Automatic Control Systems. Lecture Note 15


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1 Lecture Note 15 Modeling of Physical Systems 5 1/52
2 AC Motors AC Motors Classification i) Induction Motor (Asynchronous Motor) ii) Synchronous Motor 2/52 Advantages of AC Motors i) Costeffective ii) Convenient power source due to standard AC supply iii) No commutator and brush mechanism needed in some types iv) Lower power dissipation, lower rotor inertia, and light weight in some designs v) Virtually no electric arcing (less hazardous in chemical environments)
3 AC Motors vi) Constantspeed operation without servo control (in some synchronous machines) vii) No drift problems in AC amplifiers in supply circuits (unlike DC amplifiers) viii) High reliability Disadvantages of AC Motors i) Lower starting torque ii) Auxiliary starting device needed for some motors iii) Difficulty in variablespeed control (except when modern thyristorcontrol devices and field feedback compensation techniques are used) 3/52
4 Induction Motor(Asynchronous Motor) Induction Motor Model 1 v t a cos t p 2 v2 t a cos pt 3 4 v3 t a cos pt 3 : angular speed of a rotating field p 4/52
5 Induction Motor(Asynchronous Motor) Rotating field generates the driving torque by interacting with the rotor windings Induction Motor 5/52
6 Induction Motor(Asynchronous Motor) Rotating field speed, p f n p n : frequency of the AC supply : number of threephase winding sets used Slip rate : relative speed f m S f 6/52
7 Synchronous Motor Synchronous Motor 7/52
8 Synchronous Motor The rotor of a synchronous AC motor rotates in synchronism with a rotating field generated by the stator windings This motor has rotor windings that are energized by an external DC source. Suitable for constantspeed applications under variableload conditions Drawback : an auxiliary starter is required using a small DC motor (at steadystate, act as a DC generator) 8/52
9 Step Motor Step Motor Also called as Stepping Motor, Stepper Motor (Example1) Twostack step motor 9/52
10 Step Motor 10/52
11 Step Motor 11/52
12 Step Motor 12/52
13 Step Motor (Example2) Threephase variablereluctance step motor 13/52
14 Step Motor Fullstepping sequence for the threephase VR step motor 14/52
15 Step Motor Halfstepping sequence for the threephase VR step motor 15/52
16 Classification of step motor Step Motor 16/52
17 Step Motor Threephase singlestack VR step motor with twelve stator poles (teeth) and eight rotor teeth 17/52
18 Systems with Transportation Lags (Time Delays) Systems with Transportation Lags (Time Delays) 18/52
19 Systems with Transportation Lags (Time Delays) Time lag is given by T d d v d Ts d Ts d e b t y t T B s e Y s B s Y s time delay : seconds e Ts d 19/52
20 20/52 Systems with Transportation Lags (Time Delays) Approximation of the time delay approximat ion s T s T e e e s T s T e e s T s T e d d s T s T s T d d s T s T d d s T d d d d d d Pade : iii) ii) 2 1 i) 2 / 2 /
21 Modeling : mathematical description of physical system based on corresponding physical laws Model : differential equation, state equation, or transfer function used in simulation, analysis, and control design Real Physical System Modeling Mathematical Model i) LTI system ii) LTV system iii) Nonlinear LTI system iv) Nonlinear LTV system 21/52
22 Two approaches to derive an equation of motion i) Newtonian Mechanics : based on Newton s 2 nd law of motion ii) Lagrangian Mechanics : analytic method based on energy concept 22/52
23 Newtonian Mechanics describes rigid body motion using the balanced force relation Linear motion F ma F m a : vector sum of applied forces on a rigid body : mass of rigid body : vector of acceleration of rigid body 23/52
24 Rotational motion : T J T : sum of applied torques of rigid body J : mass moment of inertia of rigid body : angular acceleration of rigid body Note Free body diagram : net description of forces exerted on a rigid body convenient when deriving Newtonian equation of motion 24/52
