CHAPTER 2 MODELLING OF SWITCHED RELUCTANCE MOTORS

Transcription

1 9 CHAPTER 2 MODELLING OF SWITCHED RELUCTANCE MOTORS 2.1 INTRODUCTION The Switched Reluctance Motor (SRM) has a simple design with a rotor without windings and a stator with windings located at the poles. The inherent simplicity, ruggedness, and low cost of the SRM make it possess strong competition in many adjustable speed and servo-type applications. The simplicity of the motor construction promised low cost in manufacturing, which in turn, has motivated the researcher s interest. rotor stator Figure 2.1 SRM with 8/6 poles A four-phase SRM, as shown in Figure 2.1, with eight poles in the stator and six poles in the rotor, is alternatively the called 8:6 SRM. The simplicity of the motor construction, however, has a crucial disadvantage due

2 10 to the double saliency of the SRM causing its highly non-linear magnetic characteristics. Hence, understanding the motor s magnetic property is essential for proper control operation. It is important to have a knowledge of the rotor position for the good performance of the SRM, and traditionally some form of a rotor position sensor achieved it. Nowadays, there is extensive research going on to eliminate direct rotor position sensors, by indirectly determining the rotor position Working Principle of SRM The Switched Reluctance Machine operates on the tendency of the rotor, in order to move to a position where the reluctance is minimized. Switched Reluctance Motors are also named as Electronically Commutated Machines. These two names fully explain the motor's operation. Stator windings are energized at specific times to change the rotating magnetic field to move rotor poles to a position of minimized reluctance, or equivalently maximized inductance. This position is where the rotor pole is aligned with the energized stator pole. Movement in different directions and at different speeds can be achieved by exciting stator windings in a particular sequence with a particular timing. In the SRM, motion is produced by the variation of reluctance in the air gap between the stator and the rotor. When a stator winding is energized, producing a single magnetic field, a reluctance torque is produced by the tendency of the rotor to its minimum reluctance position. When a rotor pole is aligned with a stator pole as in Figure 2.2, there is no torque because field lines are orthogonal to the surfaces (considering a small gap). In this position, the inductance is maximum since reluctance is minimum. If the rotor s position is displaced, there will be torque production that will tend to bring back the rotor towards the aligned position.

3 11 Stator Rotor Figure 2.2 Aligned position L a L na Figure 2.3 Unaligned position Current is injected in the phase winding, when it is in the unaligned position, as shown in Figure 2.3; and there will be very little or no torque production. If one displaces the rotor of the unaligned position, then a torque tends to displace the rotor toward the next aligned position. The instants at which the stator currents are switched on and off are controlled basically by the observation of the rotor position by some appropriate sensor incorporated in the machine. As a current sensor is also

4 12 needed in practice, it is essential to monitor the exciting current during the low speed operation, since each phase period is of long duration and the excitation has to be chopped to restrict each phase current within the semiconductor ratings. In addition, control over the torque production is achieved by varying the mean phase current. At high speeds, current control is not necessary, since the inductance of the winding restricts the excitation to a single pulse of current. 2.2 STATIC TORQUE PRODUCTION Consider the Switched Reluctance Motor as shown in Figure 2.4; when current is passed through the phase winding of the SRM, the rotor tends to align with the stator poles, that is, it produces a torque that tends to move the rotor in a minimum reluctance position. Figure 2.4 Elementary Switched Reluctance Motor

5 13 Figure 2.5 Field energy and Co-energy In a device of the static torque production in SRM motor, the most general expression for the instantaneous torque is given as T 1 δw = i (2.1) δθ where W 1 is the co-energy defined as in Figure 2.5 W 1 = i ψ. di (2.2) 0 The relationship between the flux linkage and the current at the instant rotor position θ is a straight line whose slope is the instantaneous inductance L. Thus, ψ = L *i (2.3)

6 14 and 1 Li 2 W 1 = 2 (2.4) Therefore, the torque is given by T 1 2 dl = i (2.5) 2 dθ From the above formula it is evident that the direction of the torque produced is independent of the current. But, it depends upon the sign of dl/dθ. When the rotor poles approach the aligned position, it produces a positive monitoring torque. When the rotor poles leave the aligned position and approach the unaligned position, braking or regenerating torque is produced. Therefore, the ideal waveform is a rectangular pulse that coincides with the rising inductance. In order to produce torque at all rotor positions, the entire 360 degrees must be covered by the segments of rising inductance from different phases, and the phase currents must be commutated and sequenced to coincide with the appropriate segments. A small amount of overlap is desirable to minimize the torque ripple. The SRM has two types of characteristics; they are 1. Static characteristics 2. Dynamic characteristics Static characteristics are those, which are obtained when the motor is in the steady-state operation mode; for example, inductance versus rotor

