การวัดความเร็วมอเตอร์กระแสตรงแบบ Digital Signal Processing

การวัดความเร็วของ DC Motor แบบไมมีตัวตรวจจับโดยใชระบบประมวลผลสัญญาณเชิงเลข Speed Sensorless DC Motor Using Digital Signal Processing System

สุวิช กวีกิจวิชชา1*, พยุง เดชอยู1 , สุภรี นวรัตนธารา1, เกริกวุฒิ รังสีปญญา2 และ วิริยะ กองรัตน2 Suwit Kawikitwitcha1*, Phayung Desyoo1, Suparee Navarattara1, Krirkwut Rangseepanya2 and Wiriya Kongratana2 1 Department of Industrial Physics & Medical Instrumentation, Faculty of Applied Science, King Mongkut’s Institute of Technology North Bangkok, Bangkok 10800, Thailand; 2 Faculty of Engineering, King Mongkut’s Institute of Technology Ladkrabang, Bangkok, Thailand; E-mail address: [email protected], [email protected]

บทคัดยอ : บทความนี้นําเสนอการประมาณคาความเร็ว DC Motor แบบไมมีตัวตรวจจับโดยใชระบบประมวลผล สัญญาณเชิงเลข การออกแบบทําโดยการสรางรูปจําลองทางคณิตศาสตรของ DC motor ในรูป Discrete State space Model จากนั้นจะใชตัวประมวลผลสัญญาณเชิงเลขทําการประมาณคาความเร็วของ DC Motor ω ซึ่งเปน state variable ของระบบ ในการสราง Hardware ไดใช TMS320C31 Floating point digital signal processing เปน ตัวประมวลผล ผลการทดลองพบวาความเร็วของ DC Motor ที่ถูกประมาณคาโดยระบบประมวลผลสัญญาณเชิง เลขมีความเที่ยงตรงเมื่อเทียบกับความเร็วที่ Tacho meter Abstract : This paper proposes a new application of digital signal processing system estimate speed sensorless DC motor. In the design, the mathematical model of DC motor in discrete statespace form will be created[1, 2, 8]; the speed of DC motor which is considered as state variable and can be estimated by using digital signal processing system. In the experiment; TMS320C31 floating point digital signal processor is used for hardware implementation[3, 4]. The experimental results show the speed of DC motor which is estimated by using digital signal processing system has good accuracy when compared with the results from tacho-meter. Introduction : In the past, the motor speed is measured by using to sense. Actually, the tachometer is used for converting the speed-to-voltage; because the structure of DC motor consists of brush, commutator, mechanical components and etc. To use the tacho-meter for a long time, those components will be deteriorated and give the error in speed measurement. Also, the motor speed measurement system without mechanical components was proposed. To overcome this problem, this paper proposes the design and implementation of motor speed estimation system using digital signal processor system[6]. The design process uses the continuous time system of motor in transfer function or state-space form[1, 7, 8] where the system inputs are armature voltage and armature current, system output determines the speed. Hence, the transform of continuous time system to discrete time system [2] and implement to digital signal processor chip is considered and proposed. Theory : The transfer function of recursive discrete time system is given by

H ( z) =

b0 + b1 z −1 + b2 z −2 + ... + bN z − N 1 + a1 z −1 + a2 z −2 + ... + a N z − N

(1)

and the output response of system is rewritten in equation (2). N

N

k =1

k =0

y (n) = −∑ ak y (n − k ) + ∑ bk x(n − k )

(2)

From equation (2), can be expressed as a linear time-invariant state-space realization show in equation (3) and (4). v (n + 1) = Fv(n) + qx(n) (3)

y (n) = g t v(n) + dx(n)

(4)

Where the elements of matrix F, q, g and d are constants. Proposed design method : To estimate sensorless motor speed by using digital signal processing system is proposed as shown in Figure 1.

Linear Amplifier

Motor

ia

va ω

Digital Signal Processing System

Figure 1. Block diagram of sensorless motor speed estimation based on digital signal processing system

Figure 2. DC motor electrical model

Figure 1, Digital signal processing system is able to estimate the speed of DC motor and it uses armature voltage ( va ) and armature current ( ia ) as input of digital signal processing system for speed ( ω ) calculation. The algorithm of digital signal processing system cooperates with the others parameters of DC motor. The design has to know the characteristics and parameters of DC motor. The schematic diagram of an armature controlling by DC servomotor is shown in Figure.2. The system variables are; ea : Armature voltage eb : Back emf ia : Armature current T: Torque produced by motor θ: Angular position of motor shaft ω: Angular velocity of motor shaft The parameters of the system are; Ra : Armature resistance La : Armature inductance J: Moment of inertia of motor shaft B: Coefficient of viscous friction The system parameters (are not shown in Figure.2) are; KT : Torque constant Kb : Back emf constant Figure. 2, the armature voltages can be found as in Eq. (5); va = La

dia + Ra ia + eb dt

eb = K bω va = La

dia + Raia + K bω dt

(5)

