BLDC Machine Drives--Characteristics

This section will be devoted to the computer simulations of characteristics of brushless DC drives. Such issues include: start-up, braking and drive reversal, control of rotational speed and tracking control of the drive as well as its reaction to variable parameters of the supply and loading. Since these detailed issues can be illustrated with the aid of adequately selected results of dynamic calculations, it’s important to select motors for the demonstration of the drive in operation beside the presentation of topics devoted to mathematical modeling of the drive. For theses purposes the parameters of two BLDC motors are presented: one with a smaller and the other with the higher power and different supply voltages. A summary of the parameters is found below.

Table 3 Rated data of two PMDC motors

Rated Parameters:

Power; Voltage; Velocity of rotation; Armature current; Torque; Efficiency; Windings self-inductance; Mutual inductance; PM excitation flux; Windings resistance; Moment of inertia; Damping factor; Pulse width Commut. advance angle

Start-Up and Reversal of a Drive

Drive Start-Up

Start-up forms the basic issue associated with the motion of a drive and, hence, the motor drive and the control system have to fulfill a number of prerequisites in order to ensure the appropriate course of the process. These prerequisites include: possibility of start-up from every initial position, start-up with a required load as well as limitation of the start-up current to the values acceptable by the motor and the supply system. The process of start-up of BLDC motor is further impeded as a result of occurrence of parasitic torques, i.e. reluctance torque and cogging torque. The two effects are reduced during the process of motor design in a manner that they are not manifested too strongly during the start-up. The limitation of the start-up current can be achieved in two ways: by incremental increase of the voltage supplying the armature using PWM method or as a result of controlling the start-up current by means of PWM method as well, relative to the instantaneous value of the current. A smooth increase of the volt age during the start-up can be achieved in numerous manners. In simulation models applied for the demonstration of the start-up curves it’s achieved by exponential in crease of ku coefficient, i.e. the one denoting the active part of the pulse. The figures that follow illustrate the start-up curves for a motor with higher power, i.e. motor marked B in Table 3 as well as for a smaller motor A under a rated load. The comparison will involve the start-up curves for the d-q transformed model as well as for the untransformed motor.

+-+-+- Starting of the B (6.6 [kW]) motor with a voltage regulation: a) armature phase current b) 3-phase currents c) currents in the steady state. The results obtained from the un transformed model of BLDC.

+-+-+- The same course for: a) electromagnetic torque b) reluctance torque c) rotational speed

+-+-+- Starting of the B (6.6 [kW]) motor. Results obtained by the transformed model of BLDC: a) transformed id, iq currents, b) transformed back armature currents i_a

+-+-+- Starting of the B (6.6 [kW]) motor. Results obtained by the transformed model of BLDC: a) electromagnetic torque b) rotor's speed.

The two figures that follow present the start-up of the same motor, however, for the application of a current delimiter. The operating principle of the device involve the division or multiplication of the pulse width coefficient ku by a reduction coefficient red in the subsequent pulses depending on whether the value of the current in any of the phases exceeds or does not reach the value of the imposed limitation Ir.

+-+-+- Starting of the B (6.6 [kW]) motor with a current delimiter set for Ir = 110 [A]: a) single phase current b) 3-phase currents c) U·ku armature voltage.

+-+-+- The same course but for: a) electromagnetic torque b) rotational speed.

The illustrations that follow present the curves of the start-up of the smaller motor (motor A) gained as a result of applying untransformed and transformed models of the BLDC motor. As one can conclude, the two curves are very similar, in particular with regard to the mapping of electromechanical variables. Considerable differences are noted in terms of the current curves since in the transformed model the commutation is not as precisely modeled. In conclusion, in terms of the quality of the modeling untransformed model is a better one, while the basic advantage of the transformed model involves the 10 to 20 times decreased cost of simulations. For these reasons the transformed model of BLDC motor presents more advantages during simulations of large electromechanical systems in which a greater number of drives is present. Concurrently, the reduction of the duration of the calculations forms a considerable premise in favor of the execution of the simulations of the operation of the system.

+-+-+- Starting of the BLDC motor A (0.96 [kW]). The results of simulation by untransformed model: a) armature currents b) shape of current curves c) electromagnetic torque d) reluctance torque e) rotational speed.

+-+-+- Starting of the BLDC motor A (0.96 [kW]). The results of simulation by trans formed model: a) d-q currents b) armature currents c) shape of armature currents d) electromagnetic torque e) rotational speed.

