Simulation and Modeling of DCMotor by Current Response Model

Abstract––Pasek described a testing procedure fordetermining the dc parameters from the currentresponse to a step in the armature voltage motor. Timemoments have been introduced in automatic controlbecause of the analogy between the impulse response ofa linear system and a probability function. This basicidea has generated applications in identification, modelorder reduction and controller design. In this paper, twodeveloped identification algorithms based on themoments and Pasek’s methods, are introduced andapplied to the parameter identification of a dc motor.The simulation and experimental results are presentedand compared.

Index Terms––Identification, moments method , Pasek’s identification method, dc motor.

I. INTRODUCTION

The DC motor is the obvious provingground for advanced control algorithm inelectric drives due to the stable andstraight forward characteristicsassociated with it[1], [2]. It is alsoideally suited for trajectory controlapplications. The requirement for highperformance speed control of dc motorshas produced great research efforts inthe application of modern controltheory[1], [2], [4]. Systemidentification is the subject whichdeals with the problem of buildingmathematical models of dynamical systemsbased on the observed system data [2].The actual identification of models fromdata involves decision making on thepart of the person in search of models,as well as fairly demanding computationsto furnish bases for these decisions.There are situations when identificationis necessary even though a relatively

accurate mathematical model isavailable. For example, the dc motorparameters might be subject to sometimevariations, in the cases, a mathematicalmodel that is accurate at the time ofthe design may not accurate at a latertime. In the literature there are manyclassical methods to identify thesesparameters [1]-[6]. Time moments have been introduced in

automatic control because of the analogybetween the impulse response of a linearsystem and a probability function [6],[7]. Thus, an impulse response ischaracterized by infinity of moments,practically, only the first ones arenecessary as for a probability densityfunction. This basic idea has generatedapplications in identification, modelorder reduction and controller design,known as the method of moments.Pasek described a testing procedure

for determining the parameters of a dcmotor linear model from the currentresponse to a step in the armaturevoltage of the motor. The Pasek’sequation for the parameter determinationhas been derived in his work [8].

This paper is structured as follows.Section II, describes the dynamic ofthe separately excited dc motor. SectionIII, describes a new identificationalgorithm for dc motor

parameters by Pasek’s technique.Section IV, describes the method ofmoments and analyzes the developedidentification algorithm, applied to the

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Masanam.Sai Kiran Loshith* and V.Narendara Raj**

Department of Electricals and Electronics,Panimalar Institute of Technology,

Poonamalle,Chennai-607 123.saikiranpit@gmail.com

parameter identification of a dc motor.In section V, experimental results areillustrated and compared to thesimulation results. Finally, conclusionsof the paper are summarized in sectionVI.

II. DC MOTOR MODEL

The dynamic of the separately exciteddc motor may be expressed by thefollowing equations

(1)

(2)

Where K, Ra , La , J and f are respectively,the torque and back-EMF constant, thearmature resistance, the armatureinductance, the rotor mass moment ofinertia and the viscous frictioncoefficient. ω(t), ia(t), Ua(t) and TL(t)respectively denote the rotor angularspeed, the armature current, theterminal voltage and the load torque.

III. DEVELOPED MODEL BASED ON THE PASEK METHOD

Pasek proposed a simple method, whichrequires one test for the identificationof the dynamic model of the separatelyexcited dc motor, with the assumption:f=0 (viscous friction coefficient null)[8]. Accordingly we thought ofdeveloping a model of identificationbased on the Pasek model, by introducingthe viscous friction coefficient.It is possible from a terminal voltage

step ΔUa to determine the majority ofthe dc motor parameters that is alsopossible from the abrupt terminalvoltage variation. We measure theinitial and final currents and speeds

values shown on table III. At the steadystate we can write

(3)

(4)

Where Ua0, ia0 and ω0 respectively denotethe terminal voltage armature currentand the rotor speed at initial regimesubscripted "0". Ua1, ia1 and ω1

respectively denote the terminalvoltage, armature current and the rotorspeed at final regime subscripted "1".

According to (3) and (4) the torque andback-EMF constant K can be written as

(5)the steady state check

(6)

where  ΔUa, Δia and Δω respectively denotethe terminal voltage variation, armaturecurrent variation and rotor speedvariation.The armature resistance is given from(6) as

(7)according to equations (8) and (9) wecan obtain the two transfer functions ofthe armature current and rotor speed

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(8)

(9)the armature current transfer functionis given as

(10)

the rotor speed transfer function isgiven as

(11)

Where

Electrical

time constant

Mechanical time

constant

Usually small

coefficient

Let us pose

(12)for a terminal voltage step ΔUa the current form is given as

Where

are the

transfer function polesWith

We determine the moment t1 when thecurrent pass by its maximum, notedΔia(t1) we have

(14)By dividing the two terms of equation(14) by τe we find

(15)Let us pose

(16)Then

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(17)

Where L1, L2, L3 and L4 are shown on theappendix.

