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Electric Drives1 ELECTRIC DRIVES Ion Boldea S.A.Nasar 1998

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Electric Drives2 9. PWM INVERTER-FED INDUCTION MOTOR DRIVES 9.1. INTRODUCTION There are three main schemes to produce this linearization by intelligent manipulation of IM equations in space phasors: Vector current and voltage control - VC [1]; Direct torque and flux control - DTFC [2 - 4]; Feedback liniarization control - FLC [5 - 7] So we distinguish: Control with motion sensors; Control without motion sensors (sensorless) V 1 / f 1 with slip compensation with motion sensors or sensorless; Vector control (VC) or direct torque and flux (DTFC) or feedback linearization control (FLC) with motion sensors or sensorless.

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Electric Drives3 9.2. VECTOR CONTROL - GENERAL FLUX ORIENTATION Vector control implies independent (decoupled) control of flux - current and torque - current components of stator current through a coordinated change in the supply voltage amplitude, phase and frequency. There are three distinct flux space - phasors in the induction machine (chapter 8 equations (8.92 - 8.93)): m b - airgap flux; s b - stator flux and r b - rotor flux. Their relationships with currents are: (9.1) Vector control could be performed with respect to any of these flux space phasors by attaching the reference system d axis to the respective flux linkage space phasor direction and by keeping its amplitude under surveillance.

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Electric Drives4 In order to facilitate comparisons between the three strategies we introduce new, general, rotor variables : (9.2) and define a general flux: (9.3) From the complex variable equations (8.50 - 8.53) (with d/dt = s): (9.4)

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Electric Drives5 (9.5) (9.6) (9.7) with the slip S a defined for the general flux : (9.8) The complex variables are the stator current and the general flux. The reference system is attached to axis d along the general flux (figure 9.1): (9.9) According to (9.9), the torque expression (9.7) becomes: (9.10)

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Electric Drives6 Figure 9.1. General flux orientation axis For motoring T e >0 (for r >0) for ia >0 and thus generating is obtained with ia <0. In other words, for direct (trigonometric) motion, the stator current leads the general flux for motoring and lags it for generating. (9.11) We may consider that i qa is the torque current component (9.10) while i da is the flux current component of stator current (figure 9.1). Their decoupled control is the essence of vector control. This is so because, for constant ma, torque control means i qa control and thus flux control means i da control.

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Electric Drives7 9.3. GENERAL CURRENT DECOUPLING The input (command) variables in vector control are the reference flux ma * and the reference torque T e *. Consequently the IM is a two input system. General current decoupling means to determine the reference current space phasor based on reference flux ma * and torque T e *. For general current decoupling we purely make use of equations (9.6) - (9.11), skipping the stator voltage equation, to obtain: (9.12) (9.13) Equations ((9.11) - (9.13)) are illustarted in the general current decoupling network shown in figure 9.2.

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Electric Drives8 Figure 9.2. General current decoupling network For from ((9.1) - (9.3)) and ((9.12) - (9.13)) we obtain: (9.15)

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Electric Drives9 So only for rotor flux ( r ) orientation the current decoupling network gets simplified, to the form in figure 9.3. Figure 9.3. Current decoupling network in rotor flux orientation

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Electric Drives10 For, from ((9.1) - (9.3)): (9.16) So we end up with stator flux orientation but: (9.17) Consequently, in stator flux orientation, the current decoupling network retains the complicated form of figure 9.2. A similar situation occurs with a = 1: (9.18) that is airgap flux orientation.

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Electric Drives11 9.4. PARAMETER DETUNING EFFECTS IN ROTOR FLUX ORIENTATION CURRENT DECOUPLING Let us consider constant slip frequency: (9.19) and constant stator current: (9.20) But and, defined in ((9.21) - 9.22)) vary with magnetic saturation and rotor temperature: (9.21) (9.22)

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Electric Drives12 We may now illustrate the influence of and on T e / T e * and r / r *, for constant stator current i s * and slip frequency (S 1 ) *, by considering: (9.26) In general K 1. Introducing K in (9.16) yields: (9.27) (9.28) It is now clear that both saturation and temperature ( ) have a monotonous influence on the flux ratio while the influence of on torque ratio shows a maximum for = 1 / K: (9.29)

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Electric Drives13 Results for K = 0.5 and 1.0, = 1 an variable from 0.5 to 1.5 are shown in figure 9.4. Being monotonous, the rotor flux detuning may be used to correct the rotor time constant in rotor flux orientation indirect vector control. Figure 9.4. Actual / command torque (T e / T e * ) and rotor flux ( r / r * ) versus rotor time constant detuning ratio, for current decoupling in rotor flux orientation