Reversing DC Motor

Brushless DC machine, just as the commutator machine based one can serve very well for operation in both directions of rotation. However, it’s necessary that two fundamental conditions are met: the system of the supply and control has to be prepared for such circumstances and a particular construction of the cooling sys tem or an adequate air-flow across the machine has to be provided. A separate is sue is associated with the operation of the drive at small rotational speed and ensuring that heat is carried away in such conditions, thus, that the temperature inside the machine does not exceed the permitted limit. This may be associated with an application of a machine with independently driven fans. In order to per form the start-up of a BLDC motor in the reverse direction, it’s necessary to change the sequence of the supply of the motor phases and reverse the value of the delay angle d. A similar course of action is assumed for the case when one intends to perform reversing of a motor during its operation. The switching on of the re verse direction of rotation in accordance with the preceding description first results in a period of deceleration, named counter-current braking. After that, when the drive reaches zero speed of rotation the drive starts the operation in the reverse direction. Such a manner adopted for motor reversal generally requires the limitation of the armature current due to the conditions of the supply system and the admissible motor current. One can note at this point that exceeding by the start-up current the maximum admissible value can cause an irreversible deterioration of the machine's field of excitation originating from permanent magnets. The results of the simulation studies present the curves of the reversing of the BLDC drive without the introduction of a limitation on the current as well as during a considerable current limitation.

+-+-+- Reversing of PMDC motor (A) without current limitation: a) single phase current; b) 3-phase armature currents c) electromagnetic torque d) reluctance component of the torque e) rotational speed.

+-+-+- Reversing of PMDC motor (A) with current delimiter set on 30 [A]: a) single phase current b) 3-phase armature currents c) electromagnetic torque d) reluctance component of the torque e) angular speed.

From the comparison of the curves presented it stems that the current delimiter during the reversing of the motor operates effectively; but the reversing of the motor lasts two times longer than in the one without a de limiter if it’s considered until the time of the transfer of the speed across zero.

However, complete control over the current is present, which ensures safety of permanent magnets and the electronic commutation system supplying the drive.

Besides, one can observe that the effective operation of the delimiter, designed as the a fraction multiplier red of the PWM coefficient, considerably depends on its value. It follows the algorithm:

For the curves presented the value of his factor is set at red = 0.3.

Characteristics of BLDC Machine Drive

The term drive characteristics denotes the graphical representation of a set of points representing the operation of a drive relative to the selected parameters characterizing its operation. The parameter considered as the independent variable is found on the X-axis, while the Y-axis denotes the values of the examined variable considered as the output variable. Often the same figure contains a family of the characteristics for which the particular curve differ in terms of another parameter that is very important for the presentation of the operation of the drive, i.e. one for whose course its value is constant. If on the X-axis we find a parameter that is not time, then such a characteristic can be termed as the steady-state characteristic (curve). This characteristic forms a set of points for which the dynamic trajectory finds final steady state, if such a stationary state exists at all. The entire static characteristic informs at which point of the operation the drive is currently found after the termination of the dynamic process, i.e. for instance start-up, braking, change of the parameters of the supply or loading for a given parameter on the X axis. One should note, however, that a change in the state of the operation of the drive (dynamic trajectory) does not overlap with the static characteristic, since if this were the case, the duration of the execution of the designed trajectory would be infinitely long. One can say that the trajectory begins and ends at the static characteristic; however, its curve is different than the one for the characteristic since it occurs in a determined, finite and very often short time. The shorter the time, the further the trajectory is from the static characteristic curve. Another type of characteristic is the one in which X-axis contains time t. In such a case the curve takes the form of a time history for a given variable and generally has a different waveform for other parameters of the drive operation. It’s also relative to the initial conditions from which the curve originated. If for such a curve there is a steady state, the steady value of this state forms a component of a corresponding static characteristic. The static characteristics can be derived in a number of ways.

For a ready drive we can use a method of measurements for appropriate possibilities of variation of the parameters of the supply and load of the examined drive. If we have a mathematical model of a drive available, static characteristics can be derived by definition by conducting dynamic calculations and performing simulations until the steady state is obtained. Such calculations have to be conducted separately for each point that determines the characteristic. Concurrently, there is a possibility of assuming adequately favorable initial conditions, whose dynamic trajectory leads sufficiently fast to a steady state. Despite that, it’s a cost and time consuming enterprise. Another effective method involves the substitution of hypothetical steady states to the mathematical model in the form of differential equations of motion and converting the model into a system of algebraic equations.

