4. Electromechanical Systems. Karadeniz Technical University Department of Electrical and Electronics Engineering Trabzon, Turkey.

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Karadeniz Technical University Department of Electrical and Electronics Engineering 61080 Trabzon, Turkey Chapter 3-4-1 Modelling of Physical Systems 4.

1 Karadeniz Technical University Department of Electrical and Electronics Engineering Trabzon, Turkey Chapter Modelling of Physical Systems 4. Electromechanical Systems Bu ders notları sadece bu dersi alan öğrencilerin kullanımına açık olup, üçüncü sahıslara verilmesi, herhangi bir yöntemle çoğaltılıp başka yerlerde kullanılması, yayınlanması Prof. Dr. İsmail H. ALTAŞ ın yazılı iznine tabidir. Aksi durumlarda yasal işlem yapılacaktır. Chapter 3-4-2

2 DC motor is a very important device in an electromechanical system. It is widely used in robotic systems, computer disk drive, cdrom drive, VCR, numerical control machines, anti-aircraft radar tracking control systems, antenna positioning systems, and traction systems to name a few. Motors which are specially designed for position control systems (examples include robotics, disk drive, cd-rom drive, anti-aircraft radar tracking, antenna positioning systems) are called servomotors. Chapter Instead of a dc motor, a hydraulic or a pneumatic technologies may be used. Hydraulic systems are indicated when the required force or torque is very high. It gives a very high power to weight ratio, can develop very high peak torques and holding force, and rugged. However, its disadvantages include limited movement, oil leak, and severe nonlinearities. Pneumatic systems on the other hand are cheap, there is no danger of leakage, pressurized air is often available, but it has relatively lower power to weight ratio compared to the hydraulic system. Due to compressibility of air, its application is limited mainly to on-off (bang-bang) applications. Chapter 3-4-4

3 v The dc motor requires a power amplifier to drive it. v High speed, reliable and inexpensive solid state semi-conductor devices including diodes, thyristors (also called silicon controlled rectifiers (SCRs)), and recent devices such as gate turn-off thyristors (GTOs), bipolar function transistors (BJTs), metal oxide field effect transistors (MOSFETs) and insulated gate bipolar transistors (IGBTs) have revolutionized the DC motor drive amplifier circuits. v The SCRs are switches which can be turned on but can only be turned off when the current through the device is zero. Hence SCRs are called naturally commutated semiconductor switches. v On the other hand, the devices such as GTOs, BJTs, MOSFETS and IGBTs can be turned on or off and are called self commutated semiconductor switches. Chapter The power amplifiers The power amplifiers may be classified as follows: Linear amplifier Controlled rectifier circuit Chopper amplifier Chapter 3-4-6

4 Linear amplifier using power transistors operating in a linear mode: It is generally used in low power applications and due to linearity of the input-output model, the analysis and design of the drive system is simplified. Chapter Controlled rectifier circuit It converts ac power at constant voltage and constant frequency into a controlled dc voltage or controlled dc current. An SCR is employed to vary the delay angle at which the current to the dc motor is allowed to begin to conduct. One of the disadvantages of this simple scheme is that the power factor of the ac supply source may become low and hence it is not energy efficient. Chapter 3-4-8

Chopper amplifier The chopper is basically an electronic switch which is either fully closed or fully open, so that no power is consumed in the switching devices, and hence it is very energy

5 Chopper amplifier The chopper is basically an electronic switch which is either fully closed or fully open, so that no power is consumed in the switching devices, and hence it is very energy efficient control device. A semi conductor device such as GTO, BJT, MOSFET or IGBTO is employed. The pulse width modulation is the popular chopper amplifier based control scheme. Chapter Chopper amplifier w m Chapter

