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Lineer olamayan devreler ve sistemlerin frekans domeninde analizi

Analysis of nonlinear systems in frequency domain

  1. Tez No: 66601
  2. Yazar: İSMAİL HAKKI MARANGOZ
  3. Danışmanlar: DOÇ. DR. F. ACAR SAVACI
  4. Tez Türü: Yüksek Lisans
  5. Konular: Elektrik ve Elektronik Mühendisliği, Electrical and Electronics Engineering
  6. Anahtar Kelimeler: Belirtilmemiş.
  7. Yıl: 1997
  8. Dil: Türkçe
  9. Üniversite: İstanbul Teknik Üniversitesi
  10. Enstitü: Fen Bilimleri Enstitüsü
  11. Ana Bilim Dalı: Elektronik ve Haberleşme Mühendisliği Ana Bilim Dalı
  12. Bilim Dalı: Devreler ve Sistemler Bilim Dalı
  13. Sayfa Sayısı: 150

Özet

ÖZET Bu tezde lineer olmayan devreler ve sistemlerin frekans domeninde analizi genel olarak incelenmiştir. Çeşitli metodlar tanıtılmış ve bazı örneklere uygulanmıştır. İlk olarak lineer olmayan sistemlerin analizinde önemli bir yer tutan tarif eden fonksiyonlar (describing functions) ele alınmış ve bunlar kullanılarak sistemlerde görülebilecek periyodik çözümler, kaotik davranışlar incelenmiştir. Tarif eden fonksiyonlar temel alınarak geliştirilen ve sistemlerde kaosun belirlenmesine yarayan bir konjektür örneklerle sunulmuştur. Kaosun kontrol altına alınabilmesi, iki sinusoid girişli tarif eden fonksiyonların kullanıldığı bir yöntemle incelenmiştir. Bir yada birden fazla farklı frekansta kaynakla sürülen lineer olmayan devrelerin analizi için önerilen Ayrık Fourier Dönüşümüne dayalı yöntemler bilgisayar programlarıyla gerçeklenmiş ve sayısal integrasyon simülasyonlarıyla karşılaştırılmışlardır. Ayrıca lineer olmayan sistemlerin analitik çözümünün bulunmasında sıkça kullanılan Volterra Serileri kısaca tanıtılmış ve frekans domeni uygulamalarına yer verilmiştir. ıx

Özet (Çeviri)

