantities like power

Imtu. fluxes CIC, haJ been verified witb tb: prtical results of single-phase induction

motor mth dual exciwon. The 14 lo- on the -or side arr estimated using the

m h i m design thrary 1241. The infinite half-specc h r y is applied to the actual

rotor by modifying thc resistivity of the rotor material with the correction factors for

curvalw and cnd effccls 1141. A wmction factor IS also incarparaled for

~mqmrturc risr. l'hc interfcmcc bctwcen thc IWO input signals with regard to IIIC

windrnga ia avoided. Since thc decaicrl ~k b*wben the wind~ngs is 90'.

4 3 DEMRHTION OF THE PROBLEM

Tbc h c e d h m i i field in the rotor of poly-phase induction motor can be

made two dimensional, if it is Pllrnuacd tha! the i n d d eddycumnts in the rotor arc

in widdirection only. The rotor cm be vicwed as an iron- block, wficn its curvatm

is neglected. Such an iron -block is subjected to travelling field on its surface say in

y-z plane, it is obvious that, thm c x h m altmahg flux through any section

pafallel to x-z plane, d m w i n g a dip frrquency. Conscqucntly, the evaluation of

eddy-cwent losses in solid-iron rotor of an induction machim an be based on the

knowledge of eddycurrefit distribution in an infinite half-space of iron subjected to

pulsating ticld.

Solid-iron mlor induction motors are in use. Due to many reasons, these

motors operate from non-sinusoidal excitation. Such an excitation can be considered

to be the sum of fundamanal and the harmonic of highest magnitude for the purpose

of evaluating cddy-currcnt losses. 'herefore, for analysing cddycumnt losses in

elccuic mhna, the input is taken cu the sum of two sinusoidal signals of

comnaammtc h a q w i a . h f o n . tmwxcitation field problem is simulated using

single-phase induction mtor with two windings.

The pmblem has been formulate4 in section 2.5. But the boundary conditions

arc provided in section 3.5 (qa(3.38)) Remite the equation (2.41 ) for convenience

The consmb arc a = 2.25 Tala and y "787 Mm for the g i m mrgnctiuuion c u m

oftheMnaidurhowninfigurr4.l.

Fi4.l: B-H curve

Tht bounduy and initial conditions sre nmim here as follows:

(iii) Initial values i.c at T = 0, t i = 0 for all "xn

4 3 SOLUTION OF THE PROBLEM-PSM

In d o n 3.2, the dyt ica l solution ia developed for the problan. 'lbe same

pmblan is dso solved numerically by using Cd-Nicholson Mcthod in d o n 3.5.

But in ckptcr-Il. it has been dated that the P&Speceal Method can be a good

substitute for Cnak-Nichoh Method. Heace, in this chqm, the duel excitation

p b l a n is solved by Speclral method.

43.1 Implieit Pseudo S@ Method

In section 2.6, Pseudo Spectral Method or Chebyshev collocation metbod is

discwed elabomtely. The implicit b s l c p p i q scheme IO equation (4.1) is given by

quation (2.67) i.c.,

It may k noted thu to fud t&: values of H at (n+l) timc-step the derivatives ue

mluued at (n+l) timeslep itself. The boundary conditions of equation (4.2) me

implemented by changing the entries of first-row and lapr-row of coefficient maaix

'D' by referring section (2.61). Then Ihe equation (4.3) is solved for Wing the field

d~stnbution. Foc chc sw l rc excitation of H I S = I 1518 A/m, at u1=314.2 dps .

Hn -2303 Nm, at to? 2827.11 nd/sec. the field distribution at various layas is

shorn in figwr 4.2

43.2 Jhahtion of eddy camot ku

Once the field Wbution is abteined thc iron 11x1s ten be c a l M using the

pmculm cncW in section 3.5. In this chapter the effect of harmonics w iron

losses is alao considend.

For this purpose, along with thc f h d a m d compowm, various lmmoaic

cornponenu of field ~IX determined at various laycn, unseqwtly, the cumnt

densities M dusted wing the formula give by equation (3.41) as

Foreschvalueof j =13 : f ix h = 1.3.5. andrunk=OtoN.

