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

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**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) V1 / f1 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. Electric Drives

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**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 ( )): lmb - airgap flux; lsb - stator flux and lrb - 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. Electric Drives

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**and define a general flux: (9.3) **

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 ( ) (with d/dt = s): (9.4) Electric Drives

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**with the slip Sa defined for the general flux : (9.8) **

(9.5) (9.6) (9.7) with the slip Sa 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) Electric Drives

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**Figure 9.1. General flux orientation axis**

For motoring Te>0 (for wr>0) for dia>0 and thus generating is obtained with dia<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 iqa is the torque current component (9.10) while ida 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 lma, torque control means iqa control and thus flux control means ida control. Electric Drives

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**9.3. GENERAL CURRENT DECOUPLING**

The input (command) variables in vector control are the reference flux lma* and the reference torque Te*. Consequently the IM is a two input system. General current decoupling means to determine the reference current space phasor based on reference flux lma* and torque Te*. 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. Electric Drives

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**Figure 9.2. General current decoupling network**

For from ((9.1) - (9.3)) and ((9.12) - (9.13)) we obtain: (9.15) Electric Drives

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**Figure 9.3. Current decoupling network in rotor flux orientation**

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

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**So we end up with stator flux orientation but: (9.17)**

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. Electric Drives

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**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 a and b, defined in ((9.21) )) vary with magnetic saturation and rotor temperature: (9.21) (9.22) Electric Drives

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**In general K1. Introducing K in (9.16) yields: (9.27)**

We may now illustrate the influence of a and b on Te / Te* and lr / lr*, for constant stator current is* and slip frequency (Sw1)*, by considering: (9.26) In general K1. Introducing K in (9.16) yields: (9.27) (9.28) It is now clear that both saturation and temperature (a, b) have a monotonous influence on the flux ratio while the influence of a on torque ratio shows a maximum for a = 1 / K: (9.29) Electric Drives

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**Results for K = 0. 5 and 1. 0, b = 1 an a variable from 0. 5 to 1**

Results for K = 0.5 and 1.0, b = 1 an a 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 (Te / Te*) and rotor flux (lr / lr*) versus rotor time constant detuning ratio a, for current decoupling in rotor flux orientation Electric Drives

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**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 Electric Drives

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**9.6. A.C. VERSUS D.C. CURRENT CONTROLLERS**

Once the reference d - q currents ida*, iqa* and flux orientation angle qer+ga* 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) Electric Drives

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**Figure 9. 6. Indirect (or direct) vector current control with a. c**

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

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**D. c. current controllers serve such a solution (figure 9**

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 Electric Drives

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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 Electric Drives

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**Figure 9.9. Voltage decoupling network for stator flux orientation**

It should be noticed that the voltage decoupling network gets simplified only in stator flux orientation, when a = Ls / Lm 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 Electric Drives

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**The voltage is limited to Vsmax by the inverter: (9.34)**

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 Vsmax by the inverter: (9.34) Table 9.1. Summary of most appropriate strategies of vector control for induction motors Electric Drives

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**Figure 9. 10. Indirect (or direct) combined vector voltage and d. c**

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

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**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 wb to wmax (wmax / wb = 2 - 4) [ ]. For steady state (s = 0 in rotor flux orientation) from (9.15); (9.35) (9.36) Equation (9.35) stresses the conjecture that id is the flux current. Using equation (9.33) we may obtain: (9.37) Electric Drives

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**Figure 9.11. Stator flux and voltage in rotor flux coordinates**

(9.40) We may read equation (9.40) as if the I.M., under constant rotor flux, is a reluctance synchronous motor with Ls as d axis inductance (Ld) and Lsc as q axis inductance (Lq) : Ls>>Lsc. Figure Stator flux and voltage in rotor flux coordinates Electric Drives

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**From (9.31) and (9.39) the stator voltage becomes: (9.41) or (9.42) **

Notice that Ls = Lls + Lm (the no load inductance) is dependent on magnetic saturation through Lm for the main flux path and through Lls for leakage path. (9.43) with (9.44) where isn is the rated current. For a short time 1.5 isn or even 2.0 isn is available in most commercial drives. This, of course, means adequate motor and P.E.C. rating. Electric Drives

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**Neglecting rs (rr = 0) in (9.42) we obtain: (9.45) **

The maximum torque Tek 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. Electric Drives

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**Notice also that (9.48) means: (9.50)**

(9.51) The power factor angle j1 (figure 9.11) is: (9.52) Electric Drives

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**Figure 9.12. Current limit boundaries**

Point Ak corresponds to the peak torque Tek and current isk at base speed wb and full voltage Vsmax and: (9.53) Figure Current limit boundaries The corresponding power factor angle j1k is: (9.54) So, for maximum torque per given flux, (given voltage and frequency), the power factor is slightly below (cosj1k <0.707). Electric Drives

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**From (9.54) - (9.55) it follows that: (9.56)**

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) Electric Drives

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

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**From the equivalent circuit (chapter 8, figure 8.11), for S = 0, **

