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Voltage regulator and temperature control
1.1 Background of The Project:
The Automatic Temperature Controlled Motor Speed is a project prepared for final year project, Electrical & Electronics Engineering, Eastern University. This project is done to be able to control the motor speed by the temperature.
Generally motor speed is control manually. It is done by a normal regulator circuit which is control by outside manually. In our project we wish to make an automatic motor speed based on temperature dependency.
A voltage regulator is an electrical regulator designed to automatically maintain a constant voltage level. A voltage regulator may be a simple “feed-forward” design or may include negative feedback control loops. It may use an electromechanical mechanism, or electronic components. Depending on the design, it may be used to regulate one or more AC or DC voltages.
A simple voltage regulator can be made from a resistor in series with a diode (or series of diodes). Due to the logarithmic shape of diode V-I curves, the voltage across the diode changes only slightly due to changes in current drawn. When precise voltage control is not important, this design may work fine.
Feedback voltage regulators operate by comparing the actual output voltage to some fixed reference voltage. Any difference is amplified and used to control the regulation element in such a way as to reduce the voltage error. This forms a negative feedback control loop; increasing the open-loop gain tends to increase regulation accuracy but reduce stability (avoidance of oscillation, or ringing during step changes). There will also be a trade-off between stability and the speed of the response to changes. If the output voltage is too low (perhaps due to input voltage reducing or load current increasing), the regulation element is commanded, up to a point, to produce a higher output voltage–by dropping less of the input voltage (for linear series regulators and buck switching regulators), or to draw input current for longer periods (boost-type switching regulators); if the output voltage is too high, the regulation element will normally be commanded to produce a lower voltage. However, many regulators have over-current protection; so that they will entirely stop sourcing current (or limit the current in some way) if the output current is too high, and some regulators may also shut down if the input voltage is outside a given range
In our project there is a temperature sensor which can sense or measure the temperature by which we operate an operational amplifier. The outputs of this op-amp we drive a relay which is operate the motor automatically.
Fig 1.1: Block Diagram of Automatic Temperature Controlled Motor Speed
1.2 Objective Of The Project:
The objectives of this project are:
i. The main objective of this project is to build and examine the automatic temperature controlled motor speed.
ii. To study the circuit and determine how the different parts of the circuit function together to make the automatic temperature controlled motor speed
iii. To understand about the concept the automatic temperature controlled motor speed.
iv. To be familiar with the use of design and simulation tools in the design process. For this project the design and simulation of the FM Voice Transmitter circuit is using the Or CAD Capture CIS Software.
v. To be able to construct, analyze and test the complete project of the automatic temperature controlled motor speed. In this part of objective the students are required to solve the problem occurred since the circuit does not work as planned earlier. Some alternatives and creativities from the student were required.
1.3 Project Scope:
We have identified the scope of this project. The scope can be used as a guideline for me to conduct this project in order to complete this project in a time given and as in a plan from the earlier stage. Basically this project focused on:
i. Identifying the components and materials. In the process of identifying the components and materials going to use, We have to ensure the components and materials related in this project ready in the stock.
ii. Designing and layout. In this process we have used the Or CAD Capture CIS software, where it is provide the designing, simulation and layout design for PCB.
iii. Project Circuit Board (PCB) construction. In the part of PCB process, we have made this PCB ourselves.
iv. Testing and analyzing. In this process of testing and analyzing, we have performed the job related at Circuit Lab. The testing parts have to test using the DC fan while the analyzing processes have to do at the lab.
v. Interfacing this project we use DC fan while to testing the speed control.
1.4 Significance of the Project:
This project is an alternative for people especially for all human being who are used the fan or in any company who are used various motor for heat treatment. There are many legitimate reasons for automatic temperature control in any plant where temperature is very high and difficult to manipulate from nearby.
After completion of this project we have gathering huge knowledge about thermostat and automatic control any electrical or electronic device. By which we are able to understand about control system engineering what we studied our early semester in Electrical and Electronic Engineering.
1.5 Problem faces:
Each and every project students usually faces several problems that might occur while running and implementing its process of the project. In this project we also faces several problems occurred along the implementation of this project:
• The electronic components with the recommended value such as, thermistor could not found at accurate value, the relay could not found as required.
• To make the Project Circuit Board (PCB) we faced most difficulties. First time we could not find the Ferric Chloride which is most important materials to make PCB from the CCB.
• After making this PCB, it was also make much foolish to bore the PCB. We don’t have 0.5mm drill machine. Then we buy a 12V DC motor and 0.5mm drill bit. We placed the drill bit into a plastic body ball point pen and put it to the motor. Then we are able to bore the PCB. It was also a great experience for our study life and we are able to resolve that such problem.
1.6 Project Limitation:
Every project usually faces several limitations that might occur while running its process of the project. There are several limitations occur along the implementation of this project:
• The electronic components with the recommended value such as, thermistor could not found at accurate value, the relay could not found as required so our required output does not gives us actual as desired.
• The component we used not much sensitive as we need. In theoretically we learnt the parameter of various components but in practical it differs with the theory. process of tuning to the proper frequency to broadcast at FM radio is hard to get since the problem it’s related with the components and the circuit itself.
1.7 Report Overview:
Chapter one provides an overview of the project by giving description of the problem. Chapter one discusses about the background of the project, problem description, objective, project limitation and overview of the report.
Chapter two discusses about the literature review for the basic of the project. This chapter gives the full explanation regarding the control system which the base of this project. From the beginning of control system, its history and various types of control systems are included in this chapter.
Chapter three explained about the methodology that was used. The methodology of this project is to control some devices is based on temperature. Our project is developed with the temperature control. The main component of this temperature control is thermistor which is described in this chapter. Temperature and resistance relationship is described here.
In chapter four, we describe about another main component used in our project. Except thermistor we used transistor, operational amplifier, relay, diode, LED, resistor, capacitor etc. in our project execution. We described about transistor and operational amplifier in the chapter four. Both the devices are broadly used as amplifier in electronic circuit.
