机电一体化英文论文.doc

1、An innovative digital method for the dynamic simulation of DCelectromechanical systemsChen Chaoying a,*, P. Di Barbab, A SavinibaDepartment of Electrical Engineering, Tianjin University, 300072 Tianjin, Peoples Republic of ChinabDepartment of Electrical Engineering, University of Pavia, 27100 Pavia,

2、 ItalyAbstractIn this paper, an innovative digital simulation method, named R-K-T method, is presented. The new methodology combines RungeKutta and trapezoidal methods and possesses the advantages of both of them. The errors featuring the proposed method are analysed and their correction is worked o

3、ut. As a case study, the circuit model of a small DC motor, acting as the engine starter of a road vehicle, is considered;the proposed methodology is applied to carry out the dynamic simulation of the electromechanical device. The results are obtained efciently and with a good degree of accuracy; in

4、 particular, the numerical oscillations are suppressed.q1998 Elsevier Science Ltd. All rights reserved.Keywords:Numerical methods; Time integration; Dynamicsystems;Electromechanics; DC motor1. IntroductionSeveral digital methods, such as Euler, trapezoidal,RungeKutta and linear multistep methods are

5、 generallyused to carry out numerical integration and differentiation.The Euler method is simple, but with low accuracy; its cutoff error isO(h2), whereas that of the trapezoidal methoddecrease asO(h3). The RungeKutta method has relatively high accuracy but requires large amount of computational wor

6、k; finally, the multistep method has high accuracy, but it can not be self-started 1. Therefore, the trapezoidal method finds widespread applications in transient digital simulations. However, in DC system simulations, the trapezoidal method often introduces numerical oscillations with equal amplitu

7、des, so that its application in this case is critical. Since the backward Euler method can avoid such oscillations, in the literature 2, a damped trapezoidal method was proposed; this method introduces a damping factor into the trapezoidal method which effectively decreases the numerical oscillation

8、s but at the sacrifice of accuracy. After analysing trapezoidal and RungeKutta methods carefully, this paper presents an innovative simulation method, called R-K-T, which combines RungeKutta and trapezoidal methods ingeniously. The advantages of the new method are: the RungeKutta method can be expre

9、ssed by the companion model just like the trapezoidal method does; the numerical oscillations can be attenuated efficiently. According to frequency spectrum analysis, the errors of the method are calculated and corrected. It makes it possible to simulate DC systems accurately and efficiently.2. Nume

10、rical oscillations of trapezoidal method in DC systemsConsidering the inductive circuit shown in Fig. 1(a) the governing equation iswhere currenti is the unknown. Using the trapezoidal method for time integration, one can get:Where h is the time step of calculation.LetthenThe companion model of that

11、 depicted in Fig. 1(a) is shown in Fig. 1(b). From Eq. (1) one can also get:Fig. 1. Inductive impedance (a) and its companion models (b) and (c).whereIts companion model is shown in Fig. 1(c).Suppose, whena DC current ows through the inductive impedance. From Eq. (3) the voltage response of the indu

12、ctive branch can be calculated asIt can be seen that the oscillation of voltage is undepressed.Otherwise assume, whennk, the current is switched off, i.e. from Eq. (3) one can get:that isThe voltage response is also an undepressed oscillation.It can be proved that the backward Euler method can avoid

13、 such an oscillation. For inductive impedance it gives:It can be seen thatun11is not dependent onun, so this makes it possible to avoid numerical oscillations but greatly reduces the accuracy of backward Euler method. To solve this contradiction, the literature 2 proposes a trapezoidal method with d

14、amping. For the differential equationit givesFor the inductive impedance shown in Fig. 1 it gives:Wherea is the damping factor (0 a1).This method turns into the trapezoidal one when a=0,and becomes the backward Euler method when a=1. From Eq. (9), it can be seen that the coeficient of un is so when

15、the voltage oscillation is produced, it can be damped out quickly. The bigger the factor is, the more quickly the oscillation is reduced and the lower accuracy can be obtained by this method. Besides, the factor can be selected only according to experience: its optimum value is difcult to be determi

16、ned.3. The R-K-T methodThe RungeKutta method has higher accuracy and better stability in DC systems, but it requires the calculation of the values of a function many times during a single step; it cannot be expressed by a companion model like the trapezoidal method. If one can combine the RungeKutta

