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

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**PM AND RELUCTANCE SYNCHRONOUS MOTOR DRIVES**

11.1. INTRODUCTION PM and reluctance synchronous motors are associated with PWM voltage - source inverters in variable speed drives. While PM - SM drives enjoy by now world - wide markets, reluctance synchronous motor (RSM) drives still have a small share of the market in low power region, though recently high saliency (and performance) RSMs with q axis PMs have been successfully introduced for wide constant power speed range applications (such as spindle drives). High performance applications are in general provided with PM - SM drives. RSM drives may be used for general applications as the costs of the latter are lower than of the former and, in general, close to the costs of similar IM drives. In what follows we will deal first with PM - SM drives and after that with RSM drives. Constant power operation will be discussed at the end of the chapter for both types of synchronous motors. Electric Drives

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**11.2. PM - SM DRIVES: CLASSIFICATIONS**

Basically we may distinguish three ways to classify PM - SM drives: with respect to current waveform, voltage - frequency correlation, motion sensor presence. From the point of view of current waveform we distinguish: rectangular current control - figure 11.1.a - the so called brushless d.c. motor drive - q = 1 slot per pole per phase, surface PM rotor; sinusoidal current control - figure 11.1.b - the so called brushless a.c. drive - q 2 slots per pole per phase. From the point of view of motion sensor presence there are: drives with motion sensors; drives without motion sensors (sensorless). Finally sinusoidal current (brushless a.c.) drives may have: scalar (V / f) control - a damper cage on the rotor is required; vector control (current or current and voltage); direct torque and flux control (DTFC). The stator current waveforms - rectangular or sinusoidal (figure 11.1) - have to be synchronized with the rotor position. Electric Drives

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**Figure 11.1. Rectangular or sinusoidal current control; aa - advance angle**

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Scalar control (V / f) is related to sinusoidal current control without motion sensors (sensorless) (figure 11.2). Figure V / f (scalar) control for PM - SM (and for RSM) with torque angle increment compensation Electric Drives

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**For faster dynamics applications vector control is used (figure 11.3).**

Figure Basic vector control of PM - SM (or for RSM) 1 - with motion sensor 2 - sensorless Electric Drives

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**Figure 11.4. Direct torque and flux control (DTFC) of PM - SMs (and RSMs)**

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To simplify the motor control, the direct torque and flux control (DTFC for IMs) has been extended to PM - SM (and to RSMs) as torque vector control (TVC) in [10]. The stator flux and torque direct control leads to a table of voltage switchings (voltage vector sequence). Vector rotation has been dropped but flux and torque observers are required. While speed is to be observed, rotor position estimation is not required in sensorless driving. Again, fast flux and torque control may be obtained even in sensorless driving. Rectangular current control and sinusoidal current control (through vector control or DTFC) are going to be detailed in what follows for motion sensor and sensorless driving. Electric Drives

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**11.3. RECTANGULAR CURRENT CONTROL (BRUSHLESS D.C. MOTOR DRIVES)**

Rectangular current control is applied to nonsinusoidal (trapezoidal) PM - e.m.f. waveform - typical to concentrated coil stator windings (three slots per pole: q = 1). To reduce torque pulsations rectangular current is needed. With rectangular current, however, the reluctance torque production is not so efficient and thus nonsalient pole (surface PM) rotors is preferred. The motor phase inductance La is [1]: (11.1) (11.2) where: W1 - turns per phase, L - stack length, t - pole pitch, g - airgap, Kc - Carter coefficient, hPM - PM radial thickness, p - pole pairs, Lsl - leakage inductance. The mutual inductance between phases, Lab, is: (11.3) Thus the cycling inductance Ls (per phase) for two phase conduction is: (11.4) Electric Drives

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**11.3.1. Ideal brushless d.c. motor waveforms**

