Sāhanā Vol. 42, No. 8, August 207, pp. 285 297 DOI 0.007/s2046-07-0664-2 Ó Inian Acaemy of Sciences Experimental etermination of mechanical parameters in sensorless vector-controlle inuction motor rive V S S PAVAN KUMAR HARI, AVANISH TRIPATHI* an G NARAYANAN Department of Electrical Engineering, Inian Institute of Science, Bangalore 56002, Inia e-mail: avanish@ee.iisc.ernet.in MS receive 5 March 206; revise 20 July 206; accepte 28 September 206 Abstract. High-performance inustrial rives wiely employ inuction motors with position sensorless vector control (SLVC). The state-of-the-art SLVC is first reviewe in this paper. An improve esign proceure for current an flux controllers is propose for SLVC rives when the inverter elay is significant. The spee controller esign in such a rive is highly sensitive to the mechanical parameters of the inuction motor. These mechanical parameters change with the loa couple. This paper proposes a metho to experimentally etermine the moment of inertia an mechanical time constant of the inuction motor rive along with the loa riven. The propose metho is base on acceleration an eceleration of the motor uner constant torque, which is achieve using a sensorless vector-controlle rive itself. Experimental results from a 5-hp inuction motor rive are presente. Keywors. Inuction motor rives; fiel-oriente control; moment of inertia; frictional coefficient; parameter evaluation; sensorless vector control.. Introuction The cage rotor inuction motors (IMS) are rugge, simple an cost-effective by nature, as compare with other machines available. The spark-less operation of this motor makes it suitable for explosive an hazarous environments [ 3]. However, the ynamic spee control of IM is not so straight forwar as that of a c motor ue to couple nature of flux an torque-generating currents in an IM. This limitation has been overcome by a technique calle vector control, where the torque an flux-generating components of current are ecouple an controlle separately, in a synchronously revolving reference frame [, 2]. Vector control results in a much improve ynamic performance of the IM []. A simplifie block iagram of a vector-controlle IM is shown in figure. Vector control involves ecouple control of flux an torque as mentione earlier. The ecoupling is achieve in a synchronously rotating q reference frame, whose reference axes are shown in figure 2. While ifferent reference frames exist, the rotor flux reference frame [4] is consiere here (see figure 2). The reference axes of the stationary reference frame an the three-phase stator wining axes are also inicate in the same figure. The etails of the transformations are explaine in section 2. The controller structure inclues an inner q-axis current control loop an an outer spee control *For corresponence loop. It also inclues an inner -axis current control an an outer flux control loop. Design of current controllers in rotor flux reference frame is well known for motor rives switching at high frequencies [ 6]. Here, the inverter time elay is neglecte as compare with the other time constants [ 3]. However, the inverter elay becomes significant when the inverter switches at low frequencies. This elay is then require to be consiere uring the current controller esign [7 9]. An improve esign proceure consiering the inverter elay is presente for the esign of current controller, in section 3 of this paper. Here, the inverter is moelle as first-orer elay; the current control loop is structure to have a secon-orer response. Design of spee controller requires precise knowlege of the mechanical parameters, namely, moment of inertia (J) an coefficient of friction (B), for achieving goo spee response. Also, such precise knowlege of the parameters is require for certain applications such as computer numerical control (CNC) machine tools, where auto-tuning of controller is require [0]. These parameters also change consierably with the loa couple to the motor []. Several methos have been reporte in literature to measure an/or estimate the mechanical parameters for servo-motor rives an permanent magnet synchronous machine (PMSM)-base rives [, 0 5]. Retaration test has been suggeste for measurement of moment of inertia in []. However, the retaration test suffers from non-uniform loa 285