Thc figure 4.3 shows the profile of fundmental and harmonic c m t

densities, wbcn the surface excilations me H I S = 1 1518 Aim, at wr=3 14.2 rad/scc; aod

Ha - 2303 A h , at a11 = 2827.8 d s c c .

From Ihc figure 4.3. it is undascnd tbac the higb frequmcy signal mcnuats

fraP tha low frequency signal confuminp the validity of equation (2.38) with rrspea

IO f'rcqucacy. Monovcr. Ihc menuation is nonlinear with depth.

Fowia series is employed to scpmk the fmdamentd components at eacb

laya fmm Ihc resultant wave. Therefore, it has been assumed lhat the two finqumcy

signals uc cammmlurau at all laym. For the incommmsurate signals at the surface.

I! ir diffwult to #pantc thc fundamental components in interior of the materid. In

fm it will bc very intntsling to find the ways to separate Ihc h c n l a l

eompone~b of incommcnsmte signals from the distortad resultant fidd.

Fig. 43: Variation of cumnt densities with depth

4 3 3 Esttmrtion 01 Surface Fkkl Swcmgtlu .ad applkd voltage

For evalunting eddy current-losses by the numerical method aad compare with

h of expcrimartai values, the surface excitation must be the oame as that of rotor.

The procedure follows to estimate field strength at the rotor surface for the given the

stator current.

If D is the diameter of rotor, then the number of conductors on the mtor i s nD.

So h e ratio of vansfonnalion i s

Where N is rhc e f f d v e number of tumr of e& stator windiag.

IftbctWophrrslhvc~urlpumber~ftums*then

~*(NI+N~)/NI fortheihtwinding

5 = ( N I + N ~ ~ N ~ for the second winding

Let Hs k tbc m a p t b i q force at the d a c e of iron in Arnp/m. Ttsen the equivalent

r.m.8 cunmt of the rotor refand to the stator is

I , =- Hs (or) f i K,

Since, it has been observed practically that the magnetising cumnt draw by solid

imn rotor induction motor is as high as 30%. If one assumes the angles between

voltage & I, and vollage & I, arc 36' and 82' rrspactively, the mtor component I,' will

bc 76% of stator cumnt. Hmce

W&rc Is is IIIC stator cumn~.

For diff~tnt values of aator c m t s 11 and 12 of two W i n g s , the

ewrapoadmg velua of HIS and Ha arc found using equation (4.7). Subsequently. Ihe

flux components + I and +Z are determined using the equations (3.23) and (3.27)

mpoctively. I'hw the induced emfs arc calculated using the following equations:

W&rr NI d N2 ur: effcctivc number of turns of rapffCivc stam windings.

L :r Lncn 1 11. h,,L

Now the applied volugea are givm by

V, = E , + I , ( R , + ~ x , )

4.4 GRAPHICAL SOLUTION

In section 2.4.1, gnplaical mahod is d i s c d for single excitation problem.

Whereas in section 3.4, geometrical consauction is provided for double excitation

problem. with the wndition I s (mdm,) S 2. In this section. a similar gqhical

construction wll be explainad for (m2/mI) > 2.Thc graphical construction star& with

h c assumption of values for H and 4 at zeroth-layer. Thm Ihc values of H and 4 at

fim-lycr arc found vectorially. Having calculated the values of H and 4 at first-layer,

the vdues at second-layer arc evaluated. and so on. The above statements can be

wrinm in the mathematical form as.