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Electric Drives14 9.5. DIRECT VERSUS INDIRECT VECTOR CURRENT DECOUPLING As seen above the indirect current decoupling is either complicated (for stator flux orientation) or (and) strongly parameter dependent for rotor flux orientation. Figure 9.5. Direct (feedback) current decoupling with general flux orientation

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Electric Drives15 9.6. A.C. VERSUS D.C. CURRENT CONTROLLERS Once the reference d - q currents i da *, i qa * and flux orientation angle er + a * are known we have to translate these commands into stator currents and to use current controllers to impose these currents through the power electronic converter (P.E.C.). There are two ways to this scope: through a.c. current controllers; through d.c. (synchronous) current controllers [11] In any case Park transformation is required (chapter 8, equation (8.57)): (9.30)

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Electric Drives16 Figure 9.6. Indirect (or direct) vector current control with a.c. current controllers

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Electric Drives17 D.c. current controllers serve such a solution (figure 9.7) and are load and frequency independent [11]. Figure 9.7. Indirect (or direct) vector current control with d.c. (synchronous) current contollers

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Electric Drives18 9.7. VOLTAGE DECOUPLING Let us remember that so far we did not make use of the stator equation (9.5) summarized here for convenience: (9.31) Figure 9.8. Voltage decoupling network for general flux orientation

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Electric Drives19 It should be noticed that the voltage decoupling network gets simplified only in stator flux orientation, when a = L s / L m and. This simplified form is shown in figure 9.9. Equation (9.31) becomes: (9.32) Figure 9.9. Voltage decoupling network for stator flux orientation

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Electric Drives20 To facilitate good control above base speed voltage decoupling is required. For implementation, in general, only the motion induced voltage, E, is considered (s = 0 in figure 9.8.): (9.33) This way combined voltage - current vector control is obtained. However only d.c. current controllers (figure 9.7) allow for a practical solution (figure 9.10). The voltage is limited to V smax by the inverter: (9.34) Table 9.1. Summary of most appropriate strategies of vector control for induction motors

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Electric Drives21 Figure 9.10. Indirect (or direct) combined vector voltage and d.c. current control in general flux orientation

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Electric Drives22 9.8. VOLTAGE AND CURRENT LIMITATIONS FOR THE TORQUE AND SPEED CONTROL RANGE Both motors and power electronic converters (P.E.Cs) are voltage and current limited or kVA limited. However, in terms of speed - torque envelope, it depends on how vector control is performed to extract the most from the drive. Especially so in constant power operation for speeds from b to max ( max / b = 2 - 4) [12 - 13]. For steady state (s = 0 in rotor flux orientation) from (9.15); (9.35) (9.36) Equation (9.35) stresses the conjecture that i d is the flux current. Using equation (9.33) we may obtain: (9.37)

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Electric Drives23 (9.40) We may read equation (9.40) as if the I.M., under constant rotor flux, is a reluctance synchronous motor with L s as d axis inductance (L d ) and L sc as q axis inductance (L q ) : L s >>L sc. Figure 9.11. Stator flux and voltage in rotor flux coordinates

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Electric Drives24 From (9.31) and (9.39) the stator voltage becomes: (9.41) or (9.42) Notice that L s = L ls + L m (the no load inductance) is dependent on magnetic saturation through L m for the main flux path and through L ls for leakage path. (9.43) with (9.44) where i sn is the rated current. For a short time 1.5 i sn or even 2.0 i sn is available in most commercial drives. This, of course, means adequate motor and P.E.C. rating.

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Electric Drives25 Neglecting r s (r r = 0) in (9.42) we obtain: (9.45) The maximum torque T ek under these conditions is obtained from: (9.46) with (9.47) Finally we obtain: (9.48) (9.49) For high values of peak torques the short circuit (transient) inductance should be small by design.

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Electric Drives26 Notice also that (9.48) means: (9.50) (9.51) The power factor angle 1 (figure 9.11) is: (9.52)

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Electric Drives27 Point A k corresponds to the peak torque T ek and current i sk at base speed b and full voltage V smax and: (9.53) Figure 9.12. Current limit boundaries The corresponding power factor angle 1k is: (9.54) So, for maximum torque per given flux, (given voltage and frequency), the power factor is slightly below 0.707 (cos 1k <0.707).