286 V S S Pavan Kumar Hari et al DC Voltage Source Spee reference Voltage Source Inverter Gate rive signals Spee sensorless vector control an pulse with moulation Voltages & currents B Y R Squirrel Cage Inuction Motor online estimation requires involve computations, which may not be feasible on low-cost controller-base systems. In this paper, current control loops an flux control loop are esigne by aopting the improve proceure, which consiers the inverter elay. A spee loop is esigne consiering approximate values of J an B. The sensor-less vector control (SLVC) is implemente for a 3.7-kW IM-fe from a 0-kVA inverter controlle by a fiel programmable gate array (FPGA)-base igital platform. Initially, the rive is operate at constant spees to estimate the value of frictional coefficient (B), as explaine in section 5. Further, it is operate uner constant accelerating an ecelerating torque to estimate the combine moment of inertia (J), as escribe in section 6. Figure. Sensorless vector-controlle inuction motor rive. 2. Machine moel in rotor flux reference frame ω mr q Y b ρ Rotor flux axis R ω mr Stator R-phase axis a The axes of reference of IM moels for SLVC are illustrate in figure 2. The three-phase stator wining axes RYB are shown along with the a an b axes, which are mutually perpenicular. The a an b axes are the axes of reference in the stationary reference frame. Here the a-axis is aligne along the R-phase axis of stator wining, in stationary reference frame. Vector control of IM is carrie out in the rotor flux reference frame [] efine by an q axes shown in figure 2. Here -axis is aligne along the rotor flux space vector w r, which is efine in terms of quantities in stationary coorinates as shown: w r ¼ w ra þ jw rb ¼ L o ¼ L o i s þðþr r Þi r e je ¼ Lo i s þ i r e je ðþ Figure 2. B Axes of reference for machine moelling an control. torque ue to spee-epenent winage friction present in the rive. Reference [2] presents a spee-observer-base online metho to generate position error signal for estimation of moment of inertia. An offline metho base on time average of the prouct of torque reference an motor position for mechatronic servo systems is propose in [3]. Another online recursive least squares (RLS) estimator for a servo motor rive is presente in [4] for estimation of mechanical parameters. Reference [5] presents a PI-controller-base close-loop metho to estimate inertia an friction of servo rive. A loa-torque-observer-base metho to precisely estimate J an B for servo systems is iscusse in []. The mechanical subsystem is moelle as a secon-orer system in the aforementione methos, which is complicate to solve. Further, observer-base where i s an i r e je are stator an rotor current space vectors, respectively; is the magnetizing current corresponing to rotor flux; w ra an w rb are the components of w r along a an b axes, respectively; is the rotor inuctance an L o is the magnetizing inuctance. The ynamic moel of an IM in the rotor flux reference frame is given by [] t i s ¼ v s i s þ rl s x mr ð rþl s t rl s ð2aþ t i sq ¼ v sq rl s x mr i s ð rþl s x mr rl s ð2bþ t q ¼ x mr t ¼ R r ði s Þ ð2cþ ¼ x þ R r ¼ x þ x r ð2þ
t x ¼ ðm m L Þ P Bx J 2 ð2eþ m ¼ 2 P L o 3 2 ð þ r r Þ ¼ K m ð2fþ where v s an v sq are components of v s along an q axes, respectively; i s an are components of i s along an q axes, respectively; is j j, i.e., the magnitue of rotor flux magnetizing current; x mr is the spee of w r in electrical ra/s; x is rotor spee in electrical ra/s; x r is slip spee in electrical ra/s; q is angle between a-axis an -axis; K m is torque constant; r is total leakage coefficient; an L s are the per phase stator resistance an inuctance, respectively, an R r is the per phase rotor resistance. The ynamic equations in (2) are shown as a block iagram insie the ashe rectangle in figure 3. Experimental etermination of mechanical parameters 287 3. Controller structure Figure 3 shows the four control loops in vector control. The two inner loops are -axis current (i s ) an q-axis current ( ) control loops. The reference inputs to the inner current loops, namely, i sq an i s, are generate by the outer spee (x) an flux ( ) control loops, respectively. Spee reference x is provie externally. The reference i mr is kept constant at such a value of that the machine operates at the rate flux, since no fiel weakening operation is consiere here. Appropriate feeforwar terms e s an e sq are ae to the outputs of i s an controllers to result in the -axis an q- axis voltage references v s an v sq, respectively. Calculation of feeforwar terms will be iscusse in section 3.3. The two-phase voltage references v s an v sq in the synchronous reference frame are transforme into twophase references v sa an v sb in the stationary reference frame as shown by (3): 3. SVC This section escribes the control structure of a vectorcontrolle rive an estimation methos for feeback an fee-forwar quantities in the rive. v sa ¼ v s cos q v sq sin q; ð3aþ v sb ¼ v s sin q þ v sq cos q: ð3bþ They can be further transforme into three-phase references v RN, v YN an v BN as shown by (4): ω ω i mr PI Controller PI Controller i sq i s i s PI Controller PI Controller v sq v s e sq e s vsq vsa e jρ vs vsb cos ρ sin ρ 2-Phase 3-Phase vrn vyn vbn V p 0 V p 2V p V DC m R m Y m B Pulse With Moulation (PWM) S R S Y S B V DC 2V p V DC i R i Y i B v RN v YN v BN v RN v YN v BN 3-Phase 2-Phase v sa e jρ v sb cos ρ sin ρ v sq v s e sq e s σl s σl s i s R r R r Nr Dr ω r ω mr K m = 2 P L o 3 2 ( σ r ) e s =(σ) L s i t mr σl s ω mr e sq =( σ) L s ω mr σl s ω mr i s Π K m m m L P 2 J B ω Machine moel in rotor flux coorinates Figure 3. Vector-controlle inuction motor rive.
