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DOCUMENT OFFICE i L 36-1ooA / RESEARCH LABORATORTFEt:ECTRTI MASSACHUSETTS iNSTITUE OF TECHNOLOGY CAMBRIDGE, MASSACHUSETTS 02139, U.S.A. IMPEDANCE AND POWER TRANSFORMATIONS BY THE ISOMETRIC CIRCLE METHOD AND NON-EUCLIDEAN HYPERBOLIC GEOMETRY E. FOLKE BOLINDER TECHNICAL REPORT 312 JUNE 14, 1957 / lk / J `'--- I-I ,4 i / / . PR .' I' I, iP¢ i/ /I. ! ? MASSACHUSETTS INSTITUTE OF TECHNOLOGY RESEARCH LABORATORY OF ELECTRONICS CAMBRIDGE, MASSACHUSETTS A_ .. w. __ ._ . , - - .. , ... . ._ * _____ _ ,

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The Research Laboratory of Electronics is an interdepartmental laboratory of the Department of Electrical Engineering and the Department of Physics. The research reported in this document was made possible in part by support extended the Massachusetts Institute of Technology, Research Laboratory of Electronics, jointly by the U. S. Army (Sig- nal Corps), the U. S. Navy (Office of Naval Research), and the U. S. Air Force (Office of Scientific Research, Air Research and Develop- ment Command), under Signal Corps Contract DA36-039-sc-64637, Department of the Army Task 3-99-06-108 and Project 3-99-00-100. I I

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MASSACHUSETTS INSTITUTE OF TECHNOLOGY RESEARCH LABORATORY OF ELECTRONICS Technical Report 312 June 14, 1957 IMPEDANCE AND POWER TRANSFORMATIONS BY THE ISOMETRIC CIRCLE METHOD AND NON-EUCLIDEAN HYPERBOLIC GEOMETRY E. Folke Bolinder Abstract An introductory investigation on the means by which modern (higher) geometry can be used for solving microwave problems is presented. It is based on the use of an elementary inversion method for the linear fractional transformation, called the "iso- metric circle method," and on the use of models of non-Euclidean hyperbolic geometry. After a description of the isometric circle method, the method is applied to numerous examples of impedance and reflection-coefficient transformations through bilateral two- port networks. The method is then transferred to, and generalized in, the Cayley-Klein model of two-dimensional hyperbolic space, the "Cayley-Klein diagram," for impedance transformations through lossless two-port networks. A similar transfer and general- ization is performed in the Cayley-Klein model of three-dimensional hyperbolic space for impedance transformations through lossy two-port networks. In the Cayley-Klein models a bilateral two-port network is geometrically repre- sented by a configuration consisting of an "inner axis" and two non-Euclidean perpen- diculars to the inner axis. The position of the configuration in the models depends upon the fixed points and the multiplier of the linear fractional transformation. By using this geometric representation, an impedance transformation through a bilateral two-port net- work is performed by consecutive non-Euclidean reflections in the two perpendiculars. The Cayley-Klein model of three-dimensional hyperbolic space is used: (a) for creating a general method of analyzing bilateral two-port networks from three arbitrary impedance or reflection-coefficient measurements; (b) for creating a general method of cascading bilateral two-port networks by "the Schilling figure"; (c) for determining the efficiency of bilateral two-port networks; (d) for classifying two-port networks; (e) for splitting a two-port network into resistive and reactive parts; and (f) for com- paring the iterative impedance method and the image impedance method.