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Electric Drives28 For a new motor it seems reasonable to choose the rated conditions for maximum power factor: (9.55) From (9.54) - (9.55) it follows that: (9.56) (9.57)

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Electric Drives29 Figure 9.13. Flux weakening zone and the constant power subzone (region)

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Electric Drives30 Example 9.1. Let us consider a standard induction motor with the following parameters: r s = r r = 0.2, L ls = L lr = 0.005H, L m = 0.075H, rated line voltage 220V (rms) (star), rated frequency b = 2 60 rad/s, rated slip S n = 0.02, p = 2 pole pairs. Determine the ideal no load current i on, for, calculate i qn for rated slip, the rotor flux, stator flux components and the rated electromagnetic torque T eb and power P eb ; calculate the maximum frequency max1 for which constant electromagnetic power P eb may be produced. Solution: From the equivalent circuit (chapter 8, figure 8.11), for S = 0, (9.61) Neglecting r s : (9.62) Consequently the ideal no load phase current i on (rms) is: (9.63)

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Electric Drives31 The torque current i qn - in rotor flux orientation - is (9.15): (9.64) The rotor flux, rn (9.15), is: (9.65) The stator flux (9.39) is: (9.66) The rated electromagnetic torque T eb (9.40) is: (9.67) And the electromagnetic power P eb (9.58) is: (9.68)

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Electric Drives32 It is known that for max1 (figure 9.13) the critical value of slip frequency is reached (9.51): (9.69) The corresponding electromagnetic power P eb at max1 is (9.58): (9.70) (9.71) Notice that the base frequency and, consequently, the constant power zone covers a speed ratio max1 / b = 501.77 / 376.8 = 1.33. The corresponding currents, i dk, i qk are obtained from: (9.72)

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Electric Drives33 (9.73) (9.74) (9.75) For the base speed, the rated current i sn is: (9.76) (9.77) So even this narrow constant power speed range ratio max1 / b = 1.33 is obtained at the price of higher stator current which implies lower power factor and, perhaps, efficiency. Reducing L sc is a sure way to increase the value of max1 (9.71) and thus a wider constant power speed range is obtained. Leaving a voltage reserve at base speed or switching from star to delta winding connection in the motor are two practical methods to widen the constant power zone.

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Electric Drives34 9.9. IMPRESSING VOLTAGE AND CURRENTS THROUGH PWM 9.9.1. Switching state voltage vectors Figure 9.14. PWM voltage source inverter. One switch per leg conducting at any time

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Electric Drives35 A PWM voltage source inverter (figure 9.14) produces in the a.c. motor symmetrical rectangular voltage potentials V ap, V bp, V cp provided that the conducting P.E.S. triplet is on for 60 0 electrical degrees. This means six pulses per period or six switchings per period only. Figure 9.15. Voltage waveforms for six switchings per period a.) voltage potentials at motor terminals b.) neutral potential c.) phase voltages

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Electric Drives36 The maximum voltage fundamental V 1six - step is obtained for six pulse switching and the modulation index m is: (9.78) where from figure 9.15a: (9.79) The ideal maximum modulation index is equal to unity. Various PWM schemes allow an m max <1 which represents an important performance criterion as the inverter maximum kVA depends on the maximum voltage at motor terminals. We may use space phasors to describe the 6 non zero switching situations as: (9.80) with V an, V bn, V cn from figure 9.15.c, we obtain six space phasors, 60 0 apart (figure 9.16).

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Electric Drives37 b.) Figure 9.16. a.) Voltage space vectors b.) The corresponding phase voltages

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Electric Drives38 Timing the eight voltage space vectors V 1, …, V 8 is, in fact, the art of PWM. Impressing the voltage commands required by the vector control strategies may be done directly by open loop PWM. Impressing the current commands through the same voltage space vectors is done through closed loop PWM. Among various PWM methods - treated extensively in the power electronics literature [15 - 16] - we deal with two open loop and two closed loop PWM strategies considered here most representative. 9.9.2. Open loop space - vector PWM In space vector PWM the reference voltage space - vector of the motor is treated directly and not phase by phase. The reference voltage space vector V 1 * is sampled at a fixed clock frequency 2f s (figure 9.17a) being constructed through adequate timing of adjacent nonzero inverter voltage space vectors V 1 to V 6 and the zero voltage space vectors V 0, V 7 (figure 9.17.b): (9.81)

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Electric Drives39 (9.82) Figure 9.17. Open loop space vector PWM a.) structural diagramb.) voltage space vector in the first sector The respective timings t 1, t 2 are: (9.83) (9.84)

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Electric Drives40 In fact this tehnique produces an average of the three voltage space vectors V i, V i+1 and V 0 (V 7 ) over a subcycle T = 1/2f s. For the minimum number of commutations, with V 1 * in the first sector, the switching sequence is: (9.85) in all odd subcycles (of all 6 sectors) and (9.86) for all even subcycles of all sectors. Figure 9.18. Random PWM principle

