Dynamic Characteristics--Switched Reluctance Motor Drives (SRM)--part 3

Table __Selected data for two typical SRM

Parameters: s Z - stator's teeth number r Z - rotor's teeth number n P - rated power [kW] n n - rotor speed [rev/min] n U - rated voltage [V] n I - rated source current [A] n T - rated torque value [Nm] ph R - phase winding's resistance [?] max L - aligned inductance [mH] min L - unaligned inductance [mH] 4 3 , 3 2 ; 2 1 - - - M - mutual inductance [% max L ] 4 2 ; 3 1 - - M - mutual inductance [% max L ] ? - motor's efficiency [%] off on a a ; - switching angles [deg]

Exemplary Motors for Simulation and Tests

For the purposes of illustrating the characteristics and dynamic courses of SRM motor drives a selection of two low power motors was made. The two motors are found in the catalogues and were the subject of the research and measurements in laboratory conditions.

Starting of SRM Drive

A considerable problem is encountered during the start-up of SRM motors since the torque is relative to the initial position of the rotor, which is unfamiliar, as a rule. Incremental encoders are used in order to control the motor and determine an instantaneous position of the rotor. However, the latter don’t provide information regarding the position when it’s stalled. Another problem with the start-up, especially with regard to a motor with a small number of teeth e.g. Zs/Zr = 4/2 but also Zs/Zr = 6/4 to a certain extent, is associated with the fact that they don’t develop required start-up torque in every position of the rotor for both directions of rotation. Therefore, for the motors, which for engineering purposes are incapable of starting in every initial position of the rotor, the start-up process either occurs for a small load or the rotor is positioned prior to the starting procedure. Concurrently, the problem associated with determining the initial position of the rotor before starting the motor can be solved by:

- application of absolute encoders, which provide a reading of the initial position;

- application of resolvers, which as externally energized inductance devices that secure a precise measurement of the position in every situation. The two solutions are, however, rather expensive and they are not applied in commonly used drives.

Some other possibilities of preparing the drive to the perform the start-up include:

- positioning as a result of forcing adequately strong current flow through selected windings, i.e. its alignment prior to the start-up;

- determination of the rotor's position and selection of a starting sequence using test pulses applied to the windings prior to its starting, when the rotor is stalled.

The latter option will be examined in more detail below.

Start-Up Control for Switched Reluctance Motor by Pulse Sequence

The application of starting pulse sequence to determine the position of the rotor will be presented with regard to both motors (motor A and B). contains a summary of the test result for motor B (Zs/Zr = 6/4) within the angular range of the rotation of the rotor 0°-60°.

+-+-+- General picture of: a) test current pulses and b) respective torque response for the Zs/Zr = 6/4 motor, in the rotor position range 0°-60°

On that basis, one can conclude that the best conditions for the start up of this motor are encountered in the range from theta_r = 0°…6°, while the least optimum ones are for the angles close to theta_r = 15°. This is so because in this position the teeth on the rotor are displaced in relation to the stator teeth by ±15°,±45°, i.e. the position of the rotor in which the derivative of the rotor winding's inductance assumes a small value. In contrast, for theta_r = 0° the remaining teeth on the rotor are in the position ±30° from the axes of the phases except for the pair in the aligned position. This, in turn, offers optimum conditions for the start-up. The next figure presents a more detailed response of the motor to testing pulse for selected rotor angles.

+-+-+-SRM (motor B, m = 3) response to test voltage pulses in various rotor positions:

a) theta_r = 0° b) theta_r = 9° c) theta_r = 15° d) theta_r = 30°

On that basis, it’s possible to examine several typical start-up situations for selected rotor position angles. For the position angle theta_r = 0° there is a small current pulse in the phase '1' and two large ones in phases '2' and '3', which correspond to the reduced angles ±30°, given by the relation:

Energizing phase '2' during start-up will result in the motion in the positive direction, while energizing phase '3' in motion in the reverse direction. The situation occurring for theta_r = 15°. We have to do with small current pulses in the phase windings '1' and '2' and a strong pulse in phase '3'. This corresponds to the respective reduced rotor angles of ±15° and ±45°. The large current pulse corresponds to angle theta_r = 45°, which occurs between the axis of phase '3' and the axis of the closest tooth on the rotor, but the resulting torque is virtually nonexistent. The supply of the phase '2' leads to the rotation of the rotor in the positive direction, while of the phase '1' in the negative one. In these cases the torque is, however, three times smaller than for the angle theta_r = 0° and, hence, the start-up can be impeded. The most problematic starting conditions are encountered for theta_r = 9°, i.e. for 1,2,3 = 9°,-21°,39°, while as phases '1' or '3' are energized the negative torque is very small as it reaches around 20% of the value for angle theta_r = 0°. This results from the fact that the position angle 9° is quite small in relation to the half of the pitch of the tooth tr/2 = 45°, it’s too close to the aligned position.