6 The DC motor is based on fundamental laws of electromagnetism. Motor If a conductor of length, l, carrying a current, i, is placed in an uniform magnetic field of flux density B, it will experience a force, f, given by Ampere s law: f = B i l, when the conductor is oriented at right angles to the magnetic field. Chapter Motor In other words, if a conductor is one meter long, placed at right angles to a magnetic field of flux density one Tesla (weber/m 2 ), and carries one ampere current then the conductor will experience a force of one Newton. Chapter

7 Generator If a conductor cuts a magnetic field of flux density, B, with a velocity v then an emf, e, is induced in the conductor given by Faraday s law of electromagnetic induction Chapter If a current carrying single turn coil is placed in a magnetic field it will experience a torque, T + Chapter

8 1. a rotating cylinder called armature and 2. a field either electromagnet or a permanent magnet Chapter T e is the torque developed by the machine, T L is the load torque to be overcome, e a is the back emf, i a is the current in the armature, and i f is the field current. Chapter

9 Generally a dc motor will have N coils placed around a cylindrical rotor, termed armature. In this case the torque and emf will respectively be ( a coil will have 2 conductors) Expressing the emf, e a, in terms of the angular velocity, w m, we get (since v=r w) Expressing in terms of the diameter, d =2r, we get Chapter Mathematical model The operation of a dc motor may simply be explained using the two laws namely Ampere s law: Faraday s law of electromagnetic induction: Before we develop a mathematical model let us obtain a qualitative model which captures the operation of a dc motor. The qualitative model takes the form of a directed graph which is obtained from the cause-effect relation that exist between the various quantities that characterize the system. The dc motor operation is analyzed by considering the three distinct regimens Chapter

10 Mathematical model 1. Assume initially that the armature is stationary. Then the following are the cause-effect relationships results when the voltage is applied to the armature terminals. The voltage, v a, causes the armature current, i a The current, ia, reacting with the magnetic flux produces a force on the conductor The forces on the conductors on the rotor results in a torque The torque cause the rotor to rotate at an angular velocity w m. Chapter Mathematical model These cause effect relationship may be expressed graphically as follow current torque velocity voltage Amperes Law flux Chapter

11 Mathematical model 2. The armature rotates and the armature conductors cut the magnetic field with an angular velocity. Then the following are the cause-effect relationships An emf, termed as back emf denoted e a is induced in the conductor. Since the linear velocity, v, and the angular velocity are related by the back emf becomes Faraday s law Chapter Mathematical model 3. The emf induced will be such as to oppose the voltage which caused the motion in the first place. This is Lenz s law and is essentially the law of conservation of energy. current Amperes Law torque velocity voltage flux Combining the above mentioned three scenarios the qualitative model may be expressed by this digraph. Back emf Chapter

12 Mathematical model current Amperes Law torque velocity voltage flux Back emf Chapter Mathematical model Chapter

13 DC motor are generally used in the following applications: speed control position control torque control The dc motor control may be achieved using the following schemes armature control field control combined armature and speed control Chapter Armature-controlled DC motor: The field current, i f, is constant (and hence the flux density B is constant), and the armature voltage is varied. Chapter

14 The mathematical model can be obtained by considering the electrical system, electromagnetic interaction and mechanical system. The qualitative model expressed in the form of a digraph obtained in the previous section may be employed to derive the mathematical model. Chapter electrical system electromagnetic interaction Where B is the flux density, r is the radius of the armature, N is the number of conductors, l is the length of the conductor and mechanical system Chapter

15 The system theoretic model is given below Chapter The state-space model of armature controlled DC motor. Chapter

16 Field-controlled DC motor We can also control the dc motor speed by varying the field flux. The method of control is generally used when the motor has to run above its rated speed. To understand the operation of field control suppose that the dc motor running at a constant speed. If the field current is reduced by reducing the voltage across the field coil, the flux density will be reduced. Chapter Field-controlled DC motor Reduction in flux will reduce the back emf instantaneously and will cause armature current to increase resulting in the motor speed increasing. Consequently the back emf will increase and a new equilibrium will be established at a higher speed. With field control one can achieve as high a speed as three time the rated speed. Chapter

