Recent Developments of Electrical Drives - Part 26 pps

In fact, the effects of the faults in the external magnetic field, considering the pole number of the flux density components which are generated, can be precisely predetermined.. IMPACT

r

flux sensor

ls

L

Ls

Figure 2 Measured flux.

If the sensor has n turns and it has a rectangular shape of S area, and considering that l

r , then the h,kflux components regarding to the external flux density ones, are expressed

by:

h ,k (r ) = nS sin(hp β)

hp β ˆbn

a

Then, to obtain the component at the k ω angular frequency, it is necessary to add the different

terms which have this same frequency

k (r ) = +∞

h=−∞

The terms which compose k (r )are very numerous but many do not contribute significantly

Actually, relation (30) shows that only the terms with a small value of h will have a prominent

effect This property is enhanced by the quantity

sin(hp β)

hp β

which appears in expression (31)

The data given by the coil is a voltage which results of the Lenz law Consequently,

the ek (r) components of the corresponding measured e.m.f are obtained by the following relation:

e k (r ) = −d  k (r )

dt

e k (r ) = −kωnS +∞

h=−∞

sin(hp β)

hp β K h (r ) ˆb g

It can be noticed that the electro-motive force spectra amplify the high frequencies, what permits to distinguish the slot spectral lines more easily

244 Thailly et al.

Figure 3 Experimental device.

Principle of measurements

The synoptic of the Fig 3 shows the experimental device that allows performing the mea-surement of the radial magnetic field component The principle is to pick up the sensor signal with an analyzer via an acquisition card

A small winded flux sensor, with nS= 30 × 10−3, has been realized It is located at a

distance r equals to 100× 10−3m.

The tests are made considering a star connected 4 poles, 4 kW, 380/660 V, 50 Hz

squirrel-cage induction machine with 36 stator slots and 44 rotor bars For this machine, Rs int =

60× 10−3m, Rs

ext= 90 × 10−3m.

The stator is supplied by 380 r.m.s phase to phase voltage The supply current is equal to

I= 1.5 A In order to neglect the magnetic effects generated by the rotor, the slip is close to 0

Results

In the measured spectrum of the e.m.f only the spectral lines corresponding to kr = 0

(fundamental) and kr= ±1 will be considered for the comparison with the theory Concerning the fundamental, it is measured 5× 10−3V (r.m.s value) As the machine

is underfed, it will be supposed that the maximum of the flux density in the air-gap is 0.6 T

In that case, assuming that the flux density component at 50 Hz is mainly composed of one

p pair pole wave (h = 1, ˆb g

1,1 = 0.5 T), μ r have to be equal to 600 This low value can be justified taking into account that the frame is of cast iron and that the relative permeability

of this material is about 300

For the harmonics, Table 1 gives the relative magnitudes of the frequencies of the couple

at [1± N r(1− s)] f regarding to their fundamental.

Table 1 Relative magnitudes of the frequencies of the couple at [1± N r(1− s)] f

regarding to the fundamental

[1− N r(1− s)] f [1+ N r(1− s)] f Calculation without K hcorrection coefficient −5.49 db −5.93 db

Calculation with K hcorrection coefficient −31.72 db −22.31 db

Calculation with K hand with prominent

components

– 60

[1 – N r (1-s)]f [1 + N r (1-s)]f

Practical values Theoretical values

with K n correction coefficient

Theoretical values

without K n correction coefficient

f(Hz) – 40

–20

0db

Figure 4 Comparison between practical and theoretical toothing spectral lines in relative values

regarding to the fundamental

The two first table lines correspond to computation results where kr and ks vary from

−5 to 5 and hs varies from 1 to 13.

The third table line gives the computation results where only the prominent components depending on their number of pole pairs are considered

For the spectral line at [1− N r(1− s)] f frequency (kr = −1), the lowest number of pole pairs is obtained for hs = 1 and ks = 1; in that case, equation (8) gives |h|p = 2.

For this at [1− N r(1− s)] f frequency (kr = +1), ks = −1 and hs = −5 leads to

con-sider a prominent component such|h|p = 6.