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Table of Contents I. Introduction 1 1. 1 Scope of the Research Work 1 1.2 Two Basic Geometric Works 1 1.3 Brief Outline of the Research Work 2 II. Impedance Transformations by the Isometric Circle Method 4 2. 1 Introduction 4 2.2 The Linear Fractional Transformation 5 2. 3 The Isometric Circles 6 2.4 The Isometric Circle Method 6 2.5 Classification of Impedance Transformations through Bilateral Two- Port Networks 9 2.6 Impedance Transformations through Lossless Two-Port Networks 10 2.7 The Isometric Circle Method in Analytic Form 12 2.8 Comparison of the "Triangular Method" and the Isometric Circle Method 13 2. 9 Some Applications of the Isometric Circle Method to Impedance Transformations through Bilateral Two-Port Networks 14 a. Example of a Loxodromic Transformation 14 b. Transformation of the Right Half-Plane of the Z-Plane into the Unit Circle (Smith Chart) 16 c. Uniform Lossless Transmission Line 18 d. Lossless Transformers 19 e. A New Proof of the Weissfloch Transformer Theorem for Lossless Two-Port Networks 23 f. Cascading of a Set of Equal Lossless Two-Port Networks 25 g. Lossless Exponentially Tapered Transmission Lines 25 h. Lossless Waveguides 31 III. Impedance Transformations by the Cayley-Klein Model of Two- Dimensional Hyperbolic Space 32 3. 1 Introduction 32 3. The Cayley-Klein Diagram 32 3.3 Van Slooten's Method 35 3.4 Extension of Van Slooten's Method 37 3. 5 Transfer of the Isometric Circle Method to the Cayley- Klein Diagram 38 IV. Impedance Transformations by the Cayley-Klein Model of Three- Dimensional Hyperbolic Space 40 4.1 Introduction 40 4. 2 Stereographic Mapping of the Z- Plane on the Riemann Unit Sphere 40 4.3 Impedance and Power Transformations in Three- and Four- Dimensional Spaces 40 iii

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Table of Contents (continued) 4. 4 Transfer of the Isometric Circle Method to the Cayley-Klein Model of Three-Dimensional Hyperbolic Space 44 V. General Method of Analyzing Bilateral Two-Port Networks from Three Arbitrary Impedance or Reflection-Coefficient Measurements 48 5. 1 Introduction 48 5.2 Geometric Part of the General Method 49 a. Klein's Generalization of the Pascal Theorem 49 b. Geometric Construction of the Inner Axis 51 c. Determination of the Fixed Points and the Multiplier 51 5.3 Analytic Part of the General Method 52 a. Representation by Quadratic Equations of Lines That Cut the Sphere 52 b. Analytic Representation of a Line That is Non-Euclidean Perpendicular to Two Given Lines 52 c. Analytic Expression for the Complex Angle between Two Lines That Cut the Unit Sphere 53 d. Determination of the Fixed Points of the Transformation 54 e. Determination of the Multiplier of the Transformation 55 5.4 Calculation of Several Numerical Examples 56 a. Example 1. Attenuator 57 b. Example 2. Lossless Lowpass Network 59 c. Example 3. RLC Network 61 5.5 Comparison of the Geometric-Analytic Method with a Pure Analytic Method 63 VI. General Method of Cascading Bilateral Two-Port Networks by Means of the Schilling Figure 65 6. 1 Introduction: The Schilling Figure 65 6.2 Geometric Treatment 65 6. 3 Analytic Treatment 66 VII. Graphical Methods of Determining the Efficiency of Two-Port Networks by Means of Non-Euclidean Hyperbolic Geometry 68 7. 1 Use of Models of Two-Dimensional Hyperbolic Space 68 7.2 Use of the Cayley-Klein Model of Three-Dimensional Hyperbolic Space 70 VIII. Elementary Network Theory from an Advanced Geometric Standpoint 73 8. 1 Classification of Bilateral Two-Port Networks 73 8.2 Splitting of a Two-Port Network into Resistive and Reactive Parts 73 8.3 Comparison of the Iterative Impedance and the Image Impedance Methods 75 iv

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Table of Contents (continued) IX. Conclusion 80 Appendix 1. Models of Two- and Three-Dimensional Non-Euclidean Hyperbolic and Elliptic Spaces 81 Appendix 2. Interconnections of the Non-Euclidean Geometry Models 84 Appendix 3. Historical Note on Non-Euclidean Geometry 86 Appendix 4. Survey of the Use of Non-Euclidean Geometry in Electrical Engineering 87 Acknowledgment 90 References 91 v _ __~~~~~~_~~~~~__ ___ _____ _ _______~~~~~~~~~~-~-~--~~~~~~~~~~~~~__~~~~_~~~__

