Sampling of Discontinuous Voltage and Current Signals in Electrical Drives A System Approach

Transcription

1 /97/$0.00 (c) 997 IEEE IEEE Industy Alication Society Annual Meeting New Oleans, Louisiana, Octobe 5-9, 997 Samling of Discontinuous Voltage and Cuent Signals in Electical Dives A System Aoach Vladimi Blasko and Vikam Kaua Walte Niewiadomski Rockwell Automation - Allen Badley SueTel Technologies 6400 W. Enteise Dive th Avenue NE Mequon, WI USA Redmond, WA USA Abstact Vaious methods fo samling of voltage and cuent signals in electical dive ae analyzed and comaed. Two new methods fo samling the discontinuous voltage and cuent signals in a PWM dive ae oosed. The methods ae based on samling the instantaneous values and locally aveaged values (ove one caie eiod) of signals. Analytical exessions fo adjustment of digital cuent egulatos and achievable bandwidths with the vaious samling methods ae deived. The esults ae exeimentally veified. I. INTRODUCTION The contol boad of a moden electical dive comises of a digital comutational ocessing unit with a cetain numbe of analog and digital inuts and oututs. This basic configuation is univesal enough to suot, with softwae changes only, oeations of vaious dives such as: constant V/Hz, encodeless, vecto contol, DC and egeneative dives. All contol algoithms ae imlemented in softwae and executed on high seed Digital Signal Pocessos (DSP) o Motocontolles (MC), ocessing seed no longe being a limiting facto. Shinking contol boads with high ee of comonent integation, oweed by fast ocessos ae not without some oblems common to all digital dives. Most of the citical issues which emain to be solved ae associated with the inteaction between the comuting unit ( DSP / MC ) and the est of the dive. One of these, obably the most imotant fo dive efomance, is the convesion of the analog signals into digital signals and thei intoduction into the comuting unit. Samling of the moto voltages and cuents is aticulaly toublesome because of thei discontinuous natue caused by the PWM. These oblems have not been investigated and attemts to solve them have been eoted in few efeences only [-2]. Theoetical backgound and a actical solution fo samling of aveage value of moto cuent (locally aveaged ove a samling inteval ) is ovided in []. Using a voltage to fequency (V/F) convete, the inut signal (cuent) was fist conveted into a ulse tain with the fequency of the ulse tain ootional to the inut signal. The ulse tain was then counted / integated by a counte which was ead evey samling inteval. The value in the counte was ootional to the aveage value of the inut signal accumulated ove the samling inteval. The authos of [] wee sketical about samling the instantaneous value of cuent due to subhamonics in samled cuent signal caused by the ile cuent. On the othe hand, [2] suggested that it was ossible to detect the fundamental comonent of cuent by samling the cuent at the mid oints of zeo sace vectos in a PWM ulse atten; the instantaneous value samled at these oints being equal to the fundamental cuent comonent. This ae deals with both samling methods. It ooses oiginal and cost effective solutions fo samling instantaneous value of cuent of a PWM feed inductive load (like ac moto) and locally aveaged values of the load cuent and the load voltage. Achievable bandwidths of the synchonous efeence fame digital cuent contol loos ae used as benchmaks fo the comaison of vaious samling methods. All analytical esults ae exeimentally veified. II. SAMPLING METHODS A. Samling of instantaneous value of cuent Refeence [3] coelates the sace vecto PWM with the taditional tiangle comaison method. It shows that the mid oints of the zeo states in the sace vecto PWM coesond to the eaks of tiangle caie; shown by maximum and imum oints of u t ( at k s 0,,2...) in Fig.. Pe unit values of the caie u t, ile cuent i a and ile voltage u a ae shown fo the hase a of a thee hase inductive load diven by a tiangle comaison PWM in Fig.(c). A caie fequency f ca/(2t s ) much highe than the fequency of the PWM efeence signal was assumed in the analysis. Note that the ile comonent asses though zeo at eaks of the tiangle at k S0,,2.. Thus cuent samled at these instants coesonds to the fundamental comonent and is ile fee as suggested in [2]. Samling eiod T s can be selected equal