From two equations (15) and (17) wededuce the abacuses(Fig. 7) which gives

δ and according to λ.

Measurements of t1, Δia(t1) and Δia(2t1)define the dc motor parameters thanks tothe following step

- We calculate δ and an abacus givesλ

- The other abacus gives

- We deduce τe and λ τe = τm- Finally from τe and λ τe we deduce

respectively La and Jthe calculation of output current gaingives

(18)

from (18) we calculate

(19)

The static torque can be calculated fromsteady state as

(20)

IV. METHOD OF MOMENTS

The moments constitute the basis for anon classical representation of linearsystems. The characterization of animpulse response by its moments isequivalent to the momentcharacterization of a probabilitydensity function[7]. Impulse responsemoments are system invariants. Like fora probability density function, it isnot necessary to compute an infinity ofmoments to characterize with a goodapproximation the shape of the impulseresponse only the first ones arenecessary to perform thischaracterization.

A. Temporal Moment of a Function

Let us consider a stable linearsystem, characterized by its impulse h(t)then,

(21)H(s) can be expanded in Taylor series inthe vicinity of

(22)where

is the n th order frequency moment of h(t)for ω=ω0, notice that is complex.In the particular case ω0=0, frequencymoments correspond to classical timemoments

(23)

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they permit the characterization ofH(jω) around ω0=0, as well as that ofthe impulse response h(t). A0(h) is thearea of h(t), A1(h) defines mean time ofh(t) and A2(h) deals with the dispersionof h(t) around its mean time[7]. Equation(22) is rewritten as

(24)

Let

Then, time moments can be expressed as

(25)

B. Moments and Parameters of a Transfer Function

Let y(t) the step response of thestudied system. We proposes to identifythe system by the model

(26)

from the final value theorem, as timeapproaches infinity for a stable linearsystem, the system response approaches asteady state value K1 given by

(27)

if a step input is applied to the systemdescribed in equation (26), by takingthe Laplace transform of the normalizedresponse gives

(28)

let us consider ε(t) an error functionwith

(29)

by introducing the Laplace transform inequation (29), (26) can be written as

(30)

the development of (30) gives

(31)

Then, using (24) we can write

(32)

according to (24) and (32) we can deducethe coefficients of the transferfunction H(s) by solving the followingmatrix system

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(33)

where An(ε) is the nth order temporalmoment.

C. Dc Motor Transfer Function and its Moments

For our cases, when n=2 and m=1, thetransfer function (8) becomes

(34)

system (33) is reduced to the followingmatrix system

(35)

the resolution of this matrix system(35), gives the following coefficients:

, ,

(36)

D. Parametric Identification

After having deduced the mathematicalforms which are used for calculation ofthe transfer function coefficients andwhich enable us at the same time tocalculate the electric and mechanicalmotor parameters, we present here, thestages to be followed at the time of thedetermination of these parameters.the calculation of Ki and Kv gains of

the tow outputs ia(t) and ω(t)respectively, by taking accountequations (8) and (9) gives

(37)

(38)according to (37) and (38) we deduce fand μ

(39)

(40)

by identification of H1(s) and H2(s)denominators with H(s) denominator weobtain

(41)

(42)

according to (41) and (42) we can obtaina second order equation

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Δia(2t1)

0.0 0.1 0.2 0.3 0.4

0.0

1.0

2.0

3.0

4.0

5.0

ia1 ia0

Δia(t1)

t1 2t1

ia , A

Time, sFig. 2. Armature current response.

(43)

The resolution of the equation (43)gives two roots one is positive, theother is negative (rejected).- According to (41) and (42) we deduce

τm - The deduction of τm and τe gives La and

J.- The static torque can be calculatedfrom steady state as

(44)

V. EXPERIMENTAL RESULTS

The separately excited dc motor usedfor experimental tests, has the nominalcharacteristics shown on table IV. thefirst experiment is the determination ofelectric dc motor parameters accordingto direct tests, like armatureresistance Ra and armature inductance La,as well as the back-EMF constant K.Mechanical parameters are alsodetermined by direct tests ( statictorque Tst and viscous frictioncoefficient f ). The deceleration test,enables us to determine the moment ofinertia J. The second experiment to becarried out is to identify the dc motorparameters from the dynamic test.According to a step amplitude ΔUa ofterminal voltage applied to the armaturecircuit of the dc motor. The initial andfinal values of the armature current andthe angular speed obtained from thistest, are shown on table III. Therecording of ia(t) gives t1=0.026s wherethe current pass by its maximum. For thetwo identification methods ( developedPasek’s and moments methods), the back-EMF constant K, armature resistance Ra,and static torque Tst can