This procedure has been followed for instance during the introduction of the equivalent diagram of the induction motor. If we are capable of effectively gaining such a reduction of the differential model to an algebraic model, as a result we will obtain static characteristics of the drive in the form of functional relations between the parameters and variables. However, one has to bear in mind that not all of the obtained characteristics have to be available as a result of the termination of the dynamic curve or need not be available for arbitrary initial conditions, i.e. from any starting point. From that it stems that, as a principle, the mathematical models of electromechanical systems are non-linear.

The static characteristics of the BLDC can be gained from transformed model by algebraization after substituting constant functions for its variables:

The transformed voltages uq av, ud av that are present in this model are described by relations and account for the relation with commutation advance angle d.

As a result of the substitutions of the fixed variables in the mathematical model in the form in we obtain a system of three algebraic equations in the form: The non-linear system of algebraic equations accounts for three variables of the steady state of the drive (Iq, Id, theta_r), parameters of the supply uq av, ud av relative to U, d, parameters of the load Tl, D and engineering parameters of the drive, such as p, Lq, Ld, Ms, Ra, ?f. Such a static model makes it possible to determine the characteristics for selected variables in the subject of the examination. Important examples include mechanical characteristics theta_r, Iq, Id, Ia = f(Tl), i.e. characteristics in the function of the load torque for the remaining parameters with constant values, including parameters of the supply. The non-linear system of algebraic equations of the steady state, that are cubic in relation to variables (Iq, Id, theta_r), can be solved effectively using numerical methods, whose applications are widely found in a number of popular mathematical packages. In this case mathematical package MAPLE V was applied in order to gain the further presented characteristics. The voltages uq av, ud av of the transformed model calculated in accordance with contain phase voltages U ph, whose value for a typical supply of the BLDC motor are assumed in the form … which forms a simplification by assuming sinusoidal waveforms of the i1, i2, i3 currents in the particular phase windings of the motor. The algebraic model of the motor under of the supply of phase windings in accordance with has made it possible to determine the static characteristics in the both researched BLDC motors. For the 6.6 [kW] motor the characteristics in the function of the commutation advance angle d are presented. These are the respective families of characteristics Iq, Id, Ia = f(d) and theta_r, ? = f(d), for four values of the load torque T_l = 15, 25, 35, 45 [Nm]. More detailed descriptions are found in the captions under the figures.

From the presented curves stem a number of conclusions. Iq transformed cur rent does not depend much on advance angle d, but mainly on load torque Tl, as it decides on electromagnetic torque value at the steady state. According to, it’s equal to:

The current Id is considerably relative to the advance angle d since this corresponds to a change in the position of the axis of the brushes in a classical DC machine with a mechanical commutator.

The commutation advance angle should be set in a manner that ensures that the value of Id is close to zero, i.e. in the range of d = 25º…35º in the characteristic presented. For this case the operation of the drive occurs at a mini mum armature current Ia and maximum efficiency ?. The selection of higher values in this range makes it possible to ensure the operation of the drive for a higher rotational speed theta_r at the expense of the deterioration of efficiency.

Another group of characteristics presented shows the same variables as formerly but the results are presented is in the function of the load torque Tl. The other parameter in the figures that offers a distinction between the particular waveforms is the advance angle d = 10º, 20º, 30º, 40º.

+-+-+- Characteristics of currents for the 6.6 [kW] motor in a function of load torque T_l: a) Iq transformed current for d = 40º, 30º, 20º, 10º (top - down) b) Id transformed current for d = 40º, 30º, 20º, 10º (top - down), c) Ia armature current for d = 40º, 30º, 20º, 10º (top - down, at T_l = 10 [Nm]).

In the commentary of the information found in the sets of characteristics, one can conclude that the family of the characteristics Iq = f(Tl) generally presents the involvement of the reluctance torque in the total torque Te of the motor. The largest share of the reluctance torque T_er occurs for d = 10º and this characteristic lies the lowest in its family. This observation is confirmed by waveform Id = f(Tl), where for d = 10º, Id has negative and decreasing values, thus leading to adequately positive reluctance torque Ter. The mechanical characteristics for theta_r = f(Tl) have a hyperbolic waveform which is particularly ob served for small loads. This comes as a consequence of the demagnetizing effect of the current Id in this range, which assumes positive values there. The efficiency of the motor for d = 10º…40º and the load that is close to its rated value Tl = 20…30 [Nm] is high and exceeds ? > 90%, and reaches a maxi mum of over 92%. As one can conclude from the shape of the characteristics the curves are quite flat and even overloading of the motor two times does not result in a considerable loss of drive efficiency. The characteristics derived in an analogical manner for the smaller of the examined motors with the rated output of 0.95 [kW] (motor A) are presented in an abbreviated form.