288 V S S Pavan Kumar Hari et al v RN ¼ 2 3 v sa ; v YN ¼ 3 v sa þ p ffiffi v sb 3 ; v BN ¼ 3 v sa p ffiffi v sb 3 : ð4aþ ð4bþ ð4cþ Measure currents i R i Y i B.5 3 2 i sa i sb cos ρ e jρ sin ρ i s Three-phase sinusoial moulating signals m R, m Y an m B can be obtaine by scaling v RN, v YN an v BN, respectively, with V DC 2V p, where V DC is the DC bus voltage an V p is the peak of the bipolar triangular carrier. Gating signals for the evices in VSI can be generate base on the metho of pulse with moulation (PWM) selecte. The three-phase feeback quantities (i R, i Y an i B ) an (v RN, v YN an v BN ) nee to be transforme into the q reference frame. These transformations an also the inverse transformations require the unit vectors cos q an sin q. The unit-vector generation an estimation of other feeback quantities are iscusse in section 3.2. 3.2 Feeback estimation Estimation of the four feeback quantities in the control loops shown in figure 3, namely,, i s, an x, is iscusse in this section. The stationary three-phase feeback currents (i R, i Y an i B ) are transforme into stationary two-phase feeback currents (i sa an i sb ) as shown in figure 4a. These feeback currents are then transforme into q reference frame as i s an, which are fe back to control loops as shown in figure 4a. The corresponing equations are given in (5): i s ¼i sa cos q þ i sb sin q; ¼i sb cos q i sa sin q: ð5aþ ð5bþ Three-phase stator voltages (v RN, v YN an v BN ) are transforme into two-phase voltages (v sa an v sb ) in the stationary reference frame in the same manner as the three-phase currents are transforme into two-phase currents, presente in figure 4a. The stator fluxes (w sa an w sb ) in the stationary ab reference frame are then estimate from the stator voltages (v sa an v sb ) an stator currents (i sa an i sb ) in the stationary a b reference frame as inicate by (6) []: w sa ¼ Z t t 0 Z t ðv sa i sa Þt ¼ Z t t 0 e sa t ð6aþ w sb ¼ ðv sb i sb Þt ¼ e sb t ð6bþ t 0 t 0 where t 0 is the time at which the integration starts. The rotor fluxes (w ra an w rb ) in the a-b reference frame are, in turn, obtaine from the estimate stator fluxes (w sa an w sb ) as shown in figure 4b [4]. Z t Estimate flux (c) () ψ sa i sa σl s ψ sb i sb i s cos ρ sin ρ σl s L o L o R r R r t t ψ ra x2 y 2 ψ rb Π Π The unit vectors (cos q an sin q), require for a b to q an inverse transformations of currents an voltages, are obtaine from the estimate rotor fluxes in the a b reference frame as illustrate in figure 4b [4]. The feeback signal for the flux control loop is calculate from the values of i s an rotor time constant T r ¼ ð =R r Þ using Eq. (2c) as shown in figure 4c. The rotor spee x is the ifference between the spee of rotor flux x mr an the slip spee x r [see figure 4c an ]. The slip spee x r is estimate from the values of, an T r using Eq. (2). The spee of rotor flux x mr is estimate as inicate in figure 4 [4]. 3.3 Feeforwar estimation The mathematical moel of an IM in the rotor-flux reference frame has coupling terms as seen from (2a) an (2b). To ecouple the stator current equations, the coupling terms y x Nr Nr ψ r Dr cos ρ sin ρ ω mr Dr Dr Nr Figure 4. Determination of feeback quantities: transformation of three-phase feeback current to q reference frame feeback current, estimation of unit-vectors ðcos q an sin qþ oriente along rotor flux, (c) estimation of rotor-flux magnetizing current an () estimation of rotor spee, x. ω r ω Feeback quantities
Experimental etermination of mechanical parameters 289 i s K is ( st is ) e st v s vs v s st is st i s e s e s s ( ) σ Ls i s i sq K isq ( st isq ) e st v sq vsq v sq st isq st e sq e sq s ( ) σ Ls i mr K imr ( st imr ) st imr sτ bis st r i s i s (c) Figure 5. Sensorless vector control: -axis current control loop, q-axis current control loop an (c) flux ( ) control loop. are fe forwar to the current controller outputs. The feeforwar terms along -axis an q-axis are enote by e s an e sq [see figure 5a an b], respectively. They can be calculate from the feeback signals as e s ¼ð rþ L s T r ði s Þ rl s x mr ; ð7aþ e sq ¼ð rþl s x mr þ rl s x mr i s : ð7bþ 4. Improve esign of current an flux controllers Designs of current an flux controllers are well establishe for the cases of high-switching-frequency rives. However, in case of high-power an/or high-spee rives, the ratio of switching frequency to funamental frequency (i.e., pulse number) is low. Hence, for such cases, the inverter elay becomes significant as compare with the other time constants in the control loop. Contrary to high-switching-frequency cases, the inverter elay cannot be ignore for low-pulsenumber cases. The inverter elay is moelle as a first-orer elay for the purpose of controller esign. Further, the spee controller is esigne by the symmetric optimum metho [4] an the simulation an experimental results are presente. 4. Improve esign of current controllers The block iagrams of i s an control loops are shown in figure 5a an b, respectively. Base on the reference an feeback signals in a given sub-cycle or half-carrier cycle, the outputs of current controllers give the voltage to be applie on the machine in the next sub-cycle. Thus, there is a elay of one sub-cycle time T s ue to the controllers. Further, the voltage commane by the controllers will be applie on the machine after a elay between 0 an T s ue to the process of PWM. Hence, the average elay introuce by PWM is 0:5T s. Thus, there is an average total elay (T )of:5t s in the system. For switching frequency f sw ¼ khz, one sees that T s ¼ 500 ls an T ¼ 750 ls. Actual transfer function of the elay is given by G ðsþ ¼e st. For the esign of controllers, the transfer function of elay is approximate as G a ðsþ ¼=ð þ st Þ. The actual an approximate transfer functions of the elay are compare in figure 6 for T ¼ 750 ls. The magnitue of G a ðsþ is 3 B less than that of G ðsþ at a frequency of 22 Hz [see figure 6a]. Phase plots of both the transfer functions are quite close to each other for frequencies less than 22 Hz, as shown by figure 6b. Therefore, the approximation is vali if the total banwith of current control loop is less than 22 Hz.