17、 method and the trapezoidal method to form a new method, then it will possess the advantages of both two methods. Take the 3rd order RungeKutta method for example, to deduce the new method. For the differential equationby the 3rd order RungeKutta method, one has 3For the inductive impedance, one has

18、:whereFrom Eq. (10) it followsWhere is the voltage at the midpoint of the step, which can be found by solving the equations of the system. But we calculate it by trapezoidal method. It can be done in two different ways (A) and (B):(A) Take the average values of un and un11 and letSubstitutingun11/2

19、from Eq. (14) into Eq. (13) Eq. (10) gives:Substituting the above formula into Eq. (10), one can get:whereIt is obvious that the 3rd order RungeKutta method with un11/2 substituted by Eq. (14) may be expressed by the companion model shown in Fig. 1(b), as for the trapezoidal method; the parameters o

20、f the model are:The distinguishing feature of this method is that the coef-cients ofun11andunare not equal; their ratio A may be used to attenuate the numerical oscillation with equal amplitudes of trapezoidal method. It turns to the trapezoidal method when R=0, i.e. the formula for pure inductance

21、given by trapezoidal method:(B) TakeUsing the trapezoidal method, one has:By substituting Eq. (19) into eq, (13), one can get:By substituting the above equations into Eq. (10), it followsWhereFormula (20) may be expressed by a companion model of inductive impedance as Fig. 1(b), whereFormula (20) ha

22、s also the function of attenuating the numerical oscillations like Eq. (15), and it also turns to the trapezoidal method for pure inductance when R=0.For the 4th order RungeKutta method, it gives:Similarly one can obtain the companion model for the 4th order RungeKutta method as follows 4. (A) Takin

23、g one has:WhereIts companion model in Fig. 1(b) is(B) Takingone can getwhereIts companion model for Fig. 1(b) is:Both of the 4th order models introduced above also turn to trapezoidal method for pure inductance when R0. Thus,the RungeKutta method is combined with trapezoidal method to form a new R-K

24、-T method, which exhibits the advantages of these two methods.4. Analysis and calculation of error for the R-K-T methodIn a real-life system, voltages and currents, whatever wave forms they may have, can be analyzed by the method of frequency spectrum. The error of simulation can be analyzed for eve

25、ry frequency component; the components are then added together according to the theory of superposition to obtain the total errors.Let us assume that current and voltage of a certain system element are:Where w any one frequency component. Let us rewrite the 3rd R-K-T method (20) as follows:Substitut

26、ing Eq. (27) into Eq. (28), one can deduce:From the formula of inductive impedance, one hasThe difference of the two sides of Eq. (29) represents the error of R-K-T method for frequency component, so that the error function can be dened as:If the exciting sources contain a number of frequency compon

27、ents,e(v) should be computed for every frequency component and added together. The summation of all the e(v) gives the total error function of the 3rd order R-K-T method.5. Correction of error for the R-K-T methodFrom Eq. (29) it is clear that there is unbalance in the formula of the R-K-T method fo

28、r angular frequency; it is due to the method itself and not related to the exciting sources. If one would match the two sides of Eq. (29) by adding some items, then it could give the accurate result for frequency component. Letv0be the main angular frequency of the exciting source, in order to condu

29、ct accurate calculation forv0, it is necessary to transform Eq. (29) as follows:The coeficients of the two sides of Eq. (32) are equal w=w0. It means that it gives accurate results for w=w0.Restoring Eq. (32), one can deduce:whereEq. (33) is the formula of the R-K-T method after correction. If the e

30、xciting source of the system has a single frequency v0 , then correction can be made for this frequency. If the system has a multifrequency exciting source, then correction may be made for one of the dominant lower frequency which has higher amplitude.6. Numerical resultsTo check the accuracy of the

31、 method presented, the circuit shown in Fig. 2 has been considered. Its parameters are:The accurate expression of currenti is:The test circuit shown in Fig. 2 has been solved by each of the models stated above with time step h=0.1 ms, as well as by means of formula (34) giving a more accurate result

32、.In each case error is defined as the maximum absolute Fig. 3. Error curves for each model (T: cycle) (see Table 1).Fig. 4. Error curves for each model (T: cycle) (see Table 1).Fig. 5. Error curves for each model (T: cycle) (see Table 1).Fig. 6. Error curves for each model (T: cycle) (see Table 1).Fig. 7. Error curves for each model (T: cycle) (see Table 1).Fi