An Example to FEM Analysis of BLDC Motors

The brushless PM motor with 6 poles and 18 slots will be simulated by using Flux 2D program. The first the geometric structure of the motor should be established as shown in Fig. 2.14(a). The mesh needed for FEM analysis is generated automatically when the magnitude of mesh points is defined, as shown in Fig .2.14(b).

After defining the physical characteristics and the boundary condition, the path and the distribution of the magnetic flux can be obtain, as shown in Fig. 2.15 and Fig. 2.16.

Furthermore, the flux linkage and back-EMF can also be figured out while the external circuit and the driving algorithm are established.

Fig. 2.13 The procedure of the FEM analysis

(a) (b)

Fig. 2.14 The geometry and mesh of the brushless DC motor

Fig. 2.15 The magnetic line of force of the brushless DC motor

Fig. 2.16 The flux distribution of the brushless DC motor

Fig. 2.17 The flux linkage of the brushless DC motor

Fig. 2.18 The back-EMF of the brushless DC motor

Chapter 3

Sensorless Drive of BLDC Motors

3.1 Basic Operational Principle of BLDC Motors

The BLDC motor has a rotor with surface-mounted magnets and stator windings, which are wound for the trapezoidal back electromotive forces. The trapezoidal back-EMF implies that the mutual inductance between the stator and rotor is non-sinusoidal. Therefore, when consider the mathematic model, it is no particular advantage to transform the phase variable equations of the BLDC motor into the well-known two-axis transformation (d, q model), which is done in the case of the permanent-magnet synchronous motors with sinusoidal back-EMF.

The model of the three phases Y-connected BLDC motor consists of winding resistances, winding inductances, and back-EMF voltage sources. The inverter circuit and the equivalent model are shown in Fig. 3.1. Assume that rotor inducted currents can be neglected, no damper windings are modeled, the winding resistances are identical and the self and the mutual inductances are constant, which is independent of the rotor position. The voltage equations of the BLDC motor in phase variables can be given as

L and M self and mutual winding inductances

In state-space form the equations can be derived as

⎥⎥

and the electromagnetic torque and equation of motion are

( )

According to (3-4), the electromagnetic torque Te is dependent on the back-EMF and stator phase currents. The stator phase currents ia, ib and ic must be rectangular-shaped to produce a steady torque without pulsations due to the trapezoidal back-EMF.

Besides, the currents ia, ib and ic should flow in only two of the three phases at any instant, and there should be no torque produced near the back-EMF zero crossing. The relationship between back-EMF and stator phase currents with respect to the electrical

degree θ_e is shown in Fig. 3.2, where the electronic commutation statuses are also listed below the constant torque. With the electronic commutation statuses, the corresponding phase winding is excited to maintain the synchronization with the trapezoidal back-EMF such that the constant electromagnetic torque would be resulted. At each conduction period, one phase switch remains ON state with 120 electrical degrees and the other phase switches change states between ON, OF and short like as three-phase full bridge commutation. This drive method has lower copper utilization and more ohmic motor losses but lower cost than three-phase-ON drive.

Since there are six commutations per electrical period, the drive scheme is so-called six-step drive. The detail commutation and excited procedure is illustrated with a two poles and six slots BLDC motor as shown in Fig. 3.3.

Fig. 3.1 Schematic of the inverter and equivalent modeling for a BLDC motor.

Fig. 3.2 Torque production in a Y-connected three-phase motor.

a

b c

Rs

Ls

ea

ec

Ls

Rs

eb

Ls

Rs

S1 S2 S3

S₄ S₅ S₆

n

Fig. 3.3 The procedure of six-step drive

3.2 Concept of Sensorless Drives

Inherently, the BLDC motor is controlled electronically and requires rotor position information for properly commutating currents. In general, rotor position information is often measured by Hall sensors, which have been found to possess several disadvantages in many applications when concerning the whole system’s cost, size, and reliability. Therefore, recent investigators have paid more and more attentions to sensorless BLDC motors, which can be operated without any Hall sensors. Up to now, it is still a challenging and attractive topic. Many sensorless-related technologies have been proposed and the main methods can be primarily categorized as following: third-harmonics voltage-based position estimators [10], Back-EMF based position estimators [17], stator phase Observer-based speed and position estimators [3].

