17

CHAPTER 2

MODELLING AND PARAMETER ESTIMATION OF

BLDC SERVO SYSTEM

2.1 INTRODUCTION

Physical systems are modelled in order to analyse their behaviour under different operating conditions and employ them for various control

applications. Modelling of physical systems involves estimation of their parameters. Parameter estimation plays a vital role in perfect tuning of

controllers. This is essential to fulfill the desired performance specifications from the system.

Over the years, a great deal of research has been carried out in the computation of system parameters using genetic algorithm, fuzzy logic and

neural networks. Moment of Inertia and friction coefficient of motor alone are determined but that of load are not considered, even though various

optimization techniques, including artificial intelligence techniques and adaptive control methods are employed (Al-Qassar & Othman 2008; Babau et

al 2007; Despalatovic et al 2005; Hadef et al 2007; Kapun et al 2008). Load parameters are obtained using genetic algorithm but friction coefficient of

motor is not considered by Zhang & Bai (2008). Viscous friction coefficient of motor is determined while that of load is not considered for precise position control tasks (Campa et al 2008). The importance of estimation of

load parameters is emphasized by Lin et al (2010) but strategies for determining moment of inertia and friction coefficient at different load

conditions are not highlighted.

18

Ther

efor

e, a

sim

ple

met

hod

is p

ropo

sed

in t

his

thes

is f

or t

he

dete

rmin

atio

n of

mec

hani

cal p

aram

eter

s, vi

z. m

omen

t of

iner

tia a

nd f

rictio

n

of a

BLD

C d

rive

at d

iffer

ent l

oads

. Thi

s ch

apte

r dea

ls w

ith th

e m

odel

ling

and

com

puta

tion

of v

ario

us p

aram

eter

s of

BLD

C s

ervo

sys

tem

. The

sig

nific

ance

of

dete

rmin

atio

n of

th

e m

echa

nica

l pa

ram

eter

s at

di

ffer

ent

load

s is

emph

asis

ed. DC

pos

ition

con

trol s

yste

m e

mpl

oyin

g a

trape

zoid

al B

LDC

mot

or

was

con

side

red

in th

is th

esis

for

the

inve

stig

atio

n. T

he e

ffec

t of

load

on

the

para

met

er v

aria

tion

was

em

phas

ised

by

anal

ysin

g th

e tra

nsie

nt r

espo

nse

of a

clos

ed lo

op B

LDC

driv

e fe

d po

sitio

n co

ntro

l sys

tem

at d

iffer

ent l

oads

. In

this

chap

ter,

an a

ttem

pt is

mad

e to

ana

lyse

the

influ

ence

of p

aram

eter

var

iatio

n on

the

PID

con

trolle

r tun

ing

for t

he d

ynam

ic lo

ad v

aria

tion.

2.2

MO

DE

LL

ING

OF

BL

DC

DR

IVE

SY

STE

M

The

mat

hem

atic

al m

odel

ling

of B

LDC

driv

e sy

stem

is

desi

gned

(Kup

erm

an

&

Rab

inov

ici

2005

; Pa

rk

et

al

2003

) w

ith

the

follo

win

g

assu

mpt

ions

.

1.A

ll th

e st

ator

pha

se w

indi

ngs

have

equ

al re

sista

nce

per p

hase

and

cons

tant

self

and

mut

ual i

nduc

tanc

es.

2.Po

wer

sem

icon

duct

or d

evic

es a

re id

eal.

3.Iro

n lo

sses

are

neg

ligib

le.

4.Th

e m

otor

is u

nsat

urat

ed.

The

sche

mat

ic d

iagr

am o

f a B

LDC

driv

e sy

stem

is s

how

n in

Fig

ure

2.1.

The

thr

ee-p

hase

inpu

t vol

tage

s ar

e ex

pres

sed

as f

ollo

ws

with

the

abov

e

assu

mpt

ions

.