290 V S S Pavan Kumar Hari et al Magnitue (B) 0 5 0 G (s) G a (s) 5 0 0 0 0 2 0 3 Frequency (Hz) 0 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x bis ¼ x nis 2f 2 is þ þ 2f 2 2 is : ðþ By choosing a suitable value of f is, the gain K is of i s controller can be calculate as K is ¼ T is 4f 2 : ð2þ T is For the present work, f is is selecte as 0.6, which gives the maximum possible banwith of 200 Hz. The controller parameters for controller are K isq ¼ K is an T isq ¼ T is. Phase (egree) The time constant T is of i s controller is chosen to cancel the largest time constant in the current control loop. Thus T is ¼ rl s : ð8þ With the above choice of T is, the close-loop transfer function of the -axis current loop is given by i s ðsþ i s ðsþ ¼ 00 200 G (s) G a (s) 0 0 0 0 2 0 3 Frequency (Hz) Figure 6. Approximation of the inverter an PWM elay in current control loop by a first-orer transfer function. Comparison of magnitue plots an phase plots of the actual an approximate transfer functions. s 2 þ s T þ K is K is T is T T is T ¼ G is ðsþ: ð9þ It is to be note that Eq. (9) is a secon-orer transfer function as oppose to the first-orer one when the T is negligible. The natural frequency x nis an the amping coefficient f is of the secon-orer transfer function are as follows: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K is x nis ¼ ; ð0aþ T is T 2f is x nis ¼ T : ð0bþ The banwith x bis of the secon-orer system is given by 4.2 Improve esign of flux controller The block iagram of control loop is shown in figure 5c. The secon-orer transfer function of i s control loop G is ðsþ in Eq. (9) is approximate as a first-orer transfer function given by G isa ðsþ ¼ ; ð3aþ þ ss bis s bis ¼ 2f is x nis ¼ T is K is : ð3bþ The magnitue an phase plots of the actual an approximate transfer functions of current control loop are shown in figure 7. It can be observe that the approximation is vali if the banwith of control loop is less than 00 Hz. As before, the time constant of controller (T imr ) is chosen to cancel the lag ue to rotor time constant T r ; i.e., T imr ¼ T r. Thus, the close-loop transfer function of control loop is given by ðsþ i mr ðsþ ¼ K imr : s 2 þ s s bis þ K imr T imr s bis T imr s bis ð4þ Equation (4) is a secon-orer transfer function with natural frequency x nimr an amping coefficient f imr. The secon-orer system of control loop can be esigne following the approach iscusse in the previous section for esign of current controllers. For the present work, f imr ¼ :0, which gives a banwith of 40 Hz. The value of amping coefficient is chosen to avoi overshoots in flux. 4.3 Spee (x) controller esign A block iagram of the spee control loop is shown in figure 8a. The loa torque m L is a isturbance input to the system, an is not consiere in the esign. It is assume that is maintaine at its reference value i mr. Since the spee feeback is taken from the output of a ifferentiator (figure 4), a filter is ae in the feeback path of spee
Experimental etermination of mechanical parameters 29 Magnitue (B) Phase (egree) Figure 7. 0 0 20 30 00 50 G is (s) G isa (s) 0 0 0 0 2 0 3 Frequency (Hz) 0 50 G is (s) G isa (s) 0 0 0 0 2 0 3 Frequency (Hz) First-orer approximation of current control loop. loop. The transfer function of q-axis current loop is approximate as a first-orer transfer function given by Eq. (3). Neglecting the frictional coefficient B, the open-loop transfer function of spee loop is given by G x ðsþ ¼ K xð þ st x Þ st x ð þ ss bis Þ K mi P mr : ð5þ sj 2 þ st f The transfer function G x ðsþ has a ouble pole at origin. The magnitue plot of G x ðsþ has a slope of 40 B/ecae initially an the phase of G x ðsþ is close to 80 initially as shown in figure 8b an c, respectively. For the gain crossover to occur at a slope of 20 B/ecae, a zero is introuce before the ominant pole in the transfer function. The ratio of the ominant pole frequency to the gain crossover frequency ecies the phase margin obtaine. This metho is popularly known as the symmetric optimum metho []. If f om is the frequency of ominant pole an f bx is the gain crossover frequency, then the phase margin / m is given by / m ¼ tan f om f bx : ð6þ 2 f bx f om Further, f bx is the geometric mean of f om an the frequency of zero f z introuce by spee controller f bx ¼ pffiffiffiffiffiffiffiffiffiffi f om f z an T x ¼ 2pf z ð7þ where f bx, f om an f z are in Hz. Thus, by specifying a phase margin, the gain crossover frequency f bx an controller time constant T x can be etermine from Eqs. (6) an ω K ω ( st ω ) i sq m Π K m st ω sτ bis sj B ω m L P 2 ω st f Magnitue (B) G ω (s) 50 0 50 00 50 f z f bω f om f bis 0 2 0 0 0 0 0 2 0 3 Frequency (Hz) Phase (egree) 00 50 80 200 250 G ω (s) f z f bω f om f bis 300 0 2 0 0 0 0 0 2 0 3 Frequency (Hz) (c) φ m Figure 8. Sensorless vector control : spee control loop; magnitue an (c) phase plots of open-loop transfer function of spee control loop.
