Deerminaion of he Sampling Period Required for a Fas Dynamic Response of DC-Moors J. A. GA'EB, Deparmen of Elecrical and Compuer Eng, The Hashemie Universiy, P.O.Box 15459, Posal code 13115, Zerka, JORDAN Absrac: DC moors provide a high saring orque and play a significan role in modern indusrial drives. The speed of a dc-moor is responding o he change in load orque, armaure volage and field flux. For a sudden change in applied orque, here is a change in he moor speed. This speed change goes in ransien period for some ime before seling o he new value. To keep a consan moor speed for a sudden change in applied orque, a proporional conrol of armaure volage can be used. In his work, o have an opimum and faser speed response for he orque change a sampled- daa conroller is applied. The coninuous response of he moor speed is sampled and he acion for proporional conrol of armaure volage is aken a discree manner. A proper value of sampling frequency is deermined for a saisfacory speed response of he dc- moor. The paper discussed he resuls of dynamic response for off- ime and real ime conrols. Keywords: DC-machines, Dynamic Response, Digial Conrol, Transien Response, Real- ime Conrol, Sampled- daa Conrol, Opimum Conrol, Proporional Conrol. 1 Inroducion Boh series and separaely excied dc moors are normally used in variable- speed drives [1]. Wih he help of heir variable characerisics, i is possible o obain speed conrol over a wide range. The mos flexible conrol is obained by means of a separaely excied dc moor in which he armaure and field circuis are provided wih separae sources. This arrangemen produces speed- orque characerisics approximaely close o he ideal characerisics. For armaure volage, a conrolled recifier or a chopper is required [2]. The field curren can be also conrolled by he same provisions. Conrolled recifiers provide a variable dc oupu volage form a fixed ac volage, whereas choppers can provide a variable dc oupu volage from a dc volage. In his work, he coninuous dynamic response of he dc moor for a sudden change in load orque is sampled. The speed is measured a each sampling insan and he acion for proporional conrol of armaure volage is aken a discree manner. The mehod ha was suggesed in his work for deermining he sampling period gives a faser and opimum speed response for he dc moor. 2 DC Moor-speed conrol The seady- sae speed of a dc- moor reacs o he change in load orque, armaure volage and field flux. A separaely excied moor wih consan values of armaure volage and field curren develops a speed ha is decreased as he load orque is increased. During he ransien period for a sudden change in load orque, he addiion load orque demand is me iniially by a sudden reardaion of he roaing masses which conribue o he momen of ineria of he moor shaf. Some sudies ignore he mechanical sysem ransien by considering roor speed o be consan over he shor ime, or ignoring machine field and armaure winding inducance [3]. The dc/dc converer or chopper and phase conrolled ac/dc converer are using exclusively on he conrol of he armaure volage supply [4]. The chopper is involving semiconducor swiches o chop he consan supply volage a a fairly high repeiion rae, enabling he average value of armaure volage o be variable beween zero and he supply volage. In he essenial circuis of he chopper, he consan volage source is feeding o he hyrisor swich. This hyrisor swich chops he volage source before applied o he armaure circui. As he conducing period of he hyrisor
swich is changed, he average value of armaure volage is varied which resul in speed conrol. 3 DC Moor- Model for Transien Operaion The equivalen circui of a separaely excied dcmoor is shown in Fig.1. The field curren is assumed consan. The conrol is applied on erminal volage (V a ) which equals: V a = R a i a L a di a / d e a vol (1) and: e a = KΦω vol (2) where: e a : armaure e.m.f (vol). Φ : flux per pole (web). ω : moor speed (r/s). The air-gap or elecromechanical orque (T e ) equals: T e = J dω /d Bω T w N.m (3) Also; T e = KΦi a N.m (4) Where: J: Ineria of boh moor and driven mechanism (kg.m 2 ). B: Viscous fricion of boh moor and driven mechanism (N.m.s/r). T w : loading orque (N.m). Transien behavior of dc moor can be analyzed by ransforming he above equaions o s- domain. By aking he Laplace ransform of (1) and afer some arrangemens becomes : τ a = L a / R a (sec), is he armaure- circui ime consan. Also by aking he Laplace ransform of (3) and (4) we ge : T e(s) - T w(s) Ω (S) = (6) B (1 τ m.s) And: T e(s) = KΦI a(s) (7) τ m = J / B (sec), is he mechanical ime consan. 