Example 9.1. Let us consider a standard induction motor with the following parameters: rs = rr = 0.2W, Lls = Llr = 0.005H, Lm = 0.075H, rated line voltage 220V (rms) (star), rated frequency wb = 2p60 rad/s, rated slip Sn = 0.02, p = 2 pole pairs. Determine the ideal no load current ion, for , calculate iqn for rated slip, the rotor flux, stator flux components and the rated electromagnetic torque Teb and power Peb; calculate the maximum frequency wmax1 for which constant electromagnetic power Peb may be produced. Solution: From the equivalent circuit (chapter 8, figure 8.11), for S = 0, (9.61) Neglecting rs: (9.62) Consequently the ideal no load phase current ion (rms) is: (9.63) Electric Drives

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**The torque current iqn - in rotor flux orientation - is (9.15): (9.64) **

The rotor flux, lrn (9.15), is: (9.65) The stator flux (9.39) is: (9.66) The rated electromagnetic torque Teb (9.40) is: (9.67) And the electromagnetic power Peb (9.58) is: (9.68) Electric Drives

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**The corresponding electromagnetic power Peb at wmax1 is (9.58): (9.70)**

It is known that for wmax1 (figure 9.13) the critical value of slip frequency is reached (9.51): (9.69) The corresponding electromagnetic power Peb at wmax1 is (9.58): (9.70) (9.71) Notice that the base frequency and, consequently, the constant power zone covers a speed ratio wmax1 / wb = / = The corresponding currents, idk, iqk are obtained from: (9.72) Electric Drives

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**For the base speed, the rated current isn is: (9.76) (9.77) **

(9.73) (9.74) (9.75) For the base speed, the rated current isn is: (9.76) (9.77) So even this narrow constant power speed range ratio wmax1 / wb = 1.33 is obtained at the price of higher stator current which implies lower power factor and, perhaps, efficiency. Reducing Lsc is a sure way to increase the value of wmax1 (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. Electric Drives

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**9.9. IMPRESSING VOLTAGE AND CURRENTS THROUGH PWM**

Switching state voltage vectors Figure PWM voltage source inverter. One switch per leg conducting at any time Electric Drives

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**Voltage waveforms for six switchings per period **

A PWM voltage source inverter (figure 9.14) produces in the a.c. motor symmetrical rectangular voltage potentials Vap, Vbp, Vcp provided that the conducting P.E.S. triplet is on for 600 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 Electric Drives

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The maximum voltage fundamental V1six - 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 mmax<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 Van, Vbn, Vcn from figure 9.15.c, we obtain six space phasors, 600 apart (figure 9.16). Electric Drives

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**a.) Voltage space vectors b.) The corresponding phase voltages**

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

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**9.9.2. Open loop space - vector PWM **

Timing the eight voltage space vectors V1, …, V8 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 [ ] - we deal with two open loop and two closed loop PWM strategies considered here most representative. 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 V1* is sampled at a fixed clock frequency 2fs (figure 9.17a) being constructed through adequate timing of adjacent nonzero inverter voltage space vectors V1 to V6 and the zero voltage space vectors V0, V7 (figure 9.17.b): (9.81) Electric Drives

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**Figure 9.17. Open loop space vector PWM **

(9.82) Figure Open loop space vector PWM a.) structural diagram b.) voltage space vector in the first sector The respective timings t1, t2 are: (9.83) (9.84) Electric Drives

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**in all odd subcycles (of all 6 sectors) and (9.86) **

In fact this tehnique produces an average of the three voltage space vectors Vi, Vi+1 and V0 (V7) over a subcycle T = 1/2fs. For the minimum number of commutations, with V1* 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 Random PWM principle Electric Drives

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**Dead time: effect and compensation **

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 Td between the turn off of one P.E.S. and the turn - on of the next. Td should be larger than the maximum particle storage time of the P.E.S., Tst. The effect of the lock - out time Td is a distortion DV on the reference voltage U*. (9.88) Electric Drives

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**and is proportional to safety time Td - Tst. **

The voltage distortion DV changes sign with current space vector function: (9.89) and is proportional to safety time Td - Tst. This voltage distortion is considered by the fact that the on - time of the upper bridge arm is shortened by Td - Tst 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. Electric Drives

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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 Independent hysteresis current control a.) signal flow diagram b.) phase current waveform When the phase current error Dia = ia* - ia > +h, the upper inverter leg P.E.S., (A+), is turned on, while, when Dia < -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 V0(V7) may be applied. Electric Drives

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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 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). Electric Drives

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**Figure 9.20. Current vector hysteresis control **

a., b.) zone detection c.) switching tables (after Ref.[19]) Electric Drives

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**through Park transformation , to the final form: (9.93)**

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 w1 has to be calculated. [11, 17]. Electric Drives

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**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 The motor used for this simulation has the following parameters: Pn = 1100W, Vnf = 220V, 2p = 4, rs = 9.53W, rr = 5.619W, Lsc = 0.136H, Lr = 0.505H, Lm = 0.447H, J = kgfm2. The following figures ( ) represent the speed, torque, current and flux responses for starting transients and a load torque applied at 0.4s. Electric Drives

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

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**Figure 9.22. The a.c. current controllers**

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**Figure 9.23. The motor space phasor model**

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**Figure 9.24. Speed transient response**

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**Figure 9.25. Torque response**

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**Figure 9.26. Phase current waveform under steady state**

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**Figure 9.27. Stator flux amplitude**

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**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. Electric Drives

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