The latest table line gives the relative magnitudes of this couple of frequencies which are obtained by experimentation

The results show that considering only the prominent component which composes one spectral line is sufficient in the study of the external magnetic field

Fig 4 gives a graphic illustration of the results It can be observed that when only the

analytical expression of the electro-motive force is computed, without take the K hcorrection coefficient into account, the frequencies of the couple at [1± N r(1− s)] f have similar

magnitudes which are high toward the magnitude of the fundamental

When the K h coefficient is introduced to modify the magnitudes of the components which compose the spectral lines, it can be noticed, in one hand, that the frequencies of this same couple do not have yet the same magnitudes, and on the other hand, that the difference with practical results is reduced (the error is only about 4 dB)

Let us also precise that the b gexpression given by (7) shows that current harmonics are induced in the stator windings and the rotor bars These currents have been neglected in our

246 Thailly et al.

study but some of them contribute to define the magnitude of the considered spectral lines That can partially justify the deviations

Conclusion

It is shown that all the flux density components do not evolve, through the stator, in the

same way according to the pole pair number hp of this component So, it is determined the

correction to apply on the calculated air-gap flux density components in order to appraise the measured spectrum outside of the machine

The presented method permits to analyze a practical spectrum toward a reference one obtained by calculation, that is specific for each machine

This quantitative approach is interesting if faults are introduced in the modeling of the air-gap flux density In fact, the effects of the faults in the external magnetic field, considering the pole number of the flux density components which are generated, can be precisely predetermined

Acknowledgment

This work is part of the project “Futurelec 2” within the “Centre National de Recherche Technologique (CNRT) of Lille.”

References

[1] W.T Thomson, M Fenger, Current signature analysis to detect induction motor faults, IEEE Ind Appl Soc Meet (IAS Magazine), Vol 7, No 4, pp 26–34, 2001

[2] M.E.H Benbouzid, Bibliography on induction motors faults detection and diagnosis, IEEE Trans Energy Convers., Vol 14, No 4, pp 1065–1074, 1999

[3] H H´enao, T Assaf, G.A Capolino, “Detection of Voltage Source Dissymmetry in an Induction Motor Using the Measurement of Axial Leakage Flux”, Proceedings of International Conference

of Electrical Machines (ICEM 2000), Espoo Finland, August 2000, pp 1110–1114

[4] D Belkhayat, R Romary, M El Adnani, R Corton, J.F Brudny, Fault diagnosis in induction motors using radial field measurement with an antenna, Inst Phys Meas Sci Technol., Vol 14,

No 9, pp 1695–1700, 2003

[5] D Roger, O Ninet, S Duchesne, Wide frequency range characterization of rotating machine stator-core laminations, Eur Phys J Appl Phys., pp 103–109, 2003

[6] J.F Brudny, Etude quantitative des Harmoniques du Couple du Moteur asynchrone triphas´e d’Induction, Th`ese d’Habilitation, No H29, Lille, France, 1991

[7] J.F Brudny, Mod´elisation de la Denture des Machines asynchrones Ph´enom`ene de resonance,

J Phys III, pp 1009–1023, 1997

[8] K.J Bins, P.J Lawrenson, Analysis and Computation of Electric and Magnetic Field Problems, Oxford: Pergamon press, 1973

[9] Ph.L Alger, The Nature of Induction Machines, 2nd edition, New York: Gordon & Breach Publishers, 1970

II-10 IMPACT OF MAGNETIC

SATURATION ON THE INPUT-OUTPUT LINEARIZING TRACKING CONTROL OF

AN INDUCTION MOTOR

1Faculty of Electrical Engineering and Computer Science, University of Maribor,

Smetanova 17, SI-2000 Maribor, Slovenia

dolinar@uni-mb.si

2Ørsted DTU Automation, Technical University of Denmark, Elektrovej, Building 325, DTU,

DK-2800 Kgs Lyngby, Denmark

pl@oersted.dtu.dk

Abstract This paper deals with the tracking control design of an induction motor, based on

input-output linearization with magnetic saturation included Magnetic saturation is accounted for by the nonlinear magnetizing curve of the iron core and is used in the control design, the observer of state variables, and in the load torque estimator Experimental results show that the proposed input-output linearizing tracking control with the included saturation behaves better than the one without saturation