The static characteristics for an untransformed model of the BLDC motor, for two-phase control cannot be gained simply in the algebraic form since they are relative to the rotor's angle. Obviously, the equations with the periodically variable coefficients and solutions for the steady state in accordance with Floquet's theorem are referred to in literature with regard to mathematical models of electric machines. However, for the case of such an abbreviated mathematical model and low cost of calculations, the static characteristics can be derived as a leading to adequately positive reluctance torque Ter. The mechanical characteristics for theta_r = f(Tl) have a hyperbolic waveform which is particularly ob served for small loads. This comes as a consequence of the demagnetizing effect of the current Id in this range, which assumes positive values there. The efficiency of the motor for d = 10º…40º and the load that is close to its rated value Tl = 20…30 [Nm] is high and exceeds ? > 90%, and reaches a maxi mum of over 92%. As one can conclude from the shape of the characteristics the curves are quite flat and even overloading of the motor two times does not result in a considerable loss of drive efficiency. The characteristics derived in an analogical manner for the smaller of the examined motors with the rated output of 0.95 [kW] (motor A).

The static characteristics for an untransformed model of the BLDC motor, for two-phase control cannot be gained simply in the algebraic form since they are relative to the rotor's angle. Obviously, the equations with the periodically variable coefficients and solutions for the steady state in accordance with Floquet's theorem are referred to in literature with regard to mathematical models of electric machines. However, for the case of such an abbreviated mathematical model and low cost of calculations, the static characteristics can be derived as a set of stationary points in the dynamic state. This approach has another advantage, namely, that it presents whether a given static state of a drive is possible to achieve from a given initial state. Numerous examples indicate that this is not al ways the case, in particular with regard to BLDC motors with a higher share of the reluctance torque or cogging torque. The static characteristics of a 6.6 [kW] motor gained as a result of this method are presented.

From the comparison of characteristics derived on the basis of the transformed and untransformed models of the motor one can conclude that both of them look very similar. The only relevant difference regards the waveform marking the cur rent of the armature Ia = f(Tl). In the transformed model the values of the current for a small load are considerably higher than the ones gained on the basis of the untransformed model. This comes as a consequence of the course of the term Id in the transformed model. For higher loads the relevance of the differences starts to fade.

+-+-+-Transients following stepwise change of advance angle d: 30º?40º, for 6.6 [kW] motor, computed by untransformed model of BLDC: a) Ia currents; b) electromagnetic torque; c) rotor's speed.

At this occasion one can note that the two models applied in this case, i.e. the transformed and the untransformed ones are not completely equivalent. By definition in the transformed model an assumption is made that each of the three phases of the armature is independently supplied. In addition, the influence of commutation is disregarded. The untransformed model accounts for the constraints imposed by two-phase supply and involves commutation between the star connected windings while the switchings occur during the rotation of the rotor in the function of its position in accordance with the diagram. The advantage of the transformed model is that it’s very simple and does not pose any problems during calculations. This plays an important role in a complex regulation system in which a single BLDC motor forms one of many components of the system as a drive in one of the joints, for instance as an industrial manipulator. The static characteristics comprise the sets of the possible steady states of the drive. However, the transfer between two points on the characteristics occurs as a result of transients and, hence, transfer is not always possible since we have to do with a non-linear dynamic system. For the purposes of illustration the figures that follow present the transients for the dynamic states that occur during the change of the parameters in a system with BLDC motor. --- present transients resulting from a abrupt change of the advance angle d: 30º-40º for an untransformed and trans formed models of the motor, respectively.

+-+-+-Similar transients, but computed by transformed model: a) Iq, Id currents b) Ia armature currents c) electromagnetic torque d) rotor's speed.

For the case of the results gained from untransformed model the increase of the speed theta_r is smaller since in a two-phase motor supply the response of the drive to the change in the advance angle d is limited in comparison to the case of independent supply of three phases, which is additionally confirmed by the static characteristics of the drive presented earlier in this section. The two figures that follow, present transients for the respective transformed and untransformed model after a stepwise change in the advance angle d: 30º-20º.

