System theory

Draft version last modification: 20/02/2016

Proofread: 12/2016-1/2017

Table of contents

1.Introduction51.1.The concept of mechatronics51.2.The place of mechatronics in engineering sciences71.2.1.Physical effects and the division of signals into components121.2.2.Analytical and numerical models131.2.3.Mechatronics, as an approach131.2.4.Classical mechatronics (automated precision machines, robotics)141.2.5.Opto-mechatronics141.2.6.Vehicle-mechatronics141.2.7.HVAC-mechatronics141.2.8.Bio-mechatronics151.2.9.Agro-mechatronics151.3.The aim of system engineering studies151.4.List of symbols and abbreviations18

2.Basic concepts, mathematical description of physical phenomena222.1.Definition of the real physical system222.2.Definition of a signal222.3.Inputs and outputs232.4.Abstract definition of the system252.5.Definitions of linear and non-linear systems262.6.Description of deterministic, stochastic and chaotic systems282.7.Concept of causality292.8.Definition of parameters and variables302.9.Theory of distributed and concentrated parametric description322.9.1.Wave phenomena322.9.2.Concentrated parameter systems described with vector field models332.9.3.Distributed parametrical description of electrical phenomena with Maxwell equations352.9.4.Derivation of Kirchhoff equation (for concentrated parametric models)362.9.5.Steady-state electronic and magnetic circuits modelling two poles382.9.6.Analogies with network calculation422.9.7.Sample problems and solutions for bi- and quadripolars472.10.Deterministic description based on concentrated parameters512.10.1.Concept of time invariant and autonomous systems512.10.2.The concept of static systems532.10.3.The concept of dynamic systems532.10.4.The smoothness of the system562.11.General principles of dynamical systems562.11.1.State, state variable, state equation562.11.2.Discrete state systems and state machines602.11.3.Finite dimensional dynamical systems632.11.4.Controllability and observability662.11.5.Variable structure systems672.11.6.Linear, quantized, single in and output system682.11.7.Linear, continuous, single in and output system702.11.8.Generalized derivative722.12.State-space representation782.12.1.Similarity transformation802.12.2.Canonical (diagonal and pseudo-diagonal) form812.12.3.Controllability and observability matrix832.12.4.Kalmans LTI systems division principle:952.13.Determination of the equations that describe the system982.13.1.Newton-Euler equation for mechanical systems982.13.2.The principle of virtual work982.13.3.Second-order Lagrange-equation for mechanical systems1012.13.4.Graph methods for mixed, mechanical and electrical systems1012.13.5.Bond graph1032.13.6.Example1032.14.Sample problems and solutions connected to the concentrated parametric systems1092.15.Selection of linearly independent equations1402.15.1.The fundementals of graph theory1412.15.2.Description of circuit networks with graphs1442.15.3.Network characteristic matrices1452.15.4.Circuit calculations1472.15.5.Methods to decrease the size a matrix to be inverted1502.15.6.Sample problems and solutions for calculation of current circuits1552.16.Concept of stability1622.16.1.Statical steady state1632.16.2.Asymptotic stability1632.16.3.Ljapunov stability1642.16.4.Dynamical steady state, constant condition1642.16.5.BIBO stability1652.17.Basic test methods1653.Mathematical methods to investigate SISO LTI systems1653.1.Investigation in time domain (Decomposing into components)1663.1.1.Dirac impulse and unit step1663.1.2.Summation of the effect of an input signal decomposed into impulses1723.1.3.Sample problems and solutions for the separation of the time function into components in time-domain1803.2.Analysis in frequency domain and Laplace s-domain1813.2.1.Fourier series1853.2.2.Sample problems and solutions related to Fourier-series1903.2.3.Fourier Transform2063.2.4.Laplace transform2093.2.5.Summary table of Fourier-series expansion, Fourier and Laplace transform 2103.3.Application of the Laplace transform2113.3.1.Summary of identities2113.3.2.Final value theorems2153.3.3.The Laplace transform of some functions2153.3.4.Residuum theory (Inverse Laplace transformation of rational functions with s real coefficients)2163.3.5.Time constant2183.3.6.Sample problems and solutions for the use of the Laplace-transformation2213.4.Solving ordinary linear first-order differential equations with constant coefficients with Laplace-transformation2353.4.1.Examining some exact cases:2373.4.2.Sample problems and solutions regarding ordinary first order differential equations with constant coefficients2403.5.Switch mode2474.Examination of SISO LTI systems in Laplace-operator domain2494.1.Transfer function2494.2.Determining the transfer function with the help of the block diagram2514.2.1.Block diagrams2524.2.2.Notations2534.2.3.Solving differential equations with the analog computerized approach2544.2.4.Serial connection2554.2.5.Parallel connection2564.2.6.Feedback2564.2.7.Transformation of the block diagram2574.2.8.Exercises for the transformation of the block diagram2584.3.Application of the transfer function2654.3.1.Solving ordinary differential equations with constant coefficients using the transfer function2654.3.2.Sample problems and solutions for applying the transfer function2664.4.Relationshop between the poles of the system matrix and the eigenvalues of the system matrix2704.5.The solution of the differential equations of the state-spaces2714.5.1.Some properties of 2744.5.2.Sample problems and solutions for state-space differential equations2755.The visualization of the frequency transfer function2925.1.The connection of the transfer function and frequency transfer function2975.2.Nyquist-diagram2985.3.Bode-diagram(s)3005.4.Transfer function, Nyquist and Bode diagrams of basic terms3025.4.1.Proportional term (P)3025.4.2.Derivative term (D)3035.4.3.Integral term (I)3045.4.4.Proportional Derivative term (PD)3065.4.5.Proportional term with one storage (PT1)3085.4.6.Proportional term with two storage (PT2)3105.5.Time-lags3165.5.1.Proportional time-lags (PH)3165.5.2.Integrator element of time-lag (HI)3175.5.3.First order lag element (PHT1)3185.6.Elements that can be made out of basic elements3205.6.1.Proportional integrator element (PI)3205.6.2.Proportional integrator derivator element (PID)3215.6.3.Proportional derivator element with one storage (Real PD)3225.7.Sample problems and solutions on how to draw Bode and Nyquist plots3245.7.1.MATLAB problems3515.8.Bode plot of different filter types3585.8.1.Low-pass filter3615.8.2.High-pass filter3656.Complex numbers3726.1.Introduction to complex numbers3726.2.Operations with complex numbers3736.3.Representation of the complex numbers3756.4.Some identities related to the complex numbers3766.5.Operations with the trigonometric form3776.6.Geometric illustration of mathematic operations3826.7.Exponential form of complex numbers3846.8.Extraction of roots in the case of complex numbers3866.9.Representation of some complex functions3896.10.Sample problems3926.11.Solutions3947.Ordinary differential equations3977.1.Classification of ordinary differential equations3977.2.The solution of differential equations3987.3.Finding the solution with the trial function method3997.3.1.Solution of nth order linear homogenous differential equations with constant coefficients4007.3.2.Trial function method for nth order linear non-homogenous differential equations with constant coefficients4037.4.Examples4047.5.Technical examples4098.Laplace transformation-related deductions4118.1.Laplace transform of few simple functions4118.1.1.The Laplace transform of a unit step function4118.1.2.The Laplace transform of unit speed-step function.4118.1.3.The Laplace transform of the exponential function4128.2.Important application rules (operations)4128.2.1.LINEARITY rule4128.2.2.OFFSET rule4128.2.3.SIMILARITY rule4138.2.4.DIFFERENTIAL in the time domain4138.2.5.INTEGRATION in the time domain4148.2.6.Final value theorem414

Introduction

The concept of mechatronics

The word mechatronics was originally the trade-mark of the Japanese Yaskawa Electric Cooperation ( Kabushiki-gaisha Yasukawa Denki), which was reported in 1969, and approved in 1971. The company renounced its exclusive ownership in 1982, and the word Mechatronics has been public property since. The word originally applied to highly automated electromechanical (servo) systems. Despite the fact that in Japanese, foreign words are not written in picture-writing, the word-formation follows the rules of Japanese pictography for compound words. Most Japanese words consist of two picture-characters (kanjis). In the combination of two words, the first picture-character of the first word and the second picture of the second word are put together in the new compound word, so the new word will also have only two picture-characters. Thus, in the hungarian language the word mechatronics should be replaced with electronic-mechanics, based on Hungarian rules of compound word formation. The order of the words had to be changed, because mechatronics developed from mechanical engineering, and in this word electronics appear as an adjective and a new feature. Mechatronical aggregates are basically machines, however mechatronics is not only the combination of a mechanism and motive electronics. An impotrant part of a mechatronical system is computational controlling, so we attained to the classic, widespreadly accepted (so called EU)definition of mechatronics.

