Optimal Flux Selection of Induction Machine in the Field-Weakening Region

Wang Qingyi Faulty of Mechanical & Electronic Information

China University of Geosciences(wuhan) Wuhan, China

wangqingyi@cug.edu.cn

Liu Yang, Luohui Department of Control Science & Engineering Huazhong University of Science & Technology

Wuhan, China

Abstract—In this paper, an optimization for field current existing in field-weakening control is made. Taking the effect exerted by iron losses into consideration, a linearized compensation for it is proposed. The selection of field current is very important for indirect FOC, especially in the field-weakening region. Unreasonable selection will lead to reduction of current utilization factor and render decrease of the developed torque which in turn yields effect on loading capability, shortening the constant-power range and influencing on raising speed. This paper gives an optimization for field current, according to the principle of maximized utilization of current and voltage, and makes analysis of effect exerted by iron losses in high-speed interval, in which linearized compensation method is proposed. Finally, an experimental study is made by means of full-digital spindle drives based on digital signal processor (DSP). The experimental results show the proposed optimization solution for magnetic-field current and the compensation approach to iron losses are feasible, which can achieve noticeable enhancement of performance of the spindle drives system and is of great value in engineering application.

Keywords- induction machine; indirect vector control; field-weakening; field curren;, iron losses

I. INTRODUCTION Indirect vector control， an induction motor close loop

control strategy，is widely used all over the world at present which is implemented through decoupling stator current into two parts, field current and torque current namely [1]. For the rotor flux is proportional to the field current, rotor flux can then be regulated by the control of field current. In general, the constant flux is expected under base speed of motor. Whereas considering that the voltage can no longer be increased above the base speed, it is essential to decrease flux to increase speed, field-weakening rising speed namely. The field-weakening is very important in some application situations such as spindle drive system used in computerized numerical control (CNC), in which the motor speed is expected to operate four times higher than the base speed. The simplest way to increase speed is to render flux to vary inversely to speed, corresponding to control the reference field current inversely proportional to speed in indirect vector control [2]. However, this method doesn’t consider the optimum allocation of field current and torque current, namely the problem of the current utilization ratio. Therefore the maximum output torque cannot be obtained. The constant power region is then shortened and the speed rising ability of the system is also limited [3].

There are two restrictions in the field-weakening region: Current restriction and voltage restriction, in which current cannot be increased unlimitedly and voltage capacity supplied to the motor and endured by the motor are both limited[4].

Based on the two conditions above, an optimization solution of field current is then obtained. On the other hand, the induction motor can operate at the speed of 4-6 times higher than the base speed by using field-weakening strategy. In this situation, the iron losses cannot be ignored, for it can cause large-scale variety of motor parameters[5][6][7]. In the paper, physical experiment is carried out on the full digital spindle drive based on digital signal processor (DSP) with iron losses compensation. The experimental results demonstrate the feasibility of the scheme proposed.

II. BRIEF INTRODUCTION OF THE FULL-DIGITAL INDIRECT VECTOR CONTROL SYSTEM

The structure diagram of double closed-loop indirect vector control system is shown in Fig.1. The speed loop, current

decoupling, field orientation, current loop and SVPWM are all implemented by DSP. Fig.2 is the sketch map of current

decoupling. The rotor field position ( 1e s sθ θ θ= + ) is calculated indirectly through the real position of rotor( sθ ) and slip angle( 1sθ ) with d-axis of Park coordinate transformation

aligned to the rotor flux direction.

Fig.1. the block diagram of double closed-loop indirect vector control

978-1-4577-0547-2/12/$31.00 ©2012 IEEE

α

β

dq

eθ sθ

slθ

siα

siβdsi

qsi

maxsi

Rotor actually position

Fig.2. the sketch map of stator current decoupling in indirect vector control

In Fig.2, maxsi is the phase current peak value. siα and

siβ are the stator current components under static

coordination. dsi and qsi are the field current and torque current under rotating coordinate.