3.2.1 Back-EMF based Method

In BLDC motor, the magnitude of a back-EMF induced in the stator windings due to the magnets is position dependent. Therefore, the rotor position can be accurately estimated in real-time and can be used to control the switching pattern of the inverter circuit by accurately monitoring the back-EMF, which is the commonest method for the sensorless drive system design due to its simple concept and implementation. The main back-EMF based method will be discussed below.

In the basic BLDC motor operation, only two phases are excited at any instant and a back-EMF voltage can be measured from the terminals of the third phase conveniently. The zero-crossing method is employed to determine the switching sequence by detecting the instant where the back-EMF in the unexcited phase crosses zero. In general, when considering the inverter circuit is connected to a BLDC motor with Y-connected stator windings, as shown in Fig. 3.1, three terminal voltages va, vb

and vc can be derived as back-EMF voltages generated in the three phases. In order to describe the zero-crossing method, for example S₁ and S₅ are ON, the terminal voltages va, vb and

Ideally, it is assumed that the current applied to the winding is rectangular-shaped and the stator inductance voltage drop is negligible. The relationship of the line-to-line current iab and corresponding phase currents ia and ib can be shown as

According to (3-9), (3-10), (3-11) and (3-12), the terminal voltage of phase c can be derived as

Since the difference of 120° electrical degree in back-EMF voltage exists between any two phases, there is an instant that ec = 0 and ea = -eb. Then

Hence, the zero-crossing position of ec is independent of the load current. The terminal voltages va and vb have the same relationship as vc and vn at zero-crossing of

the back-EMF voltage when phase a or b is unexcited. After detecting the zero-crossing point of the unexcited phase, it is well know that the input signal to the unexcited phase should be given with 90° electrical degree delay to get the maximum torque, as shown in Fig. 3.4. The method can be realized by using voltage sensors and low-pass filters. However, the modulation noise is eliminated by using the low-pass filters, this causes a phase delay varies with the frequency of the excited signal for the desired rotor speed. Besides, the back-EMF voltage is zero at standstill and is proportional to the rotor speed. This method cannot be used at startup situation and works poorly at low-speed operation.

The back-EMF integration technique can be also used for detecting the rotor position of BLDC motor. The phase-to-neutral voltage of unexcited stator winding is selected to measure the desired back-EMF, which is required for position sensing as soon as the residual inductive current flowing in the unexcited winding, immediately following the removal of excitation, decays to zero. The absolute value of this back-EMF is integrated as soon as the back-EMF crosses zero. Assume the back-EMF voltages are ideal trapezoidal waveforms, that is the unexcited phase back-EMF es

Fig. 3.4 Detection of switching point P from the crossing of the neutral voltage and terminal voltage.

varies linearly as

( )

t E t

e_s = ₀ (3-15)

where E0 is the slope of the unexcited phase back-EMF. Thus, the integrated voltage value vint can be obtained as

∫ ( )

⁼

= ^t

0

2

2 k t d E

k

v_int e^s τ τ ⁰

(3-16) where k is the gain of the integrator. The next commutation signal is produced when

the integrated voltage value vint reaches a pre-set threshold value as show in Fig. 3.5.

The values of the threshold voltage and integrator gain depend on the motor and also on the alignment of the phase-current excitation waveform with the back-EMF. This method has the advantage of reducing the switch noise. Furthermore, since the back-EMF amplitude is proportional to the speed, the conduction intervals automatically scale inversely with the speed. Thus, there is an automatic adjustment of the inverter switching instants to changes in the speed.

There are still other back-EMF based position estimation methods such as the phase-locked loop method and the free-wheeling diodes conduction method. However, all of them have the same disadvantage, not suitable for a wide-speed application due to that the low signal-to-noise ratio at low-speed operation causes the accuracy problem.

Fig. 3.5 The back-EMF integration technique

C

A CB BA AC CB BA

0o 60^o 120^o 180^o 240^o 300^o 360^{o e}

θ Reset

vth

vint

ea

eb

ec

3.2.2 Third-Harmonics Voltage

As mentioned before, only two of the stator phases are excited at any instant.

The stator currents flow into one of the excited windings and out of the other with 120 electrical degree square waves at each conduction periods. For maintaining the synchronization between the stator excitation and the MMF produced by the magnet, the six-inverter switches are switched at every 60 electrical degree. The switching can be detected by monitoring the third-harmonic voltage component of the back-EMF, which is a function of the rotor magnets and the stator winding configurations. For a full pitch magnet and full pitch stator phase winding, the back-EMF voltages contains the following frequency components

(

cosωt k cos3ωt k cos5ωt k cos7ωt ...