25 Largrangian Mechanics derives equation of motion by using all the energy terms in a rigid body such as kinetic, potential, and dissipating energies Lagrange equation d T T V D Q j, j 1,2,, n dt q q j q j j qj q j V Q j : generalized coordinate, T : kinetic energy : potential energy, D : dissipating energy : nonconservative generalized force corresponding to q j 25/52
26 Note i) T, V, D are functions of generalized variable q j ii) Lagrangian : L T V iii) Lagrange equation d L L D Q j, j 1,2,, n dt q q j j qj 26/52
27 Kinetic energy T 1 2 mv J 2 m, J : mass and moment of inertia v, : linear and angular velocity Note vector equation of kinetic energy 1 T 1 T T v mv ω Jω /52
28 Dissipative friction energy b v 1 D bv 2 2 : viscous friction coefficient : velocity Note Generalized force 1. an external force as function of generalized coordinate variables 2. represents force for linear motion and torque for rotational motion, respectively 28/52
29 Example : massspringdamper system m b F : mass, k : spring constant, : damping coefficient : external force, x : displacement k x m F b 29/52 <Fig> massspringdamper system
30 i) Newtonian mechanics F ma : kx b x F m x m x b x kx F (1) <Fig> free body diagram 30/52
31 ii) Largrangian mechanics : 1 dof system( n 1) q 1 x T m x, V kx, D b x d T T V D m x, 0, kx, b x dt x x x x m x b x kx F (1) 31/52
32 Modeling of Electrical Networks Loop Method Network Equation Node Method StateVariable Method (used in modern control design) Example1 32/52
33 33/52 Voltage in L : Current in C : i) Statespace representation State : it, e t, Output : e c t yt, Input : e t dec di dt t dt y c t t di L Ri c dt t de C c dt u t 0 1 L t 1 0 e i t t c t e t et (1) it (2) 1 C e R i L t t c 1 0 u L t
34 StateDiagram optional Another statespace representation State : e t x t, e t x t, Output : t y t, Input : c e t 1 c 2 e c ut 34/52
35 (2) (1) : LC e t RC e t e t et c c c x1 t x1 t 1 R 1 u t x2 t x 2 t LC L LC 1 0 y t x t t 1 x2 35/52
36 ii) Transfer function representation if it E c E I E s 2 1 LC s 2 s R LCs RCs 1 1 is output L 1 s 1 LC s L s Cs 2 s R LCs RCs 1 1 L s 1 1 LC s s /52
37 Sensors and Encoders 37/52
38 38/52 automation sensor general sensor range sensor motor control sensor process control sensor object detection displacement position Speed/acceleration force/torque/elastic force temperature Fluid/fluid speed/fluid pressure density/thickness ph touch proximity
39 motor control sensor analog digital potentiometer linear/rotary variable differential transformer (LVDT/RVDT) resolver synchro inductive optical encoder absolute encoder laser interferometer 39/52
40 Incremental Encoder Position or velocity detecting digital output By counting the pulses or by timing the pulse width Equally spaced and identical slit areas 40/52
41 Incremental encoder (Single channel) Single channel encoder no direction information Dual channel encoder direction information detected 41/52
42 42/52
43 Absolute Encoder Many pulse tracks for position indication The pulse windows on the tracks can be organized into some pattern ( code) i) Binary Code ii) Gray Code : single bit continuous change 43/52
44 Binary Code 44/52
45 Gray Code 45/52
46 DC Motors Servo Motors (accurate motors for control purpose) i) AC Motors : cheap, robust, hard to control (due to nonlinearity) ii) DC Motors : expensive, easy to control 46/52
47 Basic Operation Principle electromagnetic force 47/52
48 f Bil ; principle of motor ( l :length of conductor) If the conductor is free to move, then it generates back electromotive force (back e.m.f.) e b e b B l v will be opposing the magnetic flux (by Lenz's law) 48/52
49 49/52
50 Actual DC Motor Schematic diagram of a DC motor 50/52
51 DC Motor Equations 51/52
52 BlockDiagram 52/52
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