2 /97/$0.00 (c) 997 IEEE u t [u] T s T ca k s t/t s i a i c ch ch A/D - T s T 2 T PWM ch8 MUX cs MC T 2 T s T PWM [u] u t u a i a cs u t i a i c i c i a ch ch2 ch2 ch ch8 u t Fig. 2. Block diagam of an 8 channel A/D convete and associated tig diagam t (c) t/t s B. Samling of aveage value of cuent o voltage The aveage value of a signal u in ove caie eiod T ca2t s is defined as: Fig.. Tiangle wave fom u t with definition of samling and caie eiod and tiangle voltage ile comonents of voltage and cuent u a and i a fo a thee hase PWM contolled load u 2T s ( 2+ ks) Ts u in kt s s dt () to (T s T ca 2T s ) o two times smalle (T s 0.5T cat s ) than the caie. A samling scheme with an A/D convete and an eight channel multilexe is shown in Fig. 2. Convesion stat signal (cs), synchonized to the eak of u t with a small advance, is issued by the moto contolle MC. This initiates ten consecutive convesions in the ode shown in Fig. 2. It is assumed that only cuent signals change substantially duing T s and that the othe signals on channels 3 to 8 ae elatively constant ove T s. cuents i a and i c ae symmetically samled befoe and afte the tiangle eak. Aveage of these symmetical samles is used fo futhe ocessing. This samling method gives the values of the cuents at the cente of the tiangle, eliating the need to use a samle and hold element in cuent feedback to ensue simultaneous samling of cuents i a and i c. Cuents samled at instant k s ae available fo use in the following inteval, actically without any delay. The esults of ocessing of contol algoithms ae witten into the PWM at k s+. As a fist aoximation, the locally aveaged value u coesonds to the fundamental comonent of the inut signal u in in the middle of the integation inteval, i.e. at the instant (k s+)t s. Because the locally aveaged signal u is available fo samling at (k s+2)t s, this method ovides vey small feedback delay of T fbt s T ca /2. Fomula () also eliates the induced noise and the need fo additional filteing in the feedback signal. Because the feedback signal exeiences a vey small delay of T s due to local aveaging, it still can be used fo cuent feedback. The method to aveage ove a samling inteval is aticulaly suitable fo samling of moe discontinuous signals like PWM ceated moto voltage. Samling of the instantaneous value of moto voltage would equie heavy filteing with advese effect on the seed of the contol. The hadwae imlementation of (), the aveage with a chage balancing cicuit to event satuation is shown in Fig. 3. The integation is caied out ove the caie eiod T ca, the samling inteval now being T s T ca 2T s. The chage on the integating caacito is balanced by alying the inveted esult of integation fom the evious samling inteval (k-)t s to the inut of the integato duing

3 /97/$0.00 (c) 997 IEEE the cuent inteval k. Fo the cicuit in the Fig. 3 the following is valid : kts uout () k ( uin + Uda) dt+ uout ( k ), (2) RC ( k ) Ts RC U ound of T u k da [ out ( )]. (3) s U da is a voltage fom D/A convete used fo chage balancing. Aveage value of the inut voltage is comuted fom (2) based on a known U da, the cuent samled value of outut and the samled evious values of the outut. To accommodate toleances of the caacito, the exact value of time constant RC T s is deteed duing owe u of the dive by alying known calibating voltage to the inut of the aveage. Note that the D/A, the A/D and the micoocesso ae ats that aleady exist in the dive iesective of the use of the aveage. The D/A convete in the chage balancing cicuit does not have to be of a aticulaly high esolution. The ound off eo due to the lack of esolution in one samling inteval will be comensated fo in the next one and will aea as non detimental andom noise on the signal. An altenate solution of chage balancing cicuit is ovided in fig. 3. U b is a fixed, known, voltage selected to be U max{ u }. Change of the duty cycle d u in u in +U b -U b U da R R R R R/2 -U b d d b C - + GND D/A MC A/D in C - + GND MC A/D u out u out by the time in MC kees the chage balance of the integato. Solution esented in fig. 3 is a modification of the solution esented in [4], to enable oeation with ac quantities. III. ADJUSTMENT OF REGULATORS A. Aoximation of tansfe functions of a lant Analysis of the synchonous samling of instantaneous values vesus synchonous samling of aveage