III. OPTIMIZED SELECTION OF FIELD CURRENT With the restrictions of busbar voltage and pulse width

modulation (PWM) Strategy, the maximal phase voltage exerted on the stator is a finite value, therefore dsV ， qsV under the d/q rotating coordinate must satisfy with the following relationship:

max

2 2 2ds qsV V V+ ≤ (1)

Meanwhile, for the output current of inverter and the current exerted on the motor are also confined to values（ maxI ）, thus dsi ， qsi under the d/q rotating coordinate must satisfy with the following relationship:

2 2 2maxds qsi i I+ ≤ (2)

Relationship (1) and (2) are two constraints that motor must comply with. The purpose of optimization for field current is to make the effective use of current so as to acquire the maximum output torque. The output torque in the indirect vector control system is shown as follows:

me qs rd

r

LT p i

L= Ψ (3)

Where p are electrical pole pairs; mL is mutual inductance between stator and rotor; rL is rotor inductance;

rdΨ is the rotor flux component in d-axis, under stable condition it can be represented as: rd m dsL iΨ = ,substituting it in (3):

2m

e qs dsr

LT p i i

L= (4)

A. Analysis of the Motor Operation under Current Restriction The current in d/q-axis satisfies with the relationship:

2 2 2maxds qsi i I+ = .

It is the prerequisite to acquire the maximum output torque. Under the condition, equation (4) can be given as:

2 2 2 2 4max maxe ds ds ds dsT ki I i k I i i= − = − (5)

Where 2

m

r

Lk p

L= is constant.

Supposing 2dsx i= ( 0x > )，the square root part in equation

(5) can be given as: 2 2maxy I x x= − (6)

From (6) we have: when2max

2I

x = , namely max

2dsI

i = , y

gets the maximum, in other words, the output torque gets the

maximum value. When dsi is located in the interval of（0，max

2I

）, y is an increasing function, while dsi is located in the

interval of ( max

2I

， + ∝ ), y is a decreasing function.

However, in fact with the existence of flux saturation, rdΨ cannot always keep the linear relationship with dsi . Equation (4) can no longer come into existence under the condition of flux saturation. For most motors, the field current ( dsni ) corresponding to the rated flux is the maximum dsi , generally,

max

2dsnI

i < . Therefore the meaningful interval for dsi is in (0，

dsni ). In this interval, the output torque eT is proportional to dsi .

From the analysis above we can conclude that under the condition of current restrictions, the greater dsi is, the greater output torque have. Accordingly, ds dsni i= can be kept under the base speed, whereas dsi is kept equally to the rated value

dsni as far as possible above the base speed. However, the back electromotive force increases with the increasing of speed, in which only both of the two restrictions are satisfied simultaneously (the current restriction and the voltage restriction) can the motor obtained the maximum output torque.

B. Analysis of the Motor Operation under the Current Restriction and the Voltage Restriction Under the stable condition, the voltage equation of

induction motor in rotating coordinates is shown as follows [1]:

ds ds s qs s eV i R i Lσ ω= − (7)

qs qs s ds s eV i R i L ω= + (8)

Where sR is stator resistance; sL is stator inductance; σ is total leakage factor.

In high-speed range, resistor voltage drop is usually ignored, thus equation (7) and (8) can be simplified as:

ds qs s eV i Lσ ω= − (9)

qs ds s eV i L ω= (10)

With satisfaction of (1) and (2) simultaneously, combined the equations (9) and (10), we get the following equations:

2 2 2max

2 2 2max( ) ( )

ds qs

qs s e ds s e

i i I

i L i L Vσ ω ω

⎧ + =⎪⎨

+ =⎪⎩ (11)

Solving the equations to obtain:

2 2 2max max

2 2dsV b I

ia b

−=

− ( 2 2 2

max maxV b I≥ ) (12)

2 2maxqs dsi I i= − (13)

Where s ea L ω= ， s eb Lσ ω= ，and a b> .