In chapter five we describe about the circuit diagram of our project. Part by part circuit operation and its description does not stated in details in this chapter. Because the operation principle of the major component of this project are stated in chapter three and four. We describe the operation principle for this project in brief.
In the last chapter, a summary of overall progress of project is presented and it will provide the conclusion of the project. The conclusion part concludes all the things related to the project including technical part, creativity and the achievement of this project. It also provides some recommendation that can be made for this project better than the previous job.
BASIC CONTROL SYSTEM THEORY
2.1 Control theory
The concept of the feedback loop to control the dynamic behavior of the system: this is negative feedback, because the sensed value is subtracted from the desired value to create the error signal, which is amplified by the controller.
Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference. When one or more output variables of a system need to follow a certain reference over time, a controller manipulates the inputs to a system to obtain the desired effect on the output of the system.
The concept of the feedback loop to control the dynamic behavior of the system: this is negative feedback, because the sensed value is subtracted from the desired value to create the error signal, which is amplified by the controller.
Control engineering or Control systems engineering is the engineering discipline that applies control theory to design systems with predictable behaviors. The practice uses sensors to measure the output performance of the device being controlled (often a vehicle) and those measurements can be used to give feedback to the input actuators that can make corrections toward desired performance. When a device is designed to perform without the need of human inputs for correction it is called automatic control (such as cruise control for regulating a car’s speed). Multi-disciplinary in nature, control systems engineering activities focus on implementation of control systems mainly derived by mathematical modeling of systems of a diverse range.
Control theory is
• a theory that deals with influencing the behavior of dynamical systems
• an interdisciplinary subfield of science, which originated in engineering and mathematics, and evolved into use by the social sciences, like psychology, sociology and criminology.
Consider a car’s cruise control, which is a device designed to maintain vehicle speed at a constant desired or reference speed provided by the driver. The controller is the cruise control, the plant is the car, and the system is the car and the cruise control. The system output is the car’s speed, and the control itself is the engine’s throttle position which determines how much power the engine generates.
A primitive way to implement cruise control is simply to lock the throttle position when the driver engages cruise control. However, if the cruise control is engaged on a stretch of flat road, then the car will travel slower going uphill and faster when going downhill. This type of controller is called an open-loop controller because no measurement of the system output (the car’s speed) is used to alter the control (the throttle position.) As a result, the controller can not compensate for changes acting on the car, like a change in the slope of the road.
In a closed-loop control system, a sensor monitors the system output (the car’s speed) and feeds the data to a controller which adjusts the control (the throttle position) as necessary to maintain the desired system output (match the car’s speed to the reference speed.) Now when the car goes uphill the decrease in speed is measured, and the throttle position changed to increase engine power, speeding the vehicle. Feedback from measuring the car’s speed has allowed the controller to dynamically compensate for changes to the car’s speed. It is from this feedback that the paradigm of the control loop arises: the control affects the system output, which in turn is measured and looped back to alter the control.
2.3 Control systems
Control engineering is the engineering discipline that focuses on the modeling of a diverse range of dynamic systems (e.g. mechanical systems) and the design of controllers that will cause these systems to behave in the desired manner. Although such controllers need not be electrical many are and hence control engineering is often viewed as a subfield of electrical engineering. However, the falling price of microprocessors is making the actual implementation of a control system essentially trivial. As a result, focus is shifting back to the mechanical engineering discipline, as intimate knowledge of the physical system being controlled is often desired.
Electrical circuits, digital signal processors and microcontrollers can all be used to implement Control systems. Control engineering has a wide range of applications from the flight and propulsion systems of commercial airliners to the cruise control present in many modern automobiles.
In most of the cases, control engineers utilize feedback when designing control systems. This is often accomplished using a PID controller system. For example, in an automobile with cruise control the vehicle’s speed is continuously monitored and fed back to the system, which adjusts the motor’s torque accordingly. Where there is regular feedback, control theory can be used to determine how the system responds to such feedback. In practically all such systems stability is important and control theory can help ensure stability is achieved.
Although feedback is an important aspect of control engineering, control engineers may also work on the control of systems without feedback. This is known as open loop control. A classic example of open loop control is a washing machine that runs through a pre-determined cycle without the use of sensors.
Centrifugal governor in a Boulton & Watt engine of 1788
Although control systems of various types date back to antiquity, a more formal analysis of the field began with a dynamics analysis of the centrifugal governor, conducted by the physicist James Clerk Maxwell in 1868 entitled On Governors. This described and analyzed the phenomenon of “hunting”, in which lags in the system can lead to overcompensation and unstable behavior. This generated a flurry of interest in the topic, during which Maxwell’s classmate Edward John Routh generalized the results of Maxwell for the general class of linear systems. Independently, Adolf Hurwitz analyzed system stability using differential equations in 1877, resulting in what is now known as the Routh-Hurwitz theorem.
A notable application of dynamic control was in the area of manned flight. The Wright brothers made their first successful test flights on December 17, 1903 and were distinguished by their ability to control their flights for substantial periods (more so than the ability to produce lift from an airfoil, which was known). Control of the airplane was necessary for safe flight.
By World War II, control theory was an important part of fire-control systems, guidance systems and electronics.
Sometimes mechanical methods are used to improve the stability of systems. For example, ship stabilizers are fins mounted beneath the waterline and emerging laterally. In contemporary vessels, they may be gyroscopically controlled active fins, which have the capacity to change their angle of attack to counteract roll caused by wind or waves acting on the ship.
The Sidewinder missile uses small control surfaces placed at the rear of the missile with spinning disks on their outer surface; these are known as rollerons. Airflow over the disk spins them to a high speed. If the missile starts to roll, the gyroscopic force of the disk drives the control surface into the airflow, cancelling the motion. Thus the Sidewinder team replaced a potentially complex control system with a simple mechanical solution.
