Lecture02.pdf
- 2. Outline • Basic System Models • Mathematical Models • Electrical System Building Blocks • System Models 2
- 3. Mathematical models • A mathematical model of a system is a description of it in terms of equations relating inputs and outputs so that outputs can be predicted from inputs. • In devising a mathematical model of a system it is necessary to make assumptions and simplifications and a balance has to be chosen between simplicity of the model and the need for it to represent the actual real- world behavior. • For example the rotation of a motor shaft controlled by a microcontroller. The speed will not immediately assume the full-speed value but will only attain that speed after some time • Any mathematical model is provided by the fundamental physical laws that govern the behavior of the system. 3
- 4. System Building Blocks • Systems can be made up from a range of building blocks. • Each building block is considered to have a single property or function. • A system is called lumped parameter system if each parameter of function is considered independently. • Example- in an electrical system the basic building blocks can be combined in different ways to make a variety of electric circuit systems. • Overall input/output relationships of the electrical system can be obtained by combining in an appropriate way, the relationships for the building blocks. • Thus a mathematical model for the system can be obtained. 4
- 5. Electrical System Building Blocks • The basic building blocks of electrical systems are inductors, capacitors and resistors. • Assumptions – resistor building block is assumed to have purely the properties of resistance, the capacitor purely that of capacitance and the inductor purely that of inductance. 5
- 6. Inductor • For an inductor the potential difference 𝑣 across it any instant depends on the rate of change of current (di/dt) through it. 𝑣 = 𝐿 𝑑𝑖 𝑑𝑡 L is the inductance. The direction of the potential difference is in the opposite direction to the potential difference used to drive the current through the inductor, hence the term back e.m.f. is used. • The equation can be rearranged to give : 𝑖 = 1 𝐿 𝑣 𝑑𝑡 6
- 7. Capacitor • For a capacitor , the potential difference across it depends on the charge 𝑞 on the capacitor plates at the instant concerned. 𝑣 = 𝑞 𝐶 C is the capacitance. • The current 𝑖 to or from the capacitor is the rate at which charge moves to or from the capacitor plates. 𝑖 = 𝑑𝑞 𝑑𝑡 Total charge 𝑞 on the plates is : 𝑞 = 𝑖 𝑑𝑡 So 𝑣 = 1 𝐶 𝑖 𝑑𝑡 Alternatively since 𝑣 = 𝑞/C then, 𝑑𝑣 𝑑𝑡 = 1 𝐶 𝑑𝑞 𝑑𝑡 = 1 𝐶 𝑖 And so 𝑖 = 𝐶 𝑑𝑣 𝑑𝑡 7
- 8. Resistor • For a resistor, the potential difference 𝑣 across it at any instant depends on the current 𝑖 through it : 𝑣 = R𝑖 R is the resistance. • Both the inductor and capacitor store energy which can be released at a later time. A resistor just dissipates it. • The energy stored by an inductor when there is current 𝑖 is : 𝐸 = 1 2 𝐿𝑖2 • The energy stored by a capacitor when there is a potential difference 𝑣 across it is : 𝐸 = 1 2 𝐶𝑣2 • The power 𝑃 dissipated by a resistor when there is a potential difference 𝑣 across it is : 𝑃 = 𝑖𝑣 = 𝑣2 𝑅 8
- 9. Summary Summary of the equations defining the characteristics of the electrical building blocks when the input is current and the output is potential difference. 9
- 10. Building up a model for an electrical system • The electrical building blocks can be combined using Kirchhoff’s Law as outlined below: • Law 1: the total current flowing towards a junction is equal to the total current flowing from the junction, i.e. the algebraic sum of the currents at the junction is zero. • Law 2 : in a closed circuit of loop, the algebraic sum of the potential difference across each part of the circuit is equal to the applied e.m.f. 10
- 11. Example • Consider a simple electrical system consisting of a resistor and capacitor in series. Applying Kirchhoff’s second law to the circuit loop : 𝑣 = 𝑣𝑅 + 𝑣𝑐 Since this is just a single loop, the current i through all the circuit elements will be the same. If the output from the circuit is the potential difference across the capacitor: Since 𝑣𝑅 = 𝑖𝑅 and 𝑖 = C (d𝑣𝑐/dt) 𝑣 = 𝑅𝐶 𝑑𝑣𝑐 𝑑𝑡 + 𝑣𝑐 the relationship between the output 𝑣𝑐 and the input 𝑣 and is a first-order differential equation. Resistor–capacitor system 11
- 12. Example • Resistor-inductor-capacitor System. If the output from the circuit is the potential difference across the capacitor, 𝑣𝑐 : Applying Kirchhoff ’s second law to this circuit loop : 𝑣 = 𝑣𝑅 + 𝑣𝐿 + 𝑣𝑐 Since there is just a single loop, the current i will be the same through all circuit elements. 𝑣 = 𝑖𝑅 + 𝐿 𝑑𝑖 𝑑𝑡 + 𝑣𝑐 But i = C(d𝑣𝑐 /dt) and so 𝑑𝑖 𝑑𝑡 = 𝐶 𝑑( Τ 𝑑𝑣𝑐 𝑑𝑡) 𝑑𝑡 = 𝑑2𝑣𝑐 𝑑𝑡2 Hence 𝑣 = 𝑅𝐶 𝑑𝑣𝑐 𝑑𝑡 + 𝐿𝐶 𝑑2𝑣𝑐 𝑑𝑡2 + 𝑣𝑐 This is a second-order differential equation 12
- 13. Example • Resistor-inductor system Relationship between the output, the potential difference across the inductor of 𝑉𝐿 and input 𝑣. Applying Kirchhoff ’s second law to the circuit loop gives, 𝑣 = 𝑣𝑅 + 𝑣𝐿 then the relationship between the input and output is 𝑣 = 𝑅 𝐿 න 𝑣𝐿𝑑𝑡 + 𝑣𝐿 13