Concurrently, for angle 39° the current is already considerably large since inductance is small but it’s too close to the limit of 45°, when we have to do with a change of the torque sense. So in consequence there is not a good option for the negative direction start-up. The final illustration in Fig 5.22d presents the effects of a cyclic power supply - the situation is such like for theta_r = 0°, only phase '2' re places phase '1', while phase '3' replaces phase '2' and, in turn phase '1' is in the place of phase '3'. The more general conclusion from the test is that in order to conduct the start up one should energize the phase winding for which there is a current pulse with the mean value or a winding for which the pulse response is as close to the mean value as possible. During energizing of the winding, whose pulse precedes the phase in which there was a pulse with the highest value, the start-up torque is positive and if we supply the phase that follows the one with the highest pulse response, the start-up torque is negative. One has to bear in mind that all responses in the form of current pulses are positive as we apply positive voltage pulses, which in the presented examples are equal to 30% of the rated voltage and are 0.5 [ms] in duration. The negative value of a current in a phase winding does not change the sense of the reluctance torque. The following figures illustrate the situations associated with pulse determination of the rotor's position and setting the start-up sequence of motor A (Zs/Zr = 8/6) in the range of the rotation angle 0°…15°.

+-+-+- Response for voltage test pulses in SRM motor A (m = 4) for rotor position

theta_r = 0°…20°: a) current peaks b) corresponding torque jerks

From +-+-+-23 it’s possible to conclude that the best start-up conditions are encountered for the rotor angle theta_r = 6°…9°. However, for 4 phase windings in every position it’s possible to conduct the start-up even under considerable motor load. The following illustrates the results of the pulse start-up test for a number of selected rotor positions.

+-+-+- SRM response (motor A, m = 4) for test voltage pulses applied to the consecutive windings in several rotor positions: a) theta_r = 0° b) theta_r = 6° c) theta_r = 12° d) theta_r = 15°.

For the motor with the phase number m = 4, the subsequent positions of the ro tor's teeth in relation to the axes of the stator teeth, i.e. the subsequent values of the reduced angle result from the relation:

+-+-+-24a contains test results for theta_r = 0° and for this case the subsequent values of the reduced angle are equal to ?1,2,3,4 = 0°,-15°,-30°,15°, respectively, and the large start-up torque is encountered for the supply of phases '2' and '4', i.e. for

theta_r = ±15°. The highest current pulse in the winding corresponds to angle theta_r = ±30° since it corresponds to the position with the minimum inductance in the winding.

+-+-+- presents the result of this test for angle theta_r = 6°. The following values of the reduced angle: ?1,2,3,4 = 6°,-9°,-24°,21° correspond to this position of the rotor teeth in relation to respective stator teeth. The highest current pulses correspond to the angles °,-24°,21° for phases '3' and '4', together with the highest values of the start-up torque. Another example is found for the angle theta_r = 12°. The values of the reduced angle of rotation for the following phases are equal to: ?1,2,3,4 = 12°,-3°,-18°,27°, and the highest current pulse is encountered while energizing phase '4', i.e. for ?4 = 27°. Concurrently, the positive start-up torque is en countered for phase '3' (?3 = -18°) while the negative one for phase '1', i.e. for the angle ?1 = 12°. The situation presented in +-+-+-24d regards angle theta_r = 15°, i.e. the one that is equal to the stroke angle. This well illustrates cyclic characteristics of the test pulse since, as one can see, the roles of the subsequent phases are shifted. The rules regarding the start-up sequence for the case of the motor with m = 4 phases are similar to the ones presented previously. In the examined case the start-up torque with a considerable value is produced during energizing the phase whose pulse response is the second of the four in terms of its value along with the one whose pulse response is most close to the previously selected one. In the example these are, respectively, phases '3' and '1', and for the case - phases '3' and '4'. The torque with the negative value is produced for the supply of the winding with the successive number (modulo m) in relation to the phase with the highest response of the current pulse and the positive torque is generated during energizing of the phase with the preceding number of the one with the largest pulse if it qualifies as the one that is closest to the previously selected one in terms of the value.

+-+-+-The outline of the algorithm for a starting sequence of SRM based on test impulses before starting.