In principle the surface PM extends over an angle aPM less than 1800 (which represents the pole pitch, figure 11.1). The two limits of aPM are 2p/3 and p. Let us suppose that the PM produces a rectangular airgap flux distribution over aPM = p (1800) (figure 11.5.a). The stator phase m.m.f. is supposed to be rectangular, a case corresponding to q = 1 (three slots per pole). Consequently the PM flux linkage in the stator winding lPM(qer) varies linearly with rotor position (figure 11.5.b). Finally the phase e.m.f. Ea is rectangular with respect to rotor position (figure 11.5.c). For zero advance angle aa = 0 (figure 11.1) - the phase currents are in phase with the e.m.f. (ia, Ea) for motoring (figure 11.5.d). At any instant, due to assumed instantaneous current commutation, only two phases are in conduction. Consequently only two thirds of the PMs are utilized. So the back core flux is 50% larger than necessary. On the other hand, if the currents would extend over 1800 (with three phases conducting at any time) for 1200 wide PMs, it would mean 50% more than necessary copper losses for a given torque. Electric Drives

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**Figure 11.5. Ideal waveforms for BLDC (brushless d.c. motor)**

a.) PM airgap flux density b.) PM flux per phase a c.) e.m.f. in phase a d.) ideal currents for motoring e.) ideal currents for generating For 1200 wide magnets delta connection is used while 1800 wide magnets require star connection. To reduce the fringing (leakage) flux between neighboring magnets their span is Electric Drives

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**The maximum flux linkage per phase lPM is: (11.6) **

Returning to the linear phase flux linkage variation with rotor position (figure 11.5.b) we have: (11.5) The maximum flux linkage per phase lPM is: (11.6) The phase e.m.f., Ea, is: (11.7) where wr is the electrical angular speed. As two phases conduct at any time the ideal torque Te is constant: (11.8) Between any two current instantaneous commutation the phase current is constant (id = idc) and thus the voltage equation is: (11.9) Electric Drives

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**Figure 11.6. Ideal speed - torque curves of BLDC**

(11.10) With (11.11) The ideal speed - torque curve is linear (figure 11.6) like for a d.c. brush PM motor. Figure Ideal speed - torque curves of BLDC Electric Drives

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**11.3.2. The rectangular current control system**

In general a rectangular current control system contains the BLDC motor, the PWM inverter, the speed and current controllers and the position (speed) sensors (or estimators, for sensorless control) and the current sensors (figure 11.7). Figure Rectangular current control of BLDC Electric Drives

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**a.) Current sequencing b.) phase connection**

The currents sequence, produced through inverter adequate control, with 1200 current waveforms in figure 11.8 shows also the position of the 6 elements of the proximity sensors with respect to the axis of the phase a for a zero advance angle aa = 0. Figure 11.8. a.) Current sequencing b.) phase connection Electric Drives

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**With two phases conducting the stator active m. m. f**

With two phases conducting the stator active m.m.f. is on from 600 to 1200 with respect to the rotor position. The ideal voltage vector (figure 11.8) also jumps 600 for any phase commutation in the inverter. Each phase is on 1200 out of 1800 for the 1200 conducting strategy. To reverse the speed the addresses (IGBTs) of the proximity sensor elements action are shifted by 1800 (P(a+) P(a-); P(b+) P(b-); P(c+) P(c-)). The proximity sensor has been located for zero advance angle to provide similar performance for direct and reverse motion. However, through electronic means, the advance angle may be increased as speed increases to reduce the peak PM flux in the stator phase and thus produce more torque, for limited voltage, at high speeds. Using the same hardware we may also provide for 1800 conduction conditions, at high speeds, when all three phases conduct at any time. Electric Drives

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**11.3.3. The hysteresis current controller**

Figure Current chopping Electric Drives

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**Figure 11.10. Conduction of phases a and b**

a.) on - time b.) off - time Electric Drives

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**During the on - time, figure 11.10.a, the a+b- equation is: (11.12) **

