Should ΔΣ Modulators Used in AC Motor Drives be Adapted to the Mechanical Load of the Motor?
SShould ΔΣ Modulators Used in AC Motor Drivesbe Adapted to the Mechanical Load of the Motor?
Sergio Callegari
University of Bologna,ARCES/DEIS, [email protected]
Federico Bizzarri
Polytechnic of MilanDEI, [email protected]
Abstract —We consider the use of ΔΣ modulators in ac motordrives, focusing on the many additional degrees of freedomthat this option offers over Pulse Width Modulation (PWM).Following some recent results, we show that it is possible tofully adapt the ΔΣ modulator Noise Transfer Function (NTF)to the rest of the drive chain and that the approach can bepushed even to a fine adaptation of the NTF to the specificmotor loading condition. We investigate whether and to whatextent the adaptation should be pursued. Using a representativetest case and extensive simulation, we conclude that a mildadaptation can be beneficial, leading to Signal to Noise Ratio(SNR) improvements in the order a few dB, while the advantagepushing the adaptation to the load tracking is likely to be minimal. I. I
NTRODUCTION
Switched-mode power conversion keeps gaining momentumdue to its efficiency and flexibility. An important application is acdrives where induction motors are fed by inverters so that bothfrequency and voltage can be finely varied. Two major approachesexist, relying on Pulse Width Modulation (PWM) or Pulse DensityModulation (PDM).In PWM, frequency and voltage control is achieved by varyingthe duty ratio of the inverter switches [1]. To this aim, a fixedframe frequency f PWM is established, so that the time axisis divided in frame intervals. For each interval, the modulatorproduces a pulse, selecting its width as needed. Conversely,in PDM voltage and frequency are controlled by varying theconcentration of pulses [2]. There is no frame concept, but athin pulse duration T is established, so that the time axis is divided in pulse intervals. As time flows, for each pulse intervalthe modulator can decide whether to generate a pulse or not, thusvarying the output density as needed.While effective, PWM is sometimes criticized for its fixedframe frequency that may lead to evident harmonic clusters [3]in voltage and current spectra. This may cause Electro-MagneticInterference (EMI) or acoustic noise [4]. Randomized schemes,where either the pulse width, position or frame frequency areperturbed may reduce this effect [5]–[7]. Yet, they may lead toother issues and certainly increase system complexity. This is a post-print version of a paper published as in the Proceedings of 2012IEEE International Conference on Electronics, Circuits, and Systems (ICECS),pp. 849-852, Dec. 2012. Available through DOI 10.1109/ICECS.2012.6463619.To cite this document please use the published version data.Copyright © 2012 IEEE. Personal use of this material is permitted. However,permission to use this material for any other purposes must be obtained fromthe IEEE by sending a request to [email protected].
This is one of the reasons why PDM, is now actively investi-gated as a PWM substitute [2]. Lacking a frame structure, it canbe inherently more compliant to EMI regulations. Furthermore,the higher apparent complication of PDM is often a myth. First ofall, very practical implementations now exist, typically relying on ΔΣ modulation [8], at times practiced with specialized quantizers(e.g., hexagonal [9] or vector type [2]). Secondly, the perceptionof complication mostly arises from the much wider set of tuningoptions that it offers. Operating in a wider design space isobviously harder. This is evident just by looking at the numberof adjustable knobs . In PWM only the frame frequency andresolution need to be set. In ΔΣ modulation, which is alreadya special, restricted kind of PDM, the full set of coefficients forthe filters inside the modulator is to be chosen by the designer,in addition to the sample rate.In this paper we deal with this extra flexibility and its exploita-tion. Particularly, we try to answer a specific question. For a ΔΣ modulator used to drive an ac motor, there is an obvious desireto set its parameters according to the specifications of the powerbridge, the motor itself and any other elements sitting between thetwo (e.g., a passive filter) i.e, to adapt the modulator to the restof the drive. Now, given that the motor behavior can vary, evensignificantly, with the mechanical load applied to its shaft, shouldone try to track these changes, making the modulator parametersvary accordingly? Or would this be a useless attempt at overoptimize the system, with negligible benefit in terms of measuredperformance?To answer this question we simulate a realistic motor, mod-eled at different loading conditions and with bridge commands synthesized by different ΔΣ modulators, some of which explicitlydesigned after the motor behavior at a specific mechanical loading.For the modulator optimization we exploit a recent result thatenables its adaptation to the user of the ΔΣ streams [10], [11].Eventually, for each loading condition, we compare the optimalbehavior with that obtained from the modulator optimized atanother operating condition.II. M ODELING
