Transient Analysis of Separately Excited DC Motor and Braking of DC Motor Using Numerical Technique

Journal homepage: www.mjret.in ISSN:2348-6953 Transient Analysis of Separately Excited DC Motor and Braking of DC Motor Using Numerical Technique Pavan R Patil, Javeed Kittur, Pavankumar M Pattar, Prajwal Reddy, Poornanand Chittal Department of Electrical and Electronics B.V.B College of Engineering and Technology Hubli, India e-mail:patil477pavan@gmail.com, pavankumarmpattar@gmail.com Abstract: This paper deals with transient analysis of separately excited DC motor and braking of DC motor where transient equations are worked out using numerical technique. The code is written for both transient analysis of separately excited DC motor and braking and thus verified using the GNU plot and also the results are verified using MATLAB / Simulink model. Keywords- Regenerative Braking; Plugging; Dynamic Braking. I. INTRODUCTION Transient Analysis is done to know the performance of motors. Basically, it tells us how one parameter is varied with respect to the other. In this paper, it tells us how current is varied with respect to time and how speed is varied with respect to time, without and with load torque of separately excited DC motor. Brakes are used to reduce or cease the speed of motors. We know that there are various types of motors available (DC motors, induction motors, synchronous motors, single phase motors, etc.) and the specialty and properties of these motors are different from each other, hence this braking method also differs from each other [1]. Breaking can be attained either mechanically or electrically. Under electrical breaking, there are three types, namely 117 P a g e

Regenerative Braking Plugging type Braking Dynamic Braking. II. NUMERICAL TECHNIQUE Numerical analysis involves the study of methods of computing numerical data. In many problems this implies producing a sequence of approximations by repeating the procedure again and again [2]. People who employ numerical methods for solving problems have to worry about the following issues: the rate of convergence (how long does it take for the method to end the answer), the accuracy (or even validity) of the answer and the completeness of the response (do other solutions, in addition to the one found, exist). Numerical methods provide approximations to the problems in question. No matter how accurate, they are, they do not, in most cases, provide the exact answer. In some instances working out the exact answer by a different approach may not be possible or may be too time consuming and it is in these cases where numerical methods are most often used [3]. Euler s method: Euler s method provides us with approximation for the solution of the differential equations. The idea behind Euler s method is to use the concept of linearity to join multiple small line segments so that they make up an approximation of the actual curve. Note: Generally, the approximation gets less accurate the further you get away from the initial point [4]. Three things needed in order to use Euler s method: 1) Initial point Starting point must be given. 2) Delta- The change in step size must be given directly or information to find it. 3) The differential equation-the slope of each individual line segment should be known so that delta is found. Let us take as an example an initial value problem in ODE (1) The forward Euler s method is one such numerical method and is explicit. Explicit methods calculate the state of the system at a later time from the state of the system at the current time without the need to solve algebraic equations. For the forward (from this point on 118 P a g e

forward Euler s method will be known as forward) method, we begin by choosing a step size or t the size of t determines the accuracy of the approximate solutions as well as the number of computations. Graphically, this method produces a series of line segments, which thereby approximates the solution curve. Fig. 1 Graphical illustration of Euler s method Let = 0, 1, 2 be a sequence in time with (2) Let, and, be the exact and the approximate solution at, respectively. To obtain from, we use the differential equation. Since the slope of the solution to the equation at the point is, the Euler method determines the point (), by assuming that it lies on the line through with the slope, Hence the formula for the slope of a line gives Or (4) As the step size or t decreases, then the error between the actual and approximation is reduced. Roughly speaking, we halve the error by halving the step size in this case. However, halving the t doubles the amount of computation. Backward Euler s method: The backward Euler s method is an implicit one which, contrary to explicit methods finds the solution by solving an equation involving the current state of the system and the later one. More precisely, we have (5) 119 P a g e