7 15 position. On the other hand, dynamic characteristics are obtained during the transient region of the motor operation. More emphasis is laid on the dynamic characteristics in this research study. These characteristics are usually, plots phase current, phase torque, inductance profile, and shaft torque with respect to time. Under normal operating conditions at a specified speed, the energy exchanges, between both incremental and total, can be determined by integrating the voltage equation and developing the conversion loop in the Ψ-I characteristics Figure 2.6 shows Ψ-I characteristics of the motor, which are considered in this study. Figure 2.6 Ψ-I Characteristics of the motor The necessary procedures were developed by Stephenson and Corda, and only the outline of their methods is given as follows.

8 16 The voltage equation is integrated in the form and is given as ψ = (v ri) dt (2.6) Through a one time step, one will get a new value of Ψ. If the speed is assumed to be constant, the integration can be done with respect to the rotor angle θ. Otherwise, the rotor angle must be determined by a simultaneous integration of the mechanical equations of motion, as in normal form of such simulations. The mechanical equations are And, dθ = ω dt (2.7) dω 1 = [T d ( θ,i) fω T L ] (2.8) dt J The initial rotor position is specified and the values of inductance and excitation are obtained from the inductance profile, and control the algorithm. These values when supplied to the circuit, give the instantaneous value of current in the three phases / four phases, from which the torque is calculated and is given as, 1 2 dl Td = i Nm. 2 dθ (2.9) To calculate the speed the mechanical equation (2.8) is used. Hence the equation (2.8) can be written as, 1 ω = [T ( θ,i) f ω T ] J d L rad/s (2.10)

9 17 and the rotor angle can be written as, θ = ωdt rad (2.11) Then, this closed loop system is repeated to obtain the values of the current torque and speed of the machine, and hence, the dynamic characterization of the SRM is obtained. 2.3 POWER CONVERTER A considerable number of converter topologies for the SRM have been published by many researchers, including Stefanovic and Vukosavic (1991), Hava et al (1992), Krishnan (1996), Mir et al (1997) and Barnes and Pollock (1995 and 1998). Among these well-known topologies, the C-dump converter has received considerable attention, because it uses only (n+1) switches to achieve independent phase magnetization and full suppressing voltage. Aiming to use the minimum number of switches, to keep the suppressing voltage independent of the dc-link voltage to the maximum extent, to enable single-pulse operation, and to maintain the capacitor voltage at lower level, a hybrid C-dump and buck-fronted converter is proposed, as shown in Figure 2.7 (a & b). Based on the buck-fronted converter, four additional passive components are added to the proposed converter circuit: capacitor C d and diodes D d1, D d2 and D d3. The C d functions as the dump capacitor, whose voltage V cd is automatically maintained at V cd. When V cd is higher than V dc, C d will discharge via the chopper, until its voltage drops to V dc. When the chopper switch is off, this converter works like the C-dump converter in Figure. 2.7 (a). The energy from the off-going phase winding is dumped into C d, and the chopper capacitor C c is bypassed; therefore, the suppressing

10 18 voltage of the phase is -V cd, instead of - (V dc -V cc ). When the chopper switch is on, this converter works like the buck-fronted converter: the energy from the off-going phase winding goes via the chopper inductor, which results in a suppressing voltage of about-(-v dc -V cc ). V cc Figure 2.7(a) Circuit Diagram of the Proposed Converter for single phase Converter Design Thus, the proposed topology is superior to the basic buck-fronted converter, in that the chopper switch is further utilized, i.e., turning off the chopper switch will bring in higher suppressing voltage. If the duty cycle of the chopper switch is not high, the effective suppressing voltage will be significantly increased. If the duty cycle of the chopper switch is very high and the higher suppressing voltage brought by the off state of the chopper switch is insufficient, one can deliberately turn off the chopper switch (or decrease the duty ratio temporarily) for a certain period after the phase switch is turned off, so that a high suppressing voltage will be maintained during this forced off- period.

