Passive Electronic Filters: Description and Design Problem

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How to read this thesis

This manuscript is addressed to readers with diﬀerent interests, such as filters design for mobile communications, frequency design of passive circuits, or simply system design. It is tried to meet these expectations by allowing diﬀerent levels of reading. Due to our approach, some advanced concepts and tools of Control and System theory are involved. Even though they are mainly based on widespread Linear Algebra, the reader may not be familiar with them, and numerous illustrative and numerical examples are then provided all along this thesis with the aim of emphasising the main underlying ideas. However, a reader not interested in technical details may skip them and only read the first and last sections of each chapter, in which the main ideas are introduced and summarised. If a method draw one’s attention, step-by-step procedures are also provided to directly implement the proposed methods. The resulting optimisation problems may be implemented using the Matlab software, and especially the LMI solvers4 of the Robust Control Toolbox.
Nonetheless, for a reader interested in very particular points of this thesis, we distinguish four presumed points of interest, detailed below.
Frequency design of passive circuits Due to the traditional fashion in Electron-ics to model passive components using ideal resistances, inductances and capacitances, engineers may be led to design equivalent RLC circuits. One with such interest should refer to Section 4.2 of Chapter 4.
Frequency (LFT) filter synthesis The frequency filter synthesis is a standard problem with particular applications in Signal Processing, for frequency filtering, and Control theory, for the computation of H∞-weights. A reader may then be interested in the synthesis method given in Section 5.3 of Chapter 5 that is eﬃcient and optimal, i.e. leading to a transfer function of minimum order. In Section 5.4 of the same chapter, one will find an eﬃcient synthesis method for transfer functions expressed as LFTs in a repeated transfer function T (s). Note that this requires to have a {x, y, z}-dissipative characterisation of T (s) (cf Section 2.5 of Chapter 2). Finally, in Section 5.5 is provided a conservative but eﬃcient method for the synthesis of 2D LFT filters.
Optimal and eﬃcient synthesis of LC, and LC-resonator ladder filters These filters appear of little use for today’s mobile applications, but may be still used, such as for instance in some low-power application [CRV11]. A reader inter-ested in a step-by-step approach for a synthesis method that is eﬃcient and optimal, i.e. with minimum number of elements, is invited to refer to Section 6.1 of Chapter 6.
Synthesis of AW -resonator ladder filters A reader interested in a synthesis procedure of reduced complexity for AW -resonator ladder filters, but which requires an initial point, should apply Algorithm 4.1 (p. 84). To achieve this, one will espe-cially needs to apply the modelling procedure of Subsection 4.2.2.2 (p. 61), in which a modelling example of AW -resonator ladder filters is provided. Furthermore, in Section 6.2 of Chapter 6 is proposed an approach to eﬃciently synthesise a suitable initial point.

Publications

Before stating and tackling the design problem of passive electronic filters, we intro-duce several concepts and tools used in this work. These concepts and tools mainly come from Control and System Theory, and are traditionally gathered into a generic framework for the analysis and the control of systems. As passive electronic filters are typically viewed open-loop systems, i.e. without the external intervention of a controller, the associated design problem does not straightly come under the scope of this framework. The objective of this chapter is then to provide an adapted design framework, which will be developed all along this manuscript.
This chapter starts with a brief explanation, in our view, of why the design of sys-tems for modern engineering applications becomes increasingly complex (Section 2.2). This especially motivates the introduction of advanced tools thereafter. Indeed, it is first concluded that a relevant use of the computer power should be made, and thus design methods need to be developed in that respect. In the light of this, convex opti-misation seems for us particularly suitable (Section 2.3). Second, modern systems are typically viewed as the interconnection of subsystems having complex models. Con-sidering this feature, appropriate tools to mathematically represent (Section 2.4) and characterise (Section 2.5) the models of these subsystems and their interconnection is introduced. In particular, the characterisation of the subsystems leads to simple criteria to check the stability of their interconnection. In addition, this even enables to provide a simple procedure for the design a stable interconnection of homogeneous subsystems. Finally, a link between these tools, convex optimisation and the design requirements is made explicit (Section 2.6) through the introduction of a unifying tool, namely the KYP Lemma, converted from an analysis tool to a design tool for the occasion.

On the complexity of developing modern system de-sign methods

As we will see throughout this manuscript, developing design methods for modern engineering applications is complex. In our view, two main reasons appear to justify this complexity: the design problem of modern systems is itself complex, while the design methods must be easily used by design engineers, and thus be also modern.
on the physical phenomena taken into account. For passive electronic filters, simplest models generally consider components as the ideal electrical interconnection of ideal inductances, capacitances and resistances. The complexity of the models increases when complex physical phenomena are included, such as electromagnetism or elec-tromechanics, or technological constraints, for instance the dimensions of the compo-nents. Some diﬀerences between a model and the physical system can not a priori be known, such as the manufacturing dispersion, and need to be added using uncertain parameters, increasing the complexity of the model even further. Usual design flows start with simple models and increase their complexity step-by-step [WL15]. There is generally a trade-oﬀ between the precision of the model and the time-eﬃciency of the design methods. Thus, from a certain step the design problem becomes too complex to be solved in a decent time. In practice, to tackle this issue, engineers add margins upstream to the requirements with the hope of including all the diﬀerences between a system and its model.
Using models allows then to address the design problem by tackling sub-problems of lower complexity. However, even lower, this complexity may be still important. As an example, consider the problem of designing a filter as a circuit made of induc-tors and capacitors, such that it removes undesirable frequency components from an input signal. The simplest associated design sub-problem is obtained by modelling inductors and capacitors by respectively ideal inductances and capacitances, their interconnection by the ideal electrical interconnection, and the objective as lower and upper bounds on the magnitude of the frequency-response of the filter. This simple form raises some important questions.
These key questions have been extensively studied. There are particular cases having positive answers to these questions, as will be illustrated in this manuscript. However, in general, and for most of interesting applications, only part of these questions have an answer. In addition, this answer may produce constraints that are too complex to be included in the design process. Therefore, the problem of system design, even for simple systems, is inherently complex.
Moreover, another source of complexity is the increasing sophistication of modern systems, which tend to provide increasingly complex models. This leads to increase the number and the complexity of steps to validate in the design process. The design flow is then longer. To illustrate this idea, take the same example as above but with other components, such as acoustic wave resonators, instead of inductors and capac-itors. As usual in Electronics, these components can be modelled by a particular interconnection of inductances and capacitances. The wanted filter is then modelled as the interconnection of interconnected inductances and capacitances. Hence, the resulting design problem is even more complex than previously, as the model of the filter is not only required to satisfy structural constraints on the global stage of inter-connection but also on the local interconnection of the inductances and capacitances.