Definition

Mechatronics is the science of intelligent machines, the synergic (when combined produce a total effect that is greater than the sum of the individual elements) integration of mechanical engineering, electrotechnics, and computational controlling in products by planning and manufacturing. (see Figure 11)

Figure 11. The basic elements of the classical mechatronics

Classically, mechanical engineering deals with the design and the complete structure of a mechatronical device. Electrotechnics provide the power supply and the motion of the device. Computerized control is responsible for signal processing and automated operation. Nowadays the concept of mechatronics has widened, so there are many other definitions of mechatronics. The common parts of these definitions are the high-level integration of functions, and the synergic (when combined produce a total effect that is greater than the sum of the individual elements) effect, with which a higher level of optimization and greater efficiency and quality can be achieved. Customarily, mechatronics is called the science of intelligent machines, it is the subtitle of the Mechatronics journal (see Figure 12v ). It also refers to the fact that mechatronics did not come from nothing, instead it was the direct consequence of the development of mechanical engineering (see Figure 13). The aim is always to create even more useful, smaller, and more intelligent aggregates to make peoples life and work easier. In mechatronics a machine or engineering system is usually the focus of attention, and it has to be provided, integrated, with electronics and informatics (artificial intelligence). That is why mechatronics teaching usually belongs to mechanical engineering, and mechanical engineering faculties.

Figure 12. Mechatronics, the science of intelligent machines

Figure 13. Development of mechanical engineering according to Isermann

The place of mechatronics in engineering sciences

The fundamental difference between mechanical, and mechatronical point of view is the following: Mechanical calculations usually occur offline and focuses mostly on construction. Mechanical engineering fundamentally examines how mechanical systems (also involving heat, and fluid-mechanical systems) react in a given output (deformation, velochity, acceleration, heat flow, etc.) to different inputs and incitements. It is shown in Figure 14, where the system is examined from a construction point of view. Offline calculations for systems result in better accuracy compared to online, or real time, calculations. The revolution of mechanical calculations and finite element methods occured at the end of the 20th century with the appearance of computers.

Figure 14. Mechanical engineering determines the responses of the mechanical system according to design considerations coming from the environment.

Mechatronics deal with intelligent engineering systems which can adapt to a variable environment. During operation the output signal is specified (for example, the displacement, temperature, or any other mechanical parameters) at any given point of the system. These aims are kept in mind in the case of operation. For this, sensors, measurements, signal-processing, artificial intelligence, and intervening actuators are needed in the processes. For the purpose of a more effective functioning, these do not appear as separate units, but are integrated into the aggregate engineering. In many cases, the original boundaries of these compounds cannot even be recognized. Thus, mechatronics contains mechanical construction activities. That is the reason why in mechatronics education strong mechanical foundations are needed, but it is also much more. It deals with the controlling of online function. That is why we can say that Mechatronics is not just mechanical engineering, as shown on Figure 15. Online, real-time, function uses different calculation methods from offline operation. The concept of online calculation functions started in the 21th century using the new equipments with higher capacity and lower price calculating (microcontroller, DSP, FPGA).

Figure 15. According to mechatronicsthe output in the same engineering system is prescribed, , sothe input has to be changed to get the wanted output

As an example, we can mention Kempelen Farkas chess automaton, which probably was operated by a man inside. A beautiful engineering task is to make such a mechanism. As shown in Figure 14 the input signal of the purely mechanical machine is the force (torque) exerted by the operators hand. The output signal is the displacement of the chessman. Somehow the operator could see the move of the chessman of the opponent. . He thought-out the counter-move, his brain gave order to his arm-muscles, so using the mechanism, he could place his chessman to the right square. We must call this automaton intelligent, because there was an intelligent person in the automaton.

If the human being can be replaced the chess automaton becomes a mechatronical system. A sensor is needed that gains information about the movement of the opponents chessman. A computer is also needed that calculates the required moves and defines how and in which order should the spur-wheels of the mechanism be turned for the given move. Finally, there have to be motors with suitable electronics to accomplish the wanted movement. Such an apparatus is rightfully called intelligent.

Figure 16. Illustration of Kempelen Farkas automatic chess

The mechanical system shown in Figure 14. is usually palpable, and it can even be observed with your own eyes (see the interior of Kempelen Farkas chess automation Figure 16). In contrary to this, mechatronics completes this palpable mechanical system with abstract, almost solely electric signals. If we take a look at a mechanical system, we see, for example, the connection of spur-wheels, we can see how the mechanism works. We may even see where and how big forces are exerted. If we took a look into the process-control box shown in Figure 15, we would find integrated circuits. If we took a look into the integrated circuits, we would find transistors, but we would not get much information about the transmitted signals. Generally, physical phenomena examined by mechanical engineers are said to be much more palpable, and can be more easily imagined, than the phenomena examined by electrical engineers. According to this, electrical engineering subjects need much more abstraction than mechanical lectures. In mechanical engineering subjects, non-linear phenomena play a more important role, and they are less capable of being discussed with linear limitations compared to electrical engineering subjects (see Figure 17). A typical example can be the comparison of electric voltage and intensity of current,or the rise of pressure and the flowing water in a pipe-network. The calculation itself in these two cases are almost completely identical. That said, imagining the electric current that goes through an electric bulb fed by an accumulator is not as intuitive as imagining the liquid circulated by a pump in a pipeline. However, linear equations are usually enough for calculating the resistance-network, in the case of pipe-networks, however, nonlinear phenomena have to be taken into consideration even in the simplest cases as well. In the field of mechatronics mechanical and electrical engineering approaches are both essential. The mechanical aspect is formed naturally, but it is important to take care on forming the electrical engineering aspect. Acquisition of the type abstract thinking needed takes a lot of energy from students.

Figure 17. The place of system technology subject in the field of engineering subjects

The following concrete example is trying to cast light upon the difference between mechanical and electrical approaches. Lets imagine a robot. When a mechanical engineer is planning a robotic arm, he wants to know the exerted forces and the possible fracture sites in a given operating position. So he wants to know where the mechanism has to be strengthened. Or, on the other hand, which parts are stressed less, which parts can be lightened, and where he can save material. In such a calculation the aim is to be more precise, and the computing time is unimportant. It is acceptable if the calculation for a given operating position even lasts for hours. When the robot is being controlled, the purpose is for example to make the robots holder move in a linear path.The position of the robots holder is measured in every milisecond, and according to the result the torque of the motive motors has to be planned. Ideally, we can calculate the required torque for each individual motive motor, so that the robot gets to the precisely given position at the next sampling point of time. The problem is that the calculation takes longer, than the sampling unit of time. In an absurd case, after one hour of counting, we precisely know what we would have to do one hour ago. During this time the arm of the robot might have caused damage in its environment, because of the long wait for the outcome of the calculation. Because we do not know a lot of things precisely, the accuracy of the calculations is restricted. For controlling the arm of the robot different equations and different viewpoints are needed from the equations that were used by planning the mechanics of the robot-arm. If we make measurements every milisecond, the command has to be given in the ten thousandth second after the measurement. It means suitable voltage has to be connected to the motors that move the arm of the robot, so the motors exert the necessary torque on the given parts of the arm, thus moving it in the right way. In this case speed is more important than precision. We compensate for the lack of precision by constantly checking the robotarm. The movement and the position of the arm is constantly corrected.