17 The armature current, i a, is kept constant and the flux density B is varied by varying i f. Chapter We will first obtain a qualitative model using digraph to help us obtain a macroscopic picture before deriving the mathematical model. The figure below give the digraph: Chapter

18 electrical system electromagnetic interaction Read the notes on the next slide. mechanical system Chapter In this modelling, it has been assumed that the armature current i a is constant and included in k f so that the electrical toque equation can be written as: However, keeping Va constant does not mean that ia becomes constant. Since i a is affected by e b =NlrBw m, it cannot be assumed as a constant parameter, because B and w m are not constant. Therefore ia should be included in this model to generate torque as T=NLrBi a. Chapter

19 The dc motor with both field and armature control Chapter We will first obtain a qualitative model using digraph to help us obtain a macroscopic picture before deriving the mathematical model. The figure below give the digraph: Chapter

20 electrical subsystem: field circuit Armature circuit mechanical subsystem: electromechanical interaction: Chapter In general there exist a nonlinear relationship between the flux density B and the current in the coil f i as shown in the figure. Chapter

21 Chapter X X Chapter

The Shunt Connected dc motor Chapter 3-4-43

22 The Shunt Connected dc motor Chapter The Shunt Connected dc motor X X Chapter

23 The block diagram is composed of : (1) integrator blocks, (2) gain blocks, (3) summing junction, and (4) nonlinear blocks (multiplier blocks). Comment: The combined field and armature control system is nonlinear. A dc motor with both field and armature control finds applications in traction systems. The inclusion of field control to the armature control helps to achieve the speed control over a wide range of load torque variations. Chapter Obtain a mathematical model of a dc generator relating the input, w and the voltage output. Current is reversed Chapter

24 We will first obtain a qualitative model using digraph to help us obtain a macroscopic picture before deriving the mathematical model. The figure below give the digraph: Chapter Mechanical subsystem where T e is the electric torque, T e = ki, k=bnlr, whose direction is such as to oppose the torque which caused the motion. Electrical subsystem where e=kw, k=bnlr, is the emf generated. Chapter

Chapter 3-4-49 v The frictional torque plays a significant role in the analysis and the design of systems where mechanical motion is involved.

25 Chapter v The frictional torque plays a significant role in the analysis and the design of systems where mechanical motion is involved. v It is challenging task to determine the effect of the friction and to compensate for their effect. v The frictional torque opposes the motion and is generally composed of : 1. Viscous friction 2. Static friction 3. Coulomb friction Chapter

26 (1) Viscous friction Chapter (2) static friction Chapter

27 (3) Coulomb friction Chapter One way to overcome the effects of static friction and coulomb friction is to keep the motor running all the time. This may be achieved, for example, by applying a constant dc voltage, v 0, to the normal operating input voltage, v op such that the total applied voltage, v given by Chapter

28 In this case, the motor speed does not change sign: Always rotates in one direction except during the transient period to change the direction of motion. Sometimes, a dither signal, a high frequency signal of very small amplitude with respect to maximum (normal) operating signal is added so that the mechanical elements are not in an idle position Chapter Identification of the mathematical model of a dc machine Determination of a mathematical model of a system from the input-output data record is termed identification of the system. The system identification plays an important role in evaluation of the system performance and the design of the controller. The identification involves v Obtaining the mathematical equations using the physical laws v Designing a set of experiments v Extracting the model parameters from the input-output data record Chapter

29 Identification of the mathematical model of a dc machine INPUT Physical System Test Mathematical Model Equations With the parameters a, b, c,...? OUTPUT identification Chapter Mathematical equations: Let us consider an armature controlled dc motor with no load except the frictional torques opposing the motion. A set of mathematical equations obtained from the physical laws are given below ( for simplicity of notation the subscript a is deleted from the armature resistance, inductance and the current): Chapter