(11.13) Note that if the turning on is advanced by aa part of the on time Ea - Eb = 0 and thus a faster current increase is possible. The solution of equation (11.12) is: (11.14) To allow for current rising Vd > E0. Above a certain speed Vd < E0 and thus current chopping is not feasable any more. The current waveform contains in this case a single on - off pulse triggered by the proximity sensor (estimator). Electric Drives

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**The solution of (11.15) with t’ = t - ton is: (11.16)**

During the off time (diodes D1 and D4 conducting, in figure b) the voltage equation is: (11.15) Vc0 is the capacitor voltage at the end of on time (11.15), or at the beginning of off - time. The solution of (11.15) with t’ = t - ton is: (11.16) with (11.17) The torque Te(t) expression is: (11.18) So if the e.m.f. is constant in time the electromagnetic torque reproduces the current pulsations between imin and imax. Electric Drives

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**b.) calculate the torque expression and plot it. Solution: **

Example A BLDC motor is fed through a PWM inverter from a 300V d.c. source (Vd = 300V). Rectangular current control is performed at an electrical speed wr = 2p10rad/s. The no load line voltage at wr is E0 = 48V = const. The cyclic inductance Ls = 0.5mH, rs = 0.1W and the stator winding has q = 1 slot per pole per phase and two poles (2p = 2). The filter capacitor Cf = 10mF, the current chopping frequency fc = 1.25kHz and ton / toff = 5/3. Determine: a.) the minimum and maximum values of current (imin, imax) during current chopping; b.) calculate the torque expression and plot it. Solution: a.) To find imin and imax we have to use equation (11.14) for i(t) = imax and t = ton and (11.16) for i(t) = imin and t’ = toff. (11.19) Notice that ton = 0.5ms and toff = 0.3ms (11.20) Electric Drives

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**From (11.19) and (11.24) we may calculate Imax and Imin: (11.25) **

Also from (11.17): (11.21) (11.22) (11.23) (11.24) From (11.19) and (11.24) we may calculate Imax and Imin: (11.25) The average current . b.) The average torque Tav is: (11.26) Electric Drives

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**Figure 11.11. Torque pulsations due to current chopping only**

The instantaneous torque (11.18) includes current pulsations (as from (11.14) and (11.16), figure 11.11). Figure Torque pulsations due to current chopping only It should be noticed that the chopping frequency is low for the chosen speed (E0 << Vd) and thus the current and torque pulsations are large. Increasing the chopping frequency will reduce these pulsations. To keep the current error band 2(Imax - Imin) whithin reasonable limits the chopping frequency should vary with speed (higher at lower speeds and lower at medium speeds) Though the high frequency torque pulsations due to current chopping are not followed by the motor speed, due to the much larger mechanical time constant, they produce flux density pulsations and, thus, notable additional core and copper losses (only the average current I0 is, in fact, useful). Electric Drives

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**11.3.4. Practical performance**

So far the phase commutation transients - current overlapping - have been neglected. They however introduce notable torque pulsations at 6wr frequency (figure 11.12) much lower than those due to current chopping. To account for them complete simulation or testing is required [3]. Figure Torque pulsations due to phase commutation Electric Drives

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**Figure 11.13. Current waveform at high speeds**

Special measures are required to reduce the cogging torque to less than 2 - 5% of rated torque for high performance drives. While at low speeds current chopping is feasable at high speeds one current pulse remains (figure 11.13). The current controller gets saturated and the required current is not reached. Figure Current waveform at high speeds As the advance angle is zero (aa = 0) there is a delay in “installing” the current and thus, as the e.m.f. is “in phase” with the reference current, a further reduction in torque occurs. Electric Drives

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**11.3.5. Extending the torque - speed domain **