Physical effects and the division of signals into components

Engineering problems are often simpified by dividing the quantity yet to be determined into components. Division of forces into components and using a perpendicular coordinate system is known from high school. It can be imagined easily (see Figure 18). The signals seen in Figure 15 can be divided into components as well. In our further studies we are going to see that time functions can be divided into components in frequency and in Laplace operator range. For this in the infinite dimensional Hilbert-space an ortonormated basis has to be selected and the time function has to be projected using a general scalar multiplication to the basis. While the infinite-dimensional space of real functions is hard to imagine, the complex space of complex functions is basically impossible to imagine. This means that to handle these concepts, an abstraction is required. This can hardly rely on every-day experience and it is based on mathematical knowledge that we could not get in the high school. To become a mechatronic engineer the most important step is to acquire the type of abstract thinking needed for the use of frequency and Laplace operators.

Figure 18. Resolution of forces into components

Analytical and numerical models

In the engineering sciences a radical change of attitude can be perceived Previously, it was believed that the exact solution for a problem is a closed mathematical solution. The advantage of this method is that the impact of each parameter or external effect on the whole system can be predicted from the closed formula. The disadvantage of the closed-form solution is that it requires a lot of simplifications and ignoring parts of the problem. Nowadays numerical methods are becoming more prevalent. Phenomena that would create an unmanageably large analytical model are modelable using numerical models. The verification of the functionality of a system with numeric simulation is thus becoming more accepted. Naturally designing a new (complicated) product starts with a simulation model. Only if the simulation results are satisfactory, can we start thinking of creating the product. The disadvantage of the numeric model, is that every significant operating status has to be calculated, we can only know about the calculated operating states, and before the creation of a simulation model, a system engineering analysis has to be done. Basically, the analytical model is usually simplified, but reveals deeper correlations. The numerical model is more detailed and realistic but finding deeper correlations is more complicated.

Mechatronics as an approach

Nowadays, it is clear that mechatronics should be viewed as an up-to-date engineering approach rather than an individual science. Mechatronics is the integration of at least three fields of studies. The classic automated precision machine meaning of mechatronics has significantly broadened in the last 40 years, new branches of mechatronics have appeared. Including but not limited to some of the most important branches of mechatronics:

Classical mechatronics (automated precision machines, robotics)

The classical ways of mechatronics dont lose their significance, but automation and robotics gain more importance, which enables many new interesting researches. Robotics used to be only remunerative in mass production. Simplifying robot programming is indispensable to get them placed in medium-sized and small enterprises. It is fundamentally going to present many new challenges, if robots will show up in healthcare and our homes.

Opto-mechatronics

The new fully automatic camera, video recorder, CD player, 2D and 3D scanners belong to this area. The importance of these is continuously growing. Lasers are said to be in the 21st century similar to what electricity was in the 20th century. The production of lasers is the domain of physicists, but almost everything after production requires opto-mechatronic knowledge.

Vehicle-mechatronics

Both in volume and innovation, the automotive industry must be called a key sector of the countrys economy. Classically the production of vehicles initiated the use of industrial robots in a wide range of applications, but nowadays the vehicles themselves have to be called mechatronics products. In a luxury car beside the primary internal combustion engine there can be more than hundred electric motors for various comfort-related tasks. Optimal operation is ensured by an on-board computer. The gas and brake pedals probably only send signals, sometimes even activate complicated ABS, ASR, ESP etc. systems. Usually the active security systems of vehicles belong to the field of mechatronics. Soon, the steering wheel wont be in direct mechanical connection with the controlled wheels.

HVAC-mechatronics

Automation and intelligent operation appear in buildings as well. The computer controlled ventilation, heat exchanger and heat pump systems are important elements of HVAC-mechatronics. The definition of HVAC-mechatronics is the unification of building automation, management and security. Academic and practical researches in this topic cover the automation of HVAC-engineering systems, supporting of building automation and the integration of building surveillance and security technology. If the building individually (without human help), or with a minimal human intervention can perform its tasks, it can be called an intelligent building.

Bio-mechatronics

Bio mechatronics combines two independently dynamic growing areas and can be further divided into more parts.

One of these parts aims to serve as a connection between engineers and doctors as an interdisciplinary area, supplementing engineering science with academic and practical medical knowledge. Mechatronic engineers can use their wide mechatronic knowledge supplemented with medical, natural science, and medical materials studies to design and research projects either in engineering, or in a medical workplace.

Biologically inspired systems also have become a new part of mechatronics. Solutions originating from nature can be put into practice during engineering design. For example, the woodpecker is naturally protected against the concussion in a really interesting way. This is the basis of one of the best vibration damper systems. If, for example, a robot has to be integrated into human society, it has to use methods taken from biological examples to know how to act properly.

Agro-mechatronics

The demand for automation in agriculture is strong as well. Currently, in this area the use of manual labour is most common, because compared to an industrial environment, in agriculture labour needs to be highly adaptible. In an assembly plant, parts arrive with exact timing and in a well-defined position on the assembly line, therefore it is easy to automate. At the same time, for example, a fruit-picker robot needs to detect the fruits with variable position (or obstructed with leaves), with variable size, after which the path to reach the fruit needs to be planned. The effectiveness of spraying can be significantly increased if the nozzle detects adaptively where the vegetation is denser of rarer. This area is expected to change siginificantly, more developments will be applied. Here, Mechatronics can become an important field.

The aim of systems engineering studies

Systems engineering belongs to engineering studies, so it has engineering vocabulary, analogous to C++ as a programming language. Exact mathematics is analogous to binary computer code. At its base, the computer only understands computer code, but it is really difficult programming in it and only very few fanatics can really do it. Similarly, only a fraction of humanity can acquire knowledge of real mathematics. It was like this previously and probably will be the same in the future. Most computer users work with high-level programming languages and keep away themselves away from the base mathematics. The sharp separation of the machine code and high-level programming languages has worked well, but a programming language that includes all levels of programming was required, so C++ was created. The object orientation (the ++) is the highest level, while the programming language C enables machine-close programming. This is not a machine code but its close to it. Similarly systems engineering gives us a higher-level engineering approach and was not written using only mathematical language, but certain parts are close to it.

During industrialization the specialization of professions was continuously growing and the age of polymaths is said to be over. Engineering was also divided into many areas but nowadays integrated knowledge gains more and more importance (see mechatronics). Complex systems have to be understood in full. Systems engineering studies support this way of thinking. Mathematics has to be used more routinely (maybe more than it is possible), so a close-mathematical engineering language is needed.

The world is constantly changing. There was a time, when mass-production reduced the engineers need for mathematics. In the 21st century mathematical knowledge is appreciated once more. High-tech industries need a strong knowledge of mathematics. (airplanes, robot shape recognition etc.). In order to improve upon the currently strange movement of humanoid robots to, for example, teach them ballet, the mathematics in control algorithms have to be deepened. The development of mathematical algorithms is needed for a robot to recognize its environment. Who would think that, for example, there is a pulling magnet in elevators for quiet and quick functioning (its purpose is to hold the cabin while the elevator is waiting on one of the floors), which usually uses a microprocessor that helps optimal functioning by continuously calculating the magnets differential equation. For safety reasons the door can be opened only with a immobile, and fixed, elevator cabin. In the past, a coil was set up and the magnetic field generated by electricity was how it worked. Of course this meant that, a crashing sound could be heard when the metal fasteners that held the cabin bumped into the supporting frame. If a company wanted to advertise itself as having the most inaudible and fastest elevator they had to use the switching of transistors to continuously control and manage the electric current, and by extension the motion of the magnetic fasteners, to ensure that the functioning of the clamp-magnets would be quick, but the cabin itself would slow down sufficiently to avoid an audible collision between the supporting frame and the fasteners. For this to happen, however, they had to continuously calculate the differential equations of the system itself. This wouldnt be a problem if all parameters were known but there is always an uncertainty, and several parameters change during operation. This meant that the parameters of the differential equation must be estimated from measurements. This means that the seemingly easy task became a really difficult mathematical problem. The price of electronics and microprocessors have fallen so much, that very difficult mathematical algorithms can appear in cheap (mass produced), everyday products. There is now a need for differential equations in everyday engineering practice. Mathematics is now needed not only for elite engineers. There must be a connection between mathematics and engineering sciences, and the most important element of this connection is systems engineering.