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Table of contents :

1 General Introduction
1.1 Motivation and purpose of this thesis
1.2 Scope and contributions
1.3 Structure
1.4 How to read this thesis
1.5 Publications
2 A Modern System Design Framework
2.1 Introduction
2.2 On the complexity of developing modern system design methods
2.3 Efficient use of computer power: convex optimisation
2.4 Modern systems representation via LFT
2.5 Dissipative systems characterisation
2.6 A unifying tool: the KYP Lemma
2.7 Conclusion
3 Passive Electronic Filters: Description and Design Problem
3.1 Introduction
3.2 Passive electronic filters: description and properties
3.2.1 Components modelling
3.2.2 Impedance and scattering descriptions
3.2.3 A particular electrical interconnection: the ladder topology
3.3 Design problem
3.3.1 Problem formulation
3.3.2 Problem simplification for lossless passive filters
3.4 Systematic filter design methods: a historical perspective
3.5 Summary
4 Design Approaches of Passive Electronic Filters
4.1 Introduction
4.2 Design approach 1: elements-values tuning
4.2.1 Introduction
4.2.2 Port-Hamiltonian Systems representations, modelling, analysis
4.2.2.1 PHS DAE representations
4.2.2.2 Modelling procedure and illustrations
4.2.2.3 Analysis of the resulting scattering matrix S
4.2.3 Design of electronic filters with a DAE PHS representation
4.2.3.1 PHS DAE representation synthesis as a BMI problem
4.2.3.2 Getting a particularly simple BMI form
4.2.4 Resolution of BMI optimisation problems
4.2.5 Application on a simple example
4.3 Design approach 2: realisable filter synthesis
4.3.1 Introduction
4.3.2 Frequency filter synthesis
4.3.3 Circuit synthesis
4.3.3.1 Realisation conditions of LC ladder filters
4.3.3.2 Realisation conditions of T-ladder filters
4.3.3.3 Realisation conditions of Ts, Tp-ladder filters
4.4 Summary and conclusion
5 Frequency LFT Filter Synthesis
5.1 Introduction
5.2 Problem formulation for LFT filters in a repeated T(s)
5.3 LFT filter synthesis with T(s) = 1s
5.3.1 Finite-dimensional convex formulation
5.3.1.1 Finite dimensional parametrisation
5.3.1.2 Convex formulation
5.3.2 Magnitude synthesis
5.3.3 Spectral factorisation
5.3.3.1 ARE, ARI and spectral factorisation
5.3.4 Synthesis procedure
5.3.4.1 Reverse parametrisation
5.3.4.2 On the choice of the representation matrices of B(s)
5.3.4.3 Explicit synthesis procedure
5.4 LFT filter synthesis with dissipative T(s)
5.4.1 Finite-dimensional convex formulation
5.4.2 Magnitude synthesis
5.4.3 Spectral factorisation
5.4.3.1 Spectral factorisation with lossless dissipative T(s)
5.4.3.2 Lossy spectral factorisation with dissipative T(s)
5.4.4 Generalised synthesis with factorisation error management
5.4.5 Synthesis procedures and numerical examples
5.4.5.1 Lossless dissipative T(s)
5.4.5.2 General dissipative T(s)
5.5 On the extension to 2D LFT filters synthesis
5.5.1 Conservative finite-dimensional convex formulation
5.5.1.1 Finite-dimensional parametrisation
5.5.1.2 Convex formulation
5.5.2 Positive rational function synthesis
5.5.3 Lossy spectral factorisation
5.5.4 Synthesis with factorisation error management
5.5.5 Synthesis procedure and numerical example
5.6 Summary
6 Design Examples
6.1 Design example of an LC-resonator ladder filter
6.1.1 Synthesis of a realisable W(T(s))
6.1.1.1 Magnitude synthesis under realisation constraints
6.1.1.2 Spectral factorisation
6.1.2 Elements value extraction
6.1.2.1 Elements value extraction
6.1.2.2 Application to the example
6.1.3 Order comparison with Butterworth and Chebyshev I filters
6.2 Design example of an AW-resonator ladder filter
6.2.1 2D LFT filter synthesis with some realisation constraints
6.2.1.1 Synthesis with factorisation error management
6.2.1.2 Spectral factorisation
6.2.2 Ts, Tp-ladder approximation
6.2.3 PHS synthesis algorithm
7 Conclusions and Perspectives
7.1 Conclusions
7.2 Perspectives
7.2.1 Further works
7.2.2 Future researches