# Theory of Linear and nonlinear circuits - Engberg J.

ISBN 0-47-94825

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Chapters 7 and 8 deal with the basic theory of non-linear noisy networks and systems. Chapter 7 presents the derivation of the theory, and chapter 8 presents examples and conclusion. Chapter 7 presents a unified method of analysis of low-level noise in non-linear networks and systems. Low-level noise refers to that the noise is a small pertubation of the deterministic signal regime. This book is the first to present a method based on Volterra series. The basic representation of noise sources is investigated. Both unmodulated (fundamental, independent) and modulated (dependent) noise sources are treated. The noise sources are represented as the noise response from a non-linear system with inputs given by a fundamental (unmodulated) noise source and one or more controlling variables. The controlling variables may be any system variables in the non-linear network. In this way it is possible to represent a wide variety of noise sources. Based on this representation a method is derived to analyse noise in general non-linear networks and systems. Expressions for the noise and deterministic response from the network are derived. To be able to determine average noise powers, expressions for the ensemble cross-correlation between Fourier series coefficients of the noise response at two arbitrary ports and at arbitrary frequencies are derived. The noise response may be determined as the dot product of a non-linear conversion vector and a noise vector at precalculated frequencies. The non-linear conversion vector is described by multi-port Volterra transfer functions determined from an equivalent circuit description of the network. Eexamples to illustrate the method are included in chapter 8. The practical applications of the method of non-linear noise analysis are expected to be in the analysis and optimization of noise in (near-sinusoidal) oscillators, in mixers with moderate local oscillator levels, and in frequency multipliers. Many oscillators are relatively weakly non-linear, but the non-linear i/f noise upconversion is still very important. Because of this a non-linear noise analysis is required. More and more mixers are being used in low-power applications, e g in portable communications equipment, which means that the Volterra series based noise analysis may be almost ideal for this purpose. Even though this type of mixers are not being switched on/off by the local oscillator signals, the sideband noise is still of importance and the loading of the mixer ports may have a significant impact on the mixer performance. Also the

Preface

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method of non-linear noise analysis is expected to be useful in the determination of noise models of non-linear devices.

Chapter 9 deals with the determination of multi-port Volterra transfer functions. In the existing literature only one-port Volterra transfer functions containing one-port non-linear elements are allowed. The method developed in the present book allows the determination of multi-port Volterra transfer functions containing multi-port uon-linear elements (subsystems). This is a fundamental requirement for the noise theory, since there are generally more than one input port, in the noise description of the networks. Moreover, in the analysis of (noise free) lion-linear circuits it has been pointed out that some models of MESFET’s should contain multi-port elements, but the theoretical methods required to determine the Volterra transfer functions did not exist. The present method has been implemented in a symbolic programming language which allows determination of the Volterra transfer functions in algebraic form. Using this it is possible to determine Volterra transfer functions up an order of about Я-Ю. Several examples are presented to illustrate the method, and comparisons with the existing literature have been made in some special cases. The work on multi-port Volterra series may be of high interest in the development of accurate non-linear models of devices with multi dimensional non-linear elements and in the analysis of systems with multi-port excitations.

J. Engberg & T. Larsen

Part I Linear systems

1

Some milestones in the development of noise theory

The first person to show the connection between ‘"Spontaneous Fluctuations”, as noise was called then, and thermodynamics experimentally was the Dutch scientist Geertruida Luberta de Haas-Lorentz [1,2]. Using very sensitive mirror galvanometers she showed that the electrons carrying the current behaved like molecules with temperature and she proved the thermal origin of noise on the basis of the thermodynamic theory which independently was developed by Albert Einstein [3,4] and Marian von Smoluchowski [5,6,7] and shown experimentally by Jean Baptiste Perrin [8j. In a third paper Albert Einstein [9] theoretically calculated a noise voltage on a capacitor. An interesting description of the development from the experiments of Robert Brown in 1827 and up to about 1906 is given by Haas-Lorentz in [2].

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