19

aedta

diL

aRi

av

(2.

1)

bedtbdi

Lb

Ribv

(

2.2)

cedtcdi

Lc

Ricv

(

2.3)

The

elec

trom

agne

tic to

rque

is e

xpre

ssed

as

)(

1ci ce

bi beai ae

eT

(2.

4)

Figu

re 2

.1 S

chem

atic

dia

gram

of a

BL

DC

dri

ve sy

stem

20

where va, vb and vc are the stator input voltages of phase a, b and c,

respectively, ea, eb and ec are the back emfs of phase a, b and c, respectively,

ia, ib and ic are the phase currents of phase a, b and c, respectively, R and L are

per phase resistance and inductance of each stator winding, TL is the load

torque, J is moment of inertia, is angular speed, B is friction coefficient, Kb

is back emf constant, KT is torque constant and is the angular position.

cebeaeE (2.6)

cibiaiI (2.7)

The block diagram of a typical position control system employing

BLDC motor is shown in Figure 2.2.

Figure 2.2 Block diagram of DC position control system

Desired angular position is fed as an input to the DC position

control system. Actual angular position (s) which is the output of the system

is fed as the feedback signal to the system. The difference between these two

positions actuates the per phase armature voltage Ea(s) to the BLDC motor

whose mathematical model is shown as an inner loop to the position control

system. Transfer function of BLDC motor is found out from its mathematical

model as

21

)])(([)()(

aaTb

T

a sLRBJsKKsK

sEs

(2.8)

where KT, Kb, Ra, La, J and B are specified in the name plate details of BLDC

motor.

2.3 ESTIMATION OF PARAMETERS OF DC POSITION

CONTROL SYSTEM

Parameters KT, Kb, Ra, La, J, B of DC position control system are

significant in assessing the transient and steady performance of DC position

control system. KT, Kb, Ra and La do not vary with respect to load. However,

J and B are found to vary appreciably with respect to load. Therefore, they

have to be estimated at different loads.

2.3.1 Estimation of Friction Coefficient B

The torque equation of the BLDC motor with load arrangement is

given by

BdtdJTe (2.9)

where J is moment of inertia of BLDC motor and load and B is friction

coefficient of motor and load.

When the speed is constant, the torque equation becomes

BTe (2.10)

60N2BiKT aTe

22

aTiKB (2.11)

where ia is the per phase armature current measured at steady state for the

given load current. B is determined for the given load current using Equation

(2.11).

2.3.2 Estimation of Moment of Inertia J

When the supply to the motor is switched off, motor speed reduces

to zero from its steady speed. Hence, the torque equation becomes

0BdtdJ

The solution for this equation obtained using the steady state speed

as the initial value of speed is expressed by

t)J/B(e eBT

(2.12)

When t = =J/B, mechanical time constant of the BLDC motor and load, the

motor speed reduces from steady state speed to 36.8% of steady state speed.

From the time constant, the moment of inertia of the motor and load is given

by,

BJ (2.13)

J is determined for the given load current by substituting the values

of B and time constant in the Equation (2.13).

23

2.3.3 Estimation of J and B of BLDC Drive System

Figure 2.3 shows SIMULINK model of a BLDC motor fed by a

six- step inverter used at different loads. The specifications of BLDC motor

used for the proposed work are given in Table 2.1.

Figure 2.3 BLDC motor fed by a six-step inverter

24

Table 2.1 Specifications of BLDC motor

Parameters Values

Rated voltage 24 V dc

Number of poles 8

Rated speed 4000 rpm

Rated torque 0.125 Nm

Torque constant 0.036 Nm/A

Moment of inertia 48 10-7 kg-m2

Friction coefficient 1 10-5 Nm-sec/rad

Armature resistance 1.08 per phase

Armature inductance 1.8 mH per phase

24V DC supply was given to the BLDC motor through a switch

controlled by a timer. The gates of the respective switch in the six-step

inverter were controlled based on hall sensor signals. The three-phase outputs

of the inverter were applied to stator windings of BLDC motor. The

specifications of BLDC motor from Table 2.1 were chosen in block

parameters of permanent magnet synchronous machine (PMSM). Load torque

(TL) was given as step input of size corresponding to the desired load setting.