Extending the torque - speed domain may be obtained (for a given drive) by advancing the phase commutation time by an angle aa dependent on speed. This phase advancing allows fast current rise before the “occurence” of the e.m.f. (assuming a PM span angle aPM < ) An approximate way to estimate the advance angle required aa, for 1200 conduction, may be based on linear current rise [4] to the value I: (11.27) where n - is the rotor speed in rps. Torque at even higher speeds may be obtained by switching from 1200 to 1800 current conduction (three phase working at any time). The current waveform changes, especially with advancing angle (figure 11.14). This time the e.m.f. is considered trapezoidal, that is close to reality. The advancing angle aa may be, for high speeds, calculated assuming sinusoidal e.m.f. [4] and current variation: (11.28) Electric Drives

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**Figure 11.14. 1800 conducting with advancing angle at high speed**

It has been demonstrated [3,4] that 1200 conduction is profitable at low to base speeds while 1800 conduction with advancing angle is profitable for high speeds (figure 11.15). Figure Torque - speed curves for various advancing angle aa A smooth transition between 1200 and 1800 conduction is required to fully exploit the torque - speed capabilities of brushless d.c. motor drives. Electric Drives

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**Example 11.2. Digital simulation: a brushless d.c. motor drive **

We will present here the simulation results on a permanent magnet brushless DC motor drive (BLDC). The motor equations (see equation (10.6) in chapter 10) are: (11.29) with (11.30) La, M self and mutual inductances. Introducing the null point of d.c. link (0), the phase voltages are: (11.31) For star connection of phases: (11.32) Here n represents the star connection point of stator windings. Va0, Vb0, Vc0 could easily be related to the d.c. link voltage and inverter switching state. Electric Drives

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**The simulation of this drive was implemented in MATLAB - SIMULINK**

The simulation of this drive was implemented in MATLAB - SIMULINK. The motor model was integrated in a block (PM_SM). The changing of motor parameters for different simulations is as simple as possible. After clicking on this block, a dialog box appears and you can change them by modifying their default values. The drive system consists of a PI speed controller (Ki = 20, Ti = 0.05s), a reference current calculation block, hysteresis controller and motor blocks. The study examines the system behavior for starting, load perturbation and speed reversal. The motor operates at desired speed and the phase currents are regulated within a hysteresis band around the reference currents (as functions of rotor position). The integration step (50ms) can be modified from the Simulink’s Simulation / Parameters. To find out the structure of each block presented above unmask it (Options/Unmask). Each masked block contains a short help describing that block (inputs / outputs / parameters). Electric Drives

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**Figure 11.16. The BLDC rectangular current - drive controller**

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**Figure 11.17. The BLDC motor block diagram**

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The drive and motor used for this simulation have the following parameters: Vdc = 220V, 2p = 2, Rs = 1W, Ls = 0.02H, M = H, J = 0.005kgm2, K = 0.763, hb (hysteresis band) = 0.2. The following figures represent the speed (figure 11.18), torque (figure 11.19) and current (figure 11.20) responses, and e.m.f. waveform (figure 11.21), for a starting process, loading (6Nm) at 0.2s, unloading at 0.4s, reversal at 0.5s and loading (6Nm) again at 0.8s. Electric Drives

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**Figure 11.18. Speed transient response**

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**Figure 11.19. Torque response**

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**Figure 11.20.Current waveform (ia)**

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**Figure 11.21.Induced voltage waveform (ea)**

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**11.4. VECTOR (SINUSOIDAL) CONTROL**

The vector (sinusoidal) control is applied both for surface PM and interior PM rotors and distributed stator windings (q 2). According to chapter 10 the torque Te expression is: (11.33) The space vector equation (chapter 10) is: (11.34) (11.35) (11.36) Electric Drives

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In general Ld Lq and thus the second (reactive) torque component is positive only if id 0. For id 0 the PM flux is diminished by the id m.m.f. (figure 11.22) but always the PM prevails to avoid complete PM demagnetization. For vector control, in general, the rotor does not have a damper cage and thus there is no rotor circuit, and, as a direct consequence, no current decoupling network is necessary, in contrast to IM case. Figure Space vector diagram of PM - SM with negative id (id < 0) In vector control the PM - SM is controlled in d - q coordinates and then the reference d - q currents (or voltages) are transformed into stator coordinates through the vector rotator (inverse Park transformation) to be then realized by PWM in the inverter. Electric Drives