values will be done fo a contol loo with a PI egulato and a lant, with tansfe functions F and F esectively [5]: F F F K +, (4) K, (5a) ( + ) ( + ) 2 K ( + ), (5b) 2 whee K and T ae gain and time constant of the egulato and K, T and T 2 ae gain and time constants of the lant, with T. The oen loo tansfe functions F o is: F o F F, (6) and closed loo tansfe function F c with a unity feedback: F c F o /(+F o ). (7) The cuent contol of a DC moto, decouled cuent contol of a synchonous efeence fame induction moto dive, o a egeneative voltage souce convete (VSC) [6], can be educed to the lant with tansfe function (5a). Tyically in the dive, one of the time constants is doant, i.e. T >>T 2. The othe, smalle time constant T 2, aoximates/combines all small time constants in the system. In the cuent loo, T 2 would tyically aoximate ocessing delays, PWM delays, samling and filteing delays in the cuent feedback signal. The tyical cuent contol loo with a PWM and inductive load is given in Fig. 4. It will be used as a basis fo the analysis that follows. The tansfe functions, consisting of cascade connection of multile lag elements with time constants T i, i n, and a tansotation delay element with time constant T s, can be aoximated as [5]: Fig. 3. Signal aveage with D/A convete and time and switch in chage balancing cicuit

4 /97/$0.00 (c) 997 IEEE i cmd + - i PI Regulato K( + ) + Samling/Feedback Delay + fb Pocessing Delay + µ RL Load KRL + RL PWM K + PWM - PWM u dis + whee T µ is the ocessing o execution time of the algoithm, T fb is the delay in the feedback loo deendent on samling method and filte, and T PWM is the dead time of the PWM. T PWM is equal to one half of the PWM udate time. Gain of the cuent contol loo lant is K /(K PWM R), whee K PWM is gain of the PWM. A standad ocedue to analyze lant in Fig. 4, which can be educed to (5a) using (0) and (), is Technical Otimum (TO). Accoding to this ocedue time constant of PI egulato is select to be equal to the doant time constant of the lant [5]: Fig. 4. Block diagam of a cuent contol loo of a PWM contolled inactive load n s e 2 s n ( + s i ) ( ) ( ) ( + i)! 2! i ; whee (8) n + ( + T) + e i n s i i T 2 T T + T. (9) e s i i The tansot delay in (8) was aoximated with a single time constant T s afte neglecting highe ode tems in Taylo seies exansion of exonential function. The esulting n+ lag elements wee elaced by a single lag element with the equivalent time constant T e which in equation (5) aeas as a time constant T 2. It can be easily veified that tansot delay and multile lag elements have faste hase and slowe amlitude esonse adation with incease of fequency than thei single, fist ode element aoximation. Theefoe the bandwidth in a eal digital system, defined by -3dB amlitude esonse fo a closed loo, is going to be substantially highe then bandwidth defined fo fall of hase to -45. B. Technical otimum In the lant of Fig. 4, the doant time constant is T T RLL/R, (0) whee L and R ae inductance and esistance of the inductive load. All othe much smalle time constants ae lumed togethe into the single equivalent time constant T 2: T 2T µ+t fb+t PWM, () T T RLT. (2) This cancels the doant ole in the oen loo tansfe function (6) and educes the closed loo tansfe function (7) to the second ode system. Fo a daming facto D 2, the gain of the PI egulato in this second ode system should be: T K (3) K 2T2 The coss ove fequency at which the oen loo tansfe function F o (ω c ), with egulato aametes T and K selected accoding to (2) and (3) is: ω c T 2 T 2 2 o bandwidth fequency f b : (4) ωc fb. (5) 2π 38. T 2 With the above settings, the egulato ovides a good esonse to a ste change of the efeence. Because of the usually high time constant T T RL, the integal time constant of the egulato T, selected accoding to (2) is also high. The consequence of such an adjustment is a elatively slow esonse to the distubance u dis. C. Symmetical otimum Fo a seed contol loo with an inetia load o a DC bus voltage contol loo with a lage caacito bank, the tansfe function (5b) of the lant can be used. Fo the cuent contol loo with L / R T >> T2, the tem jωt >> in the vicinity of coss ove fequency and the tansfe function of lant (5a) can be aoximated with the tansfe function (5b). The oen loo Bode lot of such a lant with a PI egulato is given in Fig. 5.