4 Transien Response Wih Discree Conrol The moor- mechanism model is deermined from (5), (6) and (7) and given in Fig.2. For a sudden change in applied orque (T w ), here is a change in he moor speed. This speed akes some ime before seling o new value. To keep a consan moor speed for a sudden change in applied orque, a proporional volage conrol is added. In his work, he response ime is sampled wih sampling period of (T-sec), and he acion of proporional conrol is aken a each insan of he discree period. This discree manipulaion will show a fas speed response wih less oscillaion. Va(s) 1 R a (sτ a 1) I a (s) kφ kφ T w (s) T e (s) 1 B(sτ m 1) Ω(s) V a(s) - KΦΩ (S) I a(s) = (5) R a (1 τ a.s) V a i a R a L a e a T e T w M R f Fig.1 A separaely excied dc- moor. i f V f Fig.2 DC- moor block diagram. 4.1 Off- ime Conrol The ransien response of moor speed o an unisep change in load orque wih no conrol of armaure volage is deermined and given in Fig.3. The final value of speed is changed and seled on new low value because here is no acion of armaure conrol. The conrol of armaure volage can be acivaed using discree manner. This is done by sampling he ime response of he speed a a sampling ime
of T- sec and he speed change is manipulaed a each sampling insan. The manipulaion is done a off-ime, i.e. he calculaions for new acion of armaure conrol is aken a each sample. I is imporan o selec a proper value for he sampling frequency o have a saisfacory ransien response. In his paper, he ransien response of he coninuous- daa conrol sysem is invesigaed and a proper equaion is developed for he deerminaion of sampling period (T P ). This equaion is: s T P = sec (8) (Mv - F v ) * 8 Where: T s : Seling ime of he coninuous response. M v : Maximum value of he response. F v : Final value of he response. From he coninuous uni- sep response of moor speed shown in Fig.3, he opimum sampling period is deermined and given below:.2 T P = =.6 sec. (.2666-.225)* 8 Speed response o an uni- sep change of load orque using discree conrol for he armaure volage wih differen values of sampling period -.5 -.1 -.15 -.2 -.25 wih no conrol of Va -.3.5.1.15.2.25.3.35.4 Fig.3 Speed response o an uni- sep change in T W is given in Fig.4. The sampling period is calculaed from (8) which equals.6 sec and is response shown in fig.4-a. For a larger sampling period of.1 sec, he moor-speed response has a large overshoo as shown in Fig.4-b. The response in Fig.4-c is for a smaller sampling period of.6 sec which has more oscillaion and caused speed perurbaion. Fas and opimum speed response is obained a he deermined sampling period T P. 4.2 Real-ime Conrol In he previous resuls, he calculaions and acion for new speed were done simulaneously. Bu pracically, he conroller needs some ime for calculaions before aking he new acion for he armaure volage. Therefore, he firs sample afer he change is lef for calculaions, and he acion of conroller is aken in he second sample. The speed response o an uni- sep change of load orque and for sampling period equals o ha one calculaed by (8) bu wih off- on conroller is shown in Fig.5. The response for a half value of he calculaed sampling period is shown in Fig.6. Resuls show ha he firs overshoo wih T=.3 sec is less han ha wih T =.6 sec bu wih more speed perurbaion. Pracically, real- ime discree conrol of he dcmoor speed is proposed in Fig.7, using a dcchopper of a consan frequency. Using he pulsewidh- modulaion mehod, he chopping period of he chopper can be defined, which is he same sampling period (T) of he digial conroller. Figure.7-b explains he way of generaing swiching signals for he chopper. The riangular carrier signal (c) has a frequency of (1/T Hz). The ampliude (p r ) of he recangular reference signal (r) depends on he signal of he dc-moor caused by he change of he loading orque. Differen load orques give differen ampliudes of (p r ). The conducion angle ( ) of he chopper is hen deermined by comparing he carrier signal ampliude (p c ) wih he reference signal ampliude (p r ). If he difference beween he carrier signal (c) and reference signal (r) becomes zero, he chopper will sop conducing and he reference signal ses o zero. Figure.8 shows he way of generaing differen conducion angles a differen signals.