The aim of systems engineering studies is to describe physical phenomena and working of technical objects with an unified mathematical toolkit in order to reveal the similarities between completely different physical phenomena and the working of technical objects. According to their functional similarities phenomena and technical objects can be categorised independently of their physical appearance. The advantage is that for completely different physical phenomena and technical object which belong to the same category (even during passive analysis or active regulation), a general mathematical toolkit can be used. In a technical subarea, for example in calculation of electric circuits, a new mathematical procedure is often developed. This method can be used in a different area as well (for example calculation of magnetic circuits, pipe systems, fluid flow, or thermodynamic problems). Impedance is usually used during calculations in relation with electric circuits, but it is used in the force regulation of robots as well. These examples show the importance of systems engineering. In your studies such mathematical techniques are introduced that can help in the solution of certain types of problems in engineering practice.

Because of this, the structure of the curriculum is the following. At first we review the various mathematical description methods of physical phenomena that appear in engineering practice. After which comes the mathematical toolkit that can be used for the analysis of abstract models, and with which its general connections can be revealed. One of the key parts of engineering is to find the connection between physical reality and abstract mathematics. The curriculum focuses on revealing this connection from both points of view. For example, if there is an opportunity in a mathematical equation to simplify it, what the physical background of that might be. Or, what the mathematical consequences are if the physical model changes. Our goal is to provide useful mathematical tools for the engineering practice, the curriculum gives the approriate deepness of mathematics for the right use of these tools, and understanding the conditions of and limits of their application. Like a laser telemeters terms of use doesnt explain the whole physics background of the laser, we cant cover all of the mathematics concerning this material in detail from its bases, because of the sheer volume of it (e.g.: linear algebra, real and complex functions, common and partial differential equations, functional analysis, gauge theory, distribution theory, graph theory, etc.). If we tried to, this book could be more than thousand pages which would be impractical. We assume that the reader has full mathematical knowledge taught in engineering bachelor studies. The higher mathematical knowledge will be presented in exact and simplified form with the important parts emphasized, without detailed argumentation.

Systems engineering studies belong to engineering, but are otherwise really close to mathematical studies. This demands more abstraction than most of the other engineering courses. In a certain sense they are closer to electrical engineering, because systems engineering requires more abstraction than mechanical engineering courses.

List of Symbols and Abbreviations

In different disciplines different conventions exist regarding sysmbols and abbreviations, this curriculum covers numerous fields and it is almost impossible to pay attention to all, sometimes contradictory conventions, so a letter in different sections could mean different things. We try to resolve this contradiction using indexes where it is possible. We try to keep the following convention: variables in lower case, constants in capital letters, italics for scalar and vector quantities are indicated in bold. Exceptions are only in cases where the consensus in literature more or less uniformly differs from our convention. In numerical examples our results arise from computer calculations. These contain decimalpoints instead of comma.

In abbreviation the original English word is always used. These are administered like Hungarian loanwords with their own meanings, so these can be used following the Hungarian pronunciation and inflection rules.

matrix of a system

system matrix in the state space representation

adjacency matrix in the graph theory

the amplitude of the ith sinusoidal harmonic in the Fourier series (without phase shift)

coefficient of a polynomial

magnetic flux density vector

matrix of a system

input matrix in the state space representation

incidence matrix in the graph theory

the amplitude of the ith cosinusoidal harmonic in the Fourier series (without phase shift)

coefficient of a polynomial

matrix of a system

output matrix in the state space representation

the amplitude of the ith cosinusoidale harmonic in the Fourier series (with phase shift)

displacement vector

matrix of a system

feedthrough matrix in the state space representation

the disturbance signal of a system in continuous time

in subscript: indicates a system with discrete time

electric field vector

a scalar time function in continuous time

left limit of function in the case of

right limit of function in the case of

a scalar time function in continuous time with a target set of

the value of a scalar time function in continuous time at the time instant

Fourier transform of a a scalar time function in continuous time

the value of the Fourier transform of a a scalar time function in continuous time at frequency

Laplace transform of a a scalar time function in continuous time

the value of the Laplace transform of a a scalar time function in continuous time at

a scalar time function in discrete time

a scalar time function in discrete time with a target set of

the value of a scalar time function in discrete time at the Kth step

frequency

magnetic field strength vector

i

in subscript and without subscript: running number,

with subscript: electric current

jimaginary unit

discrete time steps

set of natural numbers

ith pole of a rational transfer function

set of real numbers

complex argument of the Laplace transformed function

ttime

T0starting time of the analysis, usually T0=0

TKKth time step

Tstime step of sampling

Thtime delay

Tpiinverse of the breakpoint caused by the ith pole of a rational transfer function

Tziinverse of the breakpoint caused by the ith zero of a rational transfer function

without subscript or with a numeral subscript: control signal of a system in continuous time

without subscript or with a numeral subscript: control signal of a system in discrete time

with a subscript not indicating a numeral: voltage

step response of a system

impulse response of a system in continuous time

impulse response of a system in discrete time

state variable of a system in continuous time

state variable of a system in discrete time

X0initial value of a state variable (before switching the system on); for example, in the case of a Dirac delta input function the state variable can be noncontinuous at the time instant of the switch in this case X0 is the left-hand limit of the state variable at the time instant of the switch

output of a system in continuous time

output of a system in discrete time

ith zero of a rational transfer function

set of integers

set of positive integers

set of negative integers

a variable with a value of 0 or 1

Dirac delta function in discrete time

Dirac delta function in continuous time

permittivity

step function in discrete time

step function in continuous time

eigenvalue of a matrix

permeability

angular frequency

the time function of the angular velocity of a motor in continuous time

ARMAAutoregressive Moving Average (system)

BIBOBounded Input Bounded Output (system)

EMFElectromotive Force

FIRFinite Impulse Response

IIRInfinite Impulse Response

MIMO Multiple Input Multiple Output (system)

MISO Multiple Input Single Output (system)

LPVLinear Parameter Varying (system)

LTILinear Time Invariant (system)

LTVLinear Time Varying (system)

SIMOSingle Input Multiple Output (system)

SISOSingle Input Single Output (system)

1. Basic concepts, mathematical description of physical phenomena

1.1. Definition of the real physical system

Definition

The real physical system is such a physical object that changes in a measurable way in response to a measurable external factor.

Interpretation

In many cases we can have a technical problem where there is some external force and this results the alteration of something. The "real physical system" concept is introduced to simplify the latter terminology: it is a physical object that changes in a measurable way in response to a measurable external factor (see in Figure 21). This curriculum focuses on describing real physical systems mathematically.

Figure 21. Real physical system

Real physical systems in technical practice can be classified according to several criteria and a variety of mathematical descriptions of a real physical system can exist. The appropriate description method depends not only on the real physical system itself, it also depends on the object of study as well. For example, it is not equal whether we are interested in the process of changes (transient behaviour) of the real physical system or just in the final condition (steady state). In the case of these two different investigation of the same real physical system, different mathematical tools can be used. Therefore it is very important in engineering effort to define exactly the subject and first purpose of the investigation and only then to select a suitable mathematical tool for analyzing and solving the problem.

1.2. Definition of a signal

Definition

The signal is the abstract information content of a varying physical quantity.

Interpretation

The signal is the abstract information content of a varying physical quantity, typically a time function. Later as we will see in Section 2.11.2 in the interpretation of certain signals the concept of functions generalization must be allowed. It follows that the signal loses its original unit of physical quantity. The absolute size could be lost as well, if a per unit system was used. Signals are usually classified. Getting a relative unit the signal has to be shared with a specific value (usually with nominal value). So the value could be expressed using percentages. Systems with same funcitons but different physical capacities would become comparable. If I have two motors and I switch 100V voltage to them this shows nothing about its conditions. (One of them could break down because of overvoltage, and the other might not start because of insufficient voltage.) Excess capacities of the two motors could be compared if we know the value that they bear compared to their nominal values. Using relative units is beneficial for engineering practice.

Values are categorized from different viewpoints.This tutorial rather obviously focuses on systems, therefore the classification of signals is carried out in connection with systems classification.

1.3. Inputs and outputs

Definition

Physical inputs constraints that are time-varying and affect the real physical systems.

Real physical systems changes according to the physical constraints could be physical outputs, from these changes physical output is measured during examination.