11 19 Figure 2.7 (b) Complete Circuit Diagram of Hybrid C-Dump Converter Figure 2.8 (a to e) illustrates the five basic modes of operation of this hybrid converter, for phase 1 only. Figure 2.8(a) shows mode 1 operation, with both S c and S 1 on, the DC-voltage source charging the chopper capacitor C c and magnetizing phase 1. The energizing voltage across the phase winding is equal to the chopper capacitor voltage, which is regulated by the chopper at an appropriate value to support single-pulse operation. Figure 2.8(a) Mode 1 operation of the Proposed Converter

12 20 Figure 2.8(b) shows mode 2 operation, with both S c and S 1 on, but this time the dump capacitor C d discharges to supply the energy for the chopper and the phase winding. Whether the chopper and phase windings are fed in mode 1 or mode 2 is determined by the dump capacitor voltage; if V cd > V dc the converter works in mode 2; otherwise, the converter works in mode 1. Thus, the dump capacitor voltage is automatically maintained at V dc. Figure 2.8(b) Mode 2 operation of the Proposed Converter Mode 3 operation is shown in Figure 2.8(c), where S c is off and S 1 is on, and the phase winding is magnetized by the energy stored in the chopper components. Figure 2.8(c) Mode 3 operation of the Proposed Converter

13 21 Mode 4 and mode 5 operations are used for phase demagnetization. Figure 2.8(d) shows mode 4 operation, in which both S 1 and S c are off, and the phase current continues flowing via D 1, C d and D d2, charging the capacitor C d. The suppressing voltage is equal to V cd, at around V dc. While phase 1 is in mode 4, phase 2 may be in mode 3, if the phase currents overlap. Figure 2.8(d) Mode 4 operation of the Proposed Converter Figure 2.8(e) illustrates mode 5 operation where S 1 is off and S c is on, and the residual current flows via D 1, S c, and L c. The suppressing voltage is the voltage difference between V dc (or V cd ) and V cc. Again, while phase 1 is in mode 5, phase 2 may be in mode 1or mode2. Figure 2.8(e) Mode 5 operation of the Proposed Converter

14 22 For the dump capacitor C d, its value should be large enough to limit the voltage rise caused by the off-going phase current t d 1 V cd (1 2d) Irdt C (2.12) d 0 Where t d = current fall time; I r = residual current; d = duty ratio of the chopper switch. Under typical operation conditions t d =2ms and I r (t=0)=10 A in order to keep the voltage rise to less than 50 V; as a conservative design, 1 1 V <.t.i ( t = 0) < 50V cd d r Cd 2 Thus C d >200uf For the chopper capacitor, the concern is the voltage drop caused by the temporary phase current overlap. The voltage drop is roughly estimated as t overlap 1 Vcc = I r.dt C (2.13) c 0 where t overlap is the period when adjacent phases are on simultaneously. Using typical values, t overlap =2ms, I r =10 A, and C c =2000 uf are used to limit the voltage drop within 10V. The chopper inductor is simply required to limit the current variation of the buck chopper.

15 NON-LINEAR MODELLING OF SRM There are two possible ways to generate training data for a nonlinear modelling of the SRM. They are model-based data generation and experiment-based data generation. Model-based Data Generation: A suitable magnetization model for the associated SRM is used to generate the data. Given a proper model, flux linkage values are computed for randomly generated phase current and rotor position values so that the resulting flux linkage, phase current and rotor position values will judiciously cover the intended operating region. Experiment-based Data Generation: In this approach the motor is run from certain operating points with a shaft encoder, so that the magnetization characteristic is swept over certain regions, or, in a better approach, the motor is run from zero speed to full speed and every electric cycle of the flux linkage, phase current and rotor position are captured with a certain sampling rate. This allows a more judicious coverage of the magnetization characteristic. I Ψ Position Estimator of SRM θ Non-linear Modelling of SRM T d Figure 2.9 Block diagram of development of SRM model Figure 2.9 shows the block diagram of the SRM model development. Initially the values for phase flux linkages and phase currents (i) are given to calculate the position estimation of SRM. For various speed (ω),

16 24 the motor torque can be calculated using equations (2.7) to (2.11). In this research study, model-based data are used for 8/6 SRM and experiment based data are used for 6/4 SRM. The specifications of the SRM used in this research study is attached in the Annexure I. 2.5 CONCLUSION This chapter describes the basic theory and operating principle of the SRM along with mathematical modelling. It also gives an account of the operation of the power converter and its operating modes. It shows how, with the help of the above equations (2.1) to (2.11), the mathematical model for the SRM drive is developed.