Lct a! 314.2 nd/aec, nn~= 2827.8 ladlaec, p = 1 6 . 5 8 ~ 1 0 ~ ~ - m , B,= 2.0 T. 'lhe

d e fauor for H, and P M 37.88.2.0 md 23789 rwpcctively, f h n thc equations

(3.33), (3.34) d (3.35). AAa muhiplying the values of the table by these scale

factors. thc curves of field of stmgths, p o w losses, fluxes ctc., are drawn and

shown in fig. 4.4

TABLE:

Reaulb of computerized graphical construction

Fig. 4.4: R e d s of Braphifal mahod

4 3 EXPERIMENTAL PROCEDURE, RESULTS AND DISCUSSION

An experiment wrs conducted on a single-ph induction motor after

moving Ihe capacitor From the swing w i n d i and replacing the wnveapional rotor

by a solid-iron rotor with no air-gap. The specifications of the motor are given in the

rppmlix A4- I . The swing winding was connected lo the 50 Hz supply Where as.

h e m i n g winding was excited by 450 Hz signal. Magnetic intcrfrrrncr is avoided.

since the c l d d angle b?nvcen Ihr two windings is 909. The prrtid eddysurr~nt

loges in h e mamill (rotor) at 1&Jc fraluencies m found-out h m the measured

input p o w . Ir may k noted Iht Ihe iron and stny losses in the stator, including

yoke, are estimmd approximately u 15% of the total ( i ) losses, talring into

lccount the toW mount of irw. lk &tails of calWons an included in the

rppcsdix A4-2. The sirnulaud d meamid iron lor#s an show in fig.4.5

The ulcuiatcd applied voltages RE compllrtd with that of meamred valucs as

shown in figure 4.6.

F i i 4.6: Applied Volges

For p r a t i n g the high frequency signal, oscillator and high capacity (80 Wans)

pow amplifier arc wed. It is taken carc that the two signals s ~ s at the same time

or in ather-words that Ihc phase-shifl is adjusted to zero. But, the phase-shift does not

have much effect on chc power losses. The reason is that, h e lnquency separation

hawem thr two signals is more. Havv thc anrage power loss ur: almost

idcpcndm of phmeshifi.

Thc infinite hdfsprc lhcory is adapted for clatrical machincs. by modifying

thr rpscifa nsislana of mtM &al so as lo incoprue the corrcclion factor for

curvature rad nd d k t a [14]. F i y , a suitable corrcdion factor is ah h q m a t d

into d v i t y to acmmt for tanpsraftue rise. Tk tbaory is provided in appdix

A4-3. Tk effective nsihvity b 1.06 x 1.77 x 1.2 x 16.58 x 10' a-m.

4.6 CONCLUSIONS

A new clusical numerical method called Pseudo-Spectrai mabod is successfully

implemented to find the field disbibution in the solid-iron mtor of induction motor,

when it is subjected to two-fresucncy excitation. It is found that b e simulated powa

loam in the mtor M agreeing with the eexperimentd values. A grapbcal metbod is

also presented to evaluate the field distribution in the specimen.

A4: APPENDIX

ACI: W i h of single-pbuc induction motor:

Rated output pomr : 1.0 )Cw

lnpn voltwe (V) : 220 volts

Effective number of turns of (starhg) winding A , @I) : 334

D.C resistance of winding A ,(RI) : 8.0 Ohms

laducuncc of wrnding A, &I) : 0.045 Henrys

Effective number of turns of (running) wiading 9. ( N 2 ) : 286

D.C resistance of winding 9, &z) : 2.5 Ohms

lndunurcc of winding 9. (L2) : 0.026 Henrys

Diamacr of mtor (D) : 0.106 rn

Length of mtor (L) : 0.109 m

Spccific misturcc of h e rotor material (b) : 16.58~10~ fl-m

B-H curve of rotor material : Figure 4.1(p72)

Length of the air-gap :Zao

End rings : Nil

AC.2 :~mmto i lpd to t8 l~LntbLnta@r

Thcl~hlduItatotm

(a) c o p p t l l o s s g i n t & ~ a n d

(b) I r o n l ~ i n t h e ~

(a) Evrlmtioa of copper lou: Kaowia$ (he nsicrance of each winding, c o p

losses aI any load can be calculated.

(b) Evnlnation of iron lou: Tbest lo- can be further classified as stator teah and

core lows. The calculations of iron losses are based on the total migbt and the flux

density in the material. The values given in the brackets M a t c l y after the

formulae arc rcfemd to the specific motor, whose details an given in the

appendix A4- I.