The simulated electromagnetic torque (Te) and the speed responses of the

motor were observed. Mechanical time constant of motor and load ( ) was

obtained from the speed response.

The electromagnetic torque and speed responses obtained at no load

are shown in Figures 2.4 and 2.5, respectively. The steady state speed and the

mechanical time constant are found from Figure 2.5 as 5994 rpm and 0.462

25

sec, respectively. The electromagnetic torque at steady state speed of 5994

rpm is determined as 5.47 10-3 Nm from Figure 2.4. Using Equations (2.11)

and (2.13), friction coefficient and moment of inertia of BLDC motor with

loading arrangement at no load are found out as 8.72 10-6 Nm-sec/rad and

4.03 10-6 Kg-m2, respectively. The mechanical parameter values J and B

computed are in agreement with the specifications of the machine as it is

available in the name plate (BM=1 10-5 Nm-sec/rad and JM=4.8 10-6 Kg-m2).

This procedure to compute moment of inertia and friction coefficient was

repeated from no load to full load in steps of 10% and the results are tabulated

in Table 2.2.

Figure 2.4 Electromagnetic torque response

26

Figure 2.5 Speed response

Table 2.2 Estimation of J and B at different loads

TL

(Nm)

% of full load (rad/sec)

Te

(Nm) (sec)B

(Nm-sec/rad)J

(Kg-m2)

0 0 627.5 0.00547 0.462 8.72 10-6 4.03 10-6

0.0125 10 567.8 0.014 0.1051 2.47 10-5 2.59 10-6

0.025 20 517 0.0278 0.055 5.38 10-5 2.96 10-6

0.0375 30 472.7 0.0449 0.0353 9.50 10-5 3.35 10-6

0.05 40 433.5 0.0629 0.025 1.45 10-4 3.63 10-6

0.0625 50 398.7 0.0637 0.0187 1.54 10-4 2.94 10-6

0.075 60 366.3 0.08 0.0146 2.18 10-4 3.18 10-6

0.0875 70 338 0.0834 0.0117 2.47 10-4 2.89 10-6

0.1 80 311.2 0.0974 0.0096 3.13 10-4 3.00 10-6

0.1125 90 287.7 0.1336 0.0081 4.64 10-4 3.76 10-6

0.125 100 265 0.1218 0.0068 4.54 10-4 3.13 10-6

27

2.3.4 Experimental Determination of J and B of BLDC Drive System

The BLDC motor was fed with 24V DC supply. The motor was

loaded in steps using slotted weights up to 40% of full load. The steady state

load current was found to be 0.6A from Figure 2.7. Then the supply to the

motor was switched off and the speed and current responses were obtained as

shown in Figures 2.6 and 2.7, respectively.

Figure 2.6 Speed response at 40% of full load

28

Figure 2.7 Stator current (load current) response

Mechanical time constant ( ) was determined as 0.1 sec

corresponding to time taken for the speed to drop from 4409 rpm to 1623 rpm

(36.8% of initial speed 4409 rpm). B and J were found to be 4.68 10-5 Nm-

sec/rad and 4.68 10-6 Kg-m2, respectively, using Equations (2.11) and (2.13).

Similarly, B and J were determined at different load currents and are tabulated

in Table 2.3. B and J as estimated experimentally were used to develop

transfer function model of the position control system that is used for the

design of proposed compensator in chapter 5.