According to equation (12), dsi is related to eω . When eω is comparatively small, the result ds dsni i> calculated from the formula (12) cannot be allowed according to the previous analysis. Then supposing ds dsni i= to satisfy the current restriction, but the voltage restriction cannot be satisfied (as 2 2 2

maxds qsV V V+ < ).

With 2 2 2max maxV b I< , equation (12) cannot come into

existence, the current restrictions (equation (1)) can’t be

satisfied when max

maxe

s

VL I

ωσ

> namely, however,

2 2 2maxds qsi i I+ < the voltage restriction still works.

From these analyses, supposing that both the current and the voltage restrictions are satisfied at the same time,

max

maxe

s

VL I

ωσ

≤ namely, dsi is designated according to the

equation (12). If dsi exceeds the dsni , then setting ds dsni i= , and

qsi is designated according to equation (13).

C. Analysis of the Motor Operation under Voltage Restrictions

When max

maxe

s

VL I

ωσ

> , the current restriction is not satisfied

( 2 2 2maxds qsi i I+ < ). In this condition, only when the voltage

restrictions ( 2 2 2maxds qsV V V+ = ) is satisfied, can the motor

acquire the maximum output torque. Combined equation (5) , equation (9), and (10), thus:

2 2 2 4max ds ds

e

V i a iT

b−

= (14)

Taking the same analysis as above, we know that

when max

2dss e

Vi

L ω= the output torque gets the maximum value,

and then2 2 2

max dsqs

V a ii

b−

= .

Through the optimized strategy, current and voltage can be fully used, the output torque of the motor can be enhanced. Furthermore, the transition from voltage restriction to current restriction can be realized spontaneously, that is from constant torque region to constant power region and then to constant voltage region.

IV. EFFECT OF IRON LOSSES AND ITS LINEAR COMPENSATION

The equivalent circuit of induction machine with iron losses is shown in Fig.3[13], where feR is the equivalent resistance of iron losses.

sR ls s mL L L= − lr r mL L L= −

mL rRs

feR

fei mi

si

Fig.3 equivalent circuit of induction machine with iron losses

In indirect vector control system, magnetizing current is m dsi i= . From Fig.3, we know that current in d-axis ( di ) under

the rotating coordinates is actually composed of two components: field current ( dsi ) and iron losses current ( fei ). When motor operates under base speed, fei is very small compared to dsi and ignored generally. However, fei will increase gradually as the speed does, meanwhile dsi will decrease to meat the requirement of field-weakening control. In this case, fei can no longer be ignored compared with dsi .

If field current is not compensated, the actual rotor flux will not correspond to the reference value, furthermore it may lead to the inaccurate field orientation in indirect vector control. As a result, the current utilization ratio is reduced, resulting in adverse effect on the output torque. In a word, compensation for iron losses must be made in high-speed so as to enhance the output torque as far as possible.

In stable condition with no load, we know from equation (9) that 0dsV = , then the peak phase voltage of the motor s qs ds s eV V i L ω= = . If the field current ( dsi ) is inversely proportional to eω , then phase voltage should be constant. In the indirect vector control system shown in Fig.1, supposing the designated current in d-axis( dsrefi ) inversely proportional

to eω , voltage is almost constant when iron losses is very little relatively, whereas when iron losses is severe, then voltage will no longer be constant, the actual field current ( dsti ) can be measured through the voltage, then fe dsref dsti i i= − .

Base on the method discussed above, the test experiment about iron losses is performed. The experimental setup is full digital spindle drive. The indirect vector control strategy is used in the drive based on TI DSP TMS320F2812, power modulation uses PM75RLA120 operating with a switch frequency of 10 kHz. The spindle motor in the experiment is 7.5kW with two pole pairs. Its rated speed and field current are 1500r/min and 10A respectively, the maximum speed is 8000r/min. The motor parameters are shown in Tab.1. Fig.4 shows the relation curve of the voltage and the speed in the experiment. From the figure, we know that voltage declines considerably above 4500rpm, and more and more obviously as the speed rises, which indicates the great impact of the iron losses.

