Study on maximum torque generation for
sensorless controlled brushless DC motor with
trapezoidal back EMF
指導教授：王明賢
學 生：楊政達
南台科大電機系
OUTLINE
Abstract
 Introduction
 Generation torque analysis of brushless
DC motor
 Controller for maximum torque generation
 Simulation and experimental results
 Conclusion

2
Abstract

The sensorless controlled brushless DC motor with trapezoidal back
EMF has been studied. Since the detected position information on
the sensorless rotor has some uncertainty, the brushless DC motor
cannot be driven with a maximum torque.

To investigate the nature of torque in the sensorless controlled
brushless DC motor, the torque characteristics as a function of
commutation delay (or commutation timing error) have been
analysed. It shows that the generation torque is influenced by the
commutation delay and has a single maximum point. This maximum
point is changed by the rotating speed and load conditions.

An iterative learning algorithm and a fuzzy logic controller are
employed to drive the brushless DC motor with maximum torque.
3
Introduction



Permanent magnet brushless DC motors are appealing candidates
for many high performance applications because of their attractive
characteristics in such key categories as power density, torque-toinertia ratio, power efficiency, robustness and reliability
The motor drive system requires a rotor position sensor to provide
the proper commutation sequence. The position sensors such as
resolvers, absolute position encorders and Hall sensors increase the
cost and size of the motor.
The position detection algorithm using back EMF is a practical
method for a position sensorless controlled brushless DC motor.
4
Introduction

The rotor position detection errors can be divided into the error
deviation and group delay (average error). The error deviation
increases the torque pulsation since it causes commutation
imbalances. The group delay influences the average torque
because it makes the commutation delay.

The results show that the commutation imbalances can be regulated
and the firing angle is well adjusted to the position of the maximum
torque in any case. It is well demonstrated from these results that
the proposed control scheme provides the desirable performance of
a sensorless controlled brushless DC motor.
5
Generation torque analysis of
brushless DC motor
2.1 System description and model equation

The generation torque and the phase currents in a brushless DC
motor with trapezoidal back EMF are strongly influenced
by the rotor position detection error since it cannot commutate the
phase current at an optimal point.

As illustrated in this Figure, it is assumed that three stator phases
occupy consecutive nonoverlapping 60° (elec.) phase belts along
the stator airgap surface.
6
Generation torque analysis of
brushless DC motor
2.1 System description and model equation
Axial view of brushless motor including model for magnet
radial flux density distribution
Generation torque analysis of
brushless DC motor
2.1.1 Back EMF:


Using this average crossflux, the back EMF and generation torque
can be simply represented.
Using this model, the average crossflux of phase-A winding can be
represented as follows:

The shape of this flux is closed to the ideal trapezoidal one by
increasing the magnet pole arc as much as possible. Assuming
three-phase balanced windings, the average crossflux of each
phase is represented as follows:

where
is the average crossflux in phase A, B, C,
respectively, and Bm is the maximum airgap flux density.
Generation torque analysis of
brushless DC motor
2.1.1 Back EMF:

the back EMF in phase A can be represented as follows:

Therefore, using the shape function of the average crossflux , the
back EMF of each phase can be represented as follows:
Generation torque analysis of
brushless DC motor
2.1.2 BLDC motor drive system and model


The firing and speed controller performs two functions: (i) it can
develop a coordinated set of signals to maintain an optimum
relationship between magnet position and stator current in the motor
(ii) it converts a user generated signal from the system control to a
high frequency pulse width modulation (PWM) one to control the
actual voltage and current delivered to the motor
Generation torque analysis of
brushless DC motor
2.1.2 BLDC motor drive system and model



The delay angle from the zero crossing instant of the back EMF to
the phase current commutation instant is fixed to any angle, then the
zero crossing detection error directly affects the phase current
commutation instant.
It may be represented as the phase difference between the input
voltage and back EMF. Then, the voltage equation can be written as
follows:
where
are the rotor position detection errors of
each phase, respectively.
Generation torque analysis of
brushless DC motor
2.2 Generation torque analysis


In this analysis, the commutation of the current from phase C to
phase A is considered. This current transfer is done by switching off
+TC while switching on +TA.
Since the initial state of the back EMF is changed by the rotor
position detection error, the generation torque and phase current are
categorised into three commutation modes as shown in Fig
Generation torque analysis of
brushless DC motor
2.2.1 Typical commutation:

The shape function of the average crossflux can be
expressed as

where k is the slope of the trapezoid, typically, k =6/π.
Taking the beginning of the commutation as an angle
origin, the phase currents are represented as follows

Generation torque analysis of
brushless DC motor
2.2.1 Typical commutation:
Generation torque analysis of
brushless DC motor
2.2.2 Lead commutation:



In this analysis, it is assumed that
and the magnitude of
the rotor position detection error is larger than the commutation
period.
Therefore, the back EMF can be represented as follows:
As shown in Fig. 4, the shape function of the average crossflux in
this case can be given as
Generation torque analysis of
brushless DC motor
2.2.2 Lead commutation:

The phase currents can be represented as follows:

the average crossflux follows (18) for the time
phase currents are obtained as follows:

The generation torque becomes

the phase current with lead commutation is larger than that with
typical commutation
. Thus, the
Generation torque analysis of
brushless DC motor
2.2.3 Lag commutation:

In this case, it is also assumed that
Therefore, final values of the average crossflux are changed as
follows:

the phase currents during this period can be expressed as follows:

Generation torque analysis of
brushless DC motor
2.2.3 Lag commutation:

By assuming a 3-phase balanced operation, the final value of the
phase current, , can be used as an initial value of the
commutation current. Thus the phase current can be expressed as
follows:

the generation torque at

It shows that the phase current is increased but the generation
torque is decreased. Consequently, the torque current ratio is
decreased at lag commutation.
becomes
Generation torque analysis of
brushless DC motor
2.3 Torque–current ratio

The torque–current ratio is changed with a rotor position detection
error in a brushless DC motor.

In this analysis, the torque–current ratio as a function of the delay
timing will be analysed to find out a maximum torque operating
condition.

The detailed torque–current ratio calculation for a brushless DC
motor requires current equations and torque equations including the
effects of motor parameter.
Generation torque analysis of
brushless DC motor
2.3 Torque–current ratio


To obtain the maximum torque operating condition, it is assumed
that
.
Then, the generation torque can be simply subdivided into three
regions. Using the torque equation , the instantaneous torque in
subdivided regions can be expressed as:
Generation torque analysis of
brushless DC motor
2.3 Torque–current ratio

Therefore, the average torque is represented as follows:

The average input current becomes

the torque–current ratio can be
obtained as follows:
Piecewise linear model of phase current
Generation torque analysis of
brushless DC motor
2.3 Torque–current ratio

Figure shows the torque–current ratio obtained from using (7) and
(8). Key observations drawn from these curves and (42) include the
following:
Generation torque analysis of
brushless DC motor
2.3 Torque–current ratio

Increasing the rotor position detection error has the effect of
reducing the average torque.

The average torque will be maximised by lead commutation with
about half the phase current commutation time

The delay time for the maximum torque can be defined uniquely.
Thus, the delay time can be controlled to maximise the generation
torque.

The delay time for the maximum torque should be controlled to
increase the efficiency of a total drive system.
Controller for maximum torque
generation

The position sensorless driver contains rotor position detection,
drive vector generation and delay block.

The rotor position is estimated from the zero crossing point of back
EMF.

The drive vector is generated by the zero crossing detection signal,
and this generation scheme is shown in Fig. 10.

The drive vector can be summarised as shown in Table 1.
Controller for maximum torque
generation
Controller for maximum torque
generation

The rotor position detection errors caused by the zero crossing
detection of back EMF can be divided into error deviations and
group delay as shown in Fig. 11a. Fig. 11b shows that current
distortion has arisen from error
Controller for maximum torque
generation

It is noted here that zero crossing detection deviations can be
compensated using cn(k). Also, bn(k) and dn(k) in the Z-domain can
be expressed as:
Controller for maximum torque
generation
Controller for maximum torque
generation

It is shown in this Figure that the rotor position detection deviations
are well compensated, which results in removing the distortion of the
DC link current as shown in Fig. 14.
Controller for maximum torque
generation


A fuzzy logic controller (FLC) is used as a delay angle controller
since this controller provides not only good control characteristics
but also easy realisation
The block diagram of the proposed torque control system using FLC
is shown in Fig. 15.
Controller for maximum torque
generation

The FLC is basically composed of four principle components.

These are the fuzzification interface, knowledge base, inference
engine and defuzzification interface.

The input of FLC is modified as shown in Fig. 15. The output of the
approximate reasoning is a change of the delay angle in FLC.
Hence, this FLC is represented as follows:

The following linguistic variables are used: positive big (PB), positive
small (PS), zero (ZE), negative small (NS) and negative big (NB).
The triangular shaped functions are used as a membership function
in this application.
Controller for maximum torque
generation

The fuzzy rules are represented by a human action to adjust the
delay angle as follows:

The quantisation level and corresponding range of
are shown in Table 2,
and
Controller for maximum torque
generation
Controller for maximum torque
generation

The simulation results of the proposed maximum torque controller
are shown in Fig. 17.
Experimental results


The total block diagram of this system is shown in Fig. 19.
This control system contains the main board and control board. The
main board is composed of the digital signal processor
(DSP,TMS320C30), EPLD, and memories.
Experimental results
Experimental results
Experimental results
Experimental results
Conclusions






The torque characteristics as a function of the rotor position
detection error have been analysed. Key investigation results are
summarised in the following:
1. Increasing the rotor position detection error makes the average
torque decrease and the torque pulsation increase.
2. The average torque will be maximised by a lead commutation with
about half the phase current commutation time.
3. The commutation time is a function of machine parameters and
directly affects the rotating speed. Therefore, the timing to generate
the maximum torque is changed by the operating speed.
4. The delay angle should be controlled to overcome the adverse
effect of the rotor position detection error.
5. The delay angle for the maximum torque can be defined uniquely.
Therefore, the delay angle can be controlled to maximise the
generation torque.