The Space Race also depended on accurate spacecraft control. However, control theory also saw an increasing use in fields such as economics.
Automatic control Systems were first developed over two thousand years ago. The first feedback control device on record is thought to be the ancient water clock of Ktesibios in Alexandria Egypt around the third century B.C. It kept time by regulating the water level in a vessel and, therefore, the water flow from that vessel. This certainly was a successful device as water clocks of similar design were still being made in Baghdad when the Mongols captured the city in 1258 A.D. A variety of automatic devices have been used over the centuries to accomplish useful tasks or simply to just entertain. The latter includes the automata, popular in Europe in the 17th and 18th centuries, featuring dancing figures that would repeat the same task over and over again; these automata are examples of open-loop control. Milestones among feedback, or “closed-loop” automatic control devices, include the temperature regulator of a furnace attributed to Drebbel, circa 1620, and the centrifugal flyball governor used for regulating the speed of steam engines by James Watt in 1788.
In his 1868 paper “On Governors”, J. C. Maxwell (who discovered the Maxwell electromagnetic field equations) was able to explain instabilities exhibited by the flyball governor using differential equations to describe the control system. This demonstrated the importance and usefulness of mathematical models and methods in understanding complex phenomena, and signaled the beginning of mathematical control and systems theory. Elements of control theory had appeared earlier but not as dramatically and convincingly as in Maxwell’s analysis.
Control theory made significant strides in the next 100 years. New mathematical techniques made it possible to control, more accurately, significantly more complex dynamical systems than the original flyball governor. These techniques include developments in optimal control in the 1950s and 1960s, followed by progress in stochastic, robust, adaptive and optimal control methods in the 1970s and 1980s. Applications of control methodology have helped make possible space travel and communication satellites, safer and more efficient aircraft, cleaner auto engines, cleaner and more efficient chemical processes, to mention but a few.
Before it emerged as a unique discipline, control engineering was practiced as a part of mechanical engineering and control theory was studied as a part of electrical engineering, since electrical circuits can often be easily described using control theory techniques. In the very first control relationships, a current output was represented with a voltage control input. However, not having proper technology to implement electrical control systems, designers left with the option of less efficient and slow responding mechanical systems. A very effective mechanical controller that is still widely used in some hydro plants is the governor. Later on, previous to modern power electronics, process control systems for industrial applications were devised by mechanical engineers using pneumatic and hydraulic control devices, many of which are still in use today.
2.5 Classical control theory
To avoid the problems of the open-loop controller, control theory introduces feedback. A closed-loop controller uses feedback to control states or outputs of a dynamical system. Its name comes from the information path in the system: process inputs (e.g. voltage applied to an electric motor) have an effect on the process outputs (e.g. velocity or torque of the motor), which is measured with sensors and processed by the controller; the result (the control signal) is used as input to the process, closing the loop.
Closed-loop controllers have the following advantages over open-loop controllers:
• disturbance rejection (such as unmeasured friction in a motor)
• guaranteed performance even with model uncertainties, when the model structure does not match perfectly the real process and the model parameters are not exact
• unstable processes can be stabilized
• reduced sensitivity to parameter variations
• improved reference tracking performance
In some systems, closed-loop and open-loop control are used simultaneously. In such systems, the open-loop control is termed feed forward and serves to further improve reference tracking performance.
Common closed-loop controller architecture is the PID controller.
2.6 Closed-loop transfer function
For more details on this topic, see Closed-loop transfer function.
The output of the system y(t) is fed back through a sensor measurement F to the reference value r(t). The controller C then takes the error e (difference) between the reference and the output to change the inputs u to the system under control P. This is shown in the figure. This kind of controller is a closed-loop controller or feedback controller.
This is called a single-input-single-output (SISO) control system; MIMO (i.e. Multi-Input-Multi-Output) systems, with more than one input/output, are common. In such cases variables are represented through vectors instead of simple scalar values. For some distributed parameter systems the vectors may be infinite-dimensional (typically functions).
If we assume the controller C, the plant P, and the sensor F are linear and time-invariant (i.e.: elements of their transfer function C(s), P(s), and F(s) do not depend on time), the systems above can be analyzed using the Laplace transform on the variables. This gives the following relations:
Solving for Y(s) in terms of R(s) gives:
The expression is referred to as the closed-loop transfer function of the system. The numerator is the forward (open-loop) gain from r to y, and the denominator is one plus the gain in going around the feedback loop, the so-called loop gain. If , i.e. it has a large norm with each value of s, and if , then Y(s) is approximately equal to R(s) and the output closely tracks the reference input.
2.7 PID controller:
For more details on this topic, see PID controller.
The PID controller is probably the most-used feedback control design. PID is an acronym for Proportional-Integral-Differential, referring to the three terms operating on the error signal to produce a control signal. If u(t) is the control signal sent to the system, y(t) is the measured output and r(t) is the desired output, and tracking error e(t) = r(t) ? y(t), a PID controller has the general form
The desired closed loop dynamics is obtained by adjusting the three parameters KP, KI and KD, often iteratively by “tuning” and without specific knowledge of a plant model. Stability can often be ensured using only the proportional term. The integral term permits the rejection of a step disturbance (often a striking specification in process control). The derivative term is used to provide damping or shaping of the response. PID controllers are the most well established class of control systems: however, they cannot be used in several more complicated cases, especially if MIMO systems are considered.
Applying Laplace transformation results in the transformed PID controller equation
with the PID controller transfer function
2.8 Modern control theory:
In contrast to the frequency domain analysis of the classical control theory, modern control theory utilizes the time-domain state space representation, a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations. To abstract from the number of inputs, outputs and states, the variables are expressed as vectors and the differential and algebraic equations are written in matrix form (the latter only being possible when the dynamical system is linear). The state space representation (also known as the “time-domain approach”) provides a convenient and compact way to model and analyze systems with multiple inputs and outputs. With inputs and outputs, we would otherwise have to write down Laplace transforms to encode all the information about a system. Unlike the frequency domain approach, the use of the state space representation is not limited to systems with linear components and zero initial conditions. “State space” refers to the space whose axes are the state variables. The state of the system can be represented as a vector within that space.)