Interpretation

Physical inputs are such external forces which are affected to a real physical system and time-varying. Several types of physical input can be differeintated: directorial input (artificially changeable) and input defined by the environment (because of this condition does not known most of the time). The signal of the input is called commonly excitation. According to this classification two types of excitation can be differentiated: control signal and interfering signal. We do not have any informations from the interfering signal in most cases, but in real physical systems it is appropriate always to calculate with this type of excitation. Interfering signals can be divided in two large groups: noise and load natured signals. If the load is known it could be handled as an input. The symbol of the control signal is u and the interfering signal is d (from disturbance).

Real physical systems changes according to the physical inputs could be physical outputs, from these changes physical output is that is measured during examination. The output signal belongs to a physical quantity is called commonly response signal, its usual symbol is y. An important notice is that there arent any limitations or advice in this definition for the selection of the measurable physical quantity. We will have to deal with this problem seperately later: Can the function of the real physical system be reconstruated? In other words, does the measurement contain the information we are looking for?

The first step of the examination of the real physical system is to determine the inputs and outputs. According to the usage or the subject the system could have different inputs and outputs. See the following basic example: Our bike can turn using the handlebars. There are some differences in use. If we take the handlebars wide, we have to expose less power.By low speed it is surely enough to concentrate only on the handlebar, but it is not enough at higher velocity; we must also change the center of gravity. There are some people who can turn without grasping the handlebar. The input constraint is in their case the displacement of the center of gravity and the turning of the handlebar is a physical output quantity. Anyone who has used bicycle before, could have experienced that a scree or the roadside might modify the turning direction of the vehicle. So we can consider the roughness of the road to be an input signal (or interfering signal). The examined physical quantities (constrains) could be different if the displacement of the bicycle is being examined

DC servo motor is commonly used in mechatronics for actuation. It will come up in several examples in our curriculum, Because of its simplicity and common industrial application the DC motor is often applied in the examples of this curriculum. Examining the motor as a real physical system it is clear that we can control the operation of the motor trough the armature roll, but it is equally obvious that the motor is controlled with amperage or with voltage. The amperage depends on the regulated quantity in the case of amperage control and the armature voltage depends on the speed of the motor. If the voltage is the regulated quantity the amperage will depend on the load of the motor. The outgoing quantity could be also the torque, the speed or the change in the angle of the motor. These quantities are clearly not independent from each other. For example if a change of the angle is given then the angular velocity can be calculated and reserved.

The above examples illustrate that choosing the inputs and outputs is not always trivial, so the first step should be always an accurate analysis of the problem.

1.4. Abstract definition of the system

Definition

An abstract system is an abstract model of a real physical system that is valid on specified operating range, which keeps contact between the input and output signals.

Interpretation

The system is an abstract model of an existing physical system, which has a certain accuracy and a determined operation range and keeps connections between the input and output signals (see Figure 22). It is important to notice that an input signal can be chosen in such a way that could destroy the real physical system, or can mathematically calculate an output signal which could not be produced by the real physical system. It is important to make sure of the valid operation range of the system.

Control signal and interferential signal are distinguished in Figure 22 according to the previous chapter. Interferential signal often can be ignored. If it is not mentioned separately the input signal is equal to the control signal. There could be further signals inside the system. If the following expressions are used without the physical word in the next chapters, input signal, output signal and system, that means we are using the abstract, mathematical, meaning of those expressions.

In engineering real physical systems have to be created, which means we must know how they function, as changes in these systems must be describable. We can partially infer the working of physical processes from prior knowledge (so-called engineers intuition), and partially from the equations describing these processes. Mathematical equations alone are never enough to describe the world around us. Engineering experience, intuition is always needed. The beauty of engineering lies in the synergy of these two methods. Systems engineering only deals with the mathematical approach.

In the following sections the mathematical connections between the input and output signals will be examined.

Figure 22. The general graphical symbol of the system

Systems can be commonly grouped by the number of inputs and outputs(see Figure 23). Single Input Single Output systems (SISO) are used in the classical systems control theory. Modern control theory deals with the Multiple Input Multiple Output systems. Of course, there are existing Single Input Multiple Output (SIMO) and Multiple Input Single Output (MISO) systems as well.

Figure 23.Classification of the systems according to the number of inputs and outputs

1.5. Definitions of linear and non-linear systems

Definition

The most important property of linear systems is that the superposition principle is valid for them.

Interpretation

The validity of the superposition principle is one of the most important properties that has to be resolved about any examined system. Linear systems can be used much easier than nonlinear systems. This is because, by definition, linear systems can be written by linear equations (algebraic, ordinary and partial differential equations). The most important property of linear systems is the possibility of superposition of the input signals (see Figure 24).This property is used if linearity is not surely known about a system. For example, when an input voltage is connected to the resistance and the amperage is measured, if the quantity of the connected voltage cut in half or reduplicated and the quantity of amperage changes in the same way, then the system is linear in the examined operating range. Similarly if a spring is loaded and the force changes the strain of the spring will change in the same way. A system is often linear only in a given operating range (the principle of superposition is valid only in this given operating range). As an example, if the spring is totally squeezed or stretched change of the force does not cause more deformation.

It is easily proven that if a system is made of two parallel or serial linear systems the resultant system remains linear. This property is often utilized.

Figure 24.Definition of supeposition in the case of a SISO system

Systems that cannot be written with linear equations are not linear. Non-linear systems could be sorted depending on the non-linear properties of the system. In our further studies we are going to deal mostly with linear systems.

In the case of MIMO systems the input and output signals (Figure 24) generally are thought of vectorially. According to the test method excitation is connected only to one input and the response is examined. Then excitation is connected to the other input and the response of the system is examined as well. Finally the two inputs are used at the same time. If the system is linear, the complete response is equal to the sum of the separate responses.

Real physical systems surely have non-linear properties; think about how a big physical input signal can destroy the real physical system. Naturally, in that case the linearity is going to cease. So the linearity is valid for a certain operation range and certain accuracy criteria. In another approach, if a non-linear, real physical system has an operation range about an operational point, where the system can be considered to a linear system. See the operational point equation that keeps contact between the input of the operational point and the output of the operational point:

(2.1)

Modify the input signal around the operational point:

(2.2)

(2.3)

The system is linear in the given operational point, if:

(2.4)

In many cases the system is not linear around the operational point but it is instead approximated linearly. This is called operational point linearization and is readily used. There are some other opportunities for linearization. In the case of mechanical systems balance point is found and linearization is around this point. In vibration analysis there are a lot of similar solutions.

There are multiple reasons of non-linear behavior: saturation and hysteresis in the case of magnetic circles; friction, wobbling and limited moving at mechanical systems; switching mode at electronic; offset in electrical circuits or any irreversible change.

1.6. Description of deterministic, stochastic and chaotic systems

Definition

In a deterministic system in the case of a concrete input signal the concrete output signal can be calculated analytically.

Definition

The response of a stochastic system cannot be exactly determined only the probability distribution of it. The system contains probabilistic elements.

Definition

Chaotic systems are deterministic systems with a nonlinear dynamic. Long-term behaviour of these systems can only be described statistically.

Interpretation

It is important to note, that often the deterministic and stochastic nature of a system is not an important question, but instead what is important, is how we describe the system. Sometimes a deterministic process is very difficult to describe, and is instead easier to described stochastically. For example, dice are often used to generate random numbers. If the dice is geometrically exact , with homogenous raw material, then the dice can stop on either side with equal chance when thrown. It is a typical stochastic process. Children who are trying to manipulate the throw often turn up the six side in their hands. If the dice has to roll during the throw this method is usually not efficient. The force for a full turn-round by throwing cannot be exactly controlled by their hands. This process is stochastic because of the inaccuracy of the hand. But if we use a robot with exact information about throwing (start status, input signal, system equations) the process will be deterministic. In this case we have theoretical chance to calculate the result of the throw. If we examine a ball on a huge flat surface, the change of the initial status changes the final status in the same way. It is not so easy to predict changes in the result of throwing dice, because a small alteration of the initial status might not cause changes in the final status. But in linear systems, because of the principle of superposition changes of the initial signals will be proportional to the changes of output signals. If the alteration of initial values causes huge alteration in the later value of the signal, the system can become chaotic. A typical example of a chaotic system is weather. We can always measure the initial conditions more and more exactly, and do more and more complex calculations, to make more and more accurate short-term predictions (from which the base deterministic property of weather can be seen), but the longer in the future we try to predict, the less accurate our prediction will be.