This disadvantage to using this method is the time it takes to solve this equation. However, advantages to this method include that they are usually more numerically stable for solving a stiff equation a larger step size t can be used. Let us take following initial value problem (6) We will use forward and the backward Euler s method to approximate the solution to this problem and these approximations to the exact solution (7) In both methods we let t = 0.1 and the final time t = 0.5. (T) (FEA) (BEA) Exact (FE) (BE) 0 1 1 1 0 0 0.1 0.9 0.9441 0.925795 0.025795 0.018305 0.2 0.853 0.916 0.889504 0.036504 0.026496 0.3 0.8374 0.9049 0.876191 0.03791 0.028709 0.4 0.8398 0.9039 0.876191 0.036391 0.027709 0.5 0.8517 0.9086 0.883728 0.032028 0.024872 Where T-time, FEA- forward Euler s approximation, BEA-backward Euler s approximation, FE- forward error, BE- backward error [5]. Forward and Backward Euler s Method Compared to Exact Solution The ei error averages were also computed for both methods and the result was for the average error for forward Euler s method was 0.028105 and the average error for the backward Euler s method was 0.021015. As it can be seen in both the chart above and the ei error averages that the backward Euler s method seems to be the more accurate between the methods [5]. 120 P a g e

III. TRANSIENT ANALYSIS OF SEPARATELY EXCITED DC MOTOR Fig.2 Separately Excited DC Motor (Saber Model) Where V: applied voltage, : motor current, : induced back emf voltage, : armature winding inductance, : Armature Resistance, T: motor output torque, : motor output speed. Solution of Transient equations of the DC motor drive using Backward Euler s method 121 P a g e

Using equations (12) and (16) we can carry out the simulation in C-language. Here open source software Code blocks is used for programming. GNU PLOT: GNU PLOT is a command line program that can generate two and three dimensional plots of functions, data and data fits. It is a frequently used for publication-quality graphics as well as education. Pseudo Code: Start of Program Initialize the value Open a file to store current and speed values Start of loop execution continue till the specified time is encountered End of execution 122 P a g e

Flow Chart: START Initialize values END For(i=0;i<n,t<=n;i++,t=t+h) Displaying speed, current w.r.t time Fig. 3 Waveform showing Open loop operation of separately excited DC motor without load torque. Fig. 3 shows current and speed waveforms during open loop operation of DC motor drive. In this case, the motor is operating under no load conditions. The motor current increases 123 P a g e

initially, and reaches its maximum peak. The initial motor current is high as the back emf is negligible during starting. When the speed of the motor increases and tends towards steady speed, the value of back emf becomes considerable and thus opposes the supply voltage. This causes the motor current to decrease and tend towards zero. Fig. 4 Waveform showing the Open loop operation of separately excited DC motor with load torque. Fig. 4 shows the current and speed waveforms during open loop operation of DC motor with applied torque T= 3 N-m. It is observed that the motor takes more time to attain its steady speed as compared to that of no load condition. This is because, when the motor is connected to the supply, the initial value of the current is zero and due to the armature circuit inductance, it takes some time to reach the load current. And the armature current value remains finite. MATLAB (Simulink) model of separately excited DC Motor (without load torque): 124 P a g e

Fig. 5 Matlab(Simulink) model of separately excited DC Motor(without load torque) Simulink Plots: Fig. 6 Variation of speed with respect to time 125 P a g e

Fig. 7 Variation of current with respect to time IV. BRAKING The term braking comes from the term brake. The brake is an equipment to reduce the speed of any moving or rotating equipment, like vehicles, locomotives. The process of applying the brakes can be termed as braking. In braking, the motor works as a generator developing a negative torque, which opposes the motion [1]. Where, I a = Armature current V= Supply voltage E= Back emf R a = Armature resistance There are three types of braking Plugging type Braking Dynamic Braking Regenerative Braking 126 P a g e

