BLDC Machine w/ Permanent Magnets: Math Models

The presentation will focus on the mathematical models of a three-phase machine with typical engineering, i.e. one with cylindrical structure with armature windings in the stator and permanent magnet in the rotor. Quite self-evidently, there is a wide variety of BLDC (brushless direct current) machines, both in terms of the number of phase windings as for instance ones based on disk structure, where the major field has an axial direction. However, a cylindrical, three-phase ma chine forms the basic engineering solution and occurs as a small, medium and large power device. A simplified cross-section of such a machine for the number of pole pairs p = 1 is presented.

+-+-+- Cross-section and schematic view of a BLDC motor, for p = 1

The basic simplifying assumptions applied during the development of a mathematical model include:

- complete symmetry of the machine's construction,

- disregarding of factors affecting demagnetization of permanent magnets

during the operation (effects of armature, temperature increase)

- disregarding high order MMF harmonics of armature windings.

The remaining, more detailed assumptions associated with the development of the mathematical model will be presented during the course of its derivation. For such an electromechanical transducer and lack of elements serving for the accumulation of potential energy, the Lagrange's function is equal to kinetic co-energy:

.. which can also take the form of a matrix notation:

... - matrix of armature inductance

... - vector of the coupling between permanent magnet flux and armature windings,

... - vector of armature current.

The particular components of the inductance matrix of the armature windings account for the variable reluctance of the rotor and dissipation flux of the armature windings and for the purposes of simplification can be presented as follows:

The angle f accounts for the number of pole pairs in the machine ... while the self-inductance of the armature windings is assumed in the form which identifies the leakage inductance:

After accounting for these remarks, the inductance of the armature windings can be restated in the following form:

The equation of the mechanical motion of the machine can be derived from Lagrange's equation for a variable denoting rotation angle theta_r:

The expression in defines the electromagnetic torque of the machine and involves two terms. The first of them denotes the reluctance torque of the ma chine, which comes as a consequence of the reactions of armature current with the salient poles of the rotor with magnets:

Concurrently, the other term of the expression denotes the principal torque of the machine resulting from the interaction between armature currents with permanent magnets' excitation flux.

In addition, BLDC machines have another component of the torque, i.e. cogging torque beside the reluctance related one. It’s present as a result of the reaction of the principal flux with the armature teeth. In the presented model it is, however, not encountered since the harmonics associated with the stator slots are disregarded. This omission is admissible since the designers throughout their engineering efforts, tend to effectively aim at the minimization of this component of the torque.

Transformed Model Type d-q

The structure of the inductance matrix of the armature suggests the application of the orthogonal transformation, similar as in the case concerning a three phase induction machine. In this case we will apply transformation Tr for _c = 0.

Thus, ...The transformation of equations for the electric circuits of the armature will be conducted in the general form derived from of Lagrange's equations for electric variables: The determination of the particular expressions of the transformed voltages uq, ud is associated with the need to consider the problem of the commutation of armature currents, which occurs in the function of the angle of the rotor position. This issue will be discussed later. Concurrently, the quadratic form which, determines the electromagnetic torque can be transformed in the following manner: The expression which determines the electromagnetic torque of the motor, after the transformation, takes an uncomplicated form: the first term denoting the reluctance torque is relative to the product of axial currents id, iq and is proportional to inductance Ms associated with the basic harmonic of the reluctance of the air gap, while the other term denotes the principal torque proportional to the product of the magnetic flux and current in the transverse axis of the machine.

There is a complete analogy here to the commutator DC machine.

Untransformed Model of BLDC Machine with Electronic Commutation

The application of the model that does not involve the transformation of the coordinate system has a number of advantages. For the case of a motor with electronic commutation there is a possibility of a more realistic modeling of commutation and, thus, gaining results more precisely, including the electric variables over time. The commutation as well as the parameters of the switching transistors can be taken into consideration more precisely in a manner that is required for a specific problem of drive control. Secondly, for the lack of transformation, the modeling of the machine and drive itself can account for a number of asymmetries and differences in terms of parameters, which renders it possible to simulate the emergency states of the drive. In an untransformed model we consider that the armature windings are connected in a star, which take the form of adequate constraint equations. Here we will apply the matrices of constraints W_ir and W_ur for the respective currents and phase voltages of the motor.

As a result of the multiplication of the left-hand side of the equation in by the matrix of constraints W_ur and introducing the vector of armature currents.

The expression presents electromagnetic torque of BLDC machine in the natural coordinates i1, i3 without transformation, for the connection of three-phase armature windings in a star.