In the case of real physical systems measurements are needed. In all measurements there will always be some kind of noise. Noises and their effects can be handled as stochastic systems. So in many cases stochastic and deterministic signals appear together. Dissociation of these can be a very important task. The dissociation method can be the filtration of the signal. The filter can be handled as a system with mathematical function, where input is the noisy signal and output is the filtered signal. Many times the abstract mathematical model is fabricated before the real physical system in engineering practice. Therefore there is a need for the clarification of a new, important concept. We have to deal with deterministic chaotic systems that, is some aspects, seem to be stochastic.

1.7. Concept of causality

Definition

A system can be called causal if the output signal of the system in any T time is independent from the excitations/input signals after the given T time. So future events do not affect the present.

Interpretation

Naturally every real physical system and every signal processing algorithm is causal, they work parallel with the measurement. In the case of a follow-up signal prcessing algorithm (performed on experimental data) causality is not absolutely necessary. These algorithms are used in the past where the examined future is known. As an example, after filtering sinus signals with different frequency and amplitudes with an ideal low-pass filter, and after zero crossing, the signal sharply increases. This can happen because of two reasons: the signal can have low frequency and large amplitude or large frequency and low amplitude. In the first case the signal must be conceded, in the second case signal has to be repressed. At the zero crossing it is not easy to decide further actions, but knowing the whole signal it is easy to find out what to do with the signal.

Social systems can be anticipative, f.e. stock exchange processes are influenced by the knowledge of future circumstances.

1.8. Definition of parameters and variables

Definition

The coefficients of equations that describe the real physical systems are the parameters.

Interpretation

It has to be said in the first approximation that a parameter is a constant in a mathematical description of a real physical system, and a variable is time dependent quantity in that system. An example for parameter is the resistance, for variable is the current.

(2.5)

Many people insist for this strict differentiation, but there are cases of time-variant parameters in publications. Here, parameter are used as coefficients of the equation of a real physical system. A system with changing parameters means that the coefficients of the equation system are also changing.

There is often a consensus-based parametric description of a real physical system, that can be used with a given accuracy in a given operation range (see Ohm law eq.(2.5)). If the accuracy or the operation range is increased, the parameter, which was constant in the simpler description, will also have to change in the more accurate description. For example the resistant could change depending on the temperature of the electric circuit. Suppose that Eq. (2.5) descripts a voltage on roll of rotor of an asynchronous motor. That resistance could increase even 50% higher on operation temperature compare to the temperature of the room. So if we would like to have an accurate calculation we should introduce a variable. The temperature and value of can be calculated by a complicated thermodynamic function f. f depends on i(t) amperage, outer temperature, angular velocity of the motor and some other circumstances.

(2.6)

A large amount of effort is thus needed to determine temperature, especially if we consider to the distribution of the temperature that is not homogenous within the roll. However, Eq (2.5) is built into a longer equation describing the work of the motor. Usually we try to use linear systems. If we calculate with Eq (2.6) the equation of the motor loses its linearity. This system can be used by controlling of the motor. For example the actual amperage and speed of the motor are measured, then the terminal voltage can be calculated from the previous measurement data and the equation describing the motor to reach the desired operation status. If the system is complicated the calculation time is longer, and measurement data can change by the end of the calculation, making our end-result outdated. For example, if this motor is used in a car, it might have already hit a wall before it starts breaking. Of course, the growing capacity of computer devices helps us to calculate more complicated models in the same time, but there will be always a limit on the complexity of the system. This means that it is also possible that the calculation is not accurate enough in the given time of the measurement.

A compromise is to assume a typical operation and in offline mode the change of the R resistance is calculated. Then it is handled by a time-varying parameter.

(2.7)

The accuracy is increased compare to Eq(2.5), but linearity is not lost and computational capacity is not increased too much compared to Eq. (2.7). But the result is acceptable only if the operation doesnt deviate from typical operation.

Another advantage of the temporally changing parameter that the linearity of the system can be preserved. Eq (2.6) is a more accurate and always valid equation, but it is non-linear. (2.7) is only valid in typical operation, but it is a linear model.

1.9. Theory of distributed and concentrated parametric description

Definition

In the case of a distributed parametric description mathematical correlations are determined in all points of the space, mostly with partial differential equations.

In the case of concentrated parmetric description correlations in a given area, the examined real physical system should be summerized or averaged and, in that area, substituted with one equation.

Interpretation

This may be the first and the biggest detour, but we could not discuss it until the term of parameter was not made clear. distribution here refers to the spatial distribution. A certain mathematical relation is determined in every point of the space using a distributed parametric description. Typically wave phenomena (even mechanical or electromagnetic), heat or inner material stress distributions belong here. These equations are usually solved using the finite elements method. This topic belongs to a different subject.

Concentrated parametric description is applied if relations of the examined real physical system could be calculated by average spatial distribution and is replaceable with one equation. This simplification is very common in engineering solutions. F. e. if a rod is used as holder it can be characterized as a rigid body. If it is used as tuning fork all of its points have to be calculated separately for the wave analysis. Distributed parametes model must be used just like in the case of calculating the exact breakpoint of the rod when sufficient force is applied.

We use only concentrated parametric descriptions after this point.. Sometimes distributed parametric descriptions can also be used in some special case. The reason for a concentrated parametric description can be the short calculation time. For example, when designing a robot we would like to know only the biggest static and dynamic loads, to determine the highest stresses. In this case a distributed parametric description has to be used. For a 10 minute dynamic moving period, it may take over an hour to calculate the corresponding forces. In the design process it is allowable. Conversely, when controlling a robot we measure something and intervention has to occur within the shortest time possible, to reach the desired moving path. Calculating for too long to get the former type of intervention is not enough. Thus, sometimes the less accurate model has to be used, when calculation time is of critical importance.

1.9.1. Wave phenomena

1.9.2. Concentrated parameter systems described with vector field models

In this part the structural similarity of concentrated parameter systems or, to be more exact, network calculation methods are examined. Concentrated parametric description means any kind of approximations or simplifications. The approximations when calculating electrical circuits are minor. (even then, mostly in the case of low frequency excitations). This might be why the calculations concerning electrical circuits are extremely detailed, which means that in describing other physical phenomena often an analogy to electrical circuits is used. In this paper we deal with the details of calculating electrical circuits and these results will be generalized.

Many physical phenomena can be concisely described using vector field divergence and rotation. From these equations concentrated parameter models can be formed and from these physical variables can be derivated, networks can be created.

Definition

Physical quantities that refer to the whole system are called extensive physical quantities. Physical characteristics of the whole system are equal to the sum of the physical characteristics of the subsystems if the unification of subsystems accurately represents the entire system without overlap. Conservation laws can usually be used in these cases, but, for example, entropy is also an extensive physical quantity but it does not a law of conservation. Often, extensive physical quantities can be derived from the divergence of a vector field. Extensive physical quantites are usually different types of energy and quantities of materials. Volume (hydrostatics), impulse and impulse momentum also belong in this group.

Interpretation

Extensive quantities usually cant be interpreted locally (in one point). They can be used only to describe a finite size and distributed parametric system. In the case of distributed parametric models, values need to be calculated by summation. In distributed parametric models the conservation laws can be described as divergence equations. These express the flow of currents of different kinds of materials and energies in a given point of space. In this way, a scalar value is assigned to all points of the space with respect to the values of the sources. The differential form of the conservation law is often called as continuity equation. A usual approach is that in the case of the unification of two parts of the space the value characterized by the extensive physical variable is the summary of the physical variable values of the two spaceparts. Basically, the extensive physical variables are additive (in the case of linearity) by the unification of two spaceparts. The density of the source help us to calculate the summarized sources in a given part of the space. This value refers to the given moment, thus a new type of physical variable has to be defined.

Definition

The given rate of the extensive physical quantities is called the measurable change of a given physical quantity at a given surface over a unit of time.

Interpretation

During the description of the phenomena we are interested in the changes, more precisely, in the flows. E.g. charge flow, volume flow, heat flow.