5 /97/$0.00 (c) 997 IEEE Magnitude [db] [] 60 (-2) ωc 2π fb (-) 0 (-2) -60 T T ω [ ad / s] ω [ ad / s] 0 5 Fig. 5. Oen loo Bode diagam fo the choice of egulato aametes accoding to SO A common ocedue fo the deteation of egulato aametes is Symmetical Otimum (SO) [5]. To have maximum hase magin the gain of egulato should be: T K, (a 2.4). (6) K at 2 The integal time constant is: 2 T T a T 2, with a + cos ψ sinψ (7) whee ψ is a hase magin. Facto a diectly influences selection of time constant of egulato. It effects hase magin and daming of the system. To have daming facto D 2, facto a should be a 2.4. The coss ove and bandwidth fequencies ae: ωc, fb, fo a2.4, f at 2π at 2 2 b 5. T 2 (8) Note that while the gains of the PI egulato fom (3) and (6) ae vey simila, the TO time constants fom (2) is substantially highe then the SO time constant fom (7). Because of the smalle integal time constant fo egulato adjusted accoding to SO the distubance ejection (U dis in Fig. 4) is sueio to a egulato adjusted accoding to TO. Howeve, contol loo adjustment accoding to SO has an oveshoot of about 40% in its ste esonse due to a zeo in the noato of close loo tansfe function [5]. Aaently the otimum adjustment is a combination of these two methods deending on the imay objective: ejection of the distubance o following of the efeences signal with a small oveshoot. The above equations suggests that K can be ket constant accoding to (3) and T educed fo imoved distubance ejection with the value given by (7) as the ue limit. D. Samling methods and bandwidth of digital cuent contol loo Contol loo bandwidth will be used to benchmak the efomance of digital cuent contol loos with diffeent samling methods. The bandwidth is diectly deendent on delays in the system lumed into time constant T 2 (). The esults of the analysis ae summaized in Table I, with tyical delays in cuent feedback loo aanged into thee gous: Aveage and Symmetic PWM (SPWM) with udate ate once e caie eiod T ca. Synchonous Samling (SS) and SPWM with udate ate once e caie eiod T ca. (c) SS and Asymmetic PWM (APWM) with udate time 0.5T ca (o udate ate twice e caie). TABLE I. BANDWIDTH OF CURRENT CONTROL LOOPS WITH DIFFERENT SAMPLING METHODS AND DIFFERENT TYPES OF PWM (UPDATED ONCE AND TWICE PER CARRIER CYCLE) T µ T fb T wm T 2 f b Aveage & SPWM SS & SPWM (c) SS & APWM T ca T ca 05. T ca 05. T ca T ca 05. T ca 025. T ca 2 T ca 5. T ca 075. T ca f ca / f ca / f ca / 0. 3 The last ow of Table I elates achievable bandwidths to caie fequency fo vaious samling and PWM methods. The achievable bandwidths ae aoximately 0 to 30 times smalle then the caie fequency. Note that case (c) ovides 27.6/ times highe bandwidth than case, demonstating clea advantages of synchonous samling in the cuent feedback loo. Howeve when samling a discontinuous signal like moto voltage in an encodeless dive, otion does not have an altenative. Otion still ovides vey high bandwidth and has substantial advantages ove systems whee moto voltage is fist filteed and then samled.