)

Since the Y-connected stator windings, the third-harmonic voltages of the back-EMF produce no third-harmonic currents. The summation of the three stator phase voltages is a zero sequence which contains a dominant third-harmonic component and high frequency components, expressed as

high_freq

where v₃ is the third-harmonic voltage and v_{high_freq} is the high frequency components.

Then, the third-harmonic rotor flux λ can be obtained by integrating the _r3 third-harmonic voltage v₃ as

∫

= v dt

λ_{r3 3} (3-21)

It should be noted that the zero-crossings of the third-harmonic rotor flux occur at every 60 electrical degrees, which are exactly the switching instants mentioned above.

Thus, the third-harmonic rotor flux is input to a zero-crossing detector and the output

of the zero-crossing detector determines the switching sequence, as shown in Fig. 3.6.

The advantages of this method are simplicity of implementation, low susceptibility to electrical noise and robustness. This method has better performance in low-speed applications when comparing to the back-EMF based method. However, the sensing of the third-harmonic signal as described above requires access to the neutral connection of the stator phases or adopts an external resistance network to replace the neutral connection. Both of them have some undesirable influence. In most applications, the neutral terminal is not available due to the cost and structure constraints. The external resistance network causes the power lost and may change the characteristics of the motor.

3.2.3 Observer-based Method

The last method listed above determines rotor position and speed using observers. The rough estimations of the angular rotor position by measuring the phase voltages and currents can be refined utilizing observers. The inputs of the observers are the inputs and outputs of the actual system such as phase voltages and currents;

Fig. 3.6 Third-Harmonics Voltage technique ea

eb

ec

v3

λr3

C

A BC BA CA CB AB

0o 60^o 120^o 180^o 240^o 300^o 360^o

θe

the outputs of the observers are the immeasurable states of the actual system such as flux linkage or back-EMF voltage. The states produced by the observer are fed back into the system, as would be the actually measured variables as used in a closed loop system control with specific rules for the various types of observers. For example, consider the discrete-time state-space form of a single output system,

[ ]

k Ax

[ ]

k Bu

[ ]

k

x +1 = + (3-22)

[ ]

k Cx

[ ]

k Du

[ ]

k

y = + (3-23)

which must be completely observable. Then, determine a modal matrix P for the system. Using the modal matrix P, the system is transformed into an observable form.

[ ]

k Px

[ ]

k Q x

[ ]

k

Next, the full-order is designed for immeasurable states. The eigenvalues of the observer are usually chosen to be slightly faster than the eigenvalues of actual system such that the state estimated error approaches zero as time approaches infinity. If the eigenvalues of the observer are large enough, the estimated values will converge to the actual values within a sufficiently short time. The observer takes the form

[ ]

k Fζ

[ ]

k gy

[ ]

k Hu

[ ]

k

with the observer output equation

[ ]

k Pζ

[ ]

k Q ζ

[ ]

k

w = ′ = ⁻¹ ′ (3-29)

which allow the transformations

[ ]

k Pζ

[ ]

k Q ζ

[ ]

k

Thus, the full-order observer in x[k] is derived, meaning that the immeasurable states can be obtained form ζ

[ ]

k , which converges to x[k]. However, the full-order observer consists of more-order equations, the pole assignment is not easy and the manual tuning is time consuming. The reduced-order observers are developed for substitution. The reduced-order observers depend on that some of the system outputs are linear transformations of the system states. In motor drive application, the state, current, is readily available for measurement such that only the states, back-EMF voltages or flux linkages, need to be estimated by the lower-order observer. Assume that the sample rate of the system is fast enough; the sliding mode observers are available. The sliding mode observers, offspring of sliding mode control, utilize the varying switching function in order to confine the state estimation errors to approach toward zero on a phase-plane, sliding surface. For details, consider the state-space system

In BLCD systems, usually considering the PMSM or the fundamental of the trapezoidal BLDC, the state-space X consists of the d-q transformed stator variables.

The state observer takes the same form with the addition of a switching input.

( )

S Bu

X A

Xˆ& = ˆ + −K⋅sgn (3-35)

where

the hat symbol denoted estimated values K is the switching gain and

( )

⎪⎩

⎪⎨

⎧

<

−

= ≥

0 1

0 1

S S S

sgn

S is the vector of sliding surface

Defining S = ˆX −X =e, where e is the state estimation errors, the error equations can be derived as

( )

e Ae

e&= −Ksgn (3-36)

Consider the approach and sliding condition, let K be a large positive value to make sure the following equation hold.