Table 2.3 Determination of mechanical parameters J and B

ia(A) % of full load (rad/sec) (sec) B(Nm-sec/rad) J(Kg-m2)

0.32 20 482.97 0.2 2.39 10-5 4.78 10-6

0.6 40 461.71 0.1 4.68 10-5 4.68 10-6

1.2 60 387.46 0.045 1.11 10-4 4.88 10-6

1.6 80 374.89 0.032 1.54 10-4 4.93 10-6

29

The mechanical parameters, viz. moment of inertia and friction

coefficient of BLDC drive are found to vary with respect to load from Table

2.3 and therefore, they will certainly have an adverse effect on the

performance of the position control system. Hence, there is a need for the

determination of these parameters at different loading conditions.

2.4 EFFECT OF LOAD ON SYSTEM PERFORMANCE

The transfer function model given in Equation (2.8) is further

simplified as Equation (2.14) and Equation (2.15) for no load and full load,

respectively, using the experimentally estimated J and B parameter values.

9 3 -6 2a

(s) 0.036E (s) 7.254 10 s 4.368 10 s 0.001305s (2.14)

9 3 -6 2a

(s) 0.036E (s) 5.634 10 s 4.198 10 s 0.001786s (2.15)

The unit step response of the closed loop uncontrolled transfer

function model and the experimental hardware were obtained, as shown in

Figure 2.8 and Figure 2.9, respectively, for no load condition. Figure 2.10 and

Figure 2.11 indicate the simulated and hardware step response of the system

under full load condition respectively.

30

Figure 2.8 Simulated step response of position control system at no load

Figure 2.9 Real time step response of position control system at no load

31

Figure 2.10 Simulated step response of the system at full load

Figure 2.11 Real time step response of position control system at full load

32

The rise time and settling time obtained from the response

characteristics revealed that the time domain behaviour of the BLDC drive

based position control system depended on the load.

Unit step response of closed loop Parr (Parr 1989) tuned PID-

controlled BLDC drive based position control system at full load condition is

shown in Figure 2.12. PID controller parameters Kp, Td and Ti were obtained

as 18.55, 0.0022 sec and 0.0111 sec respectively using Equation (2.15). Time

domain specifications, viz. rise time, peak overshoot and settling time were

obtained as 2.87 msec, 17.6% and 29.3 msec, respectively from Figure 2.12.

Kp, Td and Ti were deduced as 11.2, 0.0029 sec and 0.0146 sec, respectively,

using no load transfer function of Equation (2.14). Unit step response of

closed loop Parr-tuned PID-controlled BLDC drive based position control

system at full load condition was obtained using these controller parameters at

no load and shown in Figure 2.13. Rise time, peak overshoot and settling time

were obtained as 4.52 msec, 15.8% and 43.1 msec, respectively, from

Figure 2.13.

Figure 2.12 Step response of PID-controlled system at full load

33

Figure 2.13 Step response of PID-controlled system at full load using

no-load model

These results are sluggish and not in agreement with the results of

Figure 2.12 since the no load model was used for tuning PID controller at full

load condition. It is observed from Figure 2.12 and Figure 2.13 that PID

controller parameters of the position control system have to be tuned for the

mechanical parameter variation in order to achieve better results under

dynamic load variation.

2.5 CONCLUSION

In this chapter, a simple method to compute the mechanical

parameters of a BLDC drive was developed for dynamic load variation. The

importance of estimation of mechanical parameters was emphasised. The

important observations made using the proposed approach are listed as

follows:

34

The mechanical parameters, i.e., moment of inertia and friction

coefficient vary with respect to the loading conditions and,

therefore, they have an adverse effect on the performance of

BLDC drive system.

The controller parameters for the closed loop position control

system are found to vary dynamically since the mechanical

parameters vary with respect to load.

It is essential to tune the controller parameters with respect to

load in order to achieve better position control since the load

influences the system dynamics.

There is a need to employ suitable tuning algorithm for PID

controller that can adjust itself its parameters in order to achieve

optimum results in the proposed closed loop BLDC drive fed

position control system under dynamic load variation.