TABLE I MOTOR PARAMETERS USED IN THE EXPERIMENT

0.751Ω 0.547Ω3.1mH 3.1mH 56.6mHsR lsL rR lrL mL

Fig.4 curves of RMS value of stator phase voltage and speed

（*）the relation curve of the voltage and speed without iron losses, the voltage keeps constant

（□）the relation curve of the actual voltage and speed measured in experiment

From equation (10) and the motor speed, we can figure out the actual field current ( dsti ), as reference current in d-axis ( dsrefi ), is known, and then from the equation fe dsref dsti i i= − ,

we can obtain the relation curve of fei with the speed, shown in Fig.5.

From Fig.5 we know that it is nearly a linear relationship between fei and the speed. As the effect of fei is obvious above 3000rpm, the approximate process of fei is developed specially above 3000rpm, and the following linear relationship is obtained:

0.7119210fe

ni = + ( n is the motor speed 3000n rpm≥ )(15)

Fig.5 relation curve of fei and the speed

According to equation (15), we can make the real-time compensation for current designated in d-axis. Comparison curves of the stator phase RMS value before and after compensation are shown in Fig.6.

Fig.6 curves of RMS value of stator phase voltage and speed

（*）the relation curve of voltage and speed with iron losses ignored （□）the relation curve of the actual voltage and speed measured in

experiment （△）the relation curve of voltage and speed after compensation

From Fig.6 we can see that the stator phase voltage with compensation is almost constant. That means the actual flux matches the reference flux.

V. PHYSICAL EXPERIMENT RESULTS Physical experiment is performed on the same equipment

proposed above. At first, the speed step experiment between 0 and 8000rpm is carried out, and three field-weakening control strategies are adopted respectively:

(1) dsrefi takes the optimized strategy proposed in the paper, meanwhile compensating iron losses

(2) dsrefi takes the optimized strategy proposed in the paper

(3) dsrefi is inversely proportional to speed

The correlation experiment curves are shown in Fig.7. From the figure, it can be seen that the speed rises fast obviously above the base speed after adopting the optimized strategy, especially after iron losses compensation(above

3000rpm), and the speed rises faster, only taking 1.5s at least(Fig.7a) and 2.1s at most(Fig.7c) from 0-8000rpm.

Fig.7 0-8000rpm speed step response curves

a) optimized field-weakening strategy with iron losses compensation b) optimized field-weakening strategy without iron losses compensation c) field-weakening strategy with dsrefi inversely proportional to speed

Fig.8 the relation curves between output torque and speed

(■)the output torque in theory under constant power operation (○)the output torque using field-weakening control with iron losses

compensation (△)the output torque using field-weakening control without iron losses

compensation (▲) the output torque using field-weakening control with dsrefi inversely

proportional to speed With three different field-weakening strategies above,

correlated experimental curves on load at different speed in the stable situation are shown in Fig.8.From the figure, it can be seen that the constant power region can only reach around 2000rpm by using the field-weakening control strategy with

dsrefi inversely proportional to speed, whereas with the use of optimized control strategy, it can reach 4500rpm, even over 5000rpm with iron losses compensation.

VI. SUMMARY In the paper, some problems that exist in field-weakening

control of induction motor in indirect vector control system are analyzed in detail, and a set of solution is proposed. Under the condition of the limited voltage and current, the optimized allotment of field current and torque current is made, thereby, the utilization ratio of current is improved. Through the experiments using these methods, it is proved that the rising

time of speed is shortened and the constant power region is expanded, therefore, the system control performance is then improved. The method proposed in the paper has been used in spindle drives putting into effect. It has been proved practically that the system has achieved the design requirements expected.

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