This may sound complex; thus, it’s presented in the form of an algorithm. This algorithm has been prepared for the motor with m = 4 phases, however, for the motor with three phases the algorithm takes the same form except for the limitation of the set of phases' numbers to {} 3 , 2 , 1 ? z . This algorithm involves determination of the phase number {} 4 , 3 , 2 , 1 ? z that is to be energized in the first order during motor start-up depending on the selected direction of the rotational speed:

- and on the basis of the pulse test for all 4 phases. One should note that in the algorithm the pulse 'c' determined as the one that is closest in terms of the amplitude to the second highest called 'b' could be the one with the largest value , i.e. the one described as 'a'.

Current Delimitation during Direct Motor Starting

The direct start-up of SRM motor for the case of conducting starting sequence from an adequate phase occurs effectively. Concurrently, there is a need of limiting the start-up current in the initial period of starting sequence. In case the drive has sensors of the phase currents this limitation can be easily implemented as a result of introduction of upper and lower boundaries of current fluctuations. 'Soft-chopping' is used within this range of regulation and involves de-energizing of one of the transistors of the H bridge after exceeding the upper current limit followed by a natural decay of the current until it reaches the lower boundary. Subsequently, the previously de-energized transistor switches on, the power is restored and the current in the phase winding rises again. This way of regulation accomplishes its role for a small number of the switching sequences in a pulse, which is mostly relative to the boundaries of current changes imposed by these limitations.

+-+-+- the initial period of the start-up for motor A, for motor B, for the case of imposed start-up current limitations and rated loading.

+-+-+- the complete starting range of motor A. On this example one can see series excitation effects of the SRM machine. It demonstrates in a slow long lasting speed increase in the final part of the starting course. This is associated with the gradual decrease of the current and de-excitation of the motor, thus causing the increasing speed.

+-+-+- Initial part of starting current of SRM motor A, with current limitation within the range of 100 [A]…90 [A]: a) phase currents b) electromagnetic torque

+-+-+- The whole starting course of SRM motor A with a current limitation and a nominal load: a) phase currents b) electromagnetic torque c) rotational speed

Braking and Generating by SRM

Generator regime of operation of SRM machine is not provided for either in the engineering structure of the machine itself nor due to the structure of the semi controlled H bridge from which it’s controlled. The rotor of the reluctance SRM motor is not energized and the excitation flux of the machine comes from the current in the stator windings. Hence, there is a lack of separately regulated excitation current that is typical for generators or permanent magnets that offer an excitation flux of the machine practically regardless of the machine load.

+-+-+- Starting of SRM motor B with current limitation in the range of 15 [A]…12 [A]:

a) phase currents b) electromagnetic torque

+-+-+- Excitation period and generating period of SRG machine during one switching cycle: a) current cycle b) current flow in a H bridge for one phase winging; i_e - braking current flow, i_g - generating current flow

+-+-+- Steady-state characteristics for a generating mode of SRM operation (motor A) as a function of rotor speed, for u = Un , aon = 16°, aoff = -10°,-12°: a) phase and source cur rents b) torque and overall efficiency

+-+-+- Unstable operation of SR machine as a generator: EMF, phase currents, torque and rotor's speed courses: a) under an equilibrium point of balance b) above an equilibrium point of balance.

+-+-+-Steady state characteristics of SRG (A) for different excitation level realized by ku factor: a) rotational speed b) phase and source currents as a torque functions c) torque and efficiency as ku functions d) shape of current pulses

The typical transistor-diode H bridge that energizes the machine's windings from the DC source makes it possible to control the machine's current only for the case of the motor regime of operation. The return of power from phase windings into the source can only occur after switching off both transistors and takes place in an uncontrolled manner until the magnetic field associated with the winding decays.

Moreover, the generation and braking of SRM machine is not steady, which results from the curves of the static characteristics presented and is con firmed by the waveforms presenting the unsteady operation in the vicinity of the equilibrium point in the system without feedback.

However, for adequate control using angles aon, aoff and in the system with feedback for the regulation of the output power the SRM machine is capable of performing the duties of the generator. In the generation regime of operation each pulse of the machine's current consists of two parts. In the first part, for two transistor in the ON state there is an excitation of the SR genera tor, sometimes denoted as SRG, since the machine draws current from the source and it operates as a brake or a motor depending on the sense of the torque produced in this period. The transformation of the mechanical power in this period is inconsiderable since the angle aon for which the machine operates as a generator precedes the aligned position by a few degrees and the excitation occurs for the small values of electromagnetic torque which actually changes the sense from the positive to the negative one. After the excitation period the two transistors are switched off for the angle of the rotor of aoff, which happens several degrees after the aligned position of the rotor and stator teeth is reached. Following we have to do with the generation regime of operation until the decay of the current in the winding, which in this case closes through two diodes of the H bridge and thus energy returns into the source. In this range of operation the energy conversion is performed at the expense of mechanical energy, since the electromagnetic torque is negative and to some extent due to magnetic field energy associated with winding's current. The illustration of the excitation and subsequently, generator regime operation of SRM is presented.