6 /97/$0.00 (c) 997 IEEE IV. EXPERIMENTAL RESULTS A. Synchonous samling of instantaneous and locally aveaged value of cuent The exeimental esults wee obtained on a thee hase VSC [6], with inut line voltage of 460V, DC bus voltage of 750V, and ated line cuent of 80A. Inut line eacto was 0.52mH, 2.28mΩ. The PWM caie fequency was f ca5khz. Actual cuent wavefom is shown in Fig. 6. The wavefom in Fig. 6 is the samled vesion of, obtained at the outut of the signal aveage (Fig. 3) and then clocked out fom the DSP though a D/A convete. Note that the aveaged signal accuately eesents the fundamental comonent of load cuent. Fig. 6. Actual cuent and cuent samled using signal aveage Figue 7 shows actual wavefom of load cuent, the same signals samled synchonously with caie with samling eiod T s T ca (oints,3, 5 etc. in Fig. ), and (c) cuent samled by aveage. Note small hase shift of 0.5T s between instantaneous and aveaged samled cuent values and (c) esectively. B. Bandwidth of cuent egulato with diffeent samling methods To evaluate the influence of samling methods on the seed of esonses of the egulatos, thee diffeent samling methods, to (c) fom Table wee imlemented. White noise was added to the cuent command i cmd in a synchonous efeence fame. The command i cmd, feedback cuent i, and eo (i cmd-i) wee clocked out fom the DSP though a D/A convete to the fequency esonse analyze. Oen and closed loo Bode lots wee measued. The egulato aametes wee adjusted accoding to (2) and (3) o accoding to TO method. Measuement esults ae summaized in the Table II fo the thee diffeent samling methods fom Table I. The fist ow contains calculated bandwidth f b fom (5). The measued oen loo bandwidth f b_ol and the measued closed loo bandwidth f b_cl ae given in ows 2 and 3 fo comaison uoses. The oen loo bandwidth f b_ol was measued at the oint whee the amlitude of the oen loo esonse cossed zeo. The close loo bandwidth f b_cl was measued when the hase of the close loo esonse was -45. A good ageement between the edicted (5) and the measued bandwidths can be seen at the zeo cossing of oen loo amlitude esonse. The measued bandwidth f b_cl is about 5% to 20% lowe then f b_ol. TABLE II. ESTIMATED f b AND MEASURED OPEN f b_ol AND CLOSED f b_cl LOOP BANDWIDTH FREQUENCIES f b at f ca 5kHz Aveage & SPWM SS & SPWM (c) SS & APWM f ca / f ca / f ca / Hz 242 Hz 485 Hz f b_ ol 20 Hz 250 Hz 508 Hz f b_ cl 65 Hz 99 Hz 4 Hz Fig. 7. Actual cuent, instantaneous synchonously samled cuent and (c) aveaged and then samled cuent Figue 8 shows oen loo esonses with the cuent samled at fequency of 0kHz - twice the ate of caie with egulatos adjusted accoding to TO and SO methods. The bandwidth with egulatos adjusted accoding to TO was 50, what is 2% highe then 452Hz, the value obtained

7 /97/$0.00 (c) 997 IEEE X:508 Hz Y: db X:4 Hz Y: db 64Hz X:508 Hz Y: kHz 36 /div 80 00Hz 80 00Hz X:4 Hz Y: kHz 36 /div Hz 00Hz X:452 Hz Y: db.664khz Hz X:99 Hz Y: db.608khz 64Hz /div Hz 00Hz 00Hz X:452 Hz Y: kHz.664kHz /div Hz X:99 Hz Y: Hz X:65 Hz Y: db.608khz.608khz Fig. 8. Oen loo esonses with cuent samled at fequency of 0kHz and egulatos adjusted accoding to TO and SO methods. with egulatos adjusted accoding to SO. This esult is in good ageement with theoetical esult; fom (5) and (8) it follows that that TO should give 9.3% highe bandwidth then SO method. The close loo fequency esonses of the egulatos adjusted accoding to TO with thee diffeent samling methods fom Tables I and II ae given in Fig. 9. Note that the amlitudes of the closed loo esonses ae faily flat with - 3dB fall at fequencies thee times highe than oen loo bandwidths. At the same oints the hases assume values close to -80. Aaently, the bandwidth fequency measued when amlitude falls to -3dB would be too high /div Hz X:65 Hz Y: Hz (c).608khz.608khz Fig. 9. Amlitude and hase closed loo esonses with cuent synchonously samled twice e caie once e caie and (c) cuent samled using aveage; caie fequency was 5kH

