INRUSH CURRENT. By Mario Célio Contin. R&D Department WEG Equipamentos Elétricos S.A. - Motores.

Transcription

1 1 INUSH UENT y Maro élo ontn &D Department WEG Equpamentos Elétrcos S.A. - Motores. 1. INTODUTION Any nductve system, whch can be represented by a seres-connected col wth a resstor (g.1), when energzed, may present a phenomenon known as Inrush urrent,. e., the ntal energzed transent current. Ths s caused due to the poston of the power supply voltage waveform, whch s varable wth the tme and relatve to ths tme. In other words, f the voltage waveform, varable wth the tme, at the moment of the ntal energzaton of the nductve system, s passng through zero, there wll be the hghest ampltude of such phenomenon. Here we presume that the col, before the ntal energzaton, has not storaged energy n ts netc ld. The hghest Inrush current actually occurs when the voltage, varable wth the tme, s passng through zero, exactly lke what s expressed by the nducton law d v( t) dt g.1- Inductve system wth power supply varable wth the tme. The dfrental uaton for nductve system shown n g.1 wll be d v( t) dt onsderng the power supply as a snusodal v( t) V sen ω t max the soluton for the system wll then be Max ( ) ( ω t ϕ) τ sen ] ( t) [ sen ϕ Max V Max ( π f ) e t

2 ϕ tg 1 π f τ The soluton for netzng current (t) conssts of two components, where the frst one as permanent duty and s gven by e g. Perm. ( ω ϕ ) sen t Max and the second one as transent duty whch corresponds to a D component Transt. Max e t τ senϕ If the col s connected, exactly at the moment the snusodal voltage waveform v(t) s passng through zero, at an ncreasng drecton, whch represents the worst energzaton condton, the netc flow and the netzng current () should be consdered as ncreasng (from - max and - Max, respectvely). However, as the crcut has been closed, we have 0 e 0, nstead of values as they should be regarded as normal for snusodal permanent perodcal duty. To satsfy the nducton law, the flow must be regarded as ncreasng from zero, durng the whole half voltage perod, for the second half-perod to be regarded as decreasng. Due to ths, the flow and netzng current snusodal wll reman dsplaced by a constant value (contnuous component) ual to ts ampltudes,. e., the current wll be I Max.. INUSH UENT O AN INDUTIVE SYSTEM WITHOUT SATUATION or an nductve system wthout saturaton where the nductance remans constant, the soluton for the energzaton transent (Inrush) s smpler and more determnable snce the netc permeablty remans constant. v (t) λ ωt v 0 g. Physcal meanng of angle λ

3 The soluton s demonstrated as follows: eng the angle λ that represents the voltage wave phase at the nstant the crcut s closed, the soluton of the dfrental uaton wll be ( t) [ sen( ω t λ ϕ ) τ sen( λ ϕ )] Max On the uaton above, we can notce that the term transent s ual to zero when (λ-ϕ)0, π, π, etc. As the system s hghly nductve, the rato ω to s qute bg causng ϕ to be close the lmt of π/. In such cases, the term transent s null when λ s approxmately π/, 3π/, 5π/, etc. Physcally, ths means that there are no transent efcts n hghly nductve crcuts, when the crcut s energzed at nstants on whch the voltage waveform s approxmately at ts maxmum value. The term transent on the prevous uaton s maxmum when (λ-ϕ)π/, 3π/, 5π/, etc. When ϕ s approxmately ual to π/, t s evdent that the term transent wll be maxmum when λ s approxmately close to 0, π, π, etc. Therefore, on a hghly nductve crcut, the term transent wll be maxmum when the crcut s closed, n ponts, where the voltage waveform s approxmately null. onsderng that the crcut s closed at the nstant the voltage waveform s passng through zero, the transent component and the permanent duty component become combned, then annullng the current at the nstant, as demonstrated n g.3. Ths, obvously, s the case for an nductve crcut where the current s not gong through t (swtched-off) rght before the energzaton. onsderng that there are not saturaton efcts, where and are constant, the maxmum netzng current ( (t)) wll be I Max, as demonstrated n g.3. e t 3 (t) transent component I Max v(t) ωt π -I Max Permanent duty component g.3 Transent of Inrush current wth no saturaton condton.

4 4 3. INUSH UENT O A SATUATED INDUTIVE SYSTEM As descrbed above, when the energzaton s done wth voltage wave passng through zero, there wll be the Inrush current phenomenon n ts fullness. Due to such efct, both current and netzng flow wll be sgnfcantly ncreased. or a non-saturated system, both current and flow wll have smlar ncreases (maxmum ual to I Max and Max, respectvely). However, for a system wth a rronetc core, lke on nducton machnes, wth sgnfcant netc flow ncrease, whch can reach up to Max, the rronetc core becomes saturated, then reducng the netc permeablty qute a lot and, as a consuence reducng the nductance of the system. Ths causes that the netzng current, at the nstant of energzaton, can be tenths tmes hgher than the rated value. onsderng t s an nducton motor, ths can mean that, at the nstant of the swtchng-on, the startng current on the termnals can be somewhat tmes hgher than the rated value on startng. As descrbed above, the Inrush current depends on the voltage waveform value, at the nstant the crcut s closed. Another varable that afcts sgnfcantly the Inrush current value s the saturaton condton the system s submtted to. oth varables can not be easly predcted snce the voltage waveform value, at the nstant of the swtchng-on, s random due to the hgh speed the wave vares, and also due to the fact that the saturaton condton depends on the value of ths voltage. On nducton motor desgns, t s qute normal to have nducton waves wth maxmum values up to 1.8 Teslas. onsderng that the netc flow, wth Inrush efct, can reach up to twce of ts maxmum value ( Max ) and consderng that the netc lamnatons become completely saturated wth ntrnsc nductons n the range of to.15 Teslas, t s evdent that the man netc crcut of the motors wll reman completely saturated. or Inrush calculaton takng nto account saturaton efcts, t s rured to establsh clearly the nstantaneous condtons of the voltage waveform at the nstant of the energzaton, and also to defne the saturaton condtons for the netc crcut of the system. As nstantaneous condton for voltage waveform, t s normal to establsh the worst case, whch rers to the nstant the wave passes through zero. Then, for saturaton condtons, due to what was consdered for the voltage, at the nstant of the energzaton, we can consder that the man netc flow of the motor s twce of ts rated value. 4. SATUATED MAGNETIING EATANE EATED TO INUSH UENT Wthout saturaton of the netc crcut, the Inrush netzng current wll be I MagInrush( s / sat ) Max I V max ( π f m) To take nto account the saturaton efcts of the netc crcut, the followng must be consdered: The total MM (Magnetomotrz orce) rured on the netc path, under normal condtons, wll be: I m ( 1 s1 fω1) Total a1