30 Mechanical subsystem where b is the viscous damping coefficient, T L is the Coulomb friction: and c is the Coulomb friction coefficient. Chapter Electrical subsystem The objective is to determine the model parameters: electrical parameters: L, R mechanical parameters: J, b electromechanical interaction parameter: k Coulomb friction coefficient c Chapter

31 Design of experiments Experiment I The resistance of the armature is determined by performing the following experiment. The armature of the motor is locked and, A dc voltage v is applied ( or a multimeter is connected to the terminal ) to determine the resistance of the armature circuit. In the steady-state the mathematical equation relating v and i is obtained as follows: Chapter Experiment I That is v = R I From the input-output data record of v and I, the resistance R is determined Chapter

32 CAUTION Experiment I When performing a locked rotor test, DO NOT APPLY full rated voltage. Since the rotor is locked, the speed is zero. Therefore armature current becomes too high that may cause damages. Apply an armature voltage such that the armature current does not exceed the rated current. Chapter Experiment 2 The inductance of the armature L is determined by performing the following experiment: The armature of the motor is locked and a step voltage v (or square wave) is applied to the armature terminal or a high frequency square wave is applied to the armature so that the motor does not rotate. Chapter

Since the armature is locked, w = 0. Hence the dc motor model consists essentially of the electrical subsystem: Since e = kw = 0 we get : Experiment 2 It is a first-order system.

33 Since the armature is locked, w = 0. Hence the dc motor model consists essentially of the electrical subsystem: Since e = kw = 0 we get : Experiment 2 It is a first-order system. The relation governing the step input v and the current i is Chapter The time constant t=l/r is determined from the i vs t graph. Experiment 2 After obtaining the time constant t the inductance L= R t Chapter

34 Experiment 3 The interaction parameter k, the Coulomb friction coefficient c, and the viscous friction coefficient b are determined from the following experiment. A dc voltage v is applied and wait till the motor reaches the steady-state operating condition so that both the current and the speed are constant. That is Chapter The electrical equation becomes Experiment 3 The mechanical equation (assuming w³0 ) becomes Repeat the experiment for various values of dc voltage v and obtain steady-state data record { v, i, w}. To estimate k, plot (v-ri) vs w and determine k from the slope. To estimate the Coulomb friction coefficient c and the viscous friction coefficient b, plot i vs w. Estimate b and c from the slope and the offset respectively. Chapter

Experiment 3 ki V-Ri k=slope c b=slope w w Chapter 3-4-69 Experiment 4 The mechanical parameters J is

Apply a dc voltage and when the steady state is reached (when the speed and the current reach constant

35 Experiment 3 ki V-Ri k=slope c b=slope w w Chapter Experiment 4 The mechanical parameters J is determined from the following experiment. Apply a dc voltage and when the steady state is reached (when the speed and the current reach constant values) disconnect the armature circuit so that I = 0. That is no current flows in the armature circuit and hence the torque developed by the motor, T is Chapter

36 Chapter The dc motor model consists of the mechanical subsystem: assuming w³0 we get Record the speed w with respect to time. It will be an exponentially decaying curve. Speed (rad(sec.)) Time (sec) Chapter

37 Taking the Laplace transform with initial value of the speed w 0 we get Taking the inverse Laplace transform yields From the plot of w vs t the parameter a may be determined: draw the tangent line to the exponential curve w vs t at t=0. Let the intercept of the tangent line with the time axis when w = - b will be the time constant t. Chapter It can be shown that the intercept on the time axis will be time constant of the system: Since a = b/j the inertia constant J is given by J= bt Chapter

CHAPTER THREE DC MOTOR OVERVIEW AND MATHEMATICAL MODEL

CHAPTER THREE DC MOTOR OVERVIEW AND MATHEMATICAL MODEL 3.1 Introduction Almost every mechanical movement that we see around us is accomplished by an electric motor. Electric machines are a means of converting