292 V S S Pavan Kumar Hari et al (7), respectively, if the frequency of ominant pole is known. Equating the magnitue of G x ðsþ to 0 B at f bx, the gain of spee controller K x can be calculate as K x ¼ J ð 2pf bxþ2 i mr K m P : ð8þ In the present work, the ominant pole is at the corner frequency of spee filter; f om ¼ 0 Hz. The spee controller is esigne for a phase margin of 73, which gives a banwith of.5 Hz as seen from figure 8b an c. 4.4 Simulation an experimental results Simulation of a vector-controlle rive is carrie out using MATLAB SIMULINK with the machine parameters in table an the controller constants esigne. The SLVC algorithm is implemente on an ALTERA-CycloneIIbase FPGA controller. The FPGA controller generates the gating signals for the IGBTs of a 0-kVA two-level voltage source inverter (VSI), which is connecte to the 5- hp IM. Parameters of the IM are given in table. The DC bus voltage V DC of the VSI is maintaine at 570 V. The IM is couple to a 230-V, 3-kW, 475-rpm DC generator. The fiel wining of the DC generator is excite from a separate DC source. A resistor bank containing eight parallele resistors, each rate for 75 X=4A, is use to loa the DC generator, which, in turn, loas the IM. Motor currents are sense using LA-00P Hall-effectbase current sensors from LEM. The sense currents are use for SLVC. Figure 9 shows the simulate an experimentally obtaine responses of all the currents, flux an spee PI controllers. The controllers are seen to perform satisfactorily in terms of tracking the respective reference. The simulation an experimental results are foun to be close to each other. Table. Parameters of motor, inverter an controllers. 5-hp, 400-V, 50-Hz, 3-phase inuction motor Number of poles P 4 Stator resistance per phase.62 X Rotor resistance per phase R r.62 X Mutual inuctance per phase L o 227 mh Stator leakage coefficient r s 0.042 Rotor leakage coefficient r r 0.042 Combine moment of inertia of motor an DC 0.2 kg m 2 generator, J (assume) Combine frictional coefficient of motor an DC generator, B (assume) 0.0 kg m 2 /s 2 Switching frequency of inverter khz Banwith of -axis an q-axis current controllers 00 Hz Banwith of controller 40 Hz Banwith of spee controller.5 Hz 5. Measurement of frictional coefficient (B) This section eals with etermination of frictional coefficient of the motor loa combine system. 5. Measurement of no-loa torque versus spee The sensor-less vector-controlle rive is run on no-loa at ifferent spees to fin the frictional coefficient of the combine system. The spee an flux references are set appropriately an the measurements are mae at steay state. The values of an are measure at each spee. Since flux is maintaine constant, only changes its value at ifferent spees; is maintaine at a value of 5:92 A, an is measure at ifferent spees. The noloa torque (m ;NL ) can be calculate using (2f). The experimental values of an m ;NL at ifferent spees are tabulate in table 2. The measure no-loa torque is shown plotte against spee in figure 0. It is seen that the variation of the no-loa torque is quite linear with spee. This coul also be non-linear at times. The evaluations of frictional coefficient in cases of linear frictional torque an non-linear frictional torque are iscusse in sections 5.2 an 5.3, respectively. 5.2 Linear frictional torque It is seen from Eq. (2e) that, at no-loa an uner steay state operating conition, the electromagnetic torque (m ) generate is equal to the frictional torque. In many cases, the variation of frictional torque with spee is quite linear as follows: m ;NL ¼ Bx m : ð9þ At known values of rotor spee, frictional coefficient B can be calculate straightaway as the ratio of torque generate to rotor spee. The values of B etermine at ifferent spees are also tabulate in table 2. As seen, these values are reasonably close to one another. The average value of B from ifferent measurements is consiere as the measure frictional coefficient here. For this average value of B, the no-loa torque versus spee characteristic is as shown in soli line in figure 0. 5.3 Non-linear frictional torque However, the mechanical subsystem coul be a non-linear first-orer system also. For cases where a fan is mounte on the shaft or in case of pump loas, the loa torque is a nonlinear function of spee. In such cases, one coul assume that the no-loa torque m ;NL varies with spee in a quaratic fashion as inicate by (20):