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**11.4.1. Optimum id - iq relationship**

Since the current decoupling network is missing and the drive command variable is the reference torque Te* as required by a speed (speed and position) controller, we should now choose id* and iq* from the torque equation (11.33). Evidently we need one more equation. This additional information may be obtained through an optimisation criterion such as: maximum torque per current, maximum torque per flux, maximum efficiency etc. As at low speeds the drive is current limited and above base speed wb it is flux limited, we may use these two criteria combined for a high performance drive. The maximum torque / ampere criterion is applied by using (11.34) and: (11.37) (11.38) to find: (11.39) For Ld = Lq, starting with (11.33), it follows that: (11.40) Electric Drives

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**On the other hand, for maximum torque per flux: (11.41) **

Proceeding as above we obtain a new relationship between and : (11.42) where: (11.43) For Ld = Lq (11.41) becomes (starting all over with (11.33)): (11.44) Equation (11.44) signifies complete cancellation of PM flux. The PM is still not completely demagnetized as part of the Ldid is leakage flux which does not flow through the PM. Also notice that constant current is means a circle in the id - iq plane and constant stator flux ls means an ellipse. Electric Drives

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**id - iq a.) for given current is* b.) for given flux ls***

We may now represent (11.38) and (11.40) and (11.41) and (11.42) as in figure Figure id - iq a.) for given current is* b.) for given flux ls* Electric Drives

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**Te* limit is dependent on speed (figure 11.24.b).**

So, for each value of reference torque Te*, according to one of the two optimisation criteria, unique values of id* and iq* are obtained provided the current limit is* and the flux limit ls* are not surpassed. Notice that the flux limit is related to speed wr (through 11.43). So, in fact, the torque Te* is limited with respect to stator current and flux (speed) (figure a). (11.45) (11.46) Te* limit is dependent on speed (figure b). Figure id - iq optimum relationship and torque and flux limits with speed Electric Drives

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**Calculate the torque Tei up to base speed and the base speed wb. **

Example A PM - SM with Vsn = V, isn = 20A, ld = 0.6p.u., no load voltage e0 = 0.7p.u., lq = 1.2p.u., nn = 15rps, p = 1 (pole pairs), rs = 0, is driven at a current of 10A up to base speed nb according to the criterion of maximum torque per current. Calculate the torque Tei up to base speed and the base speed wb. Solution: First we have to calculate ld, lq, e0 in absolute units Ld, Lq, Eo: (11.47) (11.48) (11.49) The PM flux lPM is: (11.50) Electric Drives

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**Making use of (11.39) with (11.50): (11.51)**

(11.52) (11.53) and the torque: (11.54) Electric Drives

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**Now we have to find the maximum speed for which this torque may be produced:**

(11.55) (11.56) Consequently for rated voltage (maximum inverter voltage) the base speed wb is: (11.57) (11.58) In a similar way we may proceed above base speed using the maximum torque / flux criterion. Electric Drives

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**11.4.2. The indirect vector current control**

Indirect vector current control means, in fact, making use of precalculated and (( (11.46)) to produce the reference d -q currents id*, iq*. Then, with Park transformation, the reference phase current controllers are used to produce PWM in the inverter. A rotor position sensor is required for position and speed feedback (11.25). Figure Indirect vector current control of PM - SM Electric Drives

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**11.4.3. Indirect voltage and current vector control**

As for the IMs the a.c. controllers may be replaced by d.c. current controllers (in rotor coordinates) to improve the performance at high speeds, especially. Though, in principle, direct vector control is possible, it is hardly practical unless the drive is sensorless. Indirect voltage and current vector control As expected, the vector current control does not account for the e.m.f. effect of slowing down the current transients with increasing speed. This problem may be solved by using the stator voltage equation for voltage decoupling: (11.59) (11.60) The d.c. current controllers may replace the first terms in (11.59) - (11.60) and thus only the motion induced voltages are feedforwarded. Then PWM is performed open loop to produce the phase voltages at the motor terminals through the PWM inverter (figure 11.26). Electric Drives