5 5 MM on the lamnaton core netc path (rronetc sheet core),. MM on the netc path through the ar (ar gap), I Magnetzng current ( I 0 ), [A]. I 0 No load current, per phase, [A]. 1 Number of turns per col. s1 Number of cols per pole and per phase. fω1 Wndng factor. a1 Number of parallel paths n the wndng per phase. M Number of wndng phases (1.s1.fω1). m Efctve number of turns per pole, [turns/pole]. ( I ) Magnetzng current per pole, [A/pole]. a1 ( m ) actor for resultant MM waveform wth m phases. The MM on the path through the ar (ar gap) wll be H 0.m.s netc nducton value on the ar gap, [T]. 0 Ar netc permeablty (vacuum), [H/m]. Ar gap, [mm]. The resultng MM on the path through ar gap ( ) wll then be ( ω ) I m s f Total a1 0 The average netc length path per pole through the ar gap ( ), wll be D e1 D1 D c1 hn 1 D 1 D D ( hn ) c π ( ) ( ) 1 1 D D D c c c Dc P D c1 Average stator yoke dameter, [mm]. D c Average rotor yoke dameter, [mm]. D e1 External stator lamnaton dameter, [mm]. D 1 Internal stator lamnaton dameter, [mm]. D Internal rotor yoke dameter, [mm]. hn 1 Total stator slot heght, [mm]. hn Total rotor slot heght, [mm]. ( P) Number of motor poles.

6 The r.m.s. value of the uvalent netc ld (H ), for normal operatng condtons, on the path through the ar gap wll be H and the r.m.s. value of the correspondng netc nducton ( ), through the curve xh of the netc lamnated core materal s f ( ) H The uvalent netc permeablty ( ), consderng the whole man netc crcut (ncludng ar gap), for normal operatng condtons, wll be H Total 0 The rated netzng nductance ( MagNom ) correspondng to wll be MagNom ph f P π P S ph Equvalent number of seres connected turns per phase [turns/phase]. S Magnetc area of a pole, [m ]. ength of the polar netc path, [m]. Smlarly, the saturated netzng nductance ( ), whch takes nto account the permeablty s, resultng from saturated efcts on the man netc crcut due to Inrush current efcts wll be gven by ph s P P The uvalent netc permeablty wth saturaton ( s ) whch takes nto account the Inrush current efcts, wll be obtaned as follows. The saturated netc nducton s (at the ar gap), by Inrush efct, s s and the netc ld (H s ) correspondng to ths nducton, wll be f H s ( ) s S obtaned from the curve xh of the rronetc materal. The resultng MM on the path through the rronetc materal wth Inrush efct ( s ), consuently, s s 10 3 H s and the MM on the path through ar gap, wll be s s 6

7 7 s nally, the uvalent netc permeablty wth saturaton ( s ), wll be gven by s s s H 0 s s s The rato between saturated netzng nductance ( ) and the rated netzng nductance ( MagNom ), wll result n Sat MagNom s MagNom The saturated netzng reactance wll consuently be MagNom s 5. INUSH UENT O INDUTION MAHINES As far as nducton motors are concerned, the uvalent crcut on startng, wth Inrush efct, wll be as shown n g.4. s s g.4 Equvalent crcut for nducton motors on startng, wth Inrush efct. The Inrush efct afcts manly the man netc crcut due to the fact that ths s bascally a transent on the netzng current. Therefore, we can consder that the saturaton efcts, caused by Inrush current, bascally do not afct sgnfcantly the dsperson reactances on startng. As a result, the dsperson reactances can be consdered wth ts rated values for the startng, whch are already afcted by the consuences of saturaton due to rotor current (on the brdge of the rotor slot). Therefore, the Inrush current, consderng that the motor has been swtched-on at the nstant the voltage wave passes through zero, wll be gven by uvalent crcut soluton shown n g.4. Ths current wll then be

8 8 ϕ ϕ j p p j A p p p p A tg 1 j p p j A ϕ A ( ) p p 1 p tg p A ϕ ϕ ϕ π A cos( ϕ ) sen( ) x y ϕ I nrush Max ( ) ( 1 y) 1 x V onsderng the power supply as a snusodal voltage wave, the result s as follows I nrush nom ( ) ( ) 1 x V 1 y where V nom.m.s. rated voltage per phase of the snusodal power supply [V]. The Inrush current can be gven n absolute peak value per phase (Amps) as calculated by the last uaton or n p.u. (per unt) wth bass on the rated current. 6. OOK EEENES 1) alcone, Áuro Glberto: Eletromecânca Transformadores e Transdutores, onversão Eletromecânca de Energa Volume 1 ap., pág. 84, 85. ) Kerchner & orcoran: rcutos de orrente Alternada pág. 606 a 611.