(i) Stator tu tb loor

T& total weight of all stator leah is given by

(wst-36*5x10" *2x10a * 0.109.7.65xld = 3Kgs)

Whm Ss- numba of stator slots

Ws, dss - width d of each stator slot, metem

LX-le@ of mor con, m a r s

&-specific weight of iron. kgfhn'

T& iron loss in the stslar tath (P,) = a*&,' *WP

(Pal= 6.5*1.5' *3 = 43.87 warns)

Where 'a' is a con sun^. la value is 6.5 for teeth and 4.7 for corc [24].

(if)Strtor are lou

The imn 1- in the stator corc me also &mated in tbc same manna ar tbat of

atator teeth loss.

Thc depth of stator corc is given by

Where D, Do arc inm and outer dinmeten of stator.

The mean diameter of stator wrc is given by

DMU = D O ~ S C

(D-f0.179-0.015 -0.164m)

H a w the weight of che stator corc is,

ww = A.D~~v.~sc .Lsc.~I

(wK= nb0.164* 0.015 *0.109 7.65xld = 6.44 kgs)

The iron loss in the stator con ( P s ) * a * b 2 *WSC

(PLw = 4.7* 1' ,644 1 30.26 W)

Thc total iron loss in rhc stator (P,) = Pia + Psc

(P, = 43.87 + 30.26 z 75 wans)

The won losses and stray losses in stator & yoke an d e n as two-times P, .

So

P - ZbP,

(P ~ 2 ~ 7 5 = 150 watts)

Tbdixe, fortlib Mehhretbetacal iron loucg an the ator side, includiislmy

IoMcSrre ISHof~inpttpoWer.

A C J : T b e ~ 1 h c t o n k r a m h m u d e 8 d c d k c t l [ 1 4 ]

(a) Clll'V8hm

When M infinite half- hwy is applii to r cylindrical coordinate

system, it bccomcs neceJsuy to makc an allowanx fot the curvature to confirm the

physical fact that actual c u m n ~ is reduced. This reduction in- depends upon the

depth of pmetntion. Thc redudion htor given in the reference is (D-2xlTJyD.

W k D is the dimera of the mtor, and xl is the depth of perneation of signal

which goes dapcr thrn Mothn signal. In this use it is the depth of pcmtration of

low frrqwncy signal. For the given surface excitation of the rotor, the stator cumnt is

reduced by the above factor i.e (D2x113)lD. In effect, the d o of transformation is

inuaKd by Ihe ~une h o t . Hence, an alkmmcc for the c u w ~ can be made by

~ncrusing the specific resistance by a hem,

Whcrc the valuc of x, is daenincd using the equation (3.25) for the maximum valuc

0 f I i 1 ~

@)Corrrtb.Cordrlkeb

In the intinite Wf-spueImrlyris, ilhskenlPsMItdthtalltheoumatsin

the rotor flow axially. In other words. ti^^ end e t W s have been ignored. It is obvious

that the end effects would kpend on the physical d i i i o n s o f the mtor, the type of

md- rings used, no end-rings used Md the rotor frequency. An empirical comction

for these cffeas is to modify the specific resistance of the rotor by r factor

When K=l. for the rotor with c~pper end rings.

K = (lhk), for the mtor with steel end rings.

K = 1.77(1+0.49S) with no end-rings,

S. L and t arc slip, rotor active length and pok pitch mpcctivcly.

KZ =K (Since S=O. UIC slip not come into picture as then is no revolving

magnetic field)

So K251.77

(c) Corrcrtk. tor b m p m h n rbe

To consider Ihc cffccl of incrrut in lanpm~c on che resistivity o f dK

mucrd. M appropriate multi f i ion frcm is manned as KJ = 1.2.

To wmmuizc. the effective ooncdion lrctw is K K I K ~ K l = I .06* 1.77*1.2 = 2.25

flmtfore, the m o d i f i nsiwvity is p *p, *~=16.58*10~*2.25 =37.33*14'0hm-m.