2.9 Topics in control theory:
The stability of a general dynamical system with no input can be described with Lyapunov stability criteria. A linear system that takes an input is called bounded-input bounded-output (BIBO) stable if its output will stay bounded for any bounded input. Stability for nonlinear systems that take an input is input-to-state stability (ISS), which combines Lyapunov stability and a notion similar to BIBO stability. For simplicity, the following descriptions focus on continuous-time and discrete-time linear systems.
Mathematically, this means that for a causal linear system to be stable all of the poles of its transfer function must satisfy some criteria depending on whether a continuous or discrete time analysis is used:
• In continuous time, the Laplace transform is used to obtain the transfer function. A system is stable if the poles of this transfer function lie strictly in the open left half of the complex plane (i.e. the real part of all the poles is less than zero).
• In discrete time the Z-transform is used. A system is stable if the poles of this transfer function lie strictly inside the unit circle. i.e. the magnitude of the poles is less than one).
When the appropriate conditions above are satisfied a system is said to be asymptotically stable: the variables of an asymptotically stable control system always decrease from their initial value and do not show permanent oscillations. Permanent oscillations occur when a pole has a real part exactly equal to zero (in the continuous time case) or a modulus equal to one (in the discrete time case). If a simply stable system response neither decays nor grows over time, and has no oscillations, it is marginally stable: in this case the system transfer function has non-repeated poles at complex plane origin (i.e. their real and complex component is zero in the continuous time case). Oscillations are present when poles with real part equal to zero have an imaginary part not equal to zero.
Differences between the two cases are not a contradiction. The Laplace transform is in Cartesian coordinates and the Z-transform is in circular coordinates, and it can be shown that:
• the negative-real part in the Laplace domain can map onto the interior of the unit circle
• the positive-real part in the Laplace domain can map onto the exterior of the unit circle
If a system in question has an impulse response of
then the Z-transform (see this example), is given by
which has a pole in z = 0.5 (zero imaginary part). This system is BIBO (asymptotically) stable since the pole is inside the unit circle.
However, if the impulse response was
then the Z-transform is
which has a pole at z = 1.5 and is not BIBO stable since the pole has a modulus strictly greater than one.
Numerous tools exist for the analysis of the poles of a system. These include graphical systems like the root locus, Bode plots or the Nyquist plots.
Mechanical changes can make equipment (and control systems) more stable. Sailors add ballast to improve the stability of ships. Cruise ships use antiroll fins that extend transversely from the side of the ship for perhaps 30 feet (10 m) and are continuously rotated about their axes to develop forces that oppose the roll.
2.10 Controllability and observability:
Controllability and observability are main issues in the analysis of a system before deciding the best control strategy to be applied, or whether it is even possible to control or stabilize the system. Controllability is related to the possibility of forcing the system into a particular state by using an appropriate control signal. If a state is not controllable, then no signal will ever be able to control the state. If a state is not controllable, but its dynamics are stable, then the state is termed Stabilizable. Observability instead is related to the possibility of “observing”, through output measurements, the state of a system. If a state is not observable, the controller will never be able to determine the behaviour of an unobservable state and hence cannot use it to stabilize the system. However, similar to the stabilizability condition above, if a state cannot be observed it might still be detectable.
From a geometrical point of view, looking at the states of each variable of the system to be controlled, every “bad” state of these variables must be controllable and observable to ensure a good behaviour in the closed-loop system. That is, if one of the eigenvalues of the system is not both controllable and observable, this part of the dynamics will remain untouched in the closed-loop system. If such an eigenvalue is not stable, the dynamics of this eigenvalue will be present in the closed-loop system which therefore will be unstable. Unobservable poles are not present in the transfer function realization of a state-space representation, which is why sometimes the latter is preferred in dynamical systems analysis.
Solutions to problems of uncontrollable or unobservable system include adding actuators and sensors.
2.11 Control specification:
Several different control strategies have been devised in the past years. These vary from extremely general ones (PID controller), to others devoted to very particular classes of systems (especially robotics or aircraft cruise control).