This is a change that is the opposite of the one that was previously presented as it results in the reduction of the rotor speed and braking in the transient period.

+-+-+- Transients following stepwise change of advance angle d: 30º-20º, for 6.6 [kW] motor, computed by untransformed model of BLDC: a) Ia currents b) electromagnetic torque c) rotor's velocity.

+-+-+- Similar transients, but computed by transformed model: a) Iq, Id transformed currents b) Ia armature currents c) electromagnetic torque d) rotor's speed.

The comparison between the results of calculations for the untransformed and transformed models indicates a greater decrease of the rotor's speed accompanied by an adequately higher increase of the speed for the case of the results gained using the trans formed model in which the windings are not connected. The same results are gained on the basis of static characteristics, for instance from the comparison of the results.

Control of Rotational Speed in BLDC Motors

The basic technique applied for the control of the DC motors, including BLDC motors, involves regulation by altering voltage U, which is currently realized with the aid of the pulse width factor ku of PWM control. For the systems without energy recuperation the change of the factor ku, which realizes the complete change of the rotor's speed, occurs approximately in the range:

... while in the systems with energy recuperation in the range ...

The difference for the both types of the control results from the fact that during the return of the energy the motor over this period is fed with a voltage with negative value -U, so that for ku = 0.5 the mean value of the supply voltage is equal to 0. The static characteristics for the control of the motor resulting from the change of the pulse width factor without the recuperation of the energy into the source for the adequate different values of the load torque Tl and various values of the advance angle d.

Similar characteristics are displayed for the control with energy recuperation; however, the change of the pulse width factor is limited in accordance. Selected characteristics for this type of control are presented.

The two present transient state, which occurs after decreasing pulse width factor from ku = 1 to ku = 0.75, change that is equivalent to the reduction of the supply to the half of the source voltage value.

+-+-+- Transients following stepwise change of ku factor: ku: 1 ? 0.75, for 6.6 [kW] motor by use of the untransformed model: a) ia armature current b) electromagnetic torque c) rotational speed

+-+-+- Similar transients, but computed by use of the transformed model of BLDC: a) iq current b) iq, id currents, c) ia armature current, d) electromagnetic torque, e) rotor speed basis of the untransformed model. This comes as a consequence of the sharp de crease of the current iq and increase of the current id, since in this manner the transformed model realizes the stepwise changes of the motor load. Physically this means a different type of motor braking for the independent supply of three phase windings without constraints than it’s the case in a three-phase system connected in a star for the two-phase supply. As it was mentioned before, obtaining static characteristics in an untransformed model is associated with the need of calculating a series of transients that finally gain a steady state. In this manner the characteristics were drawn up in the function of the machine load.

The same method was followed in order to gain the characteristics presented be low in the function of commutation advancement d for three different pulse width factor values ku = 1, 0.9, 0.8. This was conducted in a system with energy recuperation so that the anticipated values of rotational speed are found in the range that is in agreement with the formula below ...

The characteristics gained in this way are not smooth since they are formed on the basis of a limited number of points for the variable d, i.e. about 30 points and, in addition, the final steady state of the drive is difficult to determine in a comparable way for each final point. The characteristics presented, i.e. for ... armature current Ia, total losses S?P and, within a certain range, efficiency ? indicate that for a steady load these values are only slightly relative to factor ku. Concurrently, the curves, i.e. rotational speed theta_r and mechanical power Pm are considerably dependent on the value of the voltage and, consequently, on the value of factor ku, which directly affects the value of the voltage. As a result, it’s the value of the voltage supplying armature, here represented by pulse width factor ku, that is the basic variable responsible for the control of BLDC drive, while high energy efficiency is to be maintained and it’s not considerably affected during such control procedure. In addition, waveforms of armature current i_a for various values of the advance angle d.

+-+-+- BLDC drive control system with encoder and PID regulator. HSG - high side gates, LSG - low side gates

+-+-+- Stabilization of rotor speed by PID regulator (kP = 60; kI = 300; kD = 0.5) after stepwise load torque change Tl: 3 [Nm] ? 9 [Nm] and consequently 9 [Nm] ? 1 [Nm]: a) armature currents b) rotor speed c) electromagnetic torque d) speed error [%].