It also introduces smaller position and speed errors, and better motor stiffness

Introduction

The magnetic saturation phenomenon, which occurs in the induction motor (IM) iron core, has been recognized from the very beginning of IM use The models of saturated induction machines presented in the literature [1–3] show that magnetic saturation is most frequently considered by nonlinear variable mutual inductance, while the additional magnetic cross-couplings are neglected In general, leakage flux path saturation is rarely included in the models, although its role in the squirrel-cage IM with closed rotor slots can be rather high The level of idealization, introduced by neglecting magnetic cross-saturation, depends on the type of model, i.e on the selected state variables [4]

Although magnetic saturation of the IM is very important for the performances of the controlled drive, it has hardly ever been properly considered in control design because of its mathematical complexity Pioneering efforts to address the perturbing effects of magnetic saturation on the field oriented control are found in [5]

Saturated induction motor models found in the literature [1,3] are used primarily for the analysis of an induction machine operation Some exceptions in which linear cascade rotor

S Wiak, M Dems, K Kom˛eza (eds.), Recent Developments of Electrical Drives, 247–260.

2006 Springer.

248 Dolinar et al.

field oriented control is used can be found in [6–8] but they present an incomplete solution for the control algorithm An observer for state variables that includes magnetic saturation

is found in [9] Some nonlinear control approaches, based on modified IM models that include saturation, are given in [10–13] as well

In this paper an input-output linearizing tracking control design is presented It is based

on the mixed “stator current vector is, rotor flux linkage vectorΨr” saturated IM model introduced by [3] The effect of magnetic cross-saturation is totally included in the main flux path saturation The saturation of the stator and rotor leakage flux path is neglected In addition, a state variable observer and a load torque estimator were designed with included magnetic saturation

An is,Ψrsaturated IM model is presented in this paper The input-output linearization based on the introduced model is carried out, with respect to the rotor position To obtain the required unmeasurable state variables, an observer based on the inverse saturated IM model was designed The experimental results of the proposed input-output linearizing tracking control of IM with included magnetic saturation show better dynamic performances of the drive than the classical control, where magnetic saturation is not considered The main improvements are the smaller rotor position and speed errors, as well as higher stiffness and better load torque rejection, which results in a smaller stator current when the motor is loaded with a step change in the load torque [14]

Saturated induction machine model

The general approach to IM modeling incorporating magnetic saturation with different selections of state variables is presented in [3] The standard two-phase model of an IM in

the general reference frame with stator current vector isand rotor flux linkage vectorΨras state variables, is given by a nonlinear model (1)

u = A(x)dx

dt + B(x)x

J d ω r

dt = (t e − t l)− f ω r

(1)

where

T

⎡

⎢

⎢

⎢

⎢

⎣

L l− L2rl

L dd

−L2rl

L dq

1− L rl

L dd

−L rl

L dq

−L2rl

L dq

L l− L2rl

L qq

−L rl

L dq

1− L rl

L qq

⎤

⎥

⎥

⎥

⎥

⎦

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

L l− L2rl

L r

L m

L r

ω g

L l−L2rl

L r

L m

L r

0

−R r

L m

L r

L r

−ω sl

L m

L r

ω sl

R r

L r

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

t e = p L m

L r

(i sq ψ r d − i sd ψ r q)

u sd , u sq and i sd , i sq are the d- and q-axis stator voltages and currents, ψ rdandψ rqare the

rotor flux linkages, R s and R r are the stator and rotor resistances, L mis the mutual static

inductance, L s and L sl are the stator self-inductance and the stator leakage inductance,

L r and L rl are the rotor self-inductance and the rotor leakage inductances, L l = L sl + L rl

is the total leakage inductance, L dq is the cross-coupling inductance, L dd and L qqare the

inductances along d- and q-axis, ω g is the angular speed of the general reference frame,

ω r is the rotor angular speed,ω sl = ω g − ω r is the slip angular frequency, J is the drive moment of inertia, f is the coefficient of viscous friction, t e and t lare the electrical and the

load torque, and p is the number of pole pairs Inductances L dd , L qq , and L dqare given as follows:

1

L dd

L rl + L cos

L rl + L m

sin2ρ m

1

L qq

L rl + L m

cos2ρ m+ 1

L rl + L sin

2ρ m ,  m= |Ψm| 1

L dq

=

1

L rl + L −

1

L rl + L m

cosρ msinρ m , i m= |im| (2)

L m =  m

i m

, L = d  m

di m

, cos ρ m= ψ md

 m

, sin ρ m= ψ mq

 m

where i m is the modulus of the magnetizing current vector L represents the so-called

dynamic inductance,ψ md andψ mq are the d- and q-components of the magnetizing flux

linkage vectorΨm,ρ mis the angle of magnetizing flux linkage vectorΨm, while mis the modulus ofΨm L m and L are given by equations (2) and are obtained from the measured

magnetizing curve presented in Fig C1, Appendix C

Two-phase is,Ψrmodel (1) can be written in the state-space form as:

dx

dt = −A(x)−1B(x)x + A(x)−1u= Cx + ω gZx+ ω rWx + Du

(3)

d ω r

dt = p

J

L m

L r (i sq ψ r d − i sd ψ r q)− 1

J t l− f

J ω r

250 Dolinar et al.

where:

⎡

⎢

⎣

c11 c12 c13 c14

c21 c22 c23 c24

c31 0 c33 0

0 c42 0 c44

⎤

⎥

⎦ Z =

⎡

⎢

⎣

z11 z12 z13 z14

z21 z22 z23 z24

⎤

⎥

⎦

⎡

⎢

⎣

0 0 w13 w14

0 0 w23 w24

⎤

⎥

⎦ D =

⎡

⎢

⎣

d11 d12 0 0

d21 d22 0 0

⎤

⎥

⎦

Elements of matrices C, Z, W, D are given in Appendix A.

Input-output linearization

The two-phase IM model (3) is transformed to the stationary reference frameαβ where

ω g = 0 It is written in the compact form (4)

˙x = f(x) + Gu = f(x) + gα usα+ gβ usβ (4)

where x=θ r ω r i s α i s β ψ r α ψ r β

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

ω r

p L m

L r

1

J



i s β ψ r α − i s α ψ r β

− 1

J t l− f

J ω r

c11i s α + c12i s β + c13ψ r α

+ c14ψ r β + w13ω r ψ r α + w14ω r ψ r β

c21i s α + c22i s β + c23ψ r α

+ c24ψ r β + w23ω r ψ r α + w24ω r ψ r β

c31i s α + c33ψ r α + w34ω r ψ r β

+ c42i s β + c44ψ r β + w43ω r ψ r α

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

(5)

=



0 0 d11 d21 0 0

0 0 d12 d22 0 0

T

θ r is the rotor angle, u s α , su s β and i s α , i s βare the stator voltages and currents andψ r αand

ψ r α are the rotor flux linkages Coefficients c( ·), w( ·), and d( ·)are given in Appendix A The

output vector y=θ r 2

r

T

is chosen by (6), where2

r = ψ2

r α + ψ2

r β

y= φ(x) =φ1(x) φ2(x)T

=θ r 2

r

T

(6)

Since the load torque t l is not the system control input and cannot be directly measured, it

is excluded from the nominal part of the motor model and is later considered as an external disturbance The load torque is obtained by a simple load torque estimator described in [14] The input-output linearization technique is based on exact cancellation of the system’s nonlinearities in order to obtain a linear relationship between the control inputs and the system outputs in the closed loop The nonlinear control law is deduced to a successive differentiation of each output until at least one input appears in the derivative The derivative

of the first outputφ1(x)= θ ris:

˙y1= ∂φ1(x)

∂x

dx

The second and the third derivatives are given by (8) and (9)

¨y1= ∂x ∂ ˙y1

dx

dt =∂x ∂ (L f φ1)dx

dt = L2

f φ1

(8)

= p L m

L r

1

J



i s β ψ r α − i s α ψ r β

− 1

J t l− f

J ω r y

···

∂x ¨y1

dx

dt = L3

f φ1+ L g α L2f φ1u s α + L g β L2f φ1u s β (9)

Lie derivatives L3f φ1, L g α L2f φ1, and L g β L2f φ1are given in Appendix B The input voltages

u s α and u s β appeared in y···

1, therefore, the relative degree of the first subsystem is three The second output to be differentiated is the square of the rotor flux linkage modulus