+-+-+- Basic scheme of a bipolar 3-phase BLDC motor supply

Electronic Commutation of BLDC Motors

Commutation in a brushless DC machine involves the switching of the armature cur rent to particular phase windings depending on the position of the rotor angle theta_r. In a traditional brushless DC machine this occurred as a result of application of a mechanical commutator consisting of isolated copper segments with armature windings connected to them. Over this commutator the graphite brushes would slip thus receiving the current while the position of the brushes was fixed in space. In such a manner the commutation occurred naturally depending on the position of the rotor.

The electronic commutation is ensured by the converting bridge while the switching of the current between the windings also occurs in the function of the rotor's position angle, and the signal responsible for the control of the switchings is obtained from the position sensor measuring the angle of rotation. As a principle, such sensors are optical, including encoders and induction based ones, i.e., resolvers.

In any case, however, the commutation angle needs to be set at an appropriate value, which in a traditional DC machine was the role of the correct positioning of the brushes in a commutator. --- the standard transistor bridge fulfilling the role of an electronic commutator for a motor with three phase windings in the armature and bipolar supply of the windings connected in a star. The bipolar supply means that in the armature windings the current flows in both directions, i.e. the current flowing through windings is AC. The angular scheme of the commutation of BLDC motor for a positive direction of rotor motion.

+-+-+- Typical scheme of current commutation in BLDC motor's 3- phase armature in relation to rotor position angle: a) conducting of individual phase windings b) conducting of pairs of star-connected windings

A typical commutation diagram for three-phase windings connected in a star involves simultaneous conduction of two phase belts while the third one remains in OFF state. The simultaneous conduction of all three belts occurs only in very short commutation periods when we have to do with the transfer of the conduction from the belt that is about to terminate the operation to another phase belt, which in accordance with the commutation diagram takes the turn in starting commutation. A singular phase belt in a three-phase BLDC machine conducts over the period corresponding to the angle of rotation, i.e. 2p/3 and subsequently takes a break over the time corresponding to the angle of rotation, i.e. p/3. The subsequent conduction period for the angle of rotation equal to 2p/3 occurs after this break; however, for an opposite direction of the conduction followed by another break in conduction. It’s designed so that for a full turn of the rotor in a given phase winding the current that flows is AC with the breaks in the conduction corresponding to the rotation of the rotor over p/3 angle. In a complete BLDC machine with three phase windings commutation occurs every p/3 angle of the rotation of the machine's rotor. The development of a commutation diagram makes it possible to determine supply voltages ud, uq in the transformed model of the mo tor and perform detailed consideration of the commutation model to be applied for supply of the motor in modeling without transformation. In both cases the value of supply voltage is controlled as a principle by the pulse width modulation (PWM). Due to the course of commutation of the current the particular phases are switched on slightly in advance in relation to the theoretical commutation diagram presented. This advance angle denoted as d is usually in the range from 25º - 35º. The presentation that follows is concerned with the determination of the supply voltages ud, uq for the transformed model of the motor.

Supply Voltages of BLDC Motor in Transformed Model

The transformed voltages are calculated in accordance with the relation and the commutation diagram presented. The details of the relation are as follows:

where: U - supply voltage of the commutation bridge

state OFF for the 0

state conduction reverse for the 1 -

state conduction positive for the 1

- conduction factors

- pulse width factor (PWM control)

- electrical rotation angle

- conduction time within a pulse period TPWM

- recuperation time within a pulse period TPWM

- advance angle

Over the period of TPWM for the duration of the supply tp respective switches are in the ON state and the voltage Us = U is fed to the windings. Concurrently, when we have to do with control without energy recuperation, the closure of the phase takes place and the current flows through the return diode and one of the transistors of the bridge, and Us = 0. For the control with energy recuperation all transistors are in the OFF state and the energy is returned to the source through the two of the return diodes for the voltage of the motor Us = -U . This happens in the section of the control period tz. The above description of a single pulse with the period of TPWM offers an explanation to the issue of calculation of output volt age of the commutation bridge for both types of bridge control. The coefficient ku makes it possible to calculate the mean values of the voltages ud, uq.

These means are determined on the basis of formulae in while the functions of the conduction factors c1, c2, c3 are determined according to commutation scheme:

The examples of the waveforms for ud, uq are presented. In both cases we apply an angle depending PWM coefficient ku calculated from the relation:

...where:

Tu - is the angular constant of voltage increment.

The exponential character of voltage increase ud, uq offers the possibility of the smooth motor start-up.