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Electric Drives41 When the carrier signal reaches one of its peak values, its slope is reversed by a hysteresis block and a sample is taken from the random generator (figure 9.18) which triggers an additional variation on the slope. This way the duration of subcycles is obtained while only the average switching frequency remains constant. Dead time: effect and compensation In order to prevent shortcircuiting an inverter leg there should be a lock - out time T d between the turn off of one P.E.S. and the turn - on of the next. T d should be larger than the maximum particle storage time of the P.E.S., T st. The effect of the lock - out time T d is a distortion V on the reference voltage U *. (9.88)

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Electric Drives42 The voltage distortion V changes sign with current space vector function: (9.89) and is proportional to safety time T d - T st. This voltage distortion is considered by the fact that the on - time of the upper bridge arm is shortened by T d - T st for positive current and is increased by the same amount for negative sign of current. If closed loop (current controller) PWM is used, a compensator of dead time may not be required.

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Electric Drives43 9.9.3. Closed loop PWM Closed loop PWM involves, in general, current or flux closed loop control and may be left nonoptimal or real - time optimized. Figure 9.19. Independent hysteresis current control a.) signal flow diagramb.) phase current waveform When the phase current error i a = i a * - i a > +h, the upper inverter leg P.E.S., (A+), is turned on, while, when i a < -h, the lower leg P.E.S. (A-) is turned on. The same procedure is followed independently on phases b and c and. Evidently no zero voltage space vector V 0 (V 7 ) may be applied.

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Electric Drives44 The absence of zero voltage vector requires high switching frequency at low fundamental frequency (speed) - low voltage amplitude - that is low motor speeds; subharmonics may also occur. In order to reduce the switching frequency and decrease current harmonics, an appropiate nonzero or zero voltage vector of the inverter may be applied, based on current phasor error and its derivative d /dt position corroborated with the e.m.f. vector and the existing voltage vector position in one of the six 60 0 - wide sectors. A table of optimal switchings may thus be defined [19] based on the machine equation in stator coordinates: (9.90) (9.91) is when. Equation (9.90) shows that d /dt is determined solely by the choice of. The position of is one of the 6 sectors (figure 9.20) may be found knowing only the position of and of the applied. (figure 9.20.a, b).

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Electric Drives45

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Electric Drives46 Figure 9.20. Current vector hysteresis control a., b.) - zone detection c.) switching tables (after Ref.[19])

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Electric Drives47 However, a.c. controllers are shown to be load, motor parameter and frequency (speed) dependent while d.c. (d - q) current controllers are rather independent of frequency and crosscoupling effects [11]. D.c. (synchronous or d - q) current controllers are better and may be implemented in stator (a.c.) coordinates by transforming their equation (in PI form) in flux coordinates [11]: (9.92) through Park transformation, to the final form: (9.93) (9.94) These equations may be conveniently implemented, though for crosscoupling compensation, the value of the flux speed 1 has to be calculated. [11, 17].

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Electric Drives48 9.10. INDIRECT VECTOR A.C. CURRENT CONTROL - A CASE STUDY We will present here the simulation of a feedforward (indirect) vector current control system for induction motors. The example was implemented in MATLAB - SIMULINK simulation program. The motor model was integrated in two blocks, first represents the current and flux calculation module in d - q axis, the second represents the torque, speed and position computing module. The block structure of the electric drive system is presented in the figures 9.21 - 9.23. The motor used for this simulation has the following parameters: P n = 1100W, V nf = 220V, 2p = 4, r s = 9.53, r r = 5.619, L sc = 0.136H, L r = 0.505H, L m = 0.447H, J = 0.0026kgfm 2. The following figures (9.24. - 9.28) represent the speed, torque, current and flux responses for starting transients and a load torque applied at 0.4s.

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Electric Drives49 Figure 9.21. The indirect vector a.c. current control system for IMs

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Electric Drives50 Figure 9.22. The a.c. current controllers

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Electric Drives51 Figure 9.23. The motor space phasor model

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Electric Drives52 Figure 9.24. Speed transient response

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Electric Drives53 Figure 9.25. Torque response

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Electric Drives54 Figure 9.26. Phase current waveform under steady state

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Electric Drives55 Figure 9.27. Stator flux amplitude

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Electric Drives56 Figure 9.28. Rotor flux amplitude We should mention that the parameters were fully tuned. The influence of parameter detuning could be investigated also. This is beyond our scope here.

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