For adequately selected control angles aon , aoff the mean value of the generated current is considerably higher than the mean value of excitation current and the machine operates as a generator with a decent energy efficiency, which is, however, lower than for its motor operation.

From the characteristics one can see that the electromagnetic torque and phase currents decrease along with an increase of the rotor speed, which fore casts an unsteady characteristics of the machine's operation within this range. This is confirmed by its transients, which present the behavior of the generator after the balance is disturbed.

+-+-+-Linear transition from motor to generator regime of work of SRM (A), after linear change of: aon = 35°?13° and aoff = 10°?-12° , n= 1850 [rev/min]: a) phase currents b) electromagnetic torque Te c) partial torques constituting total Te torque

+-+-+-Fast change from motor to generator operation for: aon = 35°?16°, aoff = 10°? 10°, Tl = 3.0?-3.3 [Nm]: a) phase currents b) EMF in a winding c) electromagnetic torque d) rotational speed

+-+-+-Detail shape of a phase current and induced EMF in a phase winding for: a) motor; b) generator mode of SRM operation, control parameters.

The unstable equilibrium in the conditions presented occur for the speed n = 2100 [rev/min] and the load Tl = -3.0[Nm]. The generation regime can be stabilized as a result of including adequate feedback relative to the speed and acting upon the pulse width modulation coefficient ku defined for the PWM control, which in case of constant source voltage U regulates a level of an excitation.

This is so because the PWM voltage control by the ku factor is acting during transistor operation of the H bridge and this way it effects the excitation level. As a consequence, ku factor regulation is a main tool to control indirectly electromagnetic torque during the generator operation with a constant source voltage. The characteristics that indicate this possibility for motor A are presented.

A smooth transfer from the motor to generator regime of operation. In this case we have to do with linear change of the control angles aon = 35°-13° and aoff = 10°?-12° as well as a change of the torque on the machine shaft from the load of Tl = 3.2 [Nm] to the torque driving the generator Tl = -3.8 [Nm]. The energy efficiency for this state of the generator regime is equal to -g = 68%, in contrast to _s = 82.5% for the motor regime preceding the change of the operating regime.

The transfer from the motor to generator regime can also occur fast and does not pose any problems to the stability of the drive. An example of such fast change is illustrated. In the presented example we have to do with a prompt switching of the control: aon = 35°?16° and aoff = 10°?-10° and for the load torque of Tl = 3.0 [Nm]?-3.3 [Nm]. --- presents the detailed time waveform of the phase current and induced EMF in the SRM machine for motor and generator regime in the conditions defined. One can note the change in the shape of the waveform for phase current and EMF, which reminds of a reverse rotation of the machine for the motor regime.

The effectiveness of operation and energy efficiency of the SRM machine during generator regime are considerably relative to the control angles aon, aoff and ku factor controlling the excitation level. Under the assumption that energy is sup plied in an optimum way during excitation most of it can be returned into the source during generator regime and, thus, the efficiency is quite high. Under the assumption of a constant rotational speed and constant source voltage U of the drive this efficiency can be expressed by the relation: where the particular symbols denote: P1 - mechanical power output of the drive P2 - electric power used by the drive P_exc - electric power drawn form the source during excitation part of the cycle P_gen - electric power returned into the source during generation part of the cycle Tl - load torque I_exc,av - mean value of source current during excitation part of a cycle I_gen,av - mean value of source current during generation part of a cycle theta_r,av - mean rotational speed averaging the effect of pulsation.

The efficiency defined in this way, which is positive for generator regime, has a negative numerator since the mean current in the generator cycle is considerably higher than the mean excitation current as well as a negative denominator of the expression ( __) since the machine is actually driven from outside, which means that the load torque is also negative. Examples of the characteristics of the efficiency of the machine A - they illustrate clearly the effect of the control angle and ku factor respectively on their wave forms. The effects of the control angles on the operation and efficiency of SRG generator are presented in a number of bibliography items stressing the complexity of the issue. This complexity is due to the fact that during the genera tor regime the same point on the mechanical characteristics of SR generators (i.e. rotor speed and torque) can be obtained for various values of control variables aon, aoff and ku however, the current and efficiency differ considerably. Moreover, some bibliography items in this subject discuss the excitation and self-excitation of SR generators as well as generation for a high and low rotational speeds of the machine. The latter results from the potential application of SRGs in wind power stations in the engineering models involving mechanical gear and ones without it.