Experimental etermination of mechanical parameters 293 (c) (c) () () Figure 9. Dynamic response of controller, i s controller, (c) controller an () spee controller. (i) Simulation result (MATLAB) an (ii) experimental result. m ;NL ¼ B 0 þ B x m þ B 2 x 2 m : ð20þ 6. Measurement of moment of inertia (J) The coefficients B 0, B an B 2 can be etermine by a quaratic curve fit on the measure no-loa torque versus spee plot. The spee responses of the mechanical system to constant torque for the cases of linear an non-linear frictional coefficients are iscusse in the following section. The measurement of mechanical time constant, an thereby, moment of inertia of the combine motor an loa system is explaine in this section.
294 V S S Pavan Kumar Hari et al Table 2. Estimate values of frictional coefficient at ¼ 5:94A. x m B x m B (elec. ra/s) (A) (N m) ðkg m 2 /s 2 Þ (elec. ra/s) (A) (N m) ðkg m 2 /s 2 Þ 78.54 0.2026 0.3494 0.0089 235.62 0.4652 0.8022 0.0068 25.66 0.32 0.5535 0.0088 282.74 0.5347 0.922 0.0065 57.08 0.3654 0.630 0.0080 34.6 0.5928.022 0.0065 88.50 0.4073 0.7023 0.0075 6.b Non-linear frictional torque: In motor rives where the frictional torque is non-linear, the ynamic equation governing the spee response is given in (23): Figure 0. torque. Comparison of measure an average frictional loa 6. Theoretical spee response The theoretical spee response of a motor loa system, when the frictional coefficient is a linear or non-linear function of spee, is explaine here. 6.a Linear frictional torque: For linear frictional torque, the ifferential equation governing the spee response of the motor rive uner no-loa operating conition (i.e., m L ¼ 0) is given as t x m ¼ ½ J m Bx m Š: ð2þ The response of a linear first-orer system is exponential uner the influence of a constant input. Theoretically, the spee response of the system uner such conitions woul be the solution of Eq. (2), consiering an initial spee of x 0 an a constant torque m. The spee response can be expresse as shown in (22): h x m ðtþ ¼x 0 e B J t þ m i B e B J t : ð22þ The value of B is alreay known from the previous section. The estimation of the value of J is iscusse in sections 6.2 an 6.3. t x m ¼ J m B 0 B x m B 2 x 2 m : ð23þ The theoretical response of such systems is the solution of (23) as given in (24a), where K an K 2 are given by (24b) an (24c), respectively: x K K 0 K 2 x 0 K 2 x m ðtþ ¼ x 0 K x K 2 e tðk K 2 ÞB 2 J e tðk K 2 ÞB 2 J ; ð24aþ K ¼ B p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B 2 þ 4ðm B 0 ÞB 2 ; ð24bþ 2B 2 K 2 ¼ B p þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B 2 þ 4ðm B 0 ÞB 2 : ð24cþ 2B 2 Here again, x 0 is the initial spee an m is the constant torque applie. The values of B 0 -B 2 are known from the previous section. The etermination of J is explaine in sections 6.2 an 6.3. 6.2 Measure spee response The motor rive is operate with SLVC with appropriate references for flux an spee. Spee an currents are measure uring the acceleration an eceleration uner constant torque. Since torque is epenent upon an, both the currents shoul be maintaine constant in orer to keep the electromagnetic torque at a constant level; is maintaine constant by keeping the flux at constant level by the flux controller. However, to make constant, the output of the spee controller (i.e., reference, i sq ) shoul be force to the saturation level. For a large step change in spee reference, the spee controller hits the saturation level of for a short perio of time. In orer to ensure that i sq is maintaine at the saturation level for longer time perio, the limits on the spee controller output are reuce to a lower value than the nominal value. The rive is operate at no-loa so that the electromagnetic torque
Experimental etermination of mechanical parameters 295 Figure. Experimental result corresponing to acceleration from 25 to 50 Hz at a constant torque equal to 40% of the rate torque: spee reference x, spee feeback x, q-axis stator current an rotor flux magnetizing current an spee reference x, spee feeback x an measure R-phase current i R. X-scale = 00 ms for all channels. Channels an 2 (yscale = 25.7 elec. ra/s/iv); channel 3 (yscale = 4 A/iv) an channel 4 (yscale = 8 A/iv). generate is equal to the sum of accelerating torque an frictional torque. Figure a shows the reference spee, rotor spee an the q-axis current (inicate in figure) for the case of acceleration uner constant torque conition. The spee reference is change from 25 to 50 Hz while keeping the spee controller output saturation level at 40% of the rate value an at the rate value. Hence, the applie torque is kept at 40% of the rate value. The is seen to remain at a constant level for a perio