Using GNU Plots for these 3 breaking operations: Plugging type Braking: For plugging, the supply voltage of a separately excited motor is reversed so that it assists the back emf in forcing armature current in the reverse direction. A resistance R a is also connected in series with the armature to limit the current. Plugging gives fast braking due to higher average torque, even with one section of barking resistance R a. Plugging is highly inefficient because in addition to the generated power, the power supplied by the source is also wasted in resistances. Fig. 8 Plugging Circuit (Saber Model) Speed and current waveforms of DC motor during plugging is shown in Fig.9. Here motoring operation is same as that of regenerative braking. In plugging, the voltage applied to the armature is reversed so that the armature current flows in the reverse direction, therefore the torque produced is opposite to the earlier torque direction. Hence opposite torque braking is achieved. 127 P a g e

Fig. 9 Waveform showing motoring and plugging operations Dynamic Braking: Shows the current and speed waveform for dynamic braking. When the motor attains the desired speed, supply voltage to the armature circuit is cut off and an external resistance R B (10Ω) is connected across the armature. In this case, the motor acts as a generator and converts kinetic energy stored in moving parts into electrical energy and this energy is dissipated in the form of heat in the resistor R B. The braking time depends upon the value of the external resistance connected in parallel to the armature circuit. As shown in Fig.10, braking with R B = 50Ω is faster compared to that with R B = 10Ω, shown in Fig.11. Fig. 10 Waveform showing motoring and dynamic braking with R B =50 Ω 128 P a g e

Fig. 11 Waveform showing motoring and dynamic braking with R B =10 Ω Regenerative Braking: Regenerative braking operation of the DC motor drive is depicted in Fig.12.When the motor attains its constant speed, the supply voltage is cut off and switching pulses are reversed. Thus, the direction of the current will reverse. Later, the motor speed decreases to reach zero value. In this case, the motor act as a generator and the current stored in the armature inductance is fed back to the source. 129 P a g e

Fig. 12 Waveform showing motoring and regenerative braking V. CONCLUSION Transient behaviour: Plot of current and speed with respect to time is verified both using GNU plots (both with and without load torque) and MATLAB (Simulink) model (without load torque) of separately excited DC motor. All 3 types of braking are verified using GNU plots. In dynamic braking, braking time depends on the value of the resistance, higher the value of resistance less is the braking time. Appendix: Separately excited DC Motor specification: Sl. No Motor Parameters Values 01. Rated Power (P) 15 hp 02. Rated Voltage ( V) 230V 03. Armature resistance (Ra) 0.5 Ω 04. Armature inductance( La) 0.05H 05. Coefficient of Viscous friction (B) 0.02Nm/rad/sec 06. Moment of inertia (J) 2kg-m 2 ACKNOWLEDGEMENTS We wish to place on record our profound and deep sense of gratitude to our guide Asst Prof. Javeed Kittur, for his wholehearted guidance without which this endeavour would not have been possible. Our diction falls short of words to gratify our guide for being the primary source of inspiration and strength for the project study. We are grateful to Dr. A. B. Raju, our Head of the Department for their valuable guidance and encouragement with suggestions and permitting me to carry out my project work and also for their co-operation throughout the project. 130 P a g e

We are grateful to Dr. Ashok Shettar, our respected principal for having provided us the academic environment which nurtured our practical skills contributing to the success of our project. VI. REFERENCES [1] Types of Braking - Electrical4u.com [2] Steven C. Chapra & Raymond P. Canale, Numerical methods for Engineers, Fourth Edition. [3] Numeicalanalysis-www.math.niu.edu/~rusin/known-math/index/65. [4] What is NumericalMethod. www3.ul.ie/~mlc/support/compmaths2/files/ [5] Euler'sMethod_www.mathscoop.com(euler) [6] Gopal K. Dubey, Fundamentals of Electrical Drives, Second edition 2002. [7] Explicit and Implicit Methods In Solving Differential Equations digitalcommons.uconn.edu/cgi/viewcontent.cgi. [8] Balaguruswamy, Programming in ANSI C. 131 P a g e