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**Figure 11.26. Indirect voltage and current vector control of PMSM with dq (dc) current controllers**

As expected the open loop (voltage) PWM has to observe the voltage limit: (11.61) Electric Drives

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**Figure 11.27. Torque - speed envelope for id = 0 and **

At low speeds the current controllers prevail while at high speeds the voltage decoupler takes over. Results obtained with such a method [5], based only on fdi, fqi (maximum torque per current criterion) proved a remarkable enlargement of the torque - speed envelope (figure 11.27). Figure Torque - speed envelope for id = 0 and for voltage and current control Electric Drives

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**The current increment Dis is: (11.62) **

Fast response PM - SM drives: surface PM rotor motors with predictive control Interior PM - SM drives have been so far treated for vector control. As Lq is rather high in such motors, at least iq response is rather slow in comparison with IM drives. However for surface PM rotor PM - SMs, Ld = Lq = Ls is small and thus fast current (and torque) response may be obtained. The current increment Dis is: (11.62) This classical method [7] has produced spectacular results with position sensors (encoders) - figure The low speed / time linearity [7] is obtained also by using special measures to reduce all torque pulsations to less than 1.5% of rated torque. Figure Speed responses at low speed Electric Drives

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**Example 11.4. Digital simulation: Indirect vector a.c. current control **

For the PM - SM motor in example 11.3, making use of the indirect vector current control system of figure 11.25, perform digital simulations. The simulation of this drive was implemented in MATLAB - SIMULINK. The motor model was integrated in a block (PM_SM) (figures 11.29, 11.30, 11.31). Figure id*, iq* referencers Electric Drives

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**Figure 11.30. A.c. current controllers**

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**Figure 11.31. PM - SM block diagram**

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**Figure 11.32. Speed transient response**

The motor used for this simulation has the following parameters: Pn = 900W, Un = 220V, 2p = 4, n = 1700rpm; In = 3A, lPM = 0.272Wb, Rs = 4.3W, Ld = 0.027H, Lq = 0.067H, J = kgm2.The following figures represent the speed (figure 11.32), torque (figure 11.33) and current waveform (figure 11.34), for a starting process and load torque (2Nm) applied at 0.4s. Figure Speed transient response Electric Drives

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**Figure 11.33. Torque response**

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**Figure 11.34.Current waveform (ia) under steady state**

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The d - q reference current relationships to torque for maximum torque per current are given in figure Figure Electric Drives

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**Example 11. 5. Digital simulation: indirect vector d. c**

Example Digital simulation: indirect vector d.c. current control of PM - SM The simulation of this drive (figure 11.36) was implemented in MATLAB - SIMULINK simulation program. The motor model was integrated in the PM_SM block (figure 11.37). The motor used for this simulation has the following parameters: Pn = 900W, Un = 220V, 2p = 4, n = 1700rpm; In = 3A, lPM = 0.272Wb, Rs = 4.3W, Ld = 0.027H, Lq = 0.067H, J = kgm2. Electric Drives

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**Figure 11.36. The indirect vector d.c. current controller**

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Figure PM - SM model Electric Drives

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**Figure 11.38. id- iq reference currents versus torque for maximum torque / current criterion**

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**Figure 11.39. Speed transient response**

The following figures represent the speed (figure 11.39), torque (figure 11.40) and d - q current responses (figure 11.41, 11.42), for a starting process and load torque (2Nm) applied at 0.4s. Figure Speed transient response Electric Drives

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**Figure 11.40. Torque response**

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**Figure 11.41.Current waveform (id) under steady state**

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**Figure 11.42.Current waveform (iq) under steady state**

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