There are several reasons for the change of the extensive physical quantities:

Presence of a source or a sink

External impulsion

A vortical vector field

A different physical quantity can be found whose gradient causes the flow of the extensive physical quantities.

Definition

Intensive physical quantities are local, varying from place to place, physical quantities. According to the unification of the volume, they are not additive. Many times these can be originated from the nature of the vector field. Their inhomogenousity will start an equalization process. Typical intensive physical quantites are the pressure, electrical voltage and temperature.

Interpretation

Rotation refers to the vortex of the vector field, which means that it pertains to a possible reason for flow. A vector can be assigned to all points of a space, which can be handled as a generalized force, or the effect of this force to a unit extensive quantity in the given point. When the spaceparts are unified, the intensive physical quantities average out. This is the weighted average of the unified parts of the space. The intensive physical variables are additive from the source of the vortices. When a new source is created, the effects of the two sources will be summarized, along with their respective intensive physical variables as well. The analogical quantities are summarized in the first part of the table 2.2.

The concept of extensive and intensive physical quantities is linked strongly to the unification of two juxtaposed volumes, in which case the extensive physical quantities are added together, and the extensive physical quantities average out. The abstract fields can not only be juxtaposed, but also interlocked (imagine this is often not an easy task). Unification of an abstract field can be realized with the equalization of physical quantities and summarization of intensive physical quantities. A typical example can be binding the circuit elements in series or in parallel. By parallel binding the currents are summarized and the voltage is fixed, in the other case the currents are permanent and the voltages summarized. Later in the table 2.2 the intechangeability of roles of physical quantities will be visible.

1.9.3. Distributed parametrical description of electrical phenomena with Maxwell equations

(not recommended for first-grade students)

Here the electromagnetic field is presented as a distributed parametric model. The equations(2.8)(2.8), (2.9), (2.10) and (2.11) are usually called the Maxwell equations.

(2.8)

where H is the magnetic field strength and J is the current density vector, and D is the electric displacement vector. ( is the so-called electric current). What this means: The magnetic field is always vortical, a source of this is some kind of electric phenomena: displacement current or electric current.

(2.9)

where E is the electric field and B is the magnetic field vector

(2.10)

(2.11)

where is the charge density.

Assuming a lineal, isotropic, or homogenous medium

(2.12)

(2.13)

(2.14)

where is the mediums permittivity and and is its permeability (the value without an index applies for one specific medium, which can be divided into the value of the vacuum itself (which has an index of 0), and the relative value, which is the vacuum permeability of the medium relative to vacuums permeability.), is the specific electrical resistance of the medium, is the non-electrical source of electromotive force. It usually works against the electic space similar to the lifting pump of the pipe networks. The negative sign shows that the positive charges are being moved by it to the higher-potential space .

Table (2.2) Electric phenomena that have time-constant (steady-state) distributed parameters

Electrostatics

Steady-state flow field

*

Steady-state magnetic field

* Derived equation:

Stacionary phenomena can be described using static equations, and in this case we cannot speak of state variables (from systems engineering). In system engineering state variables can be connected to dynamical properties (storage element) and a differential equation can be written to the state variables. A way to describe the steady-state (stacioner) variable physical quantities and corresponding mathematical variables is needed. The table 2.1 can be complemented with non-electrical phenomena.

1.9.4. Derivation of Kirchhoff equation (for concentrated parametric models)

(not recommended for first-year students)

At point we deal we deal with phenomena that are constant in time (so-called steady-state phenomena). The aim is to create a concentrated parametric model from the given differential equations (from divided parametric model).

I. Node Equation

Divergence equations are converted to a vector surface integral using the Gauss-Osztogradszkij-thesis. The closed surface can be divided into parts and the vector can be averaged on a given surface. This is how we can infer the Kirchoff I. (nodal) thesis.

Electrostatics:

(2.15)

Steady-state (resistive) flow field

(2.16)

Steady-state magnetic field

(2.17)

II. Loop equation

Rotational equations are converted into a vector line integral using the Stokes thesis. The closed spline can be divided into sections and along the given sections the vector can simply be averaged. This how we can infer the concentrated parametric Kirchoff II. (loop) thesis.

Electrostatics

(2.18)

Steady-state (resistive) flow field

Because (by definition) does not have an electronic origin, it is not included in the Maxwell equations. Yet is the cause of the electron flow, thus the source of vortices. For example (in a hydrostatic analogy), a pump circulates the liquid in a closed network of pipes, thus it is important to include it in the equations describing the given vortex.

Using the theory of the superposition:

(2.19)

Interpretation

is the difference in voltage between the endpoints of a given lenght unit, but of the given values have to be considered to be zero. Thus the and in the summation all have electronic-esque properties, which means that the voltage from external electronic forces is in balance with the electrical voltage of the circuit. In the case of an ideal conductive , the electric field strength is zero. is, for example, typically the voltage difference that can be measured between the poles of a battery. If this battery is loaded the voltage on its poles is reduced because of the voltage of the inner resistance. can be found on the right side of the summary, so its direction is contrary to the direction of the current in the circuit. In this way electrons of the battery cannot move from the positive output to the negative output. In variable current circuits the induced voltage is similar to the .

Steady-state magnetic field

(2.20)

magnetic power is inserted to the magnetic circuit on a zero lengh part similar to the .

1.9.5. Steady-state electronic and magnetic circuits modelling two poles

In this chapter we again deal with steady-state phenomena, thus storage (batteries) are not covered in this chapter. In the previous chapter the actual vectorial quantity was averaged along a given part of the surface or a section of a spline. In this chapter, after averaging, homogenous distribution is assumed along the cross-section and the length. The quality of material is assumed to be identical in all points of the space, so the given spacepart can be characterized with one concentrated parameter.

Electrostatics

Assume that in all points of a given cylindrical part of the space and vectors point in the same direction as the cylinder. The size of these vectors is equal in all points of the cylinder, and , and the material of the cylinder is assumed to be homogenous, permittivity is identical in all points of the space. Let be the length of the cylinder, the area of its cross-section. Such a homogenous electrostatic space can be created between the armaments of a capacitor.

lh

Ah

Eh

Dh

Figure 25.Homogenous electric field

According to the use of the homogenous electronic space(2.13):

(2.21)

According to the electrical flux passing through the cylinder (2.15) and the stress (2.19) of the cylinder

;

(2.22)

Converting the (2.21) equation, we get the well-known nexus of capacitors

(2.23)

Steady-state (resistive) flow field

Suppose that in all points of a given cylindrical space and vectors are parallel to the direction of the cylinder. Its size is the same in all points of the cylinder, and and the material of the cylinder is homogenous and its conductivity is . The length of the cylinder is , the area of a perpendicular cross section is .

lh

Ah

Eh

Jh

Figure 26. Homogenous stationary flow field

The effect of the electric power has to be considered, but it is not part of the electric field, it can be handled as an external effect. Based on the strength of the electric field and the electromotive force, the density of the formed electric current is: (2.14):

(2.24)

Using the current flowing through the cylinder and the whole voltage of the cylinder

(2.25)

Converting the equation (2.24) the well-known Ohm-law can be deduced, where is the voltage of the resistance

(2.26)

The effect of the electromotive force appearing on the given part has to be considered separately, so the general switching of the replacement switching can be seen in the Figure 27.

Rh

EMFh

Ih

Uh

Figure 27. Electric replacement switching

Steady-state magnetic field

Similar equations can be written in the case of a magnetic field as well.

Suppose that in all points of a given cylindrical space and vectors are parellel to the direction of the cylinder. Its size is the same in all points of the cylinder, and and the material of the cylinder is homogenous and its permeability is . The length of the cylinder is , the area of a perpendicular cross section .

lh

Ah

Hh

Bh

Figure 28.Homogenous magnetic field

After seperating the effect of excitation ( magnetic motor power ), using only the electric field strength the size of the formed current density is:(2.13)

(2.27)

When calculated from theelectrical flux passing through the cylinder (2.17) and the stress (2.20) (2.20) of the cylinder:

(2.28)

Modifying the (2.27) equation the magnetic Ohm-law can be deduced

(2.29)

The effect of the excitation appearing on the given part has to be considered separately, so the general switching of the replacement switching can be seen in the Figure 29.