of more than 0.6 s. The uration of constant torque is inicate in the figure. Figure b presents the measure R-phase current i R for the perio of constant torque operation. The rotor spee ata uring the constant torque perio are capture for estimation of J. The experiment is repeate for eceleration case also. Figure 2a presents the reference spee, rotor spee an (inicate in figure) for the case of eceleration uner constant torque conition. The spee reference is change from 40 to 5 Hz with the same limit on the torque. The is seen to remain at a constant level for a perio of more than 0.6 s. Further, the measure current (i R ) is inicate in figure 2b along with the spee reference an rotor spee. The peak value of i R can be seen to remain constant over that uration. Consiering the time winow of 0.6 s inicate in figure a an 2a, the acceleration or eceleration occurs at a constant electromagnetic torque evelope (i.e., constant an ). These responses of spee are reprouce in figure 3a an b, corresponingly. 6.3 Estimation of J The moment of inertia J can be estimate by curve fitting the response of the mechanical subsystem uner constant torque conitions for the previously measure value of B. All parameters in the mathematical response expression are known except for the effective moment of inertia, J. If an appropriate value of J (i.e., J e ) is chosen, then the eviation Figure 2. Experimental result corresponing to eceleration from 40 to 5 Hz at a constant torque equal to 40% of the rate torque: spee reference x, spee feeback x, q-axis stator current an rotor flux magnetizing current an spee reference x, spee feeback x an measure R-phase current i R. X-scale = 00 ms for all channels. Channels an 2 (yscale = 25.7 elec. ra/s/iv); channel 3 (yscale = 4 A/iv) an channel 4 (yscale = 8 A/iv).
296 V S S Pavan Kumar Hari et al Spee of rotor ω (elec. ra/s) 300 275 250 225 200 Measure Curve fit 75 Mean square error is 0.6 elec. ra/s 50 0 0. 0.2 0.3 0.4 0.5 0.6 Time (s) Spee of rotor ω (elec. ra/s) 250 225 200 75 50 Measure Curve fit 25 Mean square error is 0.6 elec. ra/s 00 0 0. 0.2 0.3 0.4 0.5 0.6 Time (s) Figure 3. Experimentally obtaine spee an the best-fit first-orer response of the mechanical subsystem [Eq. (22)] : acceleration an eceleration at a constant torque equal to 40% of the rate torque. between the theoretical spee response an the measure response woul be very low. To state more quantitatively, the value of J e shoul be so chosen to minimize the root mean square (RMS) error between the theoretical spee response given by (22) an the measure spee response. Figure 3a an b shows Eq. (22) plotte with the best-fit value of J e, which minimizes the mean square error between the experimental response an the best fit curve, corresponing to figure an 2, respectively. As seen from the figures, the experimental response an the best-fit curves are almost inistinguishable. The mean square error between the experimental response an the best fit curve is foun to be lower than 0.6 elec. ra/s for both the cases. Such a best-fit value of J e is taken as the moment of inertia J of the system. The proceure is repeate with ifferent torque limits an the corresponing results are tabulate in table 3. The step change in spee reference for acceleration is kept from 25 to 50 Hz for all the cases. Similarly, the step change in spee for eceleration case is kept from 40 to 5 Hz for all the cases of ifferent torque limits. The values of J obtaine in the ifferent trials (i.e., with ifferent torque limits) are reasonably close to one another. The average of these values is taken as the moment of inertia of the mechanical sub-system. The mechanical time constant is usually measure using retaration test []. The IM is run on no-loa at rate voltage an frequency with the fiel wining of the DC generator fully excite. The motor supply is suenly switche off at t ¼ t 0, an then the motor generator set is allowe to ecelerate. Uner this conition, the mechanical time constant (s m ) is obtaine as s m ¼ J B ¼ xj t¼t 0 j x t j ¼ e bj t¼t0 t¼t 0þ j e b t j : ð25þ t¼t 0þ The measure armature voltage of DC generator (e b )is plotte against time in figure 4. The mechanical time Table 3. Estimate values of moment of inertia. Average value of B: 0:007 6 kg m 2 /s 2 an acceleration is from 25 to 50Hz an eceleration is from 40 to 5Hz: Operating conition Moment of inertia J (kg-m 2 ) 20% of rate torque Acceleration 0.0803 Deceleration 0.0874 30% of rate torque Acceleration 0.0858 Deceleration 0.0836 40% of rate torque Acceleration 0.0870 Deceleration 0.0823 Figure 4. Experimental result open circuit armature voltage uring no-loa eceleration of motor-generator set.