The occurred electrical torque in motor will change depending on the armature current as shown in Eq. (6); T = KT ia (6) The mechanical torque of motor is written as in Eq. (7); T=J

dω + Bω dt

(7)

For lossless motor, the electrical torque is equal to the mechanical torque. Eqs. (6) and (7) can be rewritten as in Eq. (8); KT ia = J

dω + Bω dt

Eqs. (5), (8) can be rearranged as shown in Eqs. (9) and (10);

(8)

dia Ri Kω e =− a − b + a dt La La La dω KT ia Bω = − dt J J

(9) (10)

The continuous state-space form of Eqs. (9) and (10) can be rewritten as in Eqs. (11) and (12);  dia   − R  dt   L  = a  dω   K T  dt   J

Kb  1 La   ia       + La ea B  ω     0  −  J 

−

i  yS 0 (t ) = [1 0]  a  ω 

(11)

(12)

Compare Eq. (14) with state Eq. it can obtain;  R − L F= a  KT   J

Kb  1 La   and q =  La    B  0  −  J  Thus, transform matrix F and matrix q to discrete system by using bilinear transform and −

substitute matrix F and matrix q in discrete system[5, 7] to Eq. (1) – (4) for speed estimation. Design example : The experiment uses DC motor as experimental test set from Feedback Instrument Inc., motor drive set model SA150D, power supply model PS150E and DC motor model MT150F. The block diagram of connection is shown in Figure 3.

Figure 3. Block diagram of motor drive test set The DC motor in the experiment has parameters as shown in Table 1. Table 1 Parameters of DC motor Parameter Ra La J B KT Kb

Value 3.0 Ohm 5.16 mH 3x10-5Kgm2 0.0158 0.0282 Nm/A 2.78 V/Krpm

Substitute the parameters from Table 1 into Eq. (11), the matrix F and matrix q can be obtained as shown in Eq. (13); 193.8  −581.40 −538.76  F= , q= 0  − 28.20 15.8    

(13)

Transform Eq. (13) to discrete system as shown in Eq. (14), where the given sampling frequency is 10 kHz. 0.01882   0.94344 −0.05229  , q= F=   0.00002   0.00273 0.99834 

(14)

Result : In the experiment, let the voltage command is between 7–10 volt, the obtained voltage from tacho-meter is compared with obtained voltage from digital signal processing system as shown in Table 2. Table 2 DC Comparison of motor speed Voltage Output voltage obtained from command tacho (ω ) 7.0 7.5 8.0 8.5 9.0 9.5 10.0

Output voltage obtained from digital signal processing system (ω )

1.0 1.15 1.3 1.5 1.6 1.8 2.0

1.0 1.2 1.4 1.5 1.65 1.85 1.95

Let the voltage command 7 V and 10 V is square wave signal frequency 0.1. The experimental results are measured by oscilloscope as shown in Figures 4.

Figure 4. The upper trace is motor speed from tacho-meter The middle trace is motor speed from digital signal processing system, The lower trace is motor speed comparison between tacho-meter and digital signal processing system Conclusion : The experimental results show that DC motor speed estimation by using digital signal processing system will give the results correctly in the vicinity when compared with the results from tacho-meter. Consequently, Digital signal processing system can be used to estimate the parameters in dynamic system with high performance which has the advantages for applications with instrumentation, control system and etc. Reference : [1] K. Ogata, Modern control engineering. 4th Ed., Upper Saddle River, N.J. : Prentice Hall, 2002 [2] K. Ogata, Discrete-Time Control Systems. 2nd Ed., Prentice Hall, 1995. [3] Texas Instruments, Minimizing Quantization Effects Using the TMS320 Digital Signal Processor Family. 1994. [4] Texas Instruments, TMS320C3x DSP Starter Kit. 1996. [5] R. G. Brown, P. Y. C. Hwang, Introduction to Random Signals and Applied Kalman filter. 1997. [6] F. L. Lewis, Optimal Estimation with an Introduction to Stochastic Control Theory. School of Electrical Engineering, Georgia Institute of Technology, Atlanta, Georgia, U.S.A. 1986. [7] J. V. Candy, Signal Processing The Model-Based Approach. McGraw-Hill Book Company, 1987 [8] V. Strejc, State Space Theory of Discrete Linear Control. 1981. Keyword : speed sensorless DC motor, Digital signal processing system.