A control problem can have several specifications. Stability, of course, is always present: the controller must ensure that the closed-loop system is stable, regardless of the open-loop stability. A poor choice of controller can even worsen the stability of the open-loop system, which must normally be avoided. Sometimes it would be desired to obtain particular dynamics in the closed loop: i.e. that the poles have , where a fixed value is strictly greater than zero, instead of simply asking that Re [?] < 0. Another typical specification is the rejection of a step disturbance; including an integrator in the open-loop chain (i.e. directly before the system under control) easily achieves this. Other classes of disturbances need different types of sub-systems to be included. Other “classical” control theory specifications regard the time-response of the closed-loop system: these include the rise time (the time needed by the control system to reach the desired value after a perturbation), peak overshoot (the highest value reached by the response before reaching the desired value) and others (settling time, quarter-decay). Frequency domain specifications are usually related to robustness (see after). Modern performance assessments use some variation of integrated tracking error (IAE, ISA, and CQI). 2.12 Model identification and robustness: A control system must always have some robustness property. A robust controller is such that its properties do not change much if applied to a system slightly different from the mathematical one used for its synthesis. This specification is important: no real physical system truly behaves like the series of differential equations used to represent it mathematically. Typically a simpler mathematical model is chosen in order to simplify calculations, otherwise the true system dynamics can be so complicated that a complete model is impossible. System identification The process of determining the equations that govern the model’s dynamics is called system identification. This can be done off-line: for example, executing a series of measures from which to calculate an approximated mathematical model, typically its transfer function or matrix. Such identification from the output, however, cannot take account of unobservable dynamics. Sometimes the model is built directly starting from known physical equations: for example, in the case of a mass-spring-damper system we know that . Even assuming that a “complete” model is used in designing the controller, all the parameters included in these equations (called “nominal parameters”) are never known with absolute precision; the control system will have to behave correctly even when connected to physical system with true parameter values away from nominal. Some advanced control techniques include an “on-line” identification process (see later). The parameters of the model are calculated (“identified”) while the controller itself is running: in this way, if a drastic variation of the parameters ensues (for example, if the robot’s arm releases a weight), the controller will adjust itself consequently in order to ensure the correct performance. Analysis Analysis of the robustness of a SISO control system can be performed in the frequency domain, considering the system’s transfer function and using Nyquist and Bode diagrams. Topics include gain and phase margin and amplitude margin. For MIMO and, in general, more complicated control systems one must consider the theoretical results devised for each control technique (see next section): i.e., if particular robustness qualities are needed, the engineer must shift his attention to a control technique including them in its properties. Constraints A particular robustness issue is the requirement for a control system to perform properly in the presence of input and state constraints. In the physical world every signal is limited. It could happen that a controller will send control signals that cannot be followed by the physical system: for example, trying to rotate a valve at excessive speed. This can produce undesired behavior of the closed-loop system, or even break actuators or other subsystems. Specific control techniques are available to solve the problem: model predictive control, and anti-wind up systems. The latter consists of an additional control block that ensures that the control signal never exceeds a given threshold. 2.13 System classifications: Linear Systems control For MIMO systems, pole placement can be performed mathematically using a state space representation of the open-loop system and calculating a feedback matrix assigning poles in the desired positions. In complicated systems this can require computer-assisted calculation capabilities, and cannot always ensure robustness. Furthermore, all system states are not in general measured and so observers must be included and incorporated in pole placement design. Nonlinear Systems control Processes in industries like robotics and the aerospace industry typically have strong nonlinear dynamics. In control theory it is sometimes possible to linearize such classes of systems and apply linear techniques, but in many cases it can be necessary to devise from scratch theories permitting control of nonlinear systems. These, e.g., feedback linearization, backstepping, sliding mode control, trajectory linearization control normally take advantage of results based on Lyapunov’s theory. Differential geometry has been widely used as a tool for generalizing well-known linear control concepts to the non-linear case, as well as showing the subtleties that make it a more challenging problem. Decentralized Systems When the system is controlled by multiple controllers, the problem is one of decentralized control. Decentralization is helpful in many ways, for instance, it helps control systems operate over a larger geographical area. The agents in decentralized control systems can interact using communication channels and coordinate their actions. 2.14 Main control strategies: Every control system must guarantee first the stability of the closed-loop behavior. For linear systems, this can be obtained by directly placing the poles. Non-linear control systems use specific theories (normally based on Aleksandr Lyapunov’s Theory) to ensure stability without regard to the inner dynamics of the system. The possibility to fulfill different specifications varies from the model considered and the control strategy chosen. Here a summary list of the main control techniques is shown: Adaptive control Adaptive control uses on-line identification of the process parameters, or modification of controller gains, thereby obtaining strong robustness properties. Adaptive controls were applied for the first time in the aerospace industry in the 1950s, and have found particular success in that field. Hierarchical control A Hierarchical control system is a type of Control System in which a set of devices and governing software is arranged in a hierarchical tree. When the links in the tree are implemented by a computer network, then that hierarchical control system is also a form of Networked control system. Intelligent control Intelligent control uses various AI computing approaches like neural networks, Bayesian probability, fuzzy logic, machine learning, evolutionary computation and genetic algorithms to control a dynamic system. Optimal control Optimal control is a particular control technique in which the control signal optimizes a certain “cost index”: for example, in the case of a satellite, the jet thrusts needed to bring it to desired trajectory that consume the least amount of fuel. Two optimal control design methods have been widely used in industrial applications, as it has been shown they can guarantee closed-loop stability. These are Model Predictive Control (MPC) and Linear-Quadratic-Gaussian control (LQG). The first can more explicitly take into account constraints on the signals in the system, which is an important feature in many industrial processes. However, the “optimal control” structure in MPC is only a means to achieve such a result, as it does not optimize a true performance index of the closed-loop control system. Together with PID controllers, MPC systems are the most widely used control technique in