In our analysis, we exploit a realistic motor model. Yet, weconsider an idealized setup restricted to a single phase feed-forward control as in Fig. 1. This is to decouple the matter underexam from any other second order effect, such as those possiblyarising from multiphase/multilevel quantization or feed-back. Wealso practice a direct connection between the power bridge andthe motor (i.e, we remove the filter from the architecture in Fig. 1), a r X i v : . [ c s . S Y ] O c t ignalIN switched mode drive Clock ( f Φ ) ΔΣ Modulator Gatedriver
Low depth,high-rate2 levelstream ac motor + V pwr − V pwr Explicit LP filter M Figure 1. Idealized, single phase simulation setup. leaving all the signal smoothing to the reactive effects providedby the motor itself. While this is unusual for large drives, it letsthe consequences of impedance variations associated to changesin the mechanical load be more evident, avoiding their hidingbehind other large, fixed reactive effects.For what concerns the electric machine, we start from the usualequations for a 3-phase induction motor. Under the basic hypoth-esis of writing the equations: (i) in a stationary reference frame;(ii) assuming null rotor voltages; and (iii) taking the equivalent dq -axis representation, the following differential system is derived[12] di sd dt = − R s i sd + ω m PL m L r i sq + R r L m L r i rd + ω m PL m i rq + v sd ∆ · L s di sq dt = − ω m PL m L r i sd − R s i sq − ω m PL m i rd + R r L m L r i rq + v sq ∆ · L s di rd dt = R s L m L s i sd − ω m PL m i sq − R r i rd − ω m PL r i rq − L m L s v sd ∆ · L r di rq dt = ω m PL m i sd + R s L m L s i sq + ω m PL r i rd − R r i rq − L m L s v sq ∆ · L r dω m dt = ( i sq i rd − i sd i rq ) P L m − Λ( t ) − Bω m J . (1) where: ∆ = 1 − L m / ( L s L r ) , i α,β and v α,β are the currents andvoltages referred to the motor stator or rotor (if α = s or α = r ,respectively), once projected on the β -axis for β ∈ { d, q } ; ω m isthe rotor mechanical angular speed; Λ( t ) is a time-varying load.Other parameters and symbols are explained in Tbl. I that alsoreports the values used for simulation (which are the same usedin [13], for the sake of an easy confrontation).Evidently: (i) the model is non-linear; and (ii) it depends on the load Λ . For what concerns the first point, it is worth observingthat if ω m has an almost constant value, one can approximateaway the non-linearity, obtaining the well known linear modeldescribing the steady-state dynamics of the motor working at the Table I
AC MOTOR PARAMETERS AND VALUES USED IN SIMULATION P Number of motor poles B · − N m s
Damping constant (dissipation due towindage and friction) J · − kg m Moment of inertia R s Ω Stator dissipative effects R r Ω Rotor dissipative effects L s . · − H Total 3-phase stator inductance L r . · − H Total 3-phase rotor inductance L m . · − H Magnetizing inductance f w
50 Hz
Nominal driving voltage angular frequency V w
320 V
Nominal driving voltage peak amplitude − f [Hz]10 − − − − | Y ss ( π i f ) | [ Ω − ] σ = . σ = . σ = . Figure 2. Transfer function due to the motor admittance at different slipvalues. ω e = ω m P / rotor constant electrical angular frequency. This isrelated to the angular frequency ω w = 2 πf w of the three-phasesinusoidal drive voltage (with amplitude V w ) through the slipcoefficient σ = 1 − ω e / ω w running from (no-load condition)to (blocked rotor). From this linear model, one can eventuallycompute the admittances from each of the three phase statorvoltages to the corresponding phase stator current, namely Y ss (i ω ) = R r L s L r + i σL s ω R r R s L r L s + i( R r L r + σ R s L s ) ω − σ ∆ ω . (2)Interestingly, in this linearized model, the mechanical loadingeffects seem to disappear. As a matter of fact, they do not. Theyare only hidden inside the slip parameter. A higher load requiresa higher motor torque that can only be obtained by augmentingthe motor currents or accepting a higher slip. Fig. 2 shows thefrequency response associated to Y (i2 πf ) for three different slipvalues (self-friction, σ = 0 . , nominal slip σ = 0 . and largeslip σ = 0 . ) and for the motor parameters in Tbl. I. Evidently,the change is quite