Modeling of Commutation in an Untransformed Model of BLDC

The modeling of commutation in an untransformed system for a three-phase windings of the armature can have a various degree of detail. In this section we will present a method that is considerably simplified and, subsequently, apply it in examples. It takes into consideration the fact that adequate supply voltage is connected to the particular pairs of phase windings connected in a star, i.e. winding no. 1-2, 2-3, 3-1 via a commutation bridge. During the commutation we will distinguish two states: first, when the commutation begins during the connection of the source voltage to the windings, i.e. during the active part tp of the supply pulse Tp and the other state, when the commutation begins during the passive part of the pulse while the energy is returned from the windings to the source or during the closure of the winding. In both these states it’s possible to adequately model resistances and voltages occurring in the particular electric circuits in the given state, i.e. the resistances of the motor windings, electronic switches as well as the blocking resistance Rb in the circuit of unsupplied phase winding. It will be illustrated by appropriate examples.

+-+-+- Currents flow during commutation +i1 ? +i2 for f = p / 3, and an active part tp of the period Tp.

We will consider the commutation occurring in the active part of the pulse tp, which takes place for the angle of rotation f = p / 3, where we have to do with the switching of the current from +i1 to current +i2, i.e. the termination of the conduction in the positive direction in the winding in phase 1 and commencement of the conduction in the positive direction by the winding in phase 2. During that time in the remaining winding of phase 3 the current flows continuously in the conventional negative direction. This situation is illustrated. The target circuit after the commutation supplied with voltage U is marked through transistors T3, T6 (+i2, -i3), while the decaying current in the winding of phase 1 is closed in the circuit with the return diode D2 that is antiparallel to transistor T2, since the transistor T1 has just been closed, and conducting transistor T6 connected to the phase winding 3. In the state presented in the figure the potential of point a amounts to 0, potential of point b is U and the potential of point c is 0. Hence, the voltages U12 = -U, U32 = -U. After the commutation, i.e. after the current +i1 ? 0 has decayed, the potential of the point a will change to ½ U and the respective volt ages will be U12 = -½ U, U32 = -U, while the resistance R1 = Rb, which means that it will assume the value of the blocking resistance. For a better illustration of the considerations we will additionally examine the commutation for the angle of rotation f = p. In this case the commutation involves a change +i2?+i3, while -i1 is continuing its flow. Decaying current +i2 is closed across the return diode D4 of the transistor T4 and transistor T2. The potentials of the particular points a,b,c associated with the beginning of the phase windings in this state amount to: 0, 0, U. As a consequence, line-to-line voltages supplying the windings are as follows: U = 0, U = U.

+-+-+- Currents flow during commutation +i2?i3 for f = p, and an active part tp of the period Tp

Concurrently, after the termination of the commutation the potential of the point b changes to ½U, which results in the following values of line-to-line volt ages supplying the windings U12 = -½U and U32 = ½U. The situation during the commutation over the passive section of pulse Tp can be presented for two various alternatives of control of the commutation bridge, i.e. for the case when during the commutation the energy returns to the source and the opposite one when the energy is not returned to the voltage source U. At the beginning, we will consider the first of the cases, when in the passive section of the pulse the energy is recuperated. For this case we will consider commutation +i1?i2, (f = p/3), but for the passive section of the PWM pulse. During this commutation transistor T1 is just switched off terminating supply to the phase winding 1, transistor 3 is not switched on because of the passive period and transistor T6 is switched off to facilitate recuperation of energy. The decaying current flows through diodes D2 and D5 against the voltage of the source. In this state the potentials of points a,b,c are respectively equal to 0, ½U, U and consequently U12 = -½U, U32 = ½U. After the commutation is finished potentials of all three a,b,c points are the same and equal to ½U and inter-phase voltages are U12 = U32 = 0. Concurrently, for the other version of commutation without energy return to the source during the passive part of the period Tp, the transistor T3 is not switched on, while transistor T6 is switched on - continuing conduction, and the decaying current of windings 1 and 3 flows through T6 and D2 in a shorted circuit. At that state the potentials of a,b,c points are respectively 0, ½U, 0 and inter-phase voltages are U12 = U32 = -½U. After the commutation there is no current and like in the previous case all three clam potentials are equal to ½U and in consequence U12= U32 = 0. An illustration of this is found.

+-+-+-Currents flow during commutation +i1?+i2 for f = p / 3 in a passive part of the period Tp, with energy recovery.

+-+-+-Currents flow during commutation +i1?+i2 for f = p / 3, in a passive part of the period Tp, without energy recovery..