2

r Its first derivative is given by equation (10)

˙y2 = ∂φ2(x)

∂x

dx

dt = L f φ2

(10)

= 2R r

L m

L r



i s α ψ r α + i s β ψ r β

− 2R r

L m

L r



ψ2

r α + ψ2

r β



The second derivative of2

r is given by (11)

¨y2= ∂

∂x ˙y2

dx

dt = ∂

∂x (L f φ2)dx

= L2

f φ2+ L g α L f φ2u s α + L g β L f φ2u s β

Lie derivatives L2

f φ2, L g α L f φ2, and L g β L f φ2 are given in Appendix B The input

voltages u s α and u s β appeared in ¨y2, therefore, the relative degree of the second subsystem

is two Consequently, the total relative degree of the system is unequal to the system order

n = 6, which reveals the existence of uncontrollable internal dynamics It is easy to prove (see Ref [14]) that the dynamics of the third subsystem is stable if the third outputφ3is selected asφ3= arctan ψ r β /ψ r α, as in [11]

In the next step the input-output linearization of the nominal system is done [11] The

nominal part of the IM model is written in the form of higher derivatives of outputs y1

and y2



y

···

1

¨y2



=



L3

f φ1

L2

f φ2



+ E(x)



u s α

u s β



where



L g α L2f φ1 L g β L2f φ1

L g α L f φ2 L g β L f φ2



(13)

is the decoupling matrix The decoupling matrix E(x) is nonsingular, except at the motor

start-up, whereψ2

r α + ψ2

r β = 2

r [14]

252 Dolinar et al.

The obtained model (12) is still nonlinear and coupled The linearized and decoupled IM

model (15) is obtained by an appropriate selection of the control input u=  u s α u s βT

(see equation (14))



u s α

u s β



= E −1 (x)



−



L3f φ1

L2

f φ2

 +



v α

v β



(14)

Note that v=  v α v βTin equation (15) is the new system input



y

···

1

¨y2



= D(x) + E(x)E −1 (x)D(x) + E −1 (x)v

(15)

= D(x) + E(x)u =



v α

v β



= v

The input-output behavior of the system (15) is linear, but the relationship between the

control input v and the states x is still nonlinear This nonlinear relationship is eliminated

by selecting a new set of state variables, z = [z1 z2 z3 z4 z5] introduced by the nonlinear

transformation z = T(x), defined as:

y = T(x) =

⎢

⎢

⎢

⎢

⎣

θ r

ω r

p L m

L r

1

J



i s β ψ r α − i s α ψ r β

−1

J t l− f

J ω r

2R r L m

L r



i s α ψ r α + i s β ψ r β

− 2R r L L m r



ψ2

r α + ψ2

r β



⎥

⎥

⎥

⎥

⎦

(16)

The block diagram of the decoupled and linearized IM model is given in Fig 1 This system is linearized, decoupled and unstable Both stabilization and tracking can be achieved without any concern about the stability of the internal dynamics using the linear tracking controllers designed by the pole placement [14]

The reference output vector y∗is given by (17)



y∗ 1

y∗ 2



=



∗

r

2 ∗

r



(17) where∗

r and2 ∗

r represent the reference trajectories of the rotor position and the square

of rotor flux linkage modulus The differences between the reference and the actual values

of the controlled outputs are tracking errors (18)

e1= y∗

1 − y1, e2= y∗

.

v = α y 1

v = β y 2

z1(0)

z2(0)

z3(0)

z4(0)

z5(0)

z5=z4

.

z3=z2 z2=z.1

z1 Θr=y1

z4=Ψr=y2 2

=

Figure 1 Linearized and decoupled IM model.

S Wiak, M Dems, K Kom˛eza (eds.), Recent Developments of Electrical Drives, ... the next step the input-output linearization of the nominal system is done [11] The

nominal part of the IM model is written in the form of higher derivatives of outputs y1...

1Faculty of Electrical Engineering and Computer Science, University of Maribor,

Smetanova 17, SI-2000 Maribor, Slovenia

dolinar@uni-mb.si

2Ørsted