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A0A1A2a1a2b1ω(t)6.1218180.14034940-0.000940.0556000.0014400.012524ia(t)-0.29657-0.152533-0.0005040.4705650.240323-5.962602

TABLE I1,2 AND 3 ORDER MOMENTS AND TRANSFER FUNCTION COEFFICIENTS VALUES

0.0 0.1 0.2 0.3 0.4

0.0

40.0

80.0

120.0

160.0

200.0

Δω

ω0

ω1

1

Time, s

Speed, rad/sec

Fig. 1. Rotor speed angular response

TABLE IICOMPARISON BETWEEN DIRECT TESTS, MOMENTS AND PASEK’S DEVELOPED METHODS FOR

PARAMETER IDENTIFICATION OF DC MOTOR

Ra

(Ω)La

(H)K(N.m.A-1)J(Kg.m2 )f

(N.m.s/rad)Tst

(N.m)Direct tests280.8201.340.00280.000540.127Moments method30.90.8031.3230.00310.00050.128Pasek’s developed

method30.90.4381.3230.00360.00050.128

be determined by using respectivelyequations (5), (7) and (20). the viscousfriction coefficient f, can be determinedby using equation (19) for the firstmethod (developed Pasek’s method) and byusing (39) for the second (momentsmethod). By using the first method andlet us know Δia(t1) and Δia(2t1) , δ is thencalculated from equation (16),( δ=0.79).from the abacus of Fig. 7 we deducerespectively λ = 4.5 and t1/ τe =1.82.let us know λ and τe we deduce τm andconsequently J and La.Table I shows the first, second and

third order moments values ,as well asthe transfer function coefficientsvalues successively calculated from thetrapezoids method. Let us know a1, a2 andμ then, we have the second orderequation

,the resolution of this equation gives

and ( τe2 is arather large time constant, is thusrejected). The deduction of τe and τm

enables to calculate J and La. Table IIsummarizes the values of the parameterscalculated from the two identificationmethods (direct tests, moments andPasek’s developed methods).Finally to check the precision of each

method, we have simulated the dynamictest applying a step amplitude ofterminal voltage ΔUa=188V, to thearmature circuit of dc motor, as wellas, deceleration test and mechanicalcharacteristic.According to Figs. 3, 4 and 5 the

curves simulated from the dynamic testparameters with the moment method areclose to the real curve (experimentalcurve) that those simulated from thedirect tests parameters and Pasek’sdeveloped method. With regard to thesteady state (Fig.6) the curvessimulated from the dynamic test anddirect tests parameters are almostidentical and close to real measurement(experimental curve).

8Ua(V)ia(A)ω(rpm)Initial regime600.113400Final regime2480.1671745

TABLE IIIDYNAMIC TEST

0.00 0.05 0.10 0.15 0.20 0.2 5 0.30 0.35 0.40 0.45

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Time, s

ia , A

Direct testsMoments methodExperimental curvePasek’s developed method

Fig. 3. Armature current response.

0.00 0.05 0.10 0 .15 0.20 0 .25 0.30 0.35 0.40

40.0

60.0

80.0

100.0

1 20.0

140.0

160.0

1 80.0

200.0

220.0

Time, s

Speed, rad/sec

Direct testsMoments methodExperimental curvePasek’s developed method

Fig. 4. Rotor speed angular response

VI. CONCLUSION

In this work, we tried to contributeour share in the discipline of the dcmotors modeling. This contribution,which can be classified in a very widefield of identification methods can besummarized in the following results: wehave developed a dynamic model based onthe moments and Pasek’s methods. Themoments method especially makes itpossible to have a model closer toreality in transitory mode. We haveproposed a comparison between variousmodels based on the identificationmethods(direct tests, moments andPasek’s methods), this comparison ismade on the basis of real measurementstaken in laboratory on a separatelyexcited dc motor, with 180 W of ratedpower. It shows the advantage of theonly dynamic test for identification,coupled to the moments method

REFERENCES

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Speed, rad/sec

Torque, N.m

Fig. 6. Mechanical characteristic

0 20 40 60 80 100 120 140 160 180 200

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Direct testsMoments methodExperimental curvePasek’s developed method

Rated power180 WRated speed1500 rpmArmature voltage270 VField

voltage220 VArmature current 1.1 AField current0.4 A

TABLE IVSPECIFICATION OF EXPERIMENTAL DC MOTOR

0 .00 0 .50 1.00 1.5 0 2.0 0 2.50 3.00 3.50

0.0

2 5.0

50.0

75.0

10 0.0

125.0

150.0

17 5.0

200.0

225.0

Time, s

Speed, rad/sec

Fig. 5. Deceleration test

Direct testsMoments methodExperimental curvePasek’s developed method

Fig. 7. Abacuses for the determination of DC motor parameters .

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