process control. Robust control Robust control deals explicitly with uncertainty in its approach to controller design. Controllers designed using robust control methods tend to be able to cope with small differences between the true system and the nominal model used for design. The early methods of Bode and others were fairly robust; the state-space methods invented in the 1960s and 1970s were sometimes found to lack robustness. A modern example of a robust control technique is H-infinity loop-shaping developed by Duncan McFarlane and Keith Glover of Cambridge University, United Kingdom. Robust methods aim to achieve robust performance and/or stability in the presence of small modeling errors. Stochastic control Stochastic control deals with control design with uncertainty in the model. In typical stochastic control problems, it is assumed that there exist random noise and disturbances in the model and the controller, and the control design must take into account these random deviations. THERMISTOR 3.1 Introduction: A thermistor is a type of resistor whose resistance varies significantly with temperature, more so than in standard resistors. The word is a portmanteau of thermal and resistor. Thermistors are widely used as inrush current limiters, temperature sensors, self-resetting overcurrent protectors, and self-regulating heating elements. Thermistors differ from resistance temperature detectors (RTD) in that the material used in a thermistor is generally a ceramic or polymer, while RTDs use pure metals. The temperature response is also different; RTDs are useful over larger temperature ranges, while thermistors typically achieve a higher precision within a limited temperature range [usually ?90 °C to 130 °C]. Thermistor symbol Assuming, as a first-order approximation, that the relationship between resistance and temperature is linear, then: where ?R = change in resistance ?T = change in temperature k = first-order temperature coefficient of resistance Thermistors can be classified into two types, depending on the sign of k. If k is positive, the resistance increases with increasing temperature, and the device is called a positive temperature coefficient (PTC) thermistor, or posistor. If k is negative, the resistance decreases with increasing temperature, and the device is called a negative temperature coefficient (NTC) thermistor. Resistors that are not thermistors are designed to have a k as close to zero as possible (smallest possible k), so that their resistance remains nearly constant over a wide temperature range. Instead of the temperature coefficient k, sometimes the temperature coefficient of resistance ? (alpha) or ?T is used. It is defined as For example, for the common PT100 sensor, ? = 0.00385 or 0.385 %/°C. This ?T coefficient should not be confused with the parameter below. 3.2 Steinhart-Hart equation: In practice, the linear approximation (above) works only over a small temperature range. For accurate temperature measurements, the resistance/temperature curve of the device must be described in more detail. The Steinhart-Hart equation is a widely used third-order approximation: where a, b and c are called the Steinhart-Hart parameters, and must be specified for each device. T is the temperature in kelvin and R is the resistance in ohms. To give resistance as a function of temperature, the above can be rearranged into: where and The error in the Steinhart-Hart equation is generally less than 0.02 °C in the measurement of temperature. As an example, typical values for a thermistor with a resistance of 3000 ? at room temperature (25 °C = 298.15 K) are: 3.3 B parameter equation: NTC thermistors can also be characterized with the B parameter equation, which is essentially the Steinhart Hart equation with a = (1 / T0) ? (1 / B)ln(R0), b = 1 / B and c = 0, Where the temperatures are in kelvins and R0 is the resistance at temperature T0 (usually 25 °C = 298.15 K). Solving for R yields: or, alternatively, where . This can be solved for the temperature: The B-parameter equation can also be written as . This can be used to convert the function of resistance vs. temperature of a thermistor into a linear function of lnR vs. 1/T. The average slope of this function will then yield an estimate of the value of the B parameter. 3.4 Conduction model: Many NTC thermistors are made from a pressed disc or cast chip of a semiconductor such as a sintered metal oxide. They work because raising the temperature of a semiconductor increases the number of electrons able to move about and carry charge – it promotes them into the conduction band. The more charge carriers that are available, the more current a material can conduct. This is described in the formula: I = electric current (amperes) n = density of charge carriers (count/m³) A = cross-sectional area of the material (m²) v = velocity of charge carriers (m/s) e = charge of an electron ( coulomb) The current is measured using an ammeter. Over large changes in temperature, calibration is necessary. Over small changes in temperature, if the right semiconductor is used, the resistance of the material is linearly proportional to the temperature. There are many different semiconducting thermistors with a range from about 0.01 kelvin to 2,000 kelvins (?273.14 °C to 1,700 °C). Most PTC thermistors are of the “switching” type, which means that their resistance rises suddenly at a certain critical temperature. The devices are made of a doped polycrystalline ceramic containing barium titanate (BaTiO3) and other compounds. The dielectric constant of this ferroelectric material varies with temperature. Below the Curie point temperature, the high dielectric constant prevents the formation of potential barriers between the crystal grains, leading to a low resistance. In this region the device has a small negative temperature coefficient. At the Curie point temperature, the dielectric constant drops sufficiently to allow the formation of potential barriers at the grain boundaries, and the resistance increases sharply. At even higher temperatures, the material reverts to NTC behaviour. The equations used for modeling this behaviour were derived by W. Heywang and G. H. Jonker in the 1960s. Another type of PTC thermistor is the polymer PTC, which is sold under brand names such as “Polyswitch” “Semifuse”, and “Multifuse”. This consists of a slice of plastic with carbon grains embedded in it. When the plastic is cool, the carbon grains are all in contact with each other, forming a conductive path through the device. When the plastic heats up, it expands, forcing the carbon grains apart, and causing the resistance of the device to rise rapidly. Like the BaTiO3 thermistor, this device has a highly nonlinear resistance/temperature response and is used for switching, not for proportional temperature measurement. Yet another type of thermistor is a silistor, a thermally sensitive silicon resistor. Silistors are similarly constructed and operate on the same principles as other thermistors, but employ silicon as the semiconductive component material. 3.5 Self-heating effects: When a current flows through a thermistor, it will generate heat which will raise the temperature of the thermistor above that of its environment. If the thermistor is being used to measure the temperature of the environment, this electrical heating may introduce a significant error if a correction is not made. Alternatively, this effect itself can be exploited. It can, for example, make a sensitive air-flow device employed in a sailplane rate-of-climb instrument, the electronic variometer, or serve as a timer for a relay as was formerly done in telephone exchanges. The electrical power input to the thermistor is just: where I is current and V is the voltage drop across the thermistor. This power is converted to heat, and this heat energy is transferred to the surrounding environment. The rate of transfer is well described by Newton’s law of cooling: where T(R) is the temperature of the thermistor as a function of its resistance R, T0 is the temperature of the surroundings, and K is the dissipation constant, usually expressed in units of milliwatts per degree Celsius. At equilibrium, the two rates must be equal. The current and voltage across the thermistor will depend on the particular circuit configuration. As a simple example, if the voltage across the thermistor is held fixed, then by Ohm’s Law we have I = V / R and the equilibrium equation can be solved for the ambient temperature as a function of the measured resistance of the thermistor: The dissipation constant is a measure of the thermal connection of the thermistor to its surroundings. It is generally given for the thermistor in still air, and in well-stirred oil. Typical values for a small glass bead thermistor are 1.5 mW/°C in still air and 6.0 mW/°C in stirred oil. If the temperature of the environment is known beforehand, then a thermistor may be used to measure the value of the dissipation constant. For example, the thermistor may be used as a flow rate sensor, since the dissipation constant increases with the rate of flow of a fluid past the thermistor. 