- 14. Example • Resistor –capacitor-inductor system. Consider the relationship between the output, the potential difference vC across the capacitor and the input v . Applying Kirchhoff ’s law 1 to node A gives 𝑖1 = 𝑖2 + 𝑖3 𝑖1 = 𝑣 − 𝑣𝐴 𝑅 𝑖2 = 1 𝐿 න 𝑣𝐴𝑑𝑡 𝑖3 = 𝐶 𝑑𝑣𝐴 𝑑𝑡 With some replacements and rearrangements ,the relationship between the input and output is 𝑣 = 𝑅𝐶 𝑑𝑣𝐶 𝑑𝑡 + 𝑣𝐶 + 𝑅 𝐿 න 𝑣𝐶 𝑑𝑡 14
- 15. Non Electrical Systems • There are other than electrical systems that can be mathematically modeled. • Mechanical Systems – translational, rotational • Fluid (Hydraulic ) Systems • Thermal Systems • There are many elements in mechanical, electrical, fluid and thermal systems which have similar behaviors. Thus, for example, mass in mechanical systems has similar properties to capacitance in electrical systems, capacitance in fluid systems and capacitance in thermal systems. 15
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- 17. System Models • Many systems encountered in engineering involve aspects of more than just electrical systems. • Building blocks of translational mechanical, rotational mechanical, electrical, fluid and thermal systems may need to be considered. • Example – Electric motor • Single-discipline building blocks can be combined to give models for multidiscipline systems. 17
- 18. Electro-mechanical Systems • Electromechanical devices transform electrical signals to rotational motion or vice versa. • Example – Potentiometers, Motors , Generators. • Potentiometer – has input of a rotation and an output of a potential difference. • Electric motor – has input of a potential difference and output of rotation of a shaft. • Generator – has input of rotation of a shaft and an output of a potential difference. 18
- 19. Potentiometer • The rotary potentiometer is a potential divider. • The relationship can be given by: 𝑣𝑜 𝑉 = 𝜃 𝜃𝑚𝑎𝑥 • V – potential difference across the full length of the potentiometer track. • 𝜃𝑚𝑎𝑥 - total angle swept out by the slider in being rotated from one end of the track to the other. • 𝑣𝑜 - output for the input 𝜃. 19
- 20. Direct Current Motor • In a d.c. motor a current through the armature coil of the motor results in a shaft being rotated and hence the load rotated. Motor driving a load One wire of armature coil 20
- 21. Direct Current Motor Direct current motor circuits The motor basically consists of a coil, the armature coil, which is free to rotate. This coil is located in the magnetic field provided by a current through field coils or a permanent magnet. 21
- 22. Direct Current Motors System model for armature-controlled d.c. motor. Armature is a coil rotating in a magnetic field, a voltage will be induced in it as a consequence of electromagnetic induction. This voltage will be in such a direction as to oppose the change producing it and is called the back e.m.f. Two equations that describe the conditions occurring for an armature-controlled motor are: 𝑣𝑎 − 𝑘3𝜔 = 𝐿𝑎 𝑑𝑖𝑎 𝑑𝑡 + 𝑅𝑎𝑖𝑎 and 𝐼 𝑑𝜔 𝑑𝑡 = 𝑘4𝑖𝑎 − 𝑐𝜔 The equation relating to the output 𝜔 with the input 𝑣𝑎 to the system by eliminating 𝑖𝑎 using Laplace transform. 22
- 23. Direct Current Motors System model for field-controlled d.c. motor. Here armature current is held constant and the motor is controlled by varying field voltage. The conditions occurring for a field-controlled motor are thus described by the equations : 𝑣𝑓 = 𝐿𝑓 𝑑𝑖𝑓 𝑑𝑡 + 𝑅𝑓𝑖𝑓 and 𝐼 𝑑𝜔 𝑑𝑡 = 𝑘5𝑖𝑓 − 𝑐𝜔 The equation relating the output 𝜔 with the input 𝑣𝑓 to the system can be obtained by eliminating 𝑖𝑓 using Laplace transform. 23
- 24. Linearity • Often real components are not linear. • Non-linear models are much more difficult to deal with and non- linear systems might be approximated by a linear model. • Example – A simple model for a spring assumes that force and extensions are proportional regardless of how large force was. Thus the mathematical model can be a simplification of a real spring. • In combining blocks to create models of systems it is assumed that the relationship for each block is linear. • It is required to make a linear approximation for a non-linear item. 24
- 25. Linearity • A system needs to satisfy Principle of superposition for it to be termed as linear system. • Example if an ideal spring system is given by F = kx, then if force F1 produces extension x1 and force F2 produces extension x2 , then force equal to (F1+F2) will produce an extension (x1+x2). • If force F1 produces an extension x1, then an input cF1 will produce an output cx1 ,where c is a constant multiplier 25
- 26. Linearity Ideal springs. Real springs. For many system components, linearity can be assumed for operations within a range of values of the variable about some operating point 26
- 27. Linearity Non-linear relationship. For such components the best that can be done to obtain a linear relationship is just to work with the straight line which is the slope of the graph at the operating point. 27