significant. It is worth underlining that thisbehavior, with a pole-zero couple that moves as σ is increased,is not limited to the example setup, but rather typical.The linearized model (2) enables a rapid evaluation of someaspects of the switched drive performance. Specifically, it letsone foresee the deviation of the motor windings currents fromideal sine waves. Such deviation results in high frequency com-ponents and on EMI. Furthermore, since currents are ultimately responsible for the motor torque, it may cause torque fluctuation,vibration and noise. Consequently, conformance of the windingcurrents to expected sinusoidal profiles, quantified through SNR,can be taken as an important quality figure. Formally, SNR isdefined as I N / I S , where I S is the effective value of the idealdrive and I N is the rms current value associated to the switchedactuation artifacts. One has I N = (cid:90) ∞ Ψ N (i2 πf ) | Y (i2 πf ) | df (3)where Ψ (i2 πf ) is the spectral density associated to the actuationartifacts on the voltage drive. Obviously, in Eqn. (3) there maybe an additional multiplicative term | G (i2 πf ) | whenever a filter G ( s ) is used to feed the motor, but we have chosen not toimplement it. From Eqn. (3), the importance of Y (i2 πf ) is selfevident, given that Ψ (i2 πf ) depends and can be determined fromthe drive. uantizer FF z FB z w nT x nT nT c Linearized quantizer model
Figure 3. ΔΣ modulator and its approximated linear model. III. ΔΣ MODULATOR IN THE DRIVE
One of the advantages of using a ΔΣ modulator in the drive isto exploit its noise shaping abilities adapting them to the signalsto be actuated and possibly to the drive chain. Recall that a ΔΣ modulator can be modeled by the standard architecture in Fig. 3,where T is the sample time. For an approximated analysis, themodel gets linearized as illustrated in the circle. In this case,one can identify a Noise Transfer Function (NTF), NTF ( z ) fromthe quantization noise (cid:15) ( nT ) to the output and a Signal TransferFunction (STF), STF ( z ) from the modulator input w ( nT ) to theoutput. Once the STF is taken to be unitary as it is normally thecase, the modulator filters FF ( z ) and FB ( z ) can be designed toobtain a desired NTF as long as some constraints on the NTFitself are respected [8].This is particularly interesting with reference to Eqn. (3). Infact, for a correctly operating modulator, (cid:15) ( nT ) can be assumed tobe approximately uncorrelated to the input, white, and uniformlydistributed within [ − ∆ / , ∆ / ] where ∆ is the quantization step[8]. Consequently, Ψ (i2 πf ) in (3) happens to be known andapproximately equal to ∆ sinc ( fT ) (cid:12)(cid:12)(cid:12) NTF (cid:16) e i2 π ffs (cid:17)(cid:12)(cid:12)(cid:12) , wherethe usual z ↔ e i2 π ffs substitution is adopted, f s = / T , a zero-order hold is assumed at the modulator output, and sinc indicatesthe normalized function sin( πx ) / ( πx ) . Hence, I N is I N = ∆ (cid:90) ∞ sinc ( fT ) (cid:12)(cid:12)(cid:12)(cid:12) NTF (cid:18) e i2 π ff Φ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) | Y (i2 πf ) | df. (4)From Eqn. (4), the opportunity of optimizing NTF ( z ) adaptingit to the motor transfer function Y ( s ) is quite evident. Thus, thefollowing questions arise: (i) is it worth selecting a custom NTFrather than using a conventional design? (ii) does it make sense to continuously modify the NTF to track the variations of Y (i2 πf ) due to slip changes?To ease the answer, one may first observe that thanks to thelow pass nature of Y (i2 πf ) , and as long as the modulator samplefrequency f s is sufficiently high, the integral in Eqn. (4) can safelybe approximated as I N = ∆ π (cid:90) π (cid:12)(cid:12)(cid:12) NTF (cid:16) e i ω (cid:17)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ˆ Y (cid:16) e i ω (cid:17)(cid:12)(cid:12)(cid:12) df. (5)where ˆ Y ( z ) is a discrete time version of Y ( s ) . Then, it is worthnoting, that due to the specific constraints posed by the modulatorarchitecture, the integral cannot be minimized in naive ways bynullifying the NTF or by making it capable of concentrating all the quantization noise at the single frequency where ˆ Y (cid:16) e i ω (cid:17) attenuates most. Conversely, as explained in [8], one needs toassure that (cid:12)(cid:12)(cid:12) NTF (cid:16) e i ω (cid:17)(cid:12)(cid:12)(cid:12) never exceeds certain values, is non-negligible in sufficiently large intervals and is realized by some − f [Hz]10 − − − − − − (cid:12)(cid:12)(cid:12) N T F (cid:16) e π i ff s (cid:17) (cid:12)(cid:12)(cid:12) optimized for σ = . σ = . σ = . Figure 4. Conventional (DELSIG) NTF and NTFs optimized at specificmotor loads and slips.