4.1 TRANSISTOR 4.1.1 Introduction: The bipolar junction transistor (BJT) was the first solid-state amplifier element and started the solid-state electronics revolution. Bardeen, Brattain and Shockley, while at Bell Laboratories, invented it in 1948 as part of a post-war effort to replace vacuum tubes with solid-state devices. Solid-state rectifiers were already in use at the time and were preferred over vacuum diodes because of their smaller size, lower weight and higher reliability. A solid-state replacement for a vacuum triode was expected to yield similar advantages. The work at Bell Laboratories was highly successful and culminated in Bardeen, Brattain and Shockley receiving the Nobel Prize in 1956. In this chapter we first present the structure of the bipolar transistor and show how a three-layer structure with alternating n-type and p-type regions. We then present the ideal transistor model and derive an expression for the current gain in the forward active mode of operation. Next, we discuss the configurations, the biasing and operation of the Transistor. 4.1.2 Transistor: A Transistor is a three or more element solid-state device that amplifies by controlling the flow of current carriers through its semiconductor materials. Fig 4.2: Elements of a Transistor. The THREE ELEMENTS OF A TRANSISTOR are (1) the EMITTER, which gives off current carriers, (2) the BASE, which controls the carriers, and (3) the COLLECTOR, which collects the carriers. 4.1.3 Types of Transistor: The two BASIC TYPES OF TRANSISTORS are the NPN and PNP. The only difference in symbolic between the two transistors is the direction of the arrow on the emitter. If the arrow points in, it is a PNP transistor and if it points outward, it is an NPN transistor. Fig 4.3: Types of Transistor. 4.1.4 Transistor Biasing: The PROPER BIASING OF A TRANSISTOR enables the transistor to be used as an amplifier. To function in this capacity, the emitter-to-base junction of the transistor is forward biased, while the base-to-collector junction is reverse biased. Fig 4.4: Transistor Biasing. 4.1.5 Structure and principle of operation: A bipolar junction transistor consists of two back-to-back p-n junctions, who share a thin common region with width, wB. Contacts are made to all three regions, the two outer regions called the emitter and collector and the middle region called the base. The structure of an npn bipolar transistor is shown in Figure 4.5 (a). Figure 4.1.5: (a) Structure and sign convention of a npn bipolar junction transistor. (b) Electron and hole flow under forward active bias, VBE > 0 and VBC = 0.
The device is called “bipolar” since its operation involves both types of mobile carriers, electrons and holes. Since the device consists of two back-to-back diodes, there are depletion regions between the quasi-neutral regions. The width of the quasi neutral regions in the emitter, base and collector are indicated with the symbols wE’, wB’ and wC’ and are calculated from
Where the depletion region widths are given by:
The sign convention of the currents and voltage is indicated on Figure 4.5(a). The base and collector current are positive if a positive current goes into the base or collector contact. The emitter current is positive for a current coming out of the emitter contact. This also implies that the emitter current, IE, equals the sum of the base current, IB, and the collector current, IC:
The base-emitter voltage and the base-collector voltage are positive if a positive voltage is applied to the base contact relative to the emitter and collector respectively.
Figure 4.1.5(b): Energy band diagram of a bipolar transistor biased in the forward active mode.
The total emitter current is the sum of the electron diffusion current, IE,n, the hole diffusion current, IE,p and the base-emitter depletion layer recombination current, Ir,d.
The total collector current is the electron diffusion current, IE,n, minus the base recombination current, Ir,B.
The base current is the sum of the hole diffusion current, IE,p, the base recombination current, Ir,B and the base-emitter depletion layer recombination current, Ir,d.
The transport factor, ?, is defined as the ratio of the collector and emitter current:
Using Kirchoff’s current law and the sign convention shown in Figure 3.2.1(a), we find that the base current equals the difference between the emitter and collector current. The current gain, ?, is defined as the ratio of the collector and base current and equals:
This explains how a bipolar junction transistor can provide current amplification. If the collector current is almost equal to the emitter current, the transport factor, ?, approaches one. The current gain, ?, can therefore become much larger than one.
To facilitate further analysis, we now rewrite the transport factor, ?, as the product of the emitter efficiency, ?E, the base transport factor, ?T, and the depletion layer recombination factor, ?r.
The emitter efficiency, ?E, is defined as the ratio of the electron current in the emitter, IE,n, to the sum of the electron and hole current diffusing across the base-emitter junction, IE,n + IE,p.
The base transport factor, ?T, equals the ratio of the current due to electrons injected in the collector, to the current due to electrons injected in the base.
Recombination in the depletion-region of the base-emitter junction further reduces the current gain, as it increases the emitter current without increasing the collector current. The depletion layer recombination factor, ?r, equals the ratio of the current due to electron and hole diffusion across the base-emitter junction to the total emitter current:
4.2 OPERATIONAL AMPLIFIER:
An operational amplifier (“op-amp”) is a DC-coupled high-gain electronic voltage amplifier with a differential input and, usually, a single-ended output. An op-amp produces an output voltage that is typically hundreds of thousands times larger than the voltage difference between its input terminals.
Operational amplifiers are important building blocks for a wide range of electronic circuits. They had their origins in analog computers where they were used in many linear, non-linear and frequency-dependent circuits. Their popularity in circuit design largely stems from the fact that characteristics of the final op-amp circuits with negative feedback (such as their gain) are set by external components with little dependence on temperature changes and manufacturing variations in the op-amp itself.
Op-amps are among the most widely used electronic devices today, being used in a vast array of consumer, industrial, and scientific devices. Many standard IC op-amps cost only a few cents in moderate production volume; however some integrated or hybrid operational amplifiers with special performance specifications may cost over $100 US in small quantities. Op-amps may be packaged as components, or used as elements of more complex integrated circuits.
The op-amp is one type of differential amplifier. Other types of differential amplifier include the fully differential amplifier (similar to the op-amp, but with two outputs), the instrumentation amplifier (usually built from three op-amps), the isolation amplifier (similar to the instrumentation amplifier, but with tolerance to common-mode voltages that would destroy an ordinary op-amp), and negative feedback amplifier (usually built from one or more op-amps and a resistive feedback network).