Experimental etermination of mechanical parameters 297 constant is obtaine using (25). The initial spee is foun to be 56.76 ra/s an the initial slope (first 50 ms of retaration) is foun to be 22.76 ra/s 2. Base on the measurement, the mechanical time constant obtaine from the retaration test is foun to be 7.06 s, which is 36% lower than that obtaine through the propose metho. In the conventional retaration test, the motor is ecelerate by the frictional an winage torques. This eceleration torque is assume to be proportional to spee, which might not be vali for many practical cases, as inicate in section. In the simplest case, the constant of proportionality, namely B, coul vary with spee. More realistically this ecelerating torque coul be a non-linear function of spee. This function itself might be unknown. The propose measurement proceure involves acceleration or eceleration uner a constant an precisely known value of torque. Hence, this proceure is expecte to give a better estimate of the mechanical time constant an moment of inertia. 7. Conclusions The state-of-the-art SLVC for IM rives along with controller structure is etaile in this paper. The low switching frequency of the inverter introuces significant inverter elay in the system. The inverter is moelle as a first-orer elay, an the complete control loop for current an flux are moelle as secon-orer systems. Improve esign proceures are presente for current an flux controllers for such cases. The esign of controllers is valiate on a 5-hp IM rive through simulations an experiments. Further, a metho for the etermining frictional coefficient (B) an moment of inertia (J) of an IM rive base on SLVC is propose in this paper. The propose metho is capable of fining the combine inertia an friction coefficient of the motor an loa. This metho is base on acceleration an eceleration of an IM rive uner constant torque conitions. The propose metho is utilize to etermine the values of B an J of a 5-hp IM, couple to a DC generator. These values of B an J can be use to refine the spee controller esign in the sensorless vector-controlle rive to achieve goo spee response. References [] Leonhar W 200 Control of electrical rives. Springer [2] Vas P 998 Sensorless vector an irect torque control. Oxfor University Press [3] Holtz J 2002 Sensorless control of inuction motor rives. Proc. IEEE 90(8): 359 394 [4] Poar G an Ranganathan V T 2004 Sensorless fiel-oriente control for ouble-inverter-fe woun-rotor inuction motor rive. IEEE Trans. In. Electron. 5(5): 089 096 [5] Hurst K D, Habetler T G, Griva G an Profumo F 998 Zerospee tacholess IM torque control: simply a matter of stator voltage integration. IEEE Trans. In. Appl. 34(4): 790 795 [6] Xu X, Doncker R D an Novotny D W 988 A stator flux oriente inuction machine rive. In: Proceeing of the IEEE. PESC, Power Electronics Specialists Conference, PP. 870 876 [7] Oikonomou N an Holtz J 2008 Close-loop control of meiumvoltage rives operate with synchronous optimal pulsewith moulation. IEEE Trans. In. Appl. 44(): 5 23 [8] Holtz J an Bube E 99 Fiel-oriente asynchronous pulsewith moulation for high-performance AC machine rives operating at low switching frequency. IEEE Trans. In. Appl. 27(3): 574 58 [9] Pavan Kumar Hari V S S 204 Space-vector-base pulse with moulation strategies to reuce pulsating torque in inuction motor rives. PhD Thesis. Bangalore, Inia: Inian Institute of Science [0] Feng Y, Yu X an Han F 203 High-orer terminal sliingmoe observer for parameter estimation of a permanentmagnet synchronous motor. IEEE Trans. In. Electron. 60(0): 4272 4280 [] Niu L, Xu D, Yang M, Gui X an Liu Z 205 On-line inertia ientification algorithm for pi parameters optimization in spee loop. IEEE Trans. Power Electron. 30(2): 849 859 [2] Choi J W, Lee S C an Kim H G 2006 Inertia ientification algorithm for high-performance spee control of electric motors. IEE Proc. Electr. Power Appl. 53(3): 379 386 [3] Anoh F 2007 Moment of inertia ientification using the time average of the prouct of torque reference input an motor position. IEEE Trans. Power Electron. 22(6): 2534 2542 [4] Truong N V 202 Mechanical parameter estimation of motion control systems. In: Proceeing of ICIAS, the 4th International Conference on Intelligent an Avance Systems, vol., pp. 00 04 [5] Garrio R an Concha A 204 Inertia an friction estimation of a velocity-controlle servo using position measurements. IEEE Trans. In. Electron. 6(9): 4759 4770