analytical determination of t-n characteristics

8/12/2019 Analytical Determination of T-n Characteristics

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An analytical determination of the torquespeed and efficiencyspeed characteristics

of a BLDC motor

Miroslav Markovic, Andre Hodder and Yves PerriardIntegrated Actuators Laboratory (LAI), Ecole Polytechnique Federale de Lausanne (EPFL), Switzerland

e-mail: [email protected]

AbstractThe paper presents a complete mathematical anal-ysis of a BLDC motor commutation which results in the motorstatic characteristics torquespeed and efficiencyspeed. Themotor, having a sinusoidal back emf, is supplied from a DCsource via a 6-leg inverter using the 120ON operation. Theformulas for the phase currents and voltages are derived. Thefirst harmonic of the phase voltage is determined, which is usedin combination with the classical BLDC motor theory to obtainsimple formulas for the motor torque and efficiency. The model isverified using a measurement on a small industrial BLDC motor.

I. INTRODUCTION

To accurately determine the torque, power and efficiency of

a BLDC motor with a sinusoidal back emf, supplied from a

DC source via a 6-leg inverter using the 120ON operation,

an exact analysis of the motor commutation periods should be

used, which is the goal of this paper.

In literature, there are no many papers treating the accurate

analysis of the motor commutation. In [1], the commutation

period is correctly analyzed, but the final solutions are not

given in a closed form. Many authors [2] present the method-

ology to simulate the inverter circuit it could be useful, but

it would be more important to have a mathematical model.

The references [3] and [4] give an accurate mathematical

analysis of the commutation period, but they analyze the

case with trapezoidal back emf which causes a much simpler

model than with the sinusoidal one. Finally, [5] and [6] treat

the same problem as in this paper, however the approach

and the solutions are different than here. Neither publication

reports a complete analysis leading to a simple motor static

characteristics, which is very important from the engineering

point of view.

Throughout the paper, the formulas for the phase currents

presented in [7] will be used, however the main ones will

be repeated for clarity. Then, analytical formulas for the

phase voltages will be added, their harmonic analysis will be

performed, and simple original formulas for the torquespeed

and efficiencyspeed characteristics will be the final result.

I I . PROBLEM DEFINITION

The goal is to analytically determine the torquespeed and

efficiencyspeed characteristics of a 3-phase BLDC motor

supplied from a DC source Udc via a 6-leg inverter (Fig. 1).As the transistor ON resistance cannot be neglected, it will be

incorporated into the phase resistance.

The BLDC motor is connected in star, and its back emfs

ea, eb andec are purely sinusoidal. Its mechanical speed is

Fig. 1. Three-phase inverter for BLDC motor supplying.

p times less than its electrical speed , wherep is the number

of pole pairs. The electrical period is T = 2/. The inverteroperation is 120ON. It means that a corresponding couple

of transistors is constantly held in the state ON during 60

electrical degrees (from now on: edeg).

III. BLDC MOTOR AND INVERTER MODELLING

The BLDC motor phase voltages are given by:

ux= Rix+ Li

x+ ex (1)

wherex {a,b,c},R is the phase resistance, L = LsLm isthe phase equivalent inductance (difference between the phase

self inductanceLsand phase-to-phase mutual inductanceLm),and denotesd/dt. We introduce three additional quantities:

=L/R, = arctan(L/R) and Z= R2 + 2L2.The phase back emfs are sinusoidal, with the rms value

E= K directly proportional to the motor speed via themotor back emf constant K:

ea =

2Ecos t (2)

eb,c=

2Ecos(t 2/3) (3)The motor instantaneous electromagnetic power pe and

torqueme are given by:

pe = me = eaia+ ebib+ ecic . (4)

The instantaneous input power pdc is given by:

pdc

= Udc

idc

. (5)

with idc the instantaneous source current.

IV. COMMUTATION , ITS SUB-PERIODS AND THE PHASE

CURRENTS AND VOLTAGES

According to Fig. 2, the switching logic for 120ON opera-

tion can be defined as: t= T /6(corresponding tot = 60edeg) around the maximum/minumum ofea, TA1/TA2 is ON(similarly for the phases b and c).

168978-1-4244-2893-9/09/$25.00 2009 IEEE


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TC2

TA1

TC2

TB1

TA2

TB1

TA2

TC1

TB2

TC1

TB2

TA1

ea

eb

ec

eac

ebc

eba

eca

ecb

eab

t=0 t=T

Fig. 2. Switching logic for BLDC motor.

We analyze the period 0 < t < 60 edeg, when TA1 andTC2 are ON. It is divided into two sub-periods as it will be

shown.

A. Sub-period 1

During this sub-period, TA1, DB1 and TC2 are conducting

(Fig. 3). The initial values are ia|t=0 = I1, ib|t=0 =I1and ic|t=0 = 0. This sub-period lasts for tx until ib reacheszero, and the final values are ia|t=tx = I2, ib|t=tx = 0 andic|t=tx = I2. The values I1, I2 andtx are unknowns whichwill be determined later.