8 /97/$0.00 (c) 997 IEEE Figue 0 shows the ste esonse of a synchonous efeence fame cuent loo with samling ate of 0kHz and caie fequency 5kHz. A comomise adjustment of cuent egulato combining TO and SO methods was adoted in ode to imove distubance ejection with still accetable oveshot to the ste change of efeence. The ootional gain was set accoding to TO (3). Note that both methods give simila esults fo ootional gain, equations (3) and (6). The integal time constant was abitaily set to T 0T 2, the value between esults given by TO and SO methods, equations (2) and (7). Such an adjustment esulted in imoved distubance ejection and accetable oveshot of 20% to the ste change of efeence. The solution with an aveage enables samling of the locally aveaged value of an inut signal ove a caie eiod. The aveage intoduces a delay of only one half of the caie eiod into the feedback. It can be used fo samling of load cuent and PWM tyes of voltage signals. The bandwidth of the cuent contol loo with such a feedback is 27.6 times smalle then the PWM caie fequency. Synchonous samling (synchonized with PWM caie) of instantaneous value was used as the effective altenative in a cuent feedback loo. The achievable bandwidth with synchonously samled cuent signal, with samling and PWM udate ate twice e caie, is 0.3 times smalle than the caie fequency. REFERENCES i e d_c V. CONCLUSION 200us 66A Fig. 0. Ste esonse of synchonous efeence fame cuent contol loo; i e d_c - efeence cuent, i e d - feedback cuent The achievable bandwidth of the cuent contol loos in electical dives is aoximately 0 to 30 times smalle than the caie fequency of the PWM. The bandwidth is a function of a ocessing seed and the delay in feedback signal. The delay in feedback deends on samling methods. Two cost effective methods of samling of instantaneous and locally aveaged values of signals ae esented and analyzed. i d e [] T. Matsui, T. Okuyama, J. Takahashi, T. Sekegava, K. Kamiyama, A High Accuacy Cuent Comonent Detection Method fo Fully Digital, Vecto - Contolled PWM VSI - Fed AC Dives, , PESC 88 Confeence Recod, Kyoto, Jaan, Ail -4, 988. [2] T. Yamada, Y. Yamamoto, T. Kodama, T. Ichioka, Advancement of Vaiable - Seed Technology, Meiden Review Intenational Edition, No. 3, 990, Seies No. 90. [3] Vladimi Blasko, A Hybid PWM Stategy Combining Modified Sace Vecto and Tiangle Comaison Methods, Conf. Rec. of PESC-96 Ann. Mtg, June 23-27, 996, Baveno - Italy. [4] John D. Miskin, Chage balancing cuent samle fo digital moto contol, U. S. Patent Numbe 4,804,895, Allen Badley Co., Febuay 4, 989. [5] W. Leonhad, Intoduction to Contol Engineeing and Linea Systems, Singe Intenational Student Edition, Allied Publishes Pivate Limited, New Delhi. (The oiginal Geman edition: Einfuhung in die Regelungstechnik, at Velag Fiedich VIWEG & Sohn, Baunschweig, Gemany 970). [6] V. Blasko and V. Kaua, A Novel Contol to Actively Dam Resonance in Inut LC Filte of a Thee Voltage Souce Convete, IEEE Tansactions on Industy Alications, , Vol. 33, No. 2, Mach/Ail 997.