.5.5 -.5 -.1 -.15 sped error -.5 -.1 -.15 -.2 -.2 -.25.1.2.3.4.5.6.7 a-wih discree conrol of T=.6 sec -.25 -.3.5.1.15.2.25.3.35.4 Fig.5 Real- ime speed response wih discree conrol of V a a T=.6 sec..5 speed erroe.5 -.5 -.1 -.15 -.2 -.25 -.3.1.2.3.4.5.6.7.8 b-wih discree conrol of T=.1 sec Ref -.5 -.1 -.15 -.2 -.25.5.1.15.2.25.3.35.4 Fig.6 Real- ime speed response wih discree conrol of V a a T=.3 sec. Digial Conroller Supply DC Chopper DC Moor Speed, ω 4 x 1-3 2 Digial Daa Acquisiion(T) -2-4 -6 a-block diagram of Conrol Sysem for dc-moor C, r signals pc for for conrol calculaions acion reference signal carrier signal -8-1.5.1.15.2.25.3.35.4 pr T 2T 3T 4T c- Wih discree conrol of T=.6 sec Fig.4 Moor-speed response o an uni- sep change of Load orque. O/P Volage of Chopper T Chopping period b-chopper Oupu Fig.7 Pracical chopper circui
Carrier signal pr 3 pr 2 pr 1 1 2 3 Fig.8 Differen conducion angles for chopper Conrol Sysem Technology, Vol. 5, No. 1, Jan. 1997. [8] H. Melkoe, F. Khorrami, S. Jain, and M. maic Robus Adapive Conrol of Variable Relucance Sepper Moors IEEE Trans. Conrol Sysem Technology, Vol. 7, No., March. 1999. [9] R. J. Wai, C. M. Lin, and C. F. Hsu Hybrid Conrol for Inducion Servomoor Drive IEE Proceeding. Conrol Theory and Applicaions, Vol. 149, Issue. 6, Nov. 22. [1] S. Di Gennaro Adapive Oupu feedback Conrol of Synchronous Moors Inernaional Journal of Conrol, Vol. 73, No. 16, Nov. 2. 5 Conclusions A sampled- daa conrol sysem is developed for a dc- moor o improve he dynamic response of moor speed for load orque changes. An opimum value of sampling period for a bes behavior of he moor speed hrough ransien response is deermined. The ransien response of moor speed o an uni- sep change in load orque is invesigaed for boh off- ime and real- ime conrol, and a pracical model for real- ime conrol is included. References: [1] S. B. Dewan, G. R. Slemon, A. Sraughen, Power Semiconducors Drives, John Wiley & Sons, 1984. [2] M. H. Rashid, Power Elecronics ; Circuis, Devices and Applicaions, Prenice- hall Inernaional, Inc, 1988. [3] C. B. Gray, Elecric Machines and Drive Sysems, Longman Group UK Limied, 1989. [4] P.C.Sen, Principles of Elecric Machines and Power Elecronics, John Wiley & sons, 1989. [5] A. G. Ulsoy, Conrol of Machining Processes, Journal of Dynamic Sysems, ASME, June 1993. [6] G. E. Escobar, P. maavelli and A. M. Sankovic, Reacive Power and Unbalanced Compensaion Using STATCOM wih Dissipaiviy- Based Conrol IEEE Trans. Conrol Sysem Technology, Vol. 9, No. 5, Sep. 21. [7] N. Rafee, T. Chen and O. P. Malik, Opimal Digial Redesign wih Applicaion o Conrol of a Single- Machine Power Sysem IEEE Trans.