4.2.2 Circuit notation:
Circuit diagram symbol for an op-amp
The circuit symbol for an op-amp is shown to the right, where:
• : non-inverting input
• : inverting input
• : output
• : positive power supply
• : negative power supply
The power supply pins ( and ) can be labeled in different ways (See IC power supply pins). Despite different labeling, the function remains the same — to provide additional power for amplification of the signal. Often these pins are left out of the diagram for clarity, and the power configuration is described or assumed from the circuit.
An op-amp without negative feedback (a comparator)
The amplifier’s differential inputs consist of an input and a input, and ideally the op-amp amplifies only the difference in voltage between the two, which is called the differential input voltage. The output voltage of the op-amp is given by the equation,
where is the voltage at the non-inverting terminal, is the voltage at the inverting terminal and AOL is the open-loop gain of the amplifier (the term “open-loop” refers to the absence of a feedback loop from the output to the input).
The magnitude of AOL is typically very large—10,000 or more for integrated circuit op-amps—and therefore even a quite small difference between and drives the amplifier output nearly to the supply voltage. This is called saturation of the amplifier. The magnitude of AOL is not well controlled by the manufacturing process, and so it is impractical to use an operational amplifier as a stand-alone differential amplifier. Without negative feedback, and perhaps with positive feedback for regeneration, an op-amp acts as a comparator. If the inverting input is held at ground (0 V) directly or by a resistor, and the input voltage Vin applied to the non-inverting input is positive, the output will be maximum positive; if Vin is negative, the output will be maximum negative. Since there is no feedback from the output to either input, this is an open loop circuit acting as a comparator. The circuit’s gain is just the GOL of the op-amp.
An op-amp with negative feedback (a non-inverting amplifier)
If predictable operation is desired, negative feedback is used, by applying a portion of the output voltage to the inverting input. The closed loop feedback greatly reduces the gain of the amplifier. If negative feedback is used, the circuit’s overall gain and other parameters become determined more by the feedback network than by the op-amp itself. If the feedback network is made of components with relatively constant, stable values, the unpredictability and inconstancy of the op-amp’s parameters do not seriously affect the circuit’s performance. Typically the op-amp’s very large gain is controlled by negative feedback, which largely determines the magnitude of its output (“closed-loop”) voltage gain in amplifier applications, or the transfer function required (in analog computers). High input impedance at the input terminals and low output impedance at the output terminal(s) are important typical characteristics. Adding negative feedback via the voltage divider Rf,Rg reduces the gain. Equilibrium will be established when Vout is just sufficient to reach around and “pull” the inverting input to the same voltage as Vin. The voltage gain Av of the entire circuit is determined by Rf/Rg. As a simple example, if Vin = 1?V and Rf = Rg, Vout will be 2?V, the amount required to keep V– at 1?V. Because of the feedback provided by Rf,Rg this is a closed loop circuit. Its overall gain Vout / Vin is called the closed-loop gain ACL. Because the feedback is negative, in this case ACL is less than the AOL of the op-amp.
4.2.4 Op-amp characteristics:
An equivalent circuit of an operational amplifier that models some resistive non-ideal parameters.
An ideal op-amp is usually considered to have the following properties, and they are considered to hold for all input voltages:
• Infinite open-loop gain (when doing theoretical analysis, a limit may be taken as open loop gain AOL goes to infinity).
• Infinite voltage range available at the output (vout) (in practice the voltages available from the output are limited by the supply voltages and ). The power supply sources are called rails.
• Infinite bandwidth (i.e., the frequency magnitude response is considered to be flat everywhere with zero phase shift).
• Infinite input impedance (so, in the diagram, , and zero current flows from to ).
• Zero input current (i.e., there is assumed to be no leakage or bias current into the device).
• Zero input offset voltage (i.e., when the input terminals are shorted so that , the output is a virtual ground or vout = 0).
• Infinite slew rate (i.e., the rate of change of the output voltage is unbounded) and power bandwidth (full output voltage and current available at all frequencies).
• Zero output impedance (i.e., Rout = 0, so that output voltage does not vary with output current).
• Zero noise.
• Infinite Common-mode rejection ratio (CMRR).
• Infinite Power supply rejection ratio for both power supply rails.
These ideals can be summarized by the two “golden rules”:
I. The output attempts to do whatever is necessary to make the voltage difference between the inputs zero.
II. The inputs draw no current.
The first rule only applies in the usual case where the op-amp is used in a closed-loop design (negative feedback, where there is a signal path of some sort feeding back from the output to the inverting input). These rules are commonly used as a good first approximation for analyzing or designing op-amp circuits.
In practice, none of these ideals can be perfectly realized, and various shortcomings and compromises have to be accepted. Depending on the parameters of interest, a real op-amp may be modeled to take account of some of the non-infinite or non-zero parameters using equivalent resistors and capacitors in the op-amp model. The designer can then include the effects of these undesirable, but real, effects into the overall performance of the final circuit. Some parameters may turn out to have negligible effect on the final design while others represent actual limitations of the final performance that must be evaluated.
4.2.5 Real op-amps:
Real op-amps differ from the ideal model in various respects.
Real operational amplifiers suffer from several non-ideal effects:
Open-loop gain is infinite in the ideal operational amplifier but finite in real operational amplifiers. Typical devices exhibit open-loop DC gain ranging from 100,000 to over 1 million. So long as the loop gain (i.e., the product of open-loop and feedback gains) is very large, the circuit gain will be determined entirely by the amount of negative feedback (i.e., it will be independent of open-loop gain). In cases where closed-loop gain must be very high, the feedback gain will be very low, and the low feedback gain causes low loop gain; in these cases, the operational amplifier will cease to behave ideally.
Finite input impedances
The differential input impedance of the operational amplifie