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I. INTRODUCTION 1. 1 SCOPE OF THE RESEARCH WORK The purpose of the work that is presented in this report has been to perform an introductory investigation on the means by which modern (higher) geometry can be used for solving microwave problems and simplifying solutions that are already being applied to these problems. At the beginning of this investigation at the Research Laboratory of Electronics, Massachusetts Institute of Technology, in September 1955, the writer decided to try to follow a certain plan for performing the research work. Two rules were prescribed: first, to start with simple problems and gradually extend the ideas and methods to more complex problems, and, second, to divide the treatment of the problems into three parts: a geometric part yielding a graphic picture of the problem, an analytic part con- stituting an analytic interpretation of the geometric part, and a part consisting of simple constructive examples to clarify the geometric and analytic treatments. These rules have been strictly followed. This fact, and the fact that numerous papers in mathematics, engineering, and physics, published in six languages (German, English, French, Italian, Dutch, and Swedish), were studied led, naturally, to rather slow progress in the research work. But the thoroughness of the study has resulted in the construction of a firm foundation on which future research can be built. Before a brief outline of the research work is given, two geometric works, which provide the mathematical foundation of the present work, are discussed. 1.2 TWO BASIC GEOMETRIC WORKS The branches of mathematics that are useful in dealing with impedance and power transformation problems in electrical engineering are analysis, algebra, and geometry. Of these, the first two have found extensive application. Although both engineers and physicists favor graphic representation of the problems they are trying to solve, geom- etry seems to have been applied to a limited extent. The reason seems to be that by the nature of his training, a person who is able to use both analysis and modern algebra, often considers elementary geometrical treatment difficult to understand. It is impor- tant to stress that modern (higher) geometry has advanced beyond the graphical con- structions that can be performed with ruler and compass. This will be understood by a quick glance at the geometric portions of the collected works of Gauss, Riemann, Cayley, Klein, Lie, Clifford, and Poincare. Two papers stand out as having been of fundamental importance in the development of modern geometry. These are Riemann's "Uber die Hypothesen, welche der Geometrie zu Grunde liegen," which was completed in 1854 and published in 1868 (80-82), and Klein's "Vergleichende Betrachtungen uber neuere geometrische Forschungen," published in 1872 (62). In the first paper, which initiated the development of Differential Geometry, Riemann discusses, among other things, manifolds of n dimensions of constant 1

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curvature. If this curvature is negative, and if n = 3, we have the non-Euclidean geom- etry of Gauss, Bolyai, and Lobachevsky, to which Klein gave the name "hyperbolic geometry" (63). If the curvature is zero, we have Euclidean geometry, which he called "parabolic geometry." Finally, for a positive curvature, Riemann created another non- Euclidean geometry, which Klein called "elliptic geometry." (We cannot be sure, how- ever, whether Riemann thought of his geometry as spherical or elliptical.) In elliptic geometry space is unbounded but finite. In the second paper, Klein proposed a program, the well-known "Erlangen Program," for the unification of the principal geometries. He classified geometric properties and assigned them to different geometries according to the invariant properties of corre- sponding transformation groups. The program was partially initiated by Lie's theory of transformation groups. For almost fifty years it remained unmodified, until the enunci- ation of the theory of general relativity. 1.3 BRIEF OUTLINE OF THE RESEARCH WORK The complex impedance (admittance) plane has been one of the most important tools for analyzing and synthesizing networks under stationary conditions, ever since Steinmetz, in the early days of network theory (1893), pointed out that the vectors which Bedell and Crehore had introduced into electrical engineering (1892) could be interpreted as points in a complex plane. For example, Feldtkeller (54) studied symmetric networks by using this tool and, likewise, Schulz (97) studied unsymmetric networks. With the advent of television, radar, and pulse-communication systems, the complex-frequency plane, in which transient conditions can be studied by means of the theory of the Laplace transform, took over the role of main tool. The complex impedance plane did not lose its entire significance; in fact, it had a revival of importance at higher frequencies, in the microwave region. A consistent microwave theory in the form of a circular geom- etric theory was created by Weissfloch (110) in 1942-43. In the complex impedance plane and the complex reflection-coefficient plane trans- formations are usually performed by the linear fractional transformation. The fact that it transforms configurations conformally suggests the use of graphical methods. The "isometric circle method" (55, 5) is a method of this kind. The method, an elementary inversion method, is thoroughly described and applied to some simple problems in Section II. Its name is derived from its utilization of two circles called the "isometric circles." One of the operations prescribed by the isometric circle method consists of an inversion in one of the isometric circles. But it is not practical to use this method with circles of large radii. In order to compress the complex impedance plane, the writer began the study of models of non-Euclidean hyperbolic geometry. Some of the two- and three-dimensional models are briefly described in Appendix 1, their interconnections are discussed in Appendix 2, and a short historical note on the evolution of non- Euclidean geometry is given in Appendix 3. A survey of the use of non-Euclidean 2

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