Te<0

e & (3-37)

Then, the observer states converge to the actual system states and the rotor position and the speed are estimated. Comparing to other methods, the achievement of rotor position estimation in wide speed and the real time estimation of the rotor speed are the critical grace. But the sampling rate, computational burden and the parameter variations need be considered.

Chapter 4

Sensorless Drive Modeling

4.1 Position Detection and Start-up Algorithm

The sensorless methods, which substitute for the Hall sensors or encoders to obtain the position information, have been introduced in the previous chapter. Each of them has specific advantages and drawbacks, but a common fateful fault. All of them can not detect the rotor position or speed at standstill or near zero speed with insufficient information required for estimating, such that are unable start-up. In order to overcome this fault, an extra start-up procedure, so-called “align and go”, has been proposed. As implied by the name, the algorithm is divided into two steps. First, excites any two phases winding of the stator to align the rotor to the specific position, and then accelerates the rotor according to the given firing sequences with decreasing time intervals. It should be noted that the initial position is not detected and the starting form unknown rotor position may be accompanied by a temporary reverse rotation or may cause a starting failure. These eventualities are not tolerable in many applications. Thus, a complete start-up procedure, which includes the detection of the initial rotor position and avoids the temporary reverse rotation or starting failure, should be developed [5]. The most proper method is based on the effect of the magnet position on the flux saturation, which cause the variations of the inductance. The details will be discussed below.

4.1.1 Initial Rotor Position Detection

Motors convert electrical energy to mechanical energy by magnetic field to generate desired torque, which is generated via the interactive force between windings

with electric current and magnet. The path of the magnetic flux produced by the permanent magnet and current is a loop crossing the airgap to the stator from the rotor.

Ideally, the B-H curve of the steel is linear and the inductance is constant due to the reluctance of the flux path is constant. But in practically, the B-H curve of the steel is a non-linear curve with saturation. Thus, the estimation of the rotor position can be detected by using the inductance variation due to the magnets position and the stator current. The relationship between inductance and flux linkage is shown as

+Li

= _PM

Phase λ

λ (4-1)

where λ_Phase is the summation of the flux from the permanent magnet, λ_PM, and the flux from the current i. L is the inductance of the excited phase. Supply the current with positive or negative direction to the phase, as shown in Fig. 4.1. The variations of the inductance are derived as

+ phase voltage and current to the variation of the inductance. The phase voltage equation is expressed as

a

The back-EMF can be neglect when a motor is at standstill. Then, solve the differential equation; the phase current can be derived as

⎟⎟⎠

According to (4-5), the phase current has a different transient dependent on the

inductance variation, which is determined by the relative position of the magnet and the direction of the current. It should be noted that i has a faster response than ⁺ i ⁻ due to the time constant R/L is larger than ⁺ R/L , as shown in Fig. 4.2. Therefore, ⁻ the position information can be obtained by monitoring the phase current i and ⁺ i ⁻ in an appropriate time interval.

As mention above, the inductance variation with respect to the electrical angle should be known for detecting the rotor position. This problem can be overcome by using the FEM software, Flux 2D, to drawn out the relationship between inductance

Fig. 4.1 The flux linkage with positive or negative current

Fig. 4.2 The responses of current with positive or negative direction λphase

i⁻ i⁺ i

λPM

∆λ−

∆λ+

t T

i⁺

| i⁻ | i

∆i R

v_an

and electrical angle, as shown in Fig. 4.3. Then, employing the information of inductance and (4-5) to gain the current responses with respect to the electrical angle in a time interval of 20 µs , as shown in Fig. 4.4 and Table 4.1 [5]. In Fig. 4.4, the current responses of the well-known six segments are calculated with the determined inductance in an appropriate interval of electrical degrees. Observing Fig. 4.4, it should be noted that the polarity of the difference between the positive current and negative current, ∆i , changes every 180 electrical degrees. Furthermore, the _n inherent differences with 120 electrical degrees exist in any two phases such that the polarity of one of three changes every 60 electrical degrees, which can provide the position information, as shown in Fig. 4.5 [5]. However, the position information from ∆i is not proper to drive a motor. The polarity of the difference between _n

m

n ∆i

∆i − , ∆∆i , can be effectively used to identify the rotor position due to it _k

∆i − , ∆∆i , can be effectively used to identify the rotor position due to it _k

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