Fig. 3. Equivalent circuit during the sub-period 1.

The model of the circuit in Fig. 3 is:

ia+ ib+ ic= 0 (6)

Udc= ub uc= ua uc (7)As it is shown in [7], assuming that the sub-period 1 is short

enough, the values ea and eb do not change significantlyduring this sub-period, therefore they are treated as constants.

Introducing Ea = ea

|t=0 and Eb = eb

|t=0, (2) and (3) give

Ea= 2E and Eb = 2E/2.The solution for the phase currents is:

ia= I1et/ +

Udc/3 EaR

(1 et/) (8)

ib = I1et/ + Udc/3 EbR

(1 et/) (9)and:

ic = Ea+ Eb 2Udc/3

R (1 et/) . (10)

The duration tx of the sub-period 1 is determined by thefinal condition ib|t=tx = 0, which using (9) gives the relationbetweentx and I1:

tx= ln Udc/3 EbRI1+ Udc/3 Eb . (11)

Substituting the last relation in (10) and knowing the final

condition ic|t=tx =

I2, the relation between I1 and I2 is

obtained as:

I2 = Ea+ Eb 2Udc/3RI1+ Udc/3 Eb I1 . (12)

B. Sub-period 2

During this sub-period, TA1 and TC2 are normally con-

ducting (Fig. 4). The initial values (t= tx) are ia|t=tx =I2,ib|t=tx = 0 and ic|t=tx =I2. This sub-period lasts for theremaining T /6 tx, and the final values are ia|t=T/6 = I1,ib|t=T/6= 0 and ic|t=T/6 = I1.

Fig. 4. Equivalent circuit during the sub-period 2.

The model of the circuit in Fig. 4 is:

ia+ ic= 0 (13)

Udc= ua uc . (14)

The solution for the phase currents is:

ia= ic= I2e(ttx)/ + Udc2R

(1 e(ttx)/) +

+

6E

2Z cos(tx

6 )e(ttx)/

6E

2Z cos(t

6 ) . (15)

The final condition ia|t=T/6 =I1 gives the relation betweenI1, I2 and tx:

I1 = I2e(T/6tx)/ +

Udc

2R

(1

e(T/6tx)/) +

+6E

2Z cos(tx

6 )e(T/6tx)/

6E

2Z cos(

6 ) . (16)

The problem is now completely solved. The system of 3

equations (11), (12) and (16) with 3 unknowns I1, I2 and txis solved. As the equations are complicated, we are obliged to

apply a numerical solution.

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C. Phase voltages

To complete the model, the equations for the phase voltages

will be derived for the two sub-periods.

For the sub-period 1, the voltage uc will be determinedcombining (1) and (10). After the derivations, the next result

is obtained:

uc= 2Udc3

(17)

which gives using (7):

ua= ub= Udc

3 (18)

For the sub-period 2, the voltage uc will be determinedcombining (1) and (15). After the derivations, the next result

is obtained:

uc= Udc2

2E

2 cos(t 2

3 ) = Udc

2 eb

2 (19)

which gives using (14):

ua=Udc

2

2E

2

cos(t

2

3

) = Udc

2

eb

2

(20)

and:

ub = eb =

2Ecos(t 23

) (21)

It is interesting to note, that the quantities tx, I1 and I2disappeared in the obtained formulas, which have a very sim-

ple form. However, the solution is not completely analytical,

as the duration tx of the sub-period 1 should be determinednumerically.

V. TORQUESPEED CHARACTERISTIC

The only approximation taken so far is that the phase back

emfs remain constant during the sub-period 1. In all other

aspects, the model is complete, and can be readily applied.

However, its main drawback is that the solution is numerical

and in addition the formulas for the currents are complicated.

For an engineering approach, it would be useful to introduce

some approximations which simplify the model and make it

purely analytical, and the price to pay is of course a loss of

accuracy.

The idea is as follows: as the formulas for the voltages are

simple, their harmonic analysis will be performed using an

approximation. Then, only the first harmonic will be taken,

and used with a classical motor model to obtain the torque

speed and efficiencyspeed characteristics.

A. Harmonic analysis of the phase voltages

The harmonic analysis of the phase voltages will be per-

formed by neglecting the sub-period 1 as too short in compar-

ison with the sub-period 2. This is the only way to avoid the

numerical solution. The voltage ua (given by (20)) is chosenfor the analysis, and it can be written in the form:

ua =

k=1,5,7,11...

2 Ukcos kt (22)

The rms value Uk of the k-th harmonic of the phase voltageis calculated as:

Uk = 4

2

2

0

uacos kt dt (23)

The integration interval is divided into two: for 0 < t

analytical determination of t-n characteristics

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