Rh_m

Uh_m

MMF

h

Figure 29. Homogenous magnetic field

1.9.6. Analogies with network calculation

The analogy between the two systems can be derived from the fact that the function of the systems can be described using similar equations. Harry F. Olson wrote the first paper (Dynamical Analogies, 2. vol., Van Nostrand, pp. 2729, 1958) about the opportunities of interoperability and analogies to help engineers in analysis and to determine the efficiency of systems. In many cases the examined system is bipolar, and can be described as a network from concentrated elements. All elements can be defined using a parameter and the equation describing the connection between the through and cross variables. The value of the through variable is the same in the two outputs of the bipolar, the cross variable is the difference between the two outputs.

Definition

Through variable: It is the rate of an extensive physical quantity in relation to the bipolar, where the bipolar has a incoming and outgoing surface. A typical example is electricity , which describes the flow of electrons (as an extensive physical quantity).

Interpretation

From the conservations law it is logically derivable that if in a given part of the space the value of an extensive physical quantity is not changing then on the bounding area the signed summation of the in and outflowing physical quantities is 0. In the case of distributed prametric systems the result is the same from the divergence equation of the Gauss-Osztrogradszkij thesis. The Kirchoff I. (nodal) thesis can be deduced from this. In general: for concentrated parametric models by calculating the through variables, the nodal equation (equivalent to the Kirchoff I. thesis) is used.

Definition

Cross variable: The measurable difference between the two poles of the bipolar is called the cross variable. Typical example is electric voltage.

Interpretation

Using the Stokes thesis, a line integral calculated quantity can calculated from the divided parametric rotational equation. The concentrated parametic cross variables derived from the rotation can be described using the loop equation, analogical with Kirchoff II. Vortex fields have to be handled separately, which are called usual or conservative fields.

Bipolar elements

Three different types of these are separated

Definition

Storage type A: this type of storage stores energy (generalized energy) using through variables. The time-derivate through variable measured in the bipolars is proportional to the difference of the two outputs of the bipolar. A typical example is L inductivity. The stored energy and the equation for the two poles:

(2.30)

Storage type K: this type of storage stores energy (generalized energy) uses cross variables. The derivate of the cross variable difference (measured in the bipolars) is proportional to the through variable. A typical example is C capacity. The stored energy and the equation for the two poles:

(2.31)

Storage type P: For this element the difference of the cross variable is proportional to the through variable (Proportional element). This element generally doesnt store energy only swallows it. In other words, it causes energy to be dissipated from the system. However, for electrostatic and magnetic circuits this element actually stores energy. A typical example is R resistance. The stored energy and the equation for the two poles:

(2.32)

Quadripolar elements

With the use of bipolar elements we can describe coherent systems, wherein the same type of energy spreads. If we want to make contact between parts separated from each other in some way (that have either same or different types of energy), then a quadripolar adapter can be used. Two-two poles connect to one another. Normally the through variables are directed inwards, and the cross variables in the same manner. In Figure 210 two types of an electric quadripolar elements are showm.

xxxxxxxx

Figure 210. Electic quadripolar

Among the four terminals of a quadripole the following general releationship can be given.

(2.33)

If we assume that the converter is linear, static (can be described with algebraic equation, see exact definition later) and lossless (according to control ) ,then two special cases can be distinguished.

Transformer converter

(2.34)

This group includes electrical transformers, electro-dynamic transducers, gear-toothrack connections, all types of traction motion transducers, threaded spindle nowadays most of all the ballscrew motion transducers.

Gyrator transformer

(2.35)

It provides a connection between different types of systems with different types of variables. Transducers like hydraulic and pneumatic cylinders, or piezoelectric transducers.

xxxxxx

Figure 211.Gyrator

Ideal sources

The ideal sources can be bi- and quadripolar. One of the cross or through variables of the bipolar elements is constant, while the other parts of the network will determine the other. The controlled sources are included in the group of the ideal quadripolars.

xxxxx

Figure 212.Ideal bipolar source

xxxxx

Figure 213.Ideal quadripolar source

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1

2.2 Table - Analogies

Extensive quantity

Intensive quantity

Concentrated parametric description using bipolar elements (ith element)

vector field

concentrated source

vektorfield

concentrated vortex

through variable

cross variable

P-type element

K-type storage

A-type storage

from extensive quantity

(resultant quantity)

Electrostatics

shift vector

electric charge

electric field strength

0 potentional

electric flux (not the rate of the charge)

Voltage difference

capacity, in reciprocal

not defined

not defined

Electric

circuit

shift vector

Electric

current

electric field strength

electromotive force

electric current through the element

Voltage difference

resistance

capacity

inductance roll

Magneticcircuit*

magnetic induction

zero, without source

magnetic field strength

excitation

(magnetomotoric force)

magnetic flux through the element

excitation of the element

magnetic resistance

not defined

hysteresis and vortex loss core

Hydraulic system

Incompressible fluid

velocity field

zero, without source

velocity field

effect of the pump

volume flow

pressure fall

pipe network element

(not linear)

volume tank

hydraulic mass

Pneumatic system

Compressible medium

velocity field

velocity field

effect of the pump

air volume flow

pressure fall

pipe network (fojt) element

(not linear)

volume tank

not defined

Linear mechanical system

not used

Distance is not calculated from divergance

force gradient

zero, (potentional)

Relative velocity

resultant force

dampened movement element

spring

two points associated mass

Rotary mechanical system

not used

Distance is not calculated from divergance

moment

gradient

zero, (potentional)

angular velocity

resultant moment

dampened movement element

spring

two points associated inertia

normalize

normalized Dirac impulse

impulse

acceleration field

zero, (potentional)

force

relative velocity

damped linear movement element**

two points associated mass

spring

Rotary mechanical system

normalized Dirac impulse

moment

impulse moment

angular acceleration field

zero, (potentional)

moment

relative angular velocity

damped rotary movement element **

two points associated rotary mass

spring

Clear thermic

heat

temperature gradient

heat flow

heat fall

conductive element

heat capacity

not defined

Note: as it is visible in the case of the electronic and magnetic phenomena, the physical variables are derived from two different vector fields each. Contrary to this, for flow problems we use the divergence and rotation of the same vector field. When matematically describing the electric or magnetic phenomena it is sufficient to define only one vector field for each one. In the case of vacuum there is no need to differentiate these. This separation is justifiable because of the properties of the materials. The source of the electric field is the electric charge, but in materials the separation of charges can occur in multiple ways. The reason for the differentiation of the shift vector and the electric field strength is the charge separation in materials. In the first case the charge separation is taken into consideration, in the second case it is not.

Similarly, unpaired electrons found in the material (on the given orbitals only one electron is moving it is a quantum physical problem) have magnetic momentum, which can strengthen the effect of the external magnetic field. This can be modelled with current loops within the material itself (f.e. the electron is moving around the nucleus), this effect is visible in the magnetic induction, but not in the magnetic field strength. Essentially, magnetic induction describes the whole magnetic field (the effects of the internal and external currents together). Contrary to this, the magnetic field strength is calculated using only the external currents using the excitation law, so the effect of the external currents are described.

More analogies are in the chapter of the Appendix named Failure! The reference source is not found. ( Missing link here!)

Sample problems and solutions for bi- and quadripolars

Task 21.

Task 22.

Task 23. Model of a real coil

Write down the differential equaton of a real coil! The coil can be modelled with a resistor and a clean inductor connected in series. The input signal is the cross variable, namely the voltage (the sum of the resistor voltage and inductance voltage ), the output signal is the through variable, i.e. the current of the coil .

Solution

The following Kirchhoff loop equation can be written:

(2.36)

(2.37)

where, , and

(2.38)

Task 24. Heat transfer

We load heat transfer on a body withmass, and surface, the environmental outlet heat is . The body is in intense exchange with its environment (e.g. a radiator). Assume that the initial value of temperature is zero (). Write down the temperature change equation of the body. The input signal is the through variable, i.e. heat transfer, the output signal is the cross variable , i.e. the temperature of the radiator. Additional parameters: heat transfer coefficient and c specific heat

m, c,

A

b

k

Figure 214. Heat transfer

Solution

The outlet thermal flow

(2.39)

The difference between the inlet and outlet thermal flow