SUMMARY ANALYSIS of NONLINEAR SYSTEMS in FREQUENCY DOMAIN Dynamics of nonlinear systems are investigated in this thesis by using frequency domain techniques. It is a well-known fact that many relationships among physical quantities are not quite linear, although they are often approximataed by linear equations mainly for mathematical simplicity. This simplification may be satisfactory as long as the resulting solutions are in agreement with experimental results. Nonlinear systems greatly in that the principle of superposition does not hold for the former. Nonlinear systems exhibit many phenomena that cannot be seen in linear systems, and in investigating such systems we must be familiar with this phenomena. There is no general method for dealing with nonlinear systems because nonlinear differantial equations are virtually devoid of a general method of attack. Exact solutions can be found only for certain simple nonlinear differantial equations. For many nonlinear differantial equations of practical importance, only approximate solutions are possible, and this solutions hold true only under the limited conditions. One way to analyze and design of nonlinear systems is to use equivalent linearization techniques and to solve the linearized problem. Analysis of nonlinear systems in the frequency domain can be done by using Volterra series. Volterra series has been used extensively in the analysis of nonlinear circuits and systems characterised by polynomial nonlinearities. In Section 2 a unified study of application of Volterra functional series to nonlinear system analysis is presented with special emphasis on frequency domain results. After giving a brief summary of the foundations Volterra series, some explicit formulas will be presented. These formulas show precisely which harmonic or intermodulation frequency components are generated by a Volterra kernel of a given order. Moreover, these formulas show that the frequency components generated by nth order Volterra kernel are disjoint from those generated by (n+l)th order kernel. In addition, formulation of nonlinear transfer functions for dynamical systems described by state equations will be given. Then formulation of nonlinear transfer functions from nonlinear circuits will be presented. Explicit and recursive formulas for obtaining nth order transfer functions of composite nonlinear systems will be given. A recursive method for obtaining nth order output of a nonlinear circuit by solving a lineer circuit n times will be derived. Each time different input sources are used. Recursive formulas for obtaining the nth order transfer functions of nonlinear circuits are then generated. The nth order transfer functions of some circuits will be found by this technique.In Section 3, two asymptotic methods.Harmonic Balance and Perturbation methods, for finding the analytic solutions of nonlinear circuits and systems will be given. By using harmonic balance principle the steady state solution of nonlinear systems in frequency domain can be found by solving nonlinear algebraic equations. The describing function method based upon the Harmonic Balance principle is one of the equivalent linearization methods. In this method establishing the stability criteria for nonlinear systems is simplier than the finding analytical solutions. Describing function method enables us to study the stabilty of many nonlinear systems from a frequency domain point of view. Suppose that the input to a nonlinear element is sinusoidal. Suppose that the output of the nonlinear element is not sinusoidal. Output may generally be the same period as the input. In the describing function analysis, we assume that only the fundemantel harmonic compenent of the output is significant. Such an assumption is often valid since the higher harmonics in the output of a nonlinear element are often of smaller amplitude than the amplitude of the fundemantel harmonic compenent. The describing function of a nonlinear element is defined to be the complex ratio of the fundemantel harmonic compenent of the output to the input. This analyzing method used describing functions can be called Harmonic Balance method. Moreover, two sinusoid input describing functions will be examined and some applications of them will be presented. Perturbation method can be applied to the systems which has small parameters. The analytical solution of the system can be obtained. The method will be applied to the oscilating and forced oscilating systems. The method will be explained by examples. In section 4, a method which finds the frequency domain solutions of nonautonomous nonlinear circuits by using The Discrite Fourier Transform techniques, will be presented. The method minimizes time domain calculations by introducing a criterion for selecting the varibles to be considered as unknowns and for solving the resulting nonlinear system by a new an efficent algorithm. It has exhibited the capability for handling a large number of harmonics and nonlinearities. To illustrate the generality and usefullness of the algorithm, Forced Chua circuit will be analysed with a computer program. The results of the algorithm will be compared to the real response. The optimum desing of circuits containing nonlinear elements requires an accurate technique for predicting their nonlinear performance. The most common techniques are based on the analysis of a circuit-type model which simulates the nonlinear behavior of the device. However, the high computational cost of the numerical methods used to analyze the interaction with the external circuit is the major drawback of these techniques. The circuits includes usually many linear elements and in most cases excitation is periodic. Only the steady state response is required. The harmonic balance method is preferable to time domain techniques because it avoids the numerical integration of the circuit dynamic equations, but it has a serious dissadvantege in the large number of unknown variebles. XIIn order to reduce the number of unknown variebles, nonlinear network is seperated into linear and nonlinear sub networks. In that section an analysis method is described which avoids the partitioning problem by introducing a criterion for selecting the variables to be considered as unknowns and solving the resulting nonlinear system by a new and efficient algorithm. In section 5, an efficent algorithm is given for calculating the steady state response of a nonlinear circuit driven by multi tone signals, possible made up of incommensurable frequencies wl, w2,,wp. The algorithm is particularly useful when the steady state response is not periodic, therby invalidating most existing methods. Nonlinear circuits which has almost periodic signals can be analyzed by the algorithm. That method is based upon the harmonic balance principle too. Harmonic balance has had limited application for simulating circuits, such as mixers that have a steady state response that contains almost periodic signals. The reason is that to model a nonlinear device, whose behaviour is more conveniently computed in the time domain, harmonic balance requires the transformation of signals from the frequency domain into the time domain and vice versa. For circuits that have a periodic response, The Discrete Fourier Transform (DFT) is used. Previously, no satisfactory transform existed for almost periodic signals. In that section, a different Fourier transform algorithm for almost periodic functions (APFT) is devoloped. It is both efficent and accurate. Unlike previous attempts to solve this problem, the new algorithm does not constrain the input frequencies and uses the therotical minumum number of time points. Another diffrence of the algorithm from the others is that exponantial Fourier transforms is used to consruct the almost Fourier transform pair. Moreover algorithm is not time consuming because a direct iteration is used to solve the algebraic equations, instead of using Newton iteration princible. Like the method presented in section 4, nonlinear differential equation are transformed to nonlinear algebraic equations by using harmonic balance principle. The almost periodic Fourier transform (APFT) is applied to only nonlinear elements in the circuit. A trnasform matrix is consructed between the time domain and the frequency domain. The almost periodic Fourier coefficients of an almost periodic waveform can be find from the time domain samples by using that matrix. Another problem is that the matrix is usually ill conditioned. So that the inverse of that matrix is computed wrong by computer. An algoritm is given on selecting the time sample points to establish a well conditioned matrix. For the two different input signal case, the algorithm is developed as a computer program to analyze forced Chua's circuit. Good result has been obtained for some cases. When the algorithm does not converges, the number of time points is incrased. That makes the algorithm converged. In section 6, the analysis of chaotic dynamics in nonlinear circuits will be xninvestigated by harmonic balance technique. Some methods for the prediction and control of chaos will be presented. In addition, limit cycles in Lur'e systems will be analyzed by describing function method and induction of limit cycles by forcing the system with a sinusoidal input will be examined. On the basis of harmonic balance principle, a practical method is presented for predicting and existence and the location of chaotic motions. This is formulated as a function of system prameters, when the system structure is fixed by rather general input output or state equation models. u(t)=0 + Q- L(s) N,n(.) y(t) - ? Figure 1 : The Lur'e System n(.) is the nonlinear part, L(s) is the linear part of the system. To explain the chaos model, let the output in the Figure 1 be as y(t) =A+Bsinwt. The equilibrium points of the system can be found by the formula below, y+n(y)L(0)=Q The output can be analyzed harmonic balance formulas using describing functions (N) of n(.). A(1+N0(A,B)L(0))=0 l+N,(A,B)L0'w)=O These formulas can be obtained easily by the analysis of feedback system, conjecture for the existence of chaos in that system can be written as follows. 1. Existence of a predicted limit cycle (PLC) :y(t)=A+Bsinwt. 2. Existence of a separate equilibrium point (EP) : y=E. 3. Stability properties of the PLC and EP. A stable PLC and a unstable EP. 4. Interaction between PLC and EP: y(t)=E for some t. xin5. Suitable filtering effect of the system: System should not attenuate the higher harmonics perfectly. Moreover, the chaotic motion in a system is controlled by forcing the sytem with a sinusoidal input. A forced Chua's circuit is used. The chaos in the autonomus case is converted to a stable periodic response by applying a sinusoidal current source to the circuit. Some conditions for the location of these motions in terms of the amplitude of the source can be obtained. In this thesis some kind of analysis techniques of nonlinear circuits in the frequency domain will be investigated. Some new algorithms will be given and some complex motions that can be seen in only nonlinear systems will be analyzed by the methods based on harmonic balance. xiv

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