NTF ( z ) whose impulse response has a unitary first coefficient[8]. Intuitively, in presence of such constraints, the integral canbe reduced by taking (cid:12)(cid:12)(cid:12) NTF (cid:16) e i ω (cid:17)(cid:12)(cid:12)(cid:12) to be approximately the inverse of (cid:12)(cid:12)(cid:12) ˆ Y ( z ) (cid:12)(cid:12)(cid:12) , as discussed in [14]. This may require somemanual adjustment to obtain a stable NTF ( z ) . Then, furtheradjustment may be required on gain and range (namely thedifference between maximum and minimum gain values) to satisfythe modulator requirements. Alternatively, one can choose themethod in [10], [11] to directly obtain a suitable NTF ( z ) througha better formalized path.Following one of the two approaches above, one may obtaindifferent NTFs matched to specific slip values. These can bechecked against a conventional NTF using expression (5) toevaluate the convenience of custom NTFs over conventional ones.Furthermore, one can evaluate some custom NTFs designed fordifferent slip values one against the other to see the convenienceof tracking slip changes by a continuous NTF adjustment.IV. E XPERIMENTAL RESULTS
In our experiments, we refer to the parameter set in Tbl. I.Furthermore, we assume that the actuation frequency is comprisedin a [0-50 ]Hz interval. Finally, for the ΔΣ modulator we takean over-sample rate (OSR) equal to 1000, setting the sample frequency at 100 kHz.With this, the diagram in Fig. 4 shows a standard NTF obtainedfor a 4 th order modulator through the popular design assistantDELSIG [15], together with three NTFs optimized for the sameslip values used for Fig. 2. These are 8 th order and obtainedfollowing [10], [11]. The comparison of NTFs corresponding todifferent modulator orders should not appear unfair, since: (i)conventional methodologies cannot safely go beyond order 4 forthis setup; and (ii) the difference between the NTFs in the twodiagram is mostly due to the different design methodology ratherthan the different order. Indeed, the striking contrast between thediagrams owes to the fact that conventional methodologies areonly based on the OSR, while the other NTFs also explicitly takeinto account the need to reduce the value of expression (5).The effectiveness of the various NTFs is reported in Tbl. IIwhich shows the SNR obtained from Eqn. (5), considering themotor model at the various slip conditions. Data has been obtained able IISNR IN D B OBTAINED USING THE VARIOUS
NTF
S IN COMBINATION WITHTHE MOTOR AT DIFFERENT MECHANICAL LOAD AND SLIP CONDITIONS .D ATA OBTAINED FROM E QN . (5).Mechanical Loading Condition σ = 0 . σ = 0 . σ = 0 . Standard NTF 24.72
NTF optimized for σ = 0 . ! σ = 0 . ! σ = 0 . ! Table IIISNR
IN D B OBTAINED USING THE VARIOUS
NTF
S IN COMBINATION WITHTHE MOTOR AT DIFFERENT MECHANICAL LOAD AND SLIP CONDITIONS .D ATA OBTAINED FROM TIME DOMAIN SIMULATIONS WITH A NONLINEARMODULATOR MODEL AND A LINEARIZED MOTOR MODEL .Mechanical Loading Condition σ = 0 . σ = 0 . σ = 0 . Standard NTF 25.87 σ = 0 . NTF optimized for σ = 0 . σ = 0 . ! ! ! selecting a 50 Hz operation at 60 % of the motor nominal voltage(namely 190 V peak).From the tabled data, a well perceivable advantage, quantifiablein approximately 3-5 dB (i.e. an approximate halving of the noise),seems to be achievable by using an NTF designed in accordanceto the motor model (rows 2 to 4), with respect to a standardNTF designed considering the OSR only (row 1). Furthermore,considering rows 2 to 4, note that in each column the bestvalue can always be found on the diagonal. In other words,and not surprisingly, the best performance at any slip conditionappears to be obtainable using the NTF designed according to themotor admittance at that particular slip condition. Nonetheless, theadvantage that the best setup offers over the others looks minimal.This indicates that optimizing the NTF for a specific slip may notbe truly convenient. Indeed, such a conclusion could already beexpected from the curves in Fig. 4, where the optimized NTFdoes not change much by changing the slip level for which theoptimization is done. To be extremely scrupulous, one may want not to limit theanalysis to data obtained by Eqn. (5), due to the many approxi-mations practiced to achieve such model. These basically amountto the linearization of both the motor and the modulator models.Among the two, the second one is certainly the most critical.Thus, it is worth repeating the test by means of a time domainsimulation of the modulator, based on a realistic quantizer. Thisnew data is illustrated in Tbl III and is in good accordance tothe previous one. Variations are typically within 1-1.5 dB. Thisindicates that the nonlinear effects in the modulator should notbe ignored, though. Altogether, we have a confirmation that it isadvantageous to design the NTF taking into account the motormodel, even if the more realistic simulations cause the gap overa conventional NTF to shrink to approximately 2.5 dB. This isstill a 44 % reduction in noise. Furthermore, the new test revealsthat the data fluctuations due to the nonlinear effects break the already thin advantage related to optimizing the NTF for somespecific slip level. Indeed, in the current table, the optimal valuesfor each column do not lay anymore on the diagonal of rows 2to 4. This confirms that trying to track a specific slip level withthe NTF is a useless over-optimization.V. C
ONCLUSIONS
We have considered the use of ΔΣ modulators in switched modedrives for ac motors. Specifically, we have looked at how themodulator NTF should be designed to achieve the best possibleperformance, trying to see if an NTF design practiced takinginto account the motor model and the motor loading conditioncan be advantageous. From the proposed analysis, it appearsthat designing the NTF in accordance to the motor model canbe convenient, while trying to have an NTF actively trackingthe motor slip condition is probably over-engineering. From ouranalysis, designing the NTF at a single, large slip condition isenough to capitalize some non-negligible advantage (an over 40 %reduction of the quantization noise effect over the motor windingcurrents). Clearly, the proposed analysis is based on simulationand limited to the parameter set of a specific motor. However, weexpect our conclusions to be valid for many similar setups.R EFERENCES [1] D. G. Holmes and T. A. Lipo,
Pulse Width Modulation for PowerConverters . IEEE Press, 2003.[2] B. Jacob and M. R. Baiju, “Spread spectrum modulation scheme fortwo-level inverter using vector quantised space vector-based pulse densitymodulation,”
IET Electrical Power Application , vol. 5, no. 7, pp. 589–596,2011.[3] J. T. Boys and P. G. Handley, “Harmonic analysis of space vector mod-ulated PWM waveforms,”
Electric Power Applications, IEE ProceedingsB. , vol. 137, no. 4, pp. 197–204, 1990.[4] T. F. Lowery and D. W. Petro, “Application considerations for PWMinverter-fed low-voltage induction motors,”
IEEE Trans. Ind. Appl. ,vol. 30, no. 2, pp. 286–293, 1994.[5] A. M. Stankoviç and H. Lev-Ari, “Randomized modulation in powerelectronic converter,”
Proceedings of the IEEE , vol. 90, no. 5, pp. 782–799, May 2002.[6] M. Balestra, A. Bellini, C. Callegari, R. Rovatti, and G. Setti, “Chaos-based generation of PWM-like signals for low-EMI induction motordrives: Analysis and experimental results,”
IEICE Transactions onElectronics , vol. E87-C, no. 1, pp. 66–75, Jan. 2004.[7] S. Callegari, R. Rovatti, and G. Setti, “Chaotic modulations canoutperform random ones in EMI reduction tasks,”
Electronics Letters ,vol. 38, no. 12, pp. 543–544, Jun. 2002.[8] R. Schreier and G. C. Temes,
Understanding Delta-Sigma Data Convert-ers . Wiley-IEEE Press, 2004.[9] G. Luckjiff and I. Dobson, “Hexagonal Sigma–Delta modulation,”
IEEETrans. Circuits Syst. I , vol. 50, no. 8, Aug. 2003.[10] S. Callegari and F. Bizzarri, “Output filter aware optimization of the noiseshaping properties of ΔΣ modulators via semi-definite programming,” IEEE Trans. Circuits Syst. I submitted, 2012.[11] S. Callegari, “An alternative strategy for optimizing the noise transferfunction of ΔΣ modulators,” ARCES, University of Bologna, InternalReport, 2012.[12] A. E. Fitzgerald, C. Kingsley, Jr., and A. Kusko, Electrical Machinery .McGraw Hill, 1990.[13] F. Bizzarri, S. Callegari, and G. Gruosso, “Towards a nearly optimalsynthesis of power bridge commands in the driving of AC motors,” in
Proceedings of ISCAS 2012 , Seoul, May 2012.[14] S. Callegari, F. Bizzarri, R. Rovatti, and G. Setti, “On the approximatesolution of a class of large discrete quadratic programming problemsby ∆Σ modulation: the case of circulant quadratic forms,” IEEE Trans.Signal Process. , vol. 58